Subgroups — C₂×C₆₇×#3561 #30741982

Order 17152.

7258 conjugacy classes of subgroups.

The subgroup lattice has 824 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3561	C₂	1	32	yes	maximal, direct product factor
2	C₂×#563×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#562×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#556×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#551×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×D₄×Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#548×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#546×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#543×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#542×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#542×C₆₇	C₂	1	4	yes	maximal, complemented
12	C₂×#538×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂×#536×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂²×#41×C₆₇	C₂²	1	2	yes	—
18	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×Q₈×C₆₇	C₂²	1	2	yes	complemented
20	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	#563×C₆₇	C₂²	1	32	yes	complemented
22	#562×C₆₇	C₂²	1	32	yes	complemented
23	#556×C₆₇	C₂²	1	32	yes	complemented
24	#551×C₆₇	C₂²	1	64	yes	complemented
25	D₄×Q₈×C₆₇	C₂²	1	32	yes	complemented
26	#548×C₆₇	C₂²	1	32	yes	complemented
27	#546×C₆₇	C₂²	1	32	yes	complemented
28	#543×C₆₇	C₂²	1	32	yes	complemented
29	#542×C₆₇	C₂²	1	32	yes	complemented
30	#542×C₆₇	C₂²	1	64	yes	complemented
31	#538×C₆₇	C₂²	1	32	yes	complemented
32	#536×C₆₇	C₂²	1	32	yes	complemented
33	#533×C₆₇	C₂²	1	16	yes	complemented
34	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
35	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
38	#529×C₆₇	C₂²	1	16	yes	complemented
39	C₂×#126×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#126×C₆₇	C₂²	1	4	yes	complemented
41	C₂×#126×C₆₇	C₂²	1	2	yes	complemented
42	C₂×#126×C₆₇	C₂²	1	4	yes	complemented
43	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂×#124×C₆₇	C₂²	1	2	yes	—
46	C₂×#124×C₆₇	C₂²	1	2	yes	—
47	C₂×#124×C₆₇	C₂²	1	2	yes	—
48	C₂×#124×C₆₇	C₂²	1	2	yes	—
49	C₂×#124×C₆₇	C₂²	1	2	yes	—
50	C₂×#124×C₆₇	C₂²	1	16	yes	complemented
51	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic, complemented
52	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
53	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
54	C₂×#123×C₆₇	C₂²	1	4	yes	—
55	C₂×#123×C₆₇	C₂²	1	2	yes	—
56	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
57	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
58	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
59	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
60	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
61	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
62	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
63	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
64	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
65	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
66	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
67	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
68	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
69	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
70	C₂×#121×C₆₇	C₂²	1	4	yes	—
71	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
72	C₂×#121×C₆₇	C₂²	1	2	yes	—
73	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
74	C₂×#120×C₆₇	C₂²	1	4	yes	—
75	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
76	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
77	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
78	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
79	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
80	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
81	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
82	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
83	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
84	C₂×#117×C₆₇	C₂²	1	2	yes	—
85	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
86	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
87	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
88	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
89	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
90	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
91	C₂⁵×C₆₇	C₂³	1	2	yes	—
92	C₂×#41×C₆₇	—	4	8	no	—
93	C₂×#41×C₆₇	C₂³	1	16	yes	—
94	C₂²×Q₈×C₆₇	—	4	4	no	—
95	C₂²×Q₈×C₆₇	—	4	8	no	—
96	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
97	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
98	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
100	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
101	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
102	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
103	C₂²×Q₈×C₆₇	C₂³	1	16	yes	complemented
104	C₂²×D₄×C₆₇	—	4	4	no	—
105	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
106	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
107	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
108	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
110	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
111	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
112	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
115	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
116	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
117	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
118	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
119	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
120	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
121	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
122	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
123	C₄⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
124	#126×C₆₇	C₂³	1	16	yes	complemented
125	#126×C₆₇	C₂³	1	32	yes	complemented
126	#126×C₆₇	C₂³	1	32	yes	complemented
127	#126×C₆₇	C₂³	1	32	yes	complemented
128	#125×C₆₇	C₂³	1	8	yes	complemented
129	#125×C₆₇	C₂³	1	16	yes	complemented
130	#124×C₆₇	C₂³	1	16	yes	—
131	#124×C₆₇	C₂³	1	16	yes	—
132	#124×C₆₇	C₂³	1	16	yes	—
133	#124×C₆₇	C₂³	1	16	yes	—
134	#124×C₆₇	C₂³	1	16	yes	—
135	#124×C₆₇	C₂³	1	8	yes	complemented
136	#124×C₆₇	C₂³	1	32	yes	complemented
137	#124×C₆₇	C₂³	1	32	yes	complemented
138	#123×C₆₇	C₂³	1	16	yes	—
139	#123×C₆₇	C₂³	1	32	yes	—
140	#123×C₆₇	C₂³	1	32	yes	complemented
141	#123×C₆₇	C₂³	1	32	yes	complemented
142	#123×C₆₇	C₂³	1	32	yes	complemented
143	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
144	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
145	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
146	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	—
147	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
148	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
149	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
150	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
151	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
152	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
153	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
154	#121×C₆₇	C₂³	1	8	yes	—
155	#121×C₆₇	C₂³	1	16	yes	—
156	#121×C₆₇	C₂³	1	32	yes	—
157	#121×C₆₇	C₂³	1	32	yes	complemented
158	#120×C₆₇	C₂³	1	32	yes	—
159	C₄×Q₈×C₆₇	C₂³	1	8	yes	complemented
160	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
161	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
162	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
163	C₄×D₄×C₆₇	C₂³	1	16	yes	—
164	C₄×D₄×C₆₇	C₂³	1	32	yes	—
165	C₄×D₄×C₆₇	C₂³	1	16	yes	—
166	C₄×D₄×C₆₇	C₂³	1	8	yes	complemented
167	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
168	#117×C₆₇	C₂³	1	16	yes	—
169	#117×C₆₇	C₂³	1	16	yes	complemented
170	#117×C₆₇	C₂³	1	32	yes	complemented
171	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
173	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
174	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
175	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
176	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
177	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
178	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
179	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
180	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
181	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
182	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
183	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
184	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
185	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
186	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
187	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
188	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
189	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
190	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
191	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
192	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
193	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
194	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
195	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
196	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
197	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
198	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
199	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
200	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
201	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
202	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
203	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
204	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
205	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
206	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
207	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
208	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
209	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
210	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
211	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
212	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
213	C₂⁴×C₆₇	—	2	8	no	—
214	C₂⁴×C₆₇	—	4	4	no	—
215	C₂⁴×C₆₇	—	4	4	no	—
216	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
217	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
218	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
219	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
220	#41×C₆₇	—	4	32	no	—
221	C₂×Q₈×C₆₇	—	2	8	no	—
222	C₂×Q₈×C₆₇	—	2	8	no	—
223	C₂×Q₈×C₆₇	—	4	8	no	—
224	C₂×Q₈×C₆₇	—	4	4	no	—
225	C₂×Q₈×C₆₇	—	4	2	no	—
226	C₂×Q₈×C₆₇	—	4	4	no	—
227	C₂×Q₈×C₆₇	—	4	16	no	—
228	C₂×Q₈×C₆₇	—	4	32	no	—
229	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
230	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
231	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
232	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
233	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
234	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
235	C₂×D₄×C₆₇	—	2	8	no	—
236	C₂×D₄×C₆₇	—	2	8	no	—
237	C₂×D₄×C₆₇	—	4	16	no	—
238	C₂×D₄×C₆₇	—	4	4	no	—
239	C₂×D₄×C₆₇	—	4	8	no	—
240	C₂×D₄×C₆₇	—	4	2	no	—
241	C₂×D₄×C₆₇	—	4	8	no	—
242	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
243	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
244	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
245	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
246	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
247	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
248	C₂²×C₄×C₆₇	—	2	4	no	—
249	C₂²×C₄×C₆₇	—	2	4	no	—
250	C₂²×C₄×C₆₇	—	2	8	no	—
251	C₂²×C₄×C₆₇	—	4	8	no	—
252	C₂²×C₄×C₆₇	—	4	2	no	—
253	C₂²×C₄×C₆₇	—	4	8	no	—
254	C₂²×C₄×C₆₇	—	4	4	no	—
255	C₂²×C₄×C₆₇	—	4	8	no	—
256	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
257	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
258	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
259	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
260	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
261	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
262	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
263	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
264	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
265	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
267	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
268	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
269	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
270	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
271	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
272	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
273	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
274	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
275	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
276	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
277	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
278	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
279	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
280	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
281	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
282	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
283	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
284	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
285	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
286	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
287	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
288	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
289	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
290	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
291	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
292	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
293	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
294	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
295	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
296	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
297	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
298	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
299	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
300	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
301	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
302	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
303	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
304	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
305	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
306	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
307	C₄²×C₆₇	C₂⁴	1	4	yes	—
308	C₄²×C₆₇	C₂⁴	1	4	yes	—
309	C₄²×C₆₇	C₂⁴	1	4	yes	—
310	C₄²×C₆₇	C₂⁴	1	4	yes	—
311	C₄²×C₆₇	C₂⁴	1	16	yes	—
312	C₄²×C₆₇	C₂⁴	1	8	yes	—
313	C₄²×C₆₇	C₂⁴	1	16	yes	—
314	C₂³×C₆₇	—	2	2	no	—
315	C₂³×C₆₇	—	2	2	no	—
316	C₂³×C₆₇	—	2	4	no	—
317	C₂³×C₆₇	—	2	8	no	—
318	C₂³×C₆₇	—	2	4	no	—
319	C₂³×C₆₇	—	2	8	no	—
320	C₂³×C₆₇	—	2	8	no	—
321	C₂³×C₆₇	—	2	4	no	—
322	C₂³×C₆₇	—	2	16	no	—
323	C₂³×C₆₇	—	4	8	no	—
324	C₂³×C₆₇	—	4	8	no	—
325	C₂³×C₆₇	—	8	16	no	—
326	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
327	C₂³×C₆₇	C₂⁵	1	8	yes	—
328	C₂³×C₆₇	C₂⁵	1	4	yes	—
329	C₂³×C₆₇	C₂⁵	1	2	yes	—
330	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
331	Q₈×C₆₇	—	2	16	no	—
332	Q₈×C₆₇	—	2	16	no	—
333	Q₈×C₆₇	—	4	16	no	—
334	Q₈×C₆₇	—	4	8	no	—
335	Q₈×C₆₇	—	4	4	no	—
336	Q₈×C₆₇	—	4	8	no	—
337	D₄×C₆₇	—	2	16	no	—
338	D₄×C₆₇	—	2	16	no	—
339	D₄×C₆₇	—	4	8	no	—
340	D₄×C₆₇	—	4	16	no	—
341	D₄×C₆₇	—	4	4	no	—
342	D₄×C₆₇	—	4	16	no	—
343	C₂×C₄×C₆₇	—	2	4	no	—
344	C₂×C₄×C₆₇	—	2	2	no	—
345	C₂×C₄×C₆₇	—	2	8	no	—
346	C₂×C₄×C₆₇	—	2	8	no	—
347	C₂×C₄×C₆₇	—	2	8	no	—
348	C₂×C₄×C₆₇	—	2	2	no	—
349	C₂×C₄×C₆₇	—	2	2	no	—
350	C₂×C₄×C₆₇	—	2	4	no	—
351	C₂×C₄×C₆₇	—	2	4	no	—
352	C₂×C₄×C₆₇	—	2	8	no	—
353	C₂×C₄×C₆₇	—	2	2	no	—
354	C₂×C₄×C₆₇	—	2	2	no	—
355	C₂×C₄×C₆₇	—	2	8	no	—
356	C₂×C₄×C₆₇	—	2	16	no	—
357	C₂×C₄×C₆₇	—	4	16	no	—
358	C₂×C₄×C₆₇	—	4	4	no	—
359	C₂×C₄×C₆₇	—	4	16	no	—
360	C₂×C₄×C₆₇	—	4	8	no	—
361	C₂×C₄×C₆₇	—	4	16	no	—
362	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
363	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
364	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
365	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
366	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
367	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
368	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
369	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
370	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
371	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
372	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
373	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
374	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
375	C₂²×C₆₇	—	2	2	no	—
376	C₂²×C₆₇	—	2	2	no	—
377	C₂²×C₆₇	—	2	8	no	—
378	C₂²×C₆₇	—	2	4	no	—
379	C₂²×C₆₇	—	2	8	no	—
380	C₂²×C₆₇	—	2	4	no	—
381	C₂²×C₆₇	—	2	8	no	—
382	C₂²×C₆₇	—	2	8	no	—
383	C₂²×C₆₇	—	2	4	no	—
384	C₂²×C₆₇	—	4	8	no	—
385	C₂²×C₆₇	—	4	16	no	—
386	C₂²×C₆₇	—	8	16	no	—
387	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
388	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	4	yes	—
389	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
390	C₂²×C₆₇	#585	1	2	yes	—
391	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
392	C₄×C₆₇	—	2	4	no	—
393	C₄×C₆₇	—	2	2	no	—
394	C₄×C₆₇	—	2	8	no	—
395	C₄×C₆₇	—	2	8	no	—
396	C₄×C₆₇	—	2	2	no	—
397	C₄×C₆₇	—	2	2	no	—
398	C₄×C₆₇	—	2	4	no	—
399	C₄×C₆₇	—	2	4	no	—
400	C₄×C₆₇	—	2	8	no	—
401	C₄×C₆₇	—	2	2	no	—
402	C₄×C₆₇	—	2	2	no	—
403	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
404	C₂×#3561	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
405	C₂×C₆₇	—	2	4	no	—
406	C₂×C₆₇	—	4	4	no	—
407	C₂×C₆₇	—	4	8	no	—
408	C₂×C₆₇	#3561	1	4	yes	direct product factor
409	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
410	C₂×C₆₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
411	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
412	#3561	C₂×C₆₇	1	32	yes	direct product factor
413	C₂×#563	C₂×C₆₇	1	2	yes	complemented
414	C₂×#562	C₂×C₆₇	1	2	yes	complemented
415	C₂×#556	C₂×C₆₇	1	2	yes	complemented
416	C₂×#551	C₂×C₆₇	1	4	yes	complemented
417	C₂×D₄×Q₈	C₂×C₆₇	1	2	yes	complemented
418	C₂×#548	C₂×C₆₇	1	2	yes	complemented
419	C₂×#546	C₂×C₆₇	1	2	yes	complemented
420	C₂×#543	C₂×C₆₇	1	2	yes	complemented
421	C₂×#542	C₂×C₆₇	1	2	yes	complemented
422	C₂×#542	C₂×C₆₇	1	4	yes	complemented
423	C₂×#538	C₂×C₆₇	1	2	yes	complemented
424	C₂×#536	C₂×C₆₇	1	2	yes	complemented
425	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
426	C₂×#529	C₂×C₆₇	1	1	yes	characteristic, complemented
427	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
428	C₆₇	C₂×#3561	1	1	yes	characteristic, direct product factor, Sylow
429	C₂²×#41	C₂²×C₆₇	1	2	yes	—
430	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
431	C₂³×Q₈	C₂²×C₆₇	1	2	yes	complemented
432	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
433	#563	C₂²×C₆₇	1	32	yes	complemented
434	#562	C₂²×C₆₇	1	32	yes	complemented
435	#556	C₂²×C₆₇	1	32	yes	complemented
436	#551	C₂²×C₆₇	1	64	yes	complemented
437	D₄×Q₈	C₂²×C₆₇	1	32	yes	complemented
438	#548	C₂²×C₆₇	1	32	yes	complemented
439	#546	C₂²×C₆₇	1	32	yes	complemented
440	#543	C₂²×C₆₇	1	32	yes	complemented
441	#542	C₂²×C₆₇	1	32	yes	complemented
442	#542	C₂²×C₆₇	1	64	yes	complemented
443	#538	C₂²×C₆₇	1	32	yes	complemented
444	#536	C₂²×C₆₇	1	32	yes	complemented
445	#533	C₂²×C₆₇	1	16	yes	complemented
446	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
447	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
448	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
449	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
450	#529	C₂²×C₆₇	1	16	yes	complemented
451	C₂×#126	C₂²×C₆₇	1	2	yes	complemented
452	C₂×#126	C₂²×C₆₇	1	4	yes	complemented
453	C₂×#126	C₂²×C₆₇	1	4	yes	complemented
454	C₂×#126	C₂²×C₆₇	1	4	yes	complemented
455	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
456	C₂×#125	C₂²×C₆₇	1	2	yes	complemented
457	C₂×#124	C₂²×C₆₇	1	2	yes	—
458	C₂×#124	C₂²×C₆₇	1	2	yes	—
459	C₂×#124	C₂²×C₆₇	1	2	yes	—
460	C₂×#124	C₂²×C₆₇	1	2	yes	—
461	C₂×#124	C₂²×C₆₇	1	2	yes	—
462	C₂×#124	C₂²×C₆₇	1	16	yes	complemented
463	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic, complemented
464	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
465	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
466	C₂×#123	C₂²×C₆₇	1	2	yes	—
467	C₂×#123	C₂²×C₆₇	1	4	yes	—
468	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
469	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
470	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
471	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
472	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
473	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
474	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
475	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
476	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
477	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
478	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
479	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
480	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
481	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
482	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
483	C₂×#121	C₂²×C₆₇	1	2	yes	—
484	C₂×#121	C₂²×C₆₇	1	4	yes	—
485	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
486	C₂×#120	C₂²×C₆₇	1	4	yes	—
487	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
488	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
489	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
490	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
491	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
492	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
493	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
494	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
495	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
496	C₂×#117	C₂²×C₆₇	1	2	yes	—
497	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
498	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
499	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
500	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
501	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
502	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
503	C₂⁵	C₂³×C₆₇	1	2	yes	—
504	C₂×#41	—	4	8	no	—
505	C₂×#41	C₂³×C₆₇	1	16	yes	—
506	C₂²×Q₈	—	4	4	no	—
507	C₂²×Q₈	—	4	8	no	—
508	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
509	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
510	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
511	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
512	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
513	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
514	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
515	C₂²×Q₈	C₂³×C₆₇	1	16	yes	complemented
516	C₂²×D₄	—	4	4	no	—
517	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
518	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
519	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
520	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
521	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
522	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
523	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
524	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
525	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
526	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
527	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
528	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
529	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
530	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
531	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
532	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
533	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
534	C₄⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
535	C₄⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
536	#126	C₂³×C₆₇	1	16	yes	complemented
537	#126	C₂³×C₆₇	1	32	yes	complemented
538	#126	C₂³×C₆₇	1	32	yes	complemented
539	#126	C₂³×C₆₇	1	32	yes	complemented
540	#125	C₂³×C₆₇	1	8	yes	complemented
541	#125	C₂³×C₆₇	1	16	yes	complemented
542	#124	C₂³×C₆₇	1	16	yes	—
543	#124	C₂³×C₆₇	1	16	yes	—
544	#124	C₂³×C₆₇	1	16	yes	—
545	#124	C₂³×C₆₇	1	16	yes	—
546	#124	C₂³×C₆₇	1	16	yes	—
547	#124	C₂³×C₆₇	1	8	yes	complemented
548	#124	C₂³×C₆₇	1	32	yes	complemented
549	#124	C₂³×C₆₇	1	32	yes	complemented
550	#123	C₂³×C₆₇	1	16	yes	—
551	#123	C₂³×C₆₇	1	32	yes	—
552	#123	C₂³×C₆₇	1	32	yes	complemented
553	#123	C₂³×C₆₇	1	32	yes	complemented
554	#123	C₂³×C₆₇	1	32	yes	complemented
555	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
556	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
557	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
558	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	—
559	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
560	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
561	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
562	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
563	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
564	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
565	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
566	#121	C₂³×C₆₇	1	8	yes	—
567	#121	C₂³×C₆₇	1	16	yes	—
568	#121	C₂³×C₆₇	1	32	yes	—
569	#121	C₂³×C₆₇	1	32	yes	complemented
570	#120	C₂³×C₆₇	1	32	yes	—
571	C₄×Q₈	C₂³×C₆₇	1	8	yes	complemented
572	C₄×Q₈	C₂³×C₆₇	1	16	yes	complemented
573	C₄×Q₈	C₂³×C₆₇	1	16	yes	complemented
574	C₄×Q₈	C₂³×C₆₇	1	16	yes	complemented
575	C₄×D₄	C₂³×C₆₇	1	16	yes	—
576	C₄×D₄	C₂³×C₆₇	1	32	yes	—
577	C₄×D₄	C₂³×C₆₇	1	16	yes	—
578	C₄×D₄	C₂³×C₆₇	1	8	yes	complemented
579	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
580	#117	C₂³×C₆₇	1	16	yes	—
581	#117	C₂³×C₆₇	1	16	yes	complemented
582	#117	C₂³×C₆₇	1	32	yes	complemented
583	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
584	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
585	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
586	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
587	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
588	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
589	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
590	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
591	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
592	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
593	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
594	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
595	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
596	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
597	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
598	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
599	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
600	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
601	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
602	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
603	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
604	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
605	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
606	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
607	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
608	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
609	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
610	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
611	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
612	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
613	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
614	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
615	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
616	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
617	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
618	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
619	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
620	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
621	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
622	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
623	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
624	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
625	C₂⁴	—	2	8	no	—
626	C₂⁴	—	4	4	no	—
627	C₂⁴	—	4	4	no	—
628	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
629	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
630	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
631	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
632	#41	—	4	32	no	—
633	C₂×Q₈	—	2	8	no	—
634	C₂×Q₈	—	2	8	no	—
635	C₂×Q₈	—	4	8	no	—
636	C₂×Q₈	—	4	4	no	—
637	C₂×Q₈	—	4	2	no	—
638	C₂×Q₈	—	4	4	no	—
639	C₂×Q₈	—	4	16	no	—
640	C₂×Q₈	—	4	32	no	—
641	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
642	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
643	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
644	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
645	C₂×Q₈	C₂⁴×C₆₇	1	16	yes	—
646	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
647	C₂×D₄	—	2	8	no	—
648	C₂×D₄	—	2	8	no	—
649	C₂×D₄	—	4	16	no	—
650	C₂×D₄	—	4	4	no	—
651	C₂×D₄	—	4	8	no	—
652	C₂×D₄	—	4	8	no	—
653	C₂×D₄	—	4	2	no	—
654	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
655	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
656	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
657	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
658	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
659	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
660	C₂²×C₄	—	2	4	no	—
661	C₂²×C₄	—	2	4	no	—
662	C₂²×C₄	—	2	8	no	—
663	C₂²×C₄	—	4	8	no	—
664	C₂²×C₄	—	4	4	no	—
665	C₂²×C₄	—	4	8	no	—
666	C₂²×C₄	—	4	8	no	—
667	C₂²×C₄	—	4	2	no	—
668	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
669	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
670	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
671	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
672	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
673	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
674	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
675	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
676	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
677	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
678	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
679	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
680	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
681	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
682	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
683	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
684	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
685	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
686	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
687	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
688	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
689	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
690	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
691	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
692	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
693	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
694	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
695	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
696	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
697	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
698	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
699	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
700	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
701	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
702	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
703	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
704	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
705	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
706	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
707	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
708	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
709	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
710	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
711	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
712	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
713	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
714	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
715	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
716	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
717	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
718	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
719	C₄²	C₂⁴×C₆₇	1	4	yes	—
720	C₄²	C₂⁴×C₆₇	1	4	yes	—
721	C₄²	C₂⁴×C₆₇	1	4	yes	—
722	C₄²	C₂⁴×C₆₇	1	4	yes	—
723	C₄²	C₂⁴×C₆₇	1	16	yes	—
724	C₄²	C₂⁴×C₆₇	1	8	yes	—
725	C₄²	C₂⁴×C₆₇	1	16	yes	—
726	C₂³	—	2	2	no	—
727	C₂³	—	2	2	no	—
728	C₂³	—	2	4	no	—
729	C₂³	—	2	8	no	—
730	C₂³	—	2	4	no	—
731	C₂³	—	2	8	no	—
732	C₂³	—	2	8	no	—
733	C₂³	—	2	4	no	—
734	C₂³	—	2	16	no	—
735	C₂³	—	4	8	no	—
736	C₂³	—	4	8	no	—
737	C₂³	—	8	16	no	—
738	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
739	C₂³	C₂⁵×C₆₇	1	8	yes	—
740	C₂³	C₂⁵×C₆₇	1	4	yes	—
741	C₂³	C₂⁵×C₆₇	1	2	yes	—
742	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
743	Q₈	—	2	16	no	—
744	Q₈	—	2	16	no	—
745	Q₈	—	4	16	no	—
746	Q₈	—	4	8	no	—
747	Q₈	—	4	4	no	—
748	Q₈	—	4	8	no	—
749	D₄	—	2	16	no	—
750	D₄	—	2	16	no	—
751	D₄	—	4	8	no	—
752	D₄	—	4	16	no	—
753	D₄	—	4	16	no	—
754	D₄	—	4	4	no	—
755	C₂×C₄	—	2	4	no	—
756	C₂×C₄	—	2	2	no	—
757	C₂×C₄	—	2	8	no	—
758	C₂×C₄	—	2	8	no	—
759	C₂×C₄	—	2	8	no	—
760	C₂×C₄	—	2	2	no	—
761	C₂×C₄	—	2	2	no	—
762	C₂×C₄	—	2	4	no	—
763	C₂×C₄	—	2	4	no	—
764	C₂×C₄	—	2	8	no	—
765	C₂×C₄	—	2	2	no	—
766	C₂×C₄	—	2	2	no	—
767	C₂×C₄	—	2	8	no	—
768	C₂×C₄	—	2	16	no	—
769	C₂×C₄	—	4	16	no	—
770	C₂×C₄	—	4	8	no	—
771	C₂×C₄	—	4	16	no	—
772	C₂×C₄	—	4	16	no	—
773	C₂×C₄	—	4	4	no	—
774	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
775	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
776	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
777	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
778	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
779	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
780	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
781	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
782	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
783	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
784	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
785	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
786	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
787	C₂²	—	2	8	no	—
788	C₂²	—	2	2	no	—
789	C₂²	—	2	2	no	—
790	C₂²	—	2	4	no	—
791	C₂²	—	2	8	no	—
792	C₂²	—	2	4	no	—
793	C₂²	—	2	8	no	—
794	C₂²	—	2	8	no	—
795	C₂²	—	2	4	no	—
796	C₂²	—	4	16	no	—
797	C₂²	—	4	8	no	—
798	C₂²	—	8	16	no	—
799	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
800	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	4	yes	—
801	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
802	C₂²	#585×C₆₇	1	2	yes	—
803	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
804	C₄	—	2	4	no	—
805	C₄	—	2	2	no	—
806	C₄	—	2	8	no	—
807	C₄	—	2	8	no	—
808	C₄	—	2	2	no	—
809	C₄	—	2	2	no	—
810	C₄	—	2	4	no	—
811	C₄	—	2	4	no	—
812	C₄	—	2	8	no	—
813	C₄	—	2	2	no	—
814	C₄	—	2	2	no	—
815	C₄	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
816	C₂	—	2	4	no	—
817	C₂	—	4	8	no	—
818	C₂	—	4	4	no	—
819	C₂	C₆₇×#3561	1	4	yes	direct product factor
820	C₂	C₂²×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
821	C₂	C₂²×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
822	C₂	C₂×#585×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3561	C₂	direct (N × H)
2. C₂×#563×C₆₇	C₂	semidirect
3. C₂×#562×C₆₇	C₂	semidirect
4. C₂×#556×C₆₇	C₂	semidirect
5. C₂×#551×C₆₇	C₂	semidirect
6. C₂×D₄×Q₈×C₆₇	C₂	semidirect
7. C₂×#548×C₆₇	C₂	semidirect
8. C₂×#546×C₆₇	C₂	semidirect
9. C₂×#543×C₆₇	C₂	semidirect
10. C₂×#542×C₆₇	C₂	semidirect
11. C₂×#542×C₆₇	C₂	semidirect
12. C₂×#538×C₆₇	C₂	semidirect
13. C₂×#536×C₆₇	C₂	semidirect
14. C₂×#533×C₆₇	C₂	semidirect
15. C₂×#529×C₆₇	C₂	semidirect
16. C₂²×#124×C₆₇	C₂	semidirect
19. C₂³×Q₈×C₆₇	C₂²	semidirect
21. #563×C₆₇	C₂²	semidirect
22. #562×C₆₇	C₂²	semidirect
23. #556×C₆₇	C₂²	semidirect
24. #551×C₆₇	C₂²	semidirect
25. D₄×Q₈×C₆₇	C₂²	semidirect
26. #548×C₆₇	C₂²	semidirect
27. #546×C₆₇	C₂²	semidirect
28. #543×C₆₇	C₂²	semidirect
29. #542×C₆₇	C₂²	semidirect
30. #542×C₆₇	C₂²	semidirect
31. #538×C₆₇	C₂²	semidirect
32. #536×C₆₇	C₂²	semidirect
33. #533×C₆₇	C₂²	semidirect
34. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
35. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
36. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
37. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
38. #529×C₆₇	C₂²	semidirect
39. C₂×#126×C₆₇	C₂²	semidirect
40. C₂×#126×C₆₇	C₂²	semidirect
41. C₂×#126×C₆₇	C₂²	semidirect
42. C₂×#126×C₆₇	C₂²	semidirect
43. C₂×#125×C₆₇	C₂²	semidirect
44. C₂×#125×C₆₇	C₂²	semidirect
50. C₂×#124×C₆₇	C₂²	semidirect
51. C₂×#124×C₆₇	C₂²	semidirect
52. C₂×#124×C₆₇	C₂²	semidirect
53. C₂×#124×C₆₇	C₂²	semidirect
56. C₂×#123×C₆₇	C₂²	semidirect
57. C₂×#123×C₆₇	C₂²	semidirect
58. C₂×#123×C₆₇	C₂²	semidirect
63. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
64. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
65. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
66. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
67. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
68. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
69. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
73. C₂×#121×C₆₇	C₂²	semidirect
75. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
76. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
77. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
78. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
82. C₂×C₄×D₄×C₆₇	C₂²	semidirect
83. C₂×C₄×D₄×C₆₇	C₂²	semidirect
85. C₂×#117×C₆₇	C₂²	semidirect
86. C₂×#117×C₆₇	C₂²	semidirect
103. C₂²×Q₈×C₆₇	C₂³	semidirect
120. C₄⋊Q₈×C₆₇	C₂³	semidirect
121. C₄⋊Q₈×C₆₇	C₂³	semidirect
122. C₄⋊Q₈×C₆₇	C₂³	semidirect
123. C₄⋊Q₈×C₆₇	C₂³	semidirect
124. #126×C₆₇	C₂³	semidirect
125. #126×C₆₇	C₂³	semidirect
126. #126×C₆₇	C₂³	semidirect
127. #126×C₆₇	C₂³	semidirect
128. #125×C₆₇	C₂³	semidirect
129. #125×C₆₇	C₂³	semidirect
135. #124×C₆₇	C₂³	semidirect
136. #124×C₆₇	C₂³	semidirect
137. #124×C₆₇	C₂³	semidirect
140. #123×C₆₇	C₂³	semidirect
141. #123×C₆₇	C₂³	semidirect
142. #123×C₆₇	C₂³	semidirect
147. C₂²⋊Q₈×C₆₇	C₂³	semidirect
148. C₂²⋊Q₈×C₆₇	C₂³	semidirect
149. C₂²⋊Q₈×C₆₇	C₂³	semidirect
150. C₂²⋊Q₈×C₆₇	C₂³	semidirect
151. C₂²⋊Q₈×C₆₇	C₂³	semidirect
152. C₂²⋊Q₈×C₆₇	C₂³	semidirect
153. C₂²⋊Q₈×C₆₇	C₂³	semidirect
157. #121×C₆₇	C₂³	semidirect
159. C₄×Q₈×C₆₇	C₂³	semidirect
160. C₄×Q₈×C₆₇	C₂³	semidirect
161. C₄×Q₈×C₆₇	C₂³	semidirect
162. C₄×Q₈×C₆₇	C₂³	semidirect
166. C₄×D₄×C₆₇	C₂³	semidirect
167. C₄×D₄×C₆₇	C₂³	semidirect
169. #117×C₆₇	C₂³	semidirect
170. #117×C₆₇	C₂³	semidirect
404. C₂×#3561	C₆₇	direct (N × H)
408. C₂×C₆₇	#3561	direct (N × H)
412. #3561	C₂×C₆₇	direct (N × H)
413. C₂×#563	C₂×C₆₇	semidirect
414. C₂×#562	C₂×C₆₇	semidirect
415. C₂×#556	C₂×C₆₇	semidirect
416. C₂×#551	C₂×C₆₇	semidirect
417. C₂×D₄×Q₈	C₂×C₆₇	semidirect
418. C₂×#548	C₂×C₆₇	semidirect
419. C₂×#546	C₂×C₆₇	semidirect
420. C₂×#543	C₂×C₆₇	semidirect
421. C₂×#542	C₂×C₆₇	semidirect
422. C₂×#542	C₂×C₆₇	semidirect
423. C₂×#538	C₂×C₆₇	semidirect
424. C₂×#536	C₂×C₆₇	semidirect
425. C₂×#533	C₂×C₆₇	semidirect
426. C₂×#529	C₂×C₆₇	semidirect
427. C₂²×#124	C₂×C₆₇	semidirect
428. C₆₇	C₂×#3561	direct (N × H)
431. C₂³×Q₈	C₂²×C₆₇	semidirect
433. #563	C₂²×C₆₇	semidirect
434. #562	C₂²×C₆₇	semidirect
435. #556	C₂²×C₆₇	semidirect
436. #551	C₂²×C₆₇	semidirect
437. D₄×Q₈	C₂²×C₆₇	semidirect
438. #548	C₂²×C₆₇	semidirect
439. #546	C₂²×C₆₇	semidirect
440. #543	C₂²×C₆₇	semidirect
441. #542	C₂²×C₆₇	semidirect
442. #542	C₂²×C₆₇	semidirect
443. #538	C₂²×C₆₇	semidirect
444. #536	C₂²×C₆₇	semidirect
445. #533	C₂²×C₆₇	semidirect
446. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
447. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
448. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
449. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
450. #529	C₂²×C₆₇	semidirect
451. C₂×#126	C₂²×C₆₇	semidirect
452. C₂×#126	C₂²×C₆₇	semidirect
453. C₂×#126	C₂²×C₆₇	semidirect
454. C₂×#126	C₂²×C₆₇	semidirect
455. C₂×#125	C₂²×C₆₇	semidirect
456. C₂×#125	C₂²×C₆₇	semidirect
462. C₂×#124	C₂²×C₆₇	semidirect
463. C₂×#124	C₂²×C₆₇	semidirect
464. C₂×#124	C₂²×C₆₇	semidirect
465. C₂×#124	C₂²×C₆₇	semidirect
468. C₂×#123	C₂²×C₆₇	semidirect
469. C₂×#123	C₂²×C₆₇	semidirect
470. C₂×#123	C₂²×C₆₇	semidirect
475. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
476. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
477. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
478. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
479. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
480. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
481. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
485. C₂×#121	C₂²×C₆₇	semidirect
487. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
488. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
489. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
490. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
494. C₂×C₄×D₄	C₂²×C₆₇	semidirect
495. C₂×C₄×D₄	C₂²×C₆₇	semidirect
497. C₂×#117	C₂²×C₆₇	semidirect
498. C₂×#117	C₂²×C₆₇	semidirect
515. C₂²×Q₈	C₂³×C₆₇	semidirect
532. C₄⋊Q₈	C₂³×C₆₇	semidirect
533. C₄⋊Q₈	C₂³×C₆₇	semidirect
534. C₄⋊Q₈	C₂³×C₆₇	semidirect
535. C₄⋊Q₈	C₂³×C₆₇	semidirect
536. #126	C₂³×C₆₇	semidirect
537. #126	C₂³×C₆₇	semidirect
538. #126	C₂³×C₆₇	semidirect
539. #126	C₂³×C₆₇	semidirect
540. #125	C₂³×C₆₇	semidirect
541. #125	C₂³×C₆₇	semidirect
547. #124	C₂³×C₆₇	semidirect
548. #124	C₂³×C₆₇	semidirect
549. #124	C₂³×C₆₇	semidirect
552. #123	C₂³×C₆₇	semidirect
553. #123	C₂³×C₆₇	semidirect
554. #123	C₂³×C₆₇	semidirect
559. C₂²⋊Q₈	C₂³×C₆₇	semidirect
560. C₂²⋊Q₈	C₂³×C₆₇	semidirect
561. C₂²⋊Q₈	C₂³×C₆₇	semidirect
562. C₂²⋊Q₈	C₂³×C₆₇	semidirect
563. C₂²⋊Q₈	C₂³×C₆₇	semidirect
564. C₂²⋊Q₈	C₂³×C₆₇	semidirect
565. C₂²⋊Q₈	C₂³×C₆₇	semidirect
569. #121	C₂³×C₆₇	semidirect
571. C₄×Q₈	C₂³×C₆₇	semidirect
572. C₄×Q₈	C₂³×C₆₇	semidirect
573. C₄×Q₈	C₂³×C₆₇	semidirect
574. C₄×Q₈	C₂³×C₆₇	semidirect
578. C₄×D₄	C₂³×C₆₇	semidirect
579. C₄×D₄	C₂³×C₆₇	semidirect
581. #117	C₂³×C₆₇	semidirect
582. #117	C₂³×C₆₇	semidirect
819. C₂	C₆₇×#3561	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3561	2
C₂×#563×C₆₇	2
C₂×#562×C₆₇	2
C₂×#556×C₆₇	2
C₂×#551×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₂×#548×C₆₇	2
C₂×#546×C₆₇	2
C₂×#543×C₆₇	2
C₂×#542×C₆₇	2
C₂×#538×C₆₇	2
C₂×#536×C₆₇	2
C₂×#533×C₆₇	2
C₂×#529×C₆₇	2
C₂²×#124×C₆₇	2
C₂×#3561	67

Subgroups — C2×C67×#3561 #30741982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3561 #30741982