Subgroups — C₂×C₆₇×#3506 #30742020

Order 17152.

7994 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	32	yes	maximal, direct product factor
2	C₂×#559×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×#556×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#555×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#553×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#550×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#547×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#538×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂×#535×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#525×C₆₇	C₂	1	4	yes	maximal, complemented
13	C₂×#520×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₆₇	C₂²	1	2	yes	—
16	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
17	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
19	#559×C₆₇	C₂²	1	64	yes	complemented
20	#556×C₆₇	C₂²	1	32	yes	complemented
21	#555×C₆₇	C₂²	1	32	yes	complemented
22	#553×C₆₇	C₂²	1	64	yes	complemented
23	#550×C₆₇	C₂²	1	32	yes	complemented
24	#547×C₆₇	C₂²	1	32	yes	complemented
25	#538×C₆₇	C₂²	1	64	yes	complemented
26	#535×C₆₇	C₂²	1	32	yes	complemented
27	#533×C₆₇	C₂²	1	16	yes	complemented
28	#532×C₆₇	C₂²	1	16	yes	complemented
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₂×#127×C₆₇	C₂²	1	4	yes	—
33	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×#126×C₆₇	C₂²	1	8	yes	complemented
35	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#124×C₆₇	C₂²	1	4	yes	—
37	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
38	C₂×#124×C₆₇	C₂²	1	8	yes	complemented
39	#525×C₆₇	C₂²	1	64	yes	complemented
40	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
41	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
42	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
43	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
44	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
45	C₂×#121×C₆₇	C₂²	1	2	yes	—
46	C₂×#121×C₆₇	C₂²	1	2	yes	—
47	C₂×#121×C₆₇	C₂²	1	4	yes	—
48	C₂×#121×C₆₇	C₂²	1	4	yes	—
49	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
50	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
51	C₂×#120×C₆₇	C₂²	1	8	yes	—
52	#520×C₆₇	C₂²	1	32	yes	complemented
53	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
54	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
55	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
56	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
57	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
58	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
59	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
60	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
61	C₂×C₄×D₄×C₆₇	C₂²	1	16	yes	complemented
62	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
63	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
64	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
65	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
66	C₂×#117×C₆₇	C₂²	1	2	yes	—
67	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
68	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
69	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
70	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
71	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
73	C₂⁵×C₆₇	C₂³	1	2	yes	—
74	C₂×#41×C₆₇	—	2	16	no	—
75	C₂×#41×C₆₇	—	4	16	no	—
76	C₂×#41×C₆₇	C₂³	1	16	yes	—
77	C₂×#41×C₆₇	C₂³	1	32	yes	complemented
78	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
79	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
81	C₂²×D₄×C₆₇	—	2	8	no	—
82	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
83	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
84	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
85	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
87	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
88	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
89	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
90	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
91	C₂³×C₄×C₆₇	—	4	8	no	—
92	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
94	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
95	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
97	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
98	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
99	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
100	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
102	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
103	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
104	#127×C₆₇	C₂³	1	32	yes	—
105	#127×C₆₇	C₂³	1	8	yes	complemented
106	#126×C₆₇	C₂³	1	64	yes	complemented
107	#125×C₆₇	C₂³	1	16	yes	complemented
108	#124×C₆₇	C₂³	1	32	yes	—
109	#124×C₆₇	C₂³	1	16	yes	complemented
110	#124×C₆₇	C₂³	1	64	yes	complemented
111	#123×C₆₇	C₂³	1	64	yes	complemented
112	#123×C₆₇	C₂³	1	64	yes	complemented
113	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
114	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
115	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
116	#121×C₆₇	C₂³	1	16	yes	—
117	#121×C₆₇	C₂³	1	16	yes	—
118	#121×C₆₇	C₂³	1	32	yes	—
119	#121×C₆₇	C₂³	1	32	yes	—
120	#121×C₆₇	C₂³	1	32	yes	complemented
121	#121×C₆₇	C₂³	1	64	yes	complemented
122	#120×C₆₇	C₂³	1	64	yes	—
123	C₄×Q₈×C₆₇	C₂³	1	32	yes	complemented
124	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
125	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
126	C₄×D₄×C₆₇	C₂³	1	16	yes	—
127	C₄×D₄×C₆₇	C₂³	1	16	yes	—
128	C₄×D₄×C₆₇	C₂³	1	16	yes	—
129	C₄×D₄×C₆₇	C₂³	1	16	yes	—
130	C₄×D₄×C₆₇	C₂³	1	64	yes	—
131	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
132	C₄×D₄×C₆₇	C₂³	1	64	yes	complemented
133	C₄×D₄×C₆₇	C₂³	1	16	yes	complemented
134	C₄×D₄×C₆₇	C₂³	1	16	yes	complemented
135	#117×C₆₇	C₂³	1	16	yes	—
136	#117×C₆₇	C₂³	1	32	yes	complemented
137	#117×C₆₇	C₂³	1	32	yes	complemented
138	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
139	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
141	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
143	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
144	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
147	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
148	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	complemented
151	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
152	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
153	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
154	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
155	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
156	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
157	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
158	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
159	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
160	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
161	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
162	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
164	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
165	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
166	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
167	C₂⁴×C₆₇	—	2	8	no	—
168	C₂⁴×C₆₇	—	4	4	no	—
169	C₂⁴×C₆₇	—	4	4	no	—
170	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
171	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
172	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
174	#41×C₆₇	—	2	64	no	—
175	#41×C₆₇	—	4	64	no	—
176	C₂×Q₈×C₆₇	—	2	8	no	—
177	C₂×Q₈×C₆₇	—	2	8	no	—
178	C₂×Q₈×C₆₇	—	4	4	no	—
179	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
180	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
181	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
182	C₂×D₄×C₆₇	—	2	16	no	—
183	C₂×D₄×C₆₇	—	2	8	no	—
184	C₂×D₄×C₆₇	—	2	8	no	—
185	C₂×D₄×C₆₇	—	2	8	no	—
186	C₂×D₄×C₆₇	—	2	32	no	—
187	C₂×D₄×C₆₇	—	2	8	no	—
188	C₂×D₄×C₆₇	—	4	4	no	—
189	C₂×D₄×C₆₇	—	4	16	no	—
190	C₂×D₄×C₆₇	—	4	16	no	—
191	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
192	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
193	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
194	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
195	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
196	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
197	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
198	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
199	C₂²×C₄×C₆₇	—	2	16	no	—
200	C₂²×C₄×C₆₇	—	2	4	no	—
201	C₂²×C₄×C₆₇	—	2	16	no	—
202	C₂²×C₄×C₆₇	—	4	32	no	—
203	C₂²×C₄×C₆₇	—	4	4	no	—
204	C₂²×C₄×C₆₇	—	4	8	no	—
205	C₂²×C₄×C₆₇	—	4	2	no	—
206	C₂²×C₄×C₆₇	—	4	16	no	—
207	C₂²×C₄×C₆₇	#41	1	8	yes	complemented
208	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
210	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
211	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
214	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
215	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
217	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
218	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
219	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
220	C₂²×C₄×C₆₇	C₂⁴	1	32	yes	—
221	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
222	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
223	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
224	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
225	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
226	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
227	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
228	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
229	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
230	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
231	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
232	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
233	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
234	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
235	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
236	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
237	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
238	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
239	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
240	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
241	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
242	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
243	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
244	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
245	C₄²×C₆₇	C₂⁴	1	4	yes	—
246	C₄²×C₆₇	C₂⁴	1	4	yes	—
247	C₄²×C₆₇	C₂⁴	1	4	yes	—
248	C₄²×C₆₇	C₂⁴	1	4	yes	—
249	C₄²×C₆₇	C₂⁴	1	16	yes	—
250	C₄²×C₆₇	C₂⁴	1	32	yes	—
251	C₂³×C₆₇	—	2	2	no	—
252	C₂³×C₆₇	—	2	2	no	—
253	C₂³×C₆₇	—	2	8	no	—
254	C₂³×C₆₇	—	2	16	no	—
255	C₂³×C₆₇	—	2	8	no	—
256	C₂³×C₆₇	—	2	8	no	—
257	C₂³×C₆₇	—	2	8	no	—
258	C₂³×C₆₇	—	2	8	no	—
259	C₂³×C₆₇	—	2	8	no	—
260	C₂³×C₆₇	—	4	8	no	—
261	C₂³×C₆₇	—	4	8	no	—
262	C₂³×C₆₇	—	4	32	no	—
263	C₂³×C₆₇	C₂×#41	1	4	yes	—
264	C₂³×C₆₇	C₂⁵	1	8	yes	—
265	C₂³×C₆₇	C₂⁵	1	8	yes	—
266	C₂³×C₆₇	C₂⁵	1	2	yes	—
267	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
268	Q₈×C₆₇	—	2	16	no	—
269	Q₈×C₆₇	—	2	16	no	—
270	Q₈×C₆₇	—	4	8	no	—
271	D₄×C₆₇	—	2	32	no	—
272	D₄×C₆₇	—	2	16	no	—
273	D₄×C₆₇	—	2	16	no	—
274	D₄×C₆₇	—	2	16	no	—
275	D₄×C₆₇	—	2	16	no	—
276	D₄×C₆₇	—	4	8	no	—
277	D₄×C₆₇	—	4	32	no	—
278	D₄×C₆₇	—	4	32	no	—
279	C₂×C₄×C₆₇	—	2	8	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	8	no	—
282	C₂×C₄×C₆₇	—	2	8	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	2	8	no	—
286	C₂×C₄×C₆₇	—	2	32	no	—
287	C₂×C₄×C₆₇	—	2	8	no	—
288	C₂×C₄×C₆₇	—	2	32	no	—
289	C₂×C₄×C₆₇	—	4	8	no	—
290	C₂×C₄×C₆₇	—	4	16	no	—
291	C₂×C₄×C₆₇	—	4	4	no	—
292	C₂×C₄×C₆₇	—	4	32	no	—
293	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
294	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
295	C₂×C₄×C₆₇	C₂×#41	1	16	yes	complemented
296	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
297	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
298	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
299	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
300	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
301	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
302	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
303	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
304	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
305	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
306	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
307	C₂²×C₆₇	—	2	2	no	—
308	C₂²×C₆₇	—	2	2	no	—
309	C₂²×C₆₇	—	2	8	no	—
310	C₂²×C₆₇	—	2	8	no	—
311	C₂²×C₆₇	—	2	8	no	—
312	C₂²×C₆₇	—	2	8	no	—
313	C₂²×C₆₇	—	2	8	no	—
314	C₂²×C₆₇	—	2	8	no	—
315	C₂²×C₆₇	—	2	8	no	—
316	C₂²×C₆₇	—	4	32	no	—
317	C₂²×C₆₇	—	4	16	no	—
318	C₂²×C₆₇	—	4	16	no	—
319	C₂²×C₆₇	C₂²×#41	1	4	yes	—
320	C₂²×C₆₇	C₂²×#41	1	2	yes	—
321	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
322	C₂²×C₆₇	#585	1	2	yes	—
323	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
324	C₄×C₆₇	—	2	8	no	—
325	C₄×C₆₇	—	2	2	no	—
326	C₄×C₆₇	—	2	8	no	—
327	C₄×C₆₇	—	2	8	no	—
328	C₄×C₆₇	—	2	2	no	—
329	C₄×C₆₇	—	2	2	no	—
330	C₄×C₆₇	—	2	8	no	—
331	C₄×C₆₇	C₂²×#41	1	4	yes	—
332	C₄×C₆₇	C₂²×#41	1	4	yes	—
333	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
334	C₂×#3506	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
335	C₂×C₆₇	—	2	4	no	—
336	C₂×C₆₇	—	4	8	no	—
337	C₂×C₆₇	—	4	8	no	—
338	C₂×C₆₇	#3506	1	4	yes	direct product factor
339	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
340	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
341	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
342	#3506	C₂×C₆₇	1	32	yes	direct product factor
343	C₂×#559	C₂×C₆₇	1	4	yes	complemented
344	C₂×#556	C₂×C₆₇	1	2	yes	complemented
345	C₂×#555	C₂×C₆₇	1	2	yes	complemented
346	C₂×#553	C₂×C₆₇	1	4	yes	complemented
347	C₂×#550	C₂×C₆₇	1	2	yes	complemented
348	C₂×#547	C₂×C₆₇	1	2	yes	complemented
349	C₂×#538	C₂×C₆₇	1	4	yes	complemented
350	C₂×#535	C₂×C₆₇	1	2	yes	complemented
351	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
352	C₂×#532	C₂×C₆₇	1	1	yes	characteristic, complemented
353	C₂×#525	C₂×C₆₇	1	4	yes	complemented
354	C₂×#520	C₂×C₆₇	1	2	yes	complemented
355	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
356	C₆₇	C₂×#3506	1	1	yes	characteristic, direct product factor, Sylow
357	C₂²×#41	C₂²×C₆₇	1	2	yes	—
358	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
359	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
360	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
361	#559	C₂²×C₆₇	1	64	yes	complemented
362	#556	C₂²×C₆₇	1	32	yes	complemented
363	#555	C₂²×C₆₇	1	32	yes	complemented
364	#553	C₂²×C₆₇	1	64	yes	complemented
365	#550	C₂²×C₆₇	1	32	yes	complemented
366	#547	C₂²×C₆₇	1	32	yes	complemented
367	#538	C₂²×C₆₇	1	64	yes	complemented
368	#535	C₂²×C₆₇	1	32	yes	complemented
369	#533	C₂²×C₆₇	1	16	yes	complemented
370	#532	C₂²×C₆₇	1	16	yes	complemented
371	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
372	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
373	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
374	C₂×#127	C₂²×C₆₇	1	4	yes	—
375	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
376	C₂×#126	C₂²×C₆₇	1	8	yes	complemented
377	C₂×#125	C₂²×C₆₇	1	2	yes	complemented
378	C₂×#124	C₂²×C₆₇	1	4	yes	—
379	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
380	C₂×#124	C₂²×C₆₇	1	8	yes	complemented
381	#525	C₂²×C₆₇	1	64	yes	complemented
382	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
383	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
384	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
385	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
386	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
387	C₂×#121	C₂²×C₆₇	1	2	yes	—
388	C₂×#121	C₂²×C₆₇	1	2	yes	—
389	C₂×#121	C₂²×C₆₇	1	4	yes	—
390	C₂×#121	C₂²×C₆₇	1	4	yes	—
391	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
392	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
393	C₂×#120	C₂²×C₆₇	1	8	yes	—
394	#520	C₂²×C₆₇	1	32	yes	complemented
395	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
396	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
397	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
398	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
399	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
400	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
401	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
402	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
403	C₂×C₄×D₄	C₂²×C₆₇	1	16	yes	complemented
404	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
405	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
406	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
407	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
408	C₂×#117	C₂²×C₆₇	1	2	yes	—
409	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
410	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
411	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
412	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
413	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
414	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
415	C₂⁵	C₂³×C₆₇	1	2	yes	—
416	C₂×#41	—	2	16	no	—
417	C₂×#41	—	4	16	no	—
418	C₂×#41	C₂³×C₆₇	1	16	yes	—
419	C₂×#41	C₂³×C₆₇	1	32	yes	complemented
420	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
421	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
422	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
423	C₂²×D₄	—	2	8	no	—
424	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
425	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
426	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
427	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
428	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
429	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
430	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
431	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
432	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
433	C₂³×C₄	—	4	8	no	—
434	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
435	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
436	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
437	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
438	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
439	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
440	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
441	C₂³×C₄	C₂³×C₆₇	1	16	yes	—
442	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
443	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
444	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
445	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
446	#127	C₂³×C₆₇	1	32	yes	—
447	#127	C₂³×C₆₇	1	8	yes	complemented
448	#126	C₂³×C₆₇	1	64	yes	complemented
449	#125	C₂³×C₆₇	1	16	yes	complemented
450	#124	C₂³×C₆₇	1	32	yes	—
451	#124	C₂³×C₆₇	1	16	yes	complemented
452	#124	C₂³×C₆₇	1	64	yes	complemented
453	#123	C₂³×C₆₇	1	64	yes	complemented
454	#123	C₂³×C₆₇	1	64	yes	complemented
455	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
456	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
457	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
458	#121	C₂³×C₆₇	1	16	yes	—
459	#121	C₂³×C₆₇	1	16	yes	—
460	#121	C₂³×C₆₇	1	32	yes	—
461	#121	C₂³×C₆₇	1	32	yes	—
462	#121	C₂³×C₆₇	1	32	yes	complemented
463	#121	C₂³×C₆₇	1	64	yes	complemented
464	#120	C₂³×C₆₇	1	64	yes	—
465	C₄×Q₈	C₂³×C₆₇	1	32	yes	complemented
466	C₄×Q₈	C₂³×C₆₇	1	16	yes	complemented
467	C₄×Q₈	C₂³×C₆₇	1	16	yes	complemented
468	C₄×D₄	C₂³×C₆₇	1	16	yes	—
469	C₄×D₄	C₂³×C₆₇	1	16	yes	—
470	C₄×D₄	C₂³×C₆₇	1	16	yes	—
471	C₄×D₄	C₂³×C₆₇	1	16	yes	—
472	C₄×D₄	C₂³×C₆₇	1	64	yes	—
473	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
474	C₄×D₄	C₂³×C₆₇	1	64	yes	complemented
475	C₄×D₄	C₂³×C₆₇	1	16	yes	complemented
476	C₄×D₄	C₂³×C₆₇	1	16	yes	complemented
477	#117	C₂³×C₆₇	1	16	yes	—
478	#117	C₂³×C₆₇	1	32	yes	complemented
479	#117	C₂³×C₆₇	1	32	yes	complemented
480	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
481	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
483	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
484	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
485	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
486	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
487	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
488	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
489	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
490	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
491	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
492	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	complemented
493	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
494	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
495	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
496	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
497	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
498	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
499	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
500	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
501	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
502	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
503	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
504	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
505	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
506	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
507	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
508	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
509	C₂⁴	—	2	8	no	—
510	C₂⁴	—	4	4	no	—
511	C₂⁴	—	4	4	no	—
512	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
513	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
514	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
515	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
516	#41	—	2	64	no	—
517	#41	—	4	64	no	—
518	C₂×Q₈	—	2	8	no	—
519	C₂×Q₈	—	2	8	no	—
520	C₂×Q₈	—	4	4	no	—
521	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
522	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
523	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
524	C₂×D₄	—	2	16	no	—
525	C₂×D₄	—	2	8	no	—
526	C₂×D₄	—	2	8	no	—
527	C₂×D₄	—	2	8	no	—
528	C₂×D₄	—	2	32	no	—
529	C₂×D₄	—	2	8	no	—
530	C₂×D₄	—	4	16	no	—
531	C₂×D₄	—	4	4	no	—
532	C₂×D₄	—	4	16	no	—
533	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
534	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
535	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
536	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
537	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
538	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
539	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
540	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
541	C₂²×C₄	—	2	16	no	—
542	C₂²×C₄	—	2	4	no	—
543	C₂²×C₄	—	2	16	no	—
544	C₂²×C₄	—	4	32	no	—
545	C₂²×C₄	—	4	4	no	—
546	C₂²×C₄	—	4	8	no	—
547	C₂²×C₄	—	4	16	no	—
548	C₂²×C₄	—	4	2	no	—
549	C₂²×C₄	#41×C₆₇	1	8	yes	complemented
550	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
551	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
552	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
553	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
554	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
555	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
556	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
557	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
558	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
559	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
560	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
561	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
562	C₂²×C₄	C₂⁴×C₆₇	1	32	yes	—
563	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
564	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
565	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
566	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
567	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
568	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
569	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
570	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
571	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
572	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
573	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
574	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
575	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
576	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
577	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
578	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
579	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
580	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
581	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
582	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
583	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
584	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
585	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
586	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
587	C₄²	C₂⁴×C₆₇	1	4	yes	—
588	C₄²	C₂⁴×C₆₇	1	4	yes	—
589	C₄²	C₂⁴×C₆₇	1	4	yes	—
590	C₄²	C₂⁴×C₆₇	1	4	yes	—
591	C₄²	C₂⁴×C₆₇	1	16	yes	—
592	C₄²	C₂⁴×C₆₇	1	32	yes	—
593	C₂³	—	2	2	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	2	8	no	—
596	C₂³	—	2	16	no	—
597	C₂³	—	2	8	no	—
598	C₂³	—	2	8	no	—
599	C₂³	—	2	8	no	—
600	C₂³	—	2	8	no	—
601	C₂³	—	2	8	no	—
602	C₂³	—	4	8	no	—
603	C₂³	—	4	8	no	—
604	C₂³	—	4	32	no	—
605	C₂³	C₂×#41×C₆₇	1	4	yes	—
606	C₂³	C₂⁵×C₆₇	1	8	yes	—
607	C₂³	C₂⁵×C₆₇	1	8	yes	—
608	C₂³	C₂⁵×C₆₇	1	2	yes	—
609	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
610	Q₈	—	2	16	no	—
611	Q₈	—	2	16	no	—
612	Q₈	—	4	8	no	—
613	D₄	—	2	32	no	—
614	D₄	—	2	16	no	—
615	D₄	—	2	16	no	—
616	D₄	—	2	16	no	—
617	D₄	—	2	16	no	—
618	D₄	—	4	32	no	—
619	D₄	—	4	8	no	—
620	D₄	—	4	32	no	—
621	C₂×C₄	—	2	8	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	8	no	—
624	C₂×C₄	—	2	8	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	8	no	—
628	C₂×C₄	—	2	32	no	—
629	C₂×C₄	—	2	8	no	—
630	C₂×C₄	—	2	32	no	—
631	C₂×C₄	—	4	8	no	—
632	C₂×C₄	—	4	16	no	—
633	C₂×C₄	—	4	32	no	—
634	C₂×C₄	—	4	4	no	—
635	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
636	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
637	C₂×C₄	C₂×#41×C₆₇	1	16	yes	complemented
638	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
639	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
640	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
641	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
642	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
643	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
644	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
645	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
646	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
647	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
648	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
649	C₂²	—	2	2	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	8	no	—
652	C₂²	—	2	8	no	—
653	C₂²	—	2	8	no	—
654	C₂²	—	2	8	no	—
655	C₂²	—	2	8	no	—
656	C₂²	—	2	8	no	—
657	C₂²	—	2	8	no	—
658	C₂²	—	4	32	no	—
659	C₂²	—	4	16	no	—
660	C₂²	—	4	16	no	—
661	C₂²	C₂²×#41×C₆₇	1	4	yes	—
662	C₂²	C₂²×#41×C₆₇	1	2	yes	—
663	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
664	C₂²	#585×C₆₇	1	2	yes	—
665	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
666	C₄	—	2	8	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	8	no	—
669	C₄	—	2	8	no	—
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	8	no	—
673	C₄	C₂²×#41×C₆₇	1	4	yes	—
674	C₄	C₂²×#41×C₆₇	1	4	yes	—
675	C₄	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
676	C₂	—	2	4	no	—
677	C₂	—	4	8	no	—
678	C₂	—	4	8	no	—
679	C₂	C₆₇×#3506	1	4	yes	direct product factor
680	C₂	C₂³×#41×C₆₇	1	1	yes	characteristic
681	C₂	C₂²×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
682	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×#3506	C₂	direct (N × H)
2. C₂×#559×C₆₇	C₂	semidirect
3. C₂×#556×C₆₇	C₂	semidirect
4. C₂×#555×C₆₇	C₂	semidirect
5. C₂×#553×C₆₇	C₂	semidirect
6. C₂×#550×C₆₇	C₂	semidirect
7. C₂×#547×C₆₇	C₂	semidirect
8. C₂×#538×C₆₇	C₂	semidirect
9. C₂×#535×C₆₇	C₂	semidirect
10. C₂×#533×C₆₇	C₂	semidirect
11. C₂×#532×C₆₇	C₂	semidirect
12. C₂×#525×C₆₇	C₂	semidirect
13. C₂×#520×C₆₇	C₂	semidirect
14. C₂²×C₄×D₄×C₆₇	C₂	semidirect
16. C₂²×#41×C₆₇	C₂²	semidirect
19. #559×C₆₇	C₂²	semidirect
20. #556×C₆₇	C₂²	semidirect
21. #555×C₆₇	C₂²	semidirect
22. #553×C₆₇	C₂²	semidirect
23. #550×C₆₇	C₂²	semidirect
24. #547×C₆₇	C₂²	semidirect
25. #538×C₆₇	C₂²	semidirect
26. #535×C₆₇	C₂²	semidirect
27. #533×C₆₇	C₂²	semidirect
28. #532×C₆₇	C₂²	semidirect
29. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
30. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
31. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
33. C₂×#127×C₆₇	C₂²	semidirect
34. C₂×#126×C₆₇	C₂²	semidirect
35. C₂×#125×C₆₇	C₂²	semidirect
37. C₂×#124×C₆₇	C₂²	semidirect
38. C₂×#124×C₆₇	C₂²	semidirect
39. #525×C₆₇	C₂²	semidirect
40. C₂×#123×C₆₇	C₂²	semidirect
41. C₂×#123×C₆₇	C₂²	semidirect
44. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
49. C₂×#121×C₆₇	C₂²	semidirect
50. C₂×#121×C₆₇	C₂²	semidirect
52. #520×C₆₇	C₂²	semidirect
53. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
54. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
55. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
61. C₂×C₄×D₄×C₆₇	C₂²	semidirect
62. C₂×C₄×D₄×C₆₇	C₂²	semidirect
63. C₂×C₄×D₄×C₆₇	C₂²	semidirect
64. C₂×C₄×D₄×C₆₇	C₂²	semidirect
65. C₂×C₄×D₄×C₆₇	C₂²	semidirect
67. C₂×#117×C₆₇	C₂²	semidirect
68. C₂×#117×C₆₇	C₂²	semidirect
70. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
77. C₂×#41×C₆₇	C₂³	semidirect
101. C₄⋊Q₈×C₆₇	C₂³	semidirect
102. C₄⋊Q₈×C₆₇	C₂³	semidirect
103. C₄⋊Q₈×C₆₇	C₂³	semidirect
105. #127×C₆₇	C₂³	semidirect
106. #126×C₆₇	C₂³	semidirect
107. #125×C₆₇	C₂³	semidirect
109. #124×C₆₇	C₂³	semidirect
110. #124×C₆₇	C₂³	semidirect
111. #123×C₆₇	C₂³	semidirect
112. #123×C₆₇	C₂³	semidirect
115. C₂²⋊Q₈×C₆₇	C₂³	semidirect
120. #121×C₆₇	C₂³	semidirect
121. #121×C₆₇	C₂³	semidirect
123. C₄×Q₈×C₆₇	C₂³	semidirect
124. C₄×Q₈×C₆₇	C₂³	semidirect
125. C₄×Q₈×C₆₇	C₂³	semidirect
131. C₄×D₄×C₆₇	C₂³	semidirect
132. C₄×D₄×C₆₇	C₂³	semidirect
133. C₄×D₄×C₆₇	C₂³	semidirect
134. C₄×D₄×C₆₇	C₂³	semidirect
136. #117×C₆₇	C₂³	semidirect
137. #117×C₆₇	C₂³	semidirect
150. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
207. C₂²×C₄×C₆₇	#41	semidirect
295. C₂×C₄×C₆₇	C₂×#41	semidirect
334. C₂×#3506	C₆₇	direct (N × H)
338. C₂×C₆₇	#3506	direct (N × H)
342. #3506	C₂×C₆₇	direct (N × H)
343. C₂×#559	C₂×C₆₇	semidirect
344. C₂×#556	C₂×C₆₇	semidirect
345. C₂×#555	C₂×C₆₇	semidirect
346. C₂×#553	C₂×C₆₇	semidirect
347. C₂×#550	C₂×C₆₇	semidirect
348. C₂×#547	C₂×C₆₇	semidirect
349. C₂×#538	C₂×C₆₇	semidirect
350. C₂×#535	C₂×C₆₇	semidirect
351. C₂×#533	C₂×C₆₇	semidirect
352. C₂×#532	C₂×C₆₇	semidirect
353. C₂×#525	C₂×C₆₇	semidirect
354. C₂×#520	C₂×C₆₇	semidirect
355. C₂²×C₄×D₄	C₂×C₆₇	semidirect
356. C₆₇	C₂×#3506	direct (N × H)
358. C₂²×#41	C₂²×C₆₇	semidirect
361. #559	C₂²×C₆₇	semidirect
362. #556	C₂²×C₆₇	semidirect
363. #555	C₂²×C₆₇	semidirect
364. #553	C₂²×C₆₇	semidirect
365. #550	C₂²×C₆₇	semidirect
366. #547	C₂²×C₆₇	semidirect
367. #538	C₂²×C₆₇	semidirect
368. #535	C₂²×C₆₇	semidirect
369. #533	C₂²×C₆₇	semidirect
370. #532	C₂²×C₆₇	semidirect
371. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
372. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
373. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
375. C₂×#127	C₂²×C₆₇	semidirect
376. C₂×#126	C₂²×C₆₇	semidirect
377. C₂×#125	C₂²×C₆₇	semidirect
379. C₂×#124	C₂²×C₆₇	semidirect
380. C₂×#124	C₂²×C₆₇	semidirect
381. #525	C₂²×C₆₇	semidirect
382. C₂×#123	C₂²×C₆₇	semidirect
383. C₂×#123	C₂²×C₆₇	semidirect
386. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
391. C₂×#121	C₂²×C₆₇	semidirect
392. C₂×#121	C₂²×C₆₇	semidirect
394. #520	C₂²×C₆₇	semidirect
395. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
396. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
397. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
403. C₂×C₄×D₄	C₂²×C₆₇	semidirect
404. C₂×C₄×D₄	C₂²×C₆₇	semidirect
405. C₂×C₄×D₄	C₂²×C₆₇	semidirect
406. C₂×C₄×D₄	C₂²×C₆₇	semidirect
407. C₂×C₄×D₄	C₂²×C₆₇	semidirect
409. C₂×#117	C₂²×C₆₇	semidirect
410. C₂×#117	C₂²×C₆₇	semidirect
412. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
419. C₂×#41	C₂³×C₆₇	semidirect
443. C₄⋊Q₈	C₂³×C₆₇	semidirect
444. C₄⋊Q₈	C₂³×C₆₇	semidirect
445. C₄⋊Q₈	C₂³×C₆₇	semidirect
447. #127	C₂³×C₆₇	semidirect
448. #126	C₂³×C₆₇	semidirect
449. #125	C₂³×C₆₇	semidirect
451. #124	C₂³×C₆₇	semidirect
452. #124	C₂³×C₆₇	semidirect
453. #123	C₂³×C₆₇	semidirect
454. #123	C₂³×C₆₇	semidirect
457. C₂²⋊Q₈	C₂³×C₆₇	semidirect
462. #121	C₂³×C₆₇	semidirect
463. #121	C₂³×C₆₇	semidirect
465. C₄×Q₈	C₂³×C₆₇	semidirect
466. C₄×Q₈	C₂³×C₆₇	semidirect
467. C₄×Q₈	C₂³×C₆₇	semidirect
473. C₄×D₄	C₂³×C₆₇	semidirect
474. C₄×D₄	C₂³×C₆₇	semidirect
475. C₄×D₄	C₂³×C₆₇	semidirect
476. C₄×D₄	C₂³×C₆₇	semidirect
478. #117	C₂³×C₆₇	semidirect
479. #117	C₂³×C₆₇	semidirect
492. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
549. C₂²×C₄	#41×C₆₇	semidirect
637. C₂×C₄	C₂×#41×C₆₇	semidirect
679. C₂	C₆₇×#3506	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×#3506	2
C₂×#559×C₆₇	2
C₂×#556×C₆₇	2
C₂×#555×C₆₇	2
C₂×#553×C₆₇	2
C₂×#550×C₆₇	2
C₂×#547×C₆₇	2
C₂×#538×C₆₇	2
C₂×#535×C₆₇	2
C₂×#533×C₆₇	2
C₂×#532×C₆₇	2
C₂×#525×C₆₇	2
C₂×#520×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₂×#3506	67

Subgroups — C2×C67×#3506 #30742020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3506 #30742020