Subgroups — C₆₇×#60234 #30739485

Order 17152.

2242 conjugacy classes of subgroups.

The subgroup lattice has 844 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2978	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2976	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2975	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2974	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2973	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2933	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2930	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2924	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2880	C₂	1	2	yes	maximal, complemented
11	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
17	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
18	#525×C₆₇	C₄	1	2	yes	—
19	#525×C₆₇	C₄	1	2	yes	—
20	#525×C₆₇	C₄	1	2	yes	—
21	#525×C₆₇	C₄	1	2	yes	—
22	#525×C₆₇	C₂²	1	2	yes	—
23	#525×C₆₇	C₂²	1	2	yes	—
24	#525×C₆₇	C₂²	1	2	yes	—
25	#525×C₆₇	C₂²	1	2	yes	—
26	C₂×#123×C₆₇	C₄	1	2	yes	—
27	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
28	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
29	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
30	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
31	C₂×#120×C₆₇	C₄	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
35	#436×C₆₇	C₂²	1	8	yes	complemented
36	#436×C₆₇	C₂²	1	8	yes	complemented
37	#436×C₆₇	C₂²	1	8	yes	complemented
38	#435×C₆₇	C₂²	1	8	yes	complemented
39	#435×C₆₇	C₂²	1	8	yes	complemented
40	#435×C₆₇	C₂²	1	8	yes	complemented
41	#435×C₆₇	C₂²	1	8	yes	complemented
42	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
43	#433×C₆₇	C₂²	1	8	yes	complemented
44	#432×C₆₇	C₂²	1	4	yes	complemented
45	#431×C₆₇	C₂²	1	4	yes	complemented
46	#424×C₆₇	C₂²	1	4	yes	complemented
47	#408×C₆₇	C₂²	1	4	yes	complemented
48	#408×C₆₇	C₂²	1	8	yes	complemented
49	#407×C₆₇	C₂²	1	2	yes	—
50	#407×C₆₇	C₂²	1	1	yes	characteristic
51	#407×C₆₇	C₂²	1	2	yes	—
52	#407×C₆₇	C₂²	1	2	yes	—
53	#407×C₆₇	C₂²	1	1	yes	characteristic
54	#407×C₆₇	C₂²	1	2	yes	complemented
55	#407×C₆₇	C₂²	1	2	yes	complemented
56	#407×C₆₇	C₂²	1	2	yes	complemented
57	#407×C₆₇	C₂²	1	2	yes	complemented
58	#407×C₆₇	C₂²	1	4	yes	complemented
59	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
60	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
61	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
62	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
63	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
64	#405×C₆₇	C₂²	1	4	yes	complemented
65	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×#41×C₆₇	—	4	4	no	—
67	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×#41×C₆₇	C₂³	1	2	yes	—
70	C₂×#41×C₆₇	C₂³	1	2	yes	—
71	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
74	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	—	4	4	no	—
77	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
90	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	#131×C₆₇	—	4	8	no	—
93	C₂×M₁₆×C₆₇	—	4	4	no	—
94	C₂×M₁₆×C₆₇	—	4	4	no	—
95	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
96	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
97	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
98	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
99	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
100	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
101	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
102	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
103	C₂²×C₈×C₆₇	C₂³	1	8	yes	complemented
104	#123×C₆₇	C₂×C₄	1	4	yes	—
105	#123×C₆₇	C₂×C₄	1	4	yes	—
106	#123×C₆₇	C₂×C₄	1	4	yes	—
107	#123×C₆₇	C₂×C₄	1	4	yes	—
108	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
109	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
110	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
111	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
112	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
113	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
114	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
115	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
116	#121×C₆₇	C₂×C₄	1	2	yes	—
117	#121×C₆₇	C₂×C₄	1	2	yes	—
118	#121×C₆₇	C₂×C₄	1	2	yes	—
119	#121×C₆₇	C₂×C₄	1	2	yes	—
120	#121×C₆₇	C₂×C₄	1	2	yes	—
121	#121×C₆₇	C₂×C₄	1	2	yes	—
122	#121×C₆₇	C₂×C₄	1	2	yes	—
123	#121×C₆₇	C₂×C₄	1	2	yes	—
124	#120×C₆₇	C₂×C₄	1	4	yes	—
125	#120×C₆₇	C₂×C₄	1	4	yes	—
126	#120×C₆₇	C₂×C₄	1	4	yes	—
127	#120×C₆₇	C₂×C₄	1	4	yes	—
128	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
129	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
130	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
131	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
132	C₄×D₄×C₆₇	C₂³	1	4	yes	—
133	C₄×D₄×C₆₇	C₂³	1	4	yes	—
134	C₄×D₄×C₆₇	C₂³	1	4	yes	—
135	C₄×D₄×C₆₇	C₂³	1	4	yes	—
136	#117×C₆₇	C₂×C₄	1	2	yes	—
137	#117×C₆₇	C₂×C₄	1	2	yes	—
138	#117×C₆₇	C₂³	1	2	yes	—
139	#117×C₆₇	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
143	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
144	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
145	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
146	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
147	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
148	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
149	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
150	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
151	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
152	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
153	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
154	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
155	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
156	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
157	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
158	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
159	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
160	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
161	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
162	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
163	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
164	#97×C₆₇	C₂³	1	8	yes	complemented
165	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
166	C₂⁴×C₆₇	—	4	2	no	—
167	C₂⁴×C₆₇	—	4	2	no	—
168	C₂⁴×C₆₇	—	4	2	no	—
169	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
174	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
176	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
177	#41×C₆₇	—	2	8	no	—
178	#41×C₆₇	—	4	4	no	—
179	#41×C₆₇	—	4	4	no	—
180	#41×C₆₇	—	4	4	no	—
181	C₂×Q₈×C₆₇	—	4	2	no	—
182	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
183	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
184	C₂×D₄×C₆₇	—	4	2	no	—
185	C₂×D₄×C₆₇	—	4	4	no	—
186	C₂×D₄×C₆₇	—	4	4	no	—
187	C₂×D₄×C₆₇	—	4	4	no	—
188	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
189	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
190	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
191	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
192	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
193	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
194	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
195	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	2	4	no	—
198	C₂²×C₄×C₆₇	—	2	4	no	—
199	C₂²×C₄×C₆₇	—	4	2	no	—
200	C₂²×C₄×C₆₇	—	4	2	no	—
201	C₂²×C₄×C₆₇	—	4	2	no	—
202	C₂²×C₄×C₆₇	—	4	2	no	—
203	C₂²×C₄×C₆₇	—	4	2	no	—
204	C₂²×C₄×C₆₇	—	4	2	no	—
205	C₂²×C₄×C₆₇	—	4	2	no	—
206	C₂²×C₄×C₆₇	—	4	2	no	—
207	C₂²×C₄×C₆₇	—	4	4	no	—
208	C₂²×C₄×C₆₇	—	4	2	no	—
209	C₂²×C₄×C₆₇	—	4	2	no	—
210	C₂²×C₄×C₆₇	—	4	2	no	—
211	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
212	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
214	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
215	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
217	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
218	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
219	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
220	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
221	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
222	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
223	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
224	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
225	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
226	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
227	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
228	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
229	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
230	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
231	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
232	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
233	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
234	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
235	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
236	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
237	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
238	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
239	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
240	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
241	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
242	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
243	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
244	M₁₆×C₆₇	—	2	8	no	—
245	M₁₆×C₆₇	—	4	2	no	—
246	M₁₆×C₆₇	—	4	4	no	—
247	M₁₆×C₆₇	—	4	2	no	—
248	M₁₆×C₆₇	—	4	4	no	—
249	M₁₆×C₆₇	—	4	8	no	—
250	C₂×C₈×C₆₇	—	2	8	no	—
251	C₂×C₈×C₆₇	—	4	4	no	—
252	C₂×C₈×C₆₇	—	4	8	no	—
253	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
254	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
255	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
256	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
257	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
258	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
259	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
260	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
261	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
262	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
263	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
264	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
265	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
266	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
267	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
268	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
269	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
270	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
271	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
272	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
273	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
274	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
275	C₄²×C₆₇	C₂⁴	1	4	yes	—
276	C₂³×C₆₇	—	2	2	no	—
277	C₂³×C₆₇	—	2	2	no	—
278	C₂³×C₆₇	—	2	2	no	—
279	C₂³×C₆₇	—	2	2	no	—
280	C₂³×C₆₇	—	2	2	no	—
281	C₂³×C₆₇	—	2	2	no	—
282	C₂³×C₆₇	—	2	2	no	—
283	C₂³×C₆₇	—	2	2	no	—
284	C₂³×C₆₇	—	2	2	no	—
285	C₂³×C₆₇	—	2	4	no	—
286	C₂³×C₆₇	—	4	1	no	—
287	C₂³×C₆₇	—	4	1	no	—
288	C₂³×C₆₇	—	4	1	no	—
289	C₂³×C₆₇	—	4	1	no	—
290	C₂³×C₆₇	—	4	1	no	—
291	C₂³×C₆₇	—	4	1	no	—
292	C₂³×C₆₇	—	4	1	no	—
293	C₂³×C₆₇	—	4	1	no	—
294	C₂³×C₆₇	—	4	1	no	—
295	C₂³×C₆₇	—	4	1	no	—
296	C₂³×C₆₇	—	4	1	no	—
297	C₂³×C₆₇	—	4	1	no	—
298	C₂³×C₆₇	—	4	2	no	—
299	C₂³×C₆₇	—	4	2	no	—
300	C₂³×C₆₇	—	4	16	no	—
301	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
302	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
303	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
304	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
305	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
306	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
307	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
308	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
309	Q₈×C₆₇	—	2	4	no	—
310	Q₈×C₆₇	—	4	2	no	—
311	D₄×C₆₇	—	2	8	no	—
312	D₄×C₆₇	—	2	8	no	—
313	D₄×C₆₇	—	2	4	no	—
314	D₄×C₆₇	—	4	4	no	—
315	D₄×C₆₇	—	4	4	no	—
316	D₄×C₆₇	—	4	4	no	—
317	D₄×C₆₇	—	4	2	no	—
318	D₄×C₆₇	—	4	4	no	—
319	C₂×C₄×C₆₇	—	2	2	no	—
320	C₂×C₄×C₆₇	—	2	2	no	—
321	C₂×C₄×C₆₇	—	2	2	no	—
322	C₂×C₄×C₆₇	—	2	4	no	—
323	C₂×C₄×C₆₇	—	2	2	no	—
324	C₂×C₄×C₆₇	—	2	2	no	—
325	C₂×C₄×C₆₇	—	2	2	no	—
326	C₂×C₄×C₆₇	—	2	2	no	—
327	C₂×C₄×C₆₇	—	2	2	no	—
328	C₂×C₄×C₆₇	—	2	2	no	—
329	C₂×C₄×C₆₇	—	2	2	no	—
330	C₂×C₄×C₆₇	—	2	8	no	—
331	C₂×C₄×C₆₇	—	2	2	no	—
332	C₂×C₄×C₆₇	—	2	2	no	—
333	C₂×C₄×C₆₇	—	2	2	no	—
334	C₂×C₄×C₆₇	—	2	2	no	—
335	C₂×C₄×C₆₇	—	2	2	no	—
336	C₂×C₄×C₆₇	—	4	1	no	—
337	C₂×C₄×C₆₇	—	4	1	no	—
338	C₂×C₄×C₆₇	—	4	1	no	—
339	C₂×C₄×C₆₇	—	4	1	no	—
340	C₂×C₄×C₆₇	—	4	1	no	—
341	C₂×C₄×C₆₇	—	4	1	no	—
342	C₂×C₄×C₆₇	—	4	1	no	—
343	C₂×C₄×C₆₇	—	4	1	no	—
344	C₂×C₄×C₆₇	—	4	1	no	—
345	C₂×C₄×C₆₇	—	4	1	no	—
346	C₂×C₄×C₆₇	—	4	1	no	—
347	C₂×C₄×C₆₇	—	4	1	no	—
348	C₂×C₄×C₆₇	—	4	2	no	—
349	C₂×C₄×C₆₇	—	4	4	no	—
350	C₂×C₄×C₆₇	—	4	2	no	—
351	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
352	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
353	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
354	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
355	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
356	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
357	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
358	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
359	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
360	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
361	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
362	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
363	C₈×C₆₇	—	2	4	no	—
364	C₈×C₆₇	—	2	4	no	—
365	C₈×C₆₇	—	2	8	no	—
366	C₂²×C₆₇	—	2	1	no	—
367	C₂²×C₆₇	—	2	1	no	—
368	C₂²×C₆₇	—	2	1	no	—
369	C₂²×C₆₇	—	2	1	no	—
370	C₂²×C₆₇	—	2	1	no	—
371	C₂²×C₆₇	—	2	1	no	—
372	C₂²×C₆₇	—	2	1	no	—
373	C₂²×C₆₇	—	2	1	no	—
374	C₂²×C₆₇	—	2	1	no	—
375	C₂²×C₆₇	—	2	1	no	—
376	C₂²×C₆₇	—	2	4	no	—
377	C₂²×C₆₇	—	2	4	no	—
378	C₂²×C₆₇	—	2	1	no	—
379	C₂²×C₆₇	—	2	1	no	—
380	C₂²×C₆₇	—	2	1	no	—
381	C₂²×C₆₇	—	2	4	no	—
382	C₂²×C₆₇	—	2	1	no	—
383	C₂²×C₆₇	—	2	1	no	—
384	C₂²×C₆₇	—	2	1	no	—
385	C₂²×C₆₇	—	2	1	no	—
386	C₂²×C₆₇	—	2	1	no	—
387	C₂²×C₆₇	—	4	4	no	—
388	C₂²×C₆₇	—	4	4	no	—
389	C₂²×C₆₇	—	4	4	no	—
390	C₂²×C₆₇	—	4	4	no	—
391	C₂²×C₆₇	—	4	4	no	—
392	C₂²×C₆₇	—	4	4	no	—
393	C₂²×C₆₇	—	4	4	no	—
394	C₂²×C₆₇	—	8	4	no	—
395	C₂²×C₆₇	#518	1	2	yes	—
396	C₂²×C₆₇	#568	1	2	yes	—
397	C₂²×C₆₇	#568	1	2	yes	—
398	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
399	C₄×C₆₇	—	2	1	no	—
400	C₄×C₆₇	—	2	1	no	—
401	C₄×C₆₇	—	2	1	no	—
402	C₄×C₆₇	—	2	1	no	—
403	C₄×C₆₇	—	2	1	no	—
404	C₄×C₆₇	—	2	1	no	—
405	C₄×C₆₇	—	2	4	no	—
406	C₄×C₆₇	—	2	4	no	—
407	C₄×C₆₇	—	2	4	no	—
408	C₄×C₆₇	#518	1	2	yes	—
409	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
410	#60234	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
411	C₂×C₆₇	—	2	2	no	—
412	C₂×C₆₇	—	2	2	no	—
413	C₂×C₆₇	—	2	2	no	—
414	C₂×C₆₇	—	4	1	no	—
415	C₂×C₆₇	—	4	1	no	—
416	C₂×C₆₇	—	4	1	no	—
417	C₂×C₆₇	—	4	1	no	—
418	C₂×C₆₇	—	4	4	no	—
419	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
420	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
421	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
422	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
423	#2978	C₂×C₆₇	1	4	yes	complemented
424	#2976	C₂×C₆₇	1	4	yes	complemented
425	#2975	C₂×C₆₇	1	2	yes	complemented
426	#2974	C₂×C₆₇	1	4	yes	complemented
427	#2973	C₂×C₆₇	1	2	yes	complemented
428	#2933	C₂×C₆₇	1	4	yes	complemented
429	#2930	C₂×C₆₇	1	4	yes	complemented
430	#2924	C₂×C₆₇	1	2	yes	complemented
431	#2880	C₂×C₆₇	1	2	yes	complemented
432	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
433	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
434	C₆₇	#60234	1	1	yes	characteristic, direct product factor, Sylow
435	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
436	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
437	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
438	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
439	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
440	#525	C₄×C₆₇	1	2	yes	—
441	#525	C₄×C₆₇	1	2	yes	—
442	#525	C₄×C₆₇	1	2	yes	—
443	#525	C₄×C₆₇	1	2	yes	—
444	#525	C₂²×C₆₇	1	2	yes	—
445	#525	C₂²×C₆₇	1	2	yes	—
446	#525	C₂²×C₆₇	1	2	yes	—
447	#525	C₂²×C₆₇	1	2	yes	—
448	C₂×#123	C₄×C₆₇	1	2	yes	—
449	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
450	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
451	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
452	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
453	C₂×#120	C₄×C₆₇	1	2	yes	—
454	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
455	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
456	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
457	#436	C₂²×C₆₇	1	8	yes	complemented
458	#436	C₂²×C₆₇	1	8	yes	complemented
459	#436	C₂²×C₆₇	1	8	yes	complemented
460	#435	C₂²×C₆₇	1	8	yes	complemented
461	#435	C₂²×C₆₇	1	8	yes	complemented
462	#435	C₂²×C₆₇	1	8	yes	complemented
463	#435	C₂²×C₆₇	1	8	yes	complemented
464	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
465	#433	C₂²×C₆₇	1	8	yes	complemented
466	#432	C₂²×C₆₇	1	4	yes	complemented
467	#431	C₂²×C₆₇	1	4	yes	complemented
468	#424	C₂²×C₆₇	1	4	yes	complemented
469	#408	C₂²×C₆₇	1	8	yes	complemented
470	#408	C₂²×C₆₇	1	4	yes	complemented
471	#407	C₂²×C₆₇	1	2	yes	—
472	#407	C₂²×C₆₇	1	1	yes	characteristic
473	#407	C₂²×C₆₇	1	2	yes	—
474	#407	C₂²×C₆₇	1	2	yes	—
475	#407	C₂²×C₆₇	1	1	yes	characteristic
476	#407	C₂²×C₆₇	1	2	yes	complemented
477	#407	C₂²×C₆₇	1	2	yes	complemented
478	#407	C₂²×C₆₇	1	2	yes	complemented
479	#407	C₂²×C₆₇	1	2	yes	complemented
480	#407	C₂²×C₆₇	1	4	yes	complemented
481	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
482	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
483	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
484	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
485	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
486	#405	C₂²×C₆₇	1	4	yes	complemented
487	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
488	C₂×#41	—	4	4	no	—
489	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
490	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
491	C₂×#41	C₂³×C₆₇	1	2	yes	—
492	C₂×#41	C₂³×C₆₇	1	2	yes	—
493	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
494	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
495	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
496	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
497	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
498	C₂³×C₄	—	4	4	no	—
499	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
500	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
501	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
502	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
503	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
504	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
505	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
506	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
507	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
508	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
509	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
510	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
511	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
512	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
513	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
514	#131	—	4	8	no	—
515	C₂×M₁₆	—	4	4	no	—
516	C₂×M₁₆	—	4	4	no	—
517	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
518	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
519	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
520	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
521	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
522	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
523	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
524	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
525	C₂²×C₈	C₂³×C₆₇	1	8	yes	complemented
526	#123	C₂×C₄×C₆₇	1	4	yes	—
527	#123	C₂×C₄×C₆₇	1	4	yes	—
528	#123	C₂×C₄×C₆₇	1	4	yes	—
529	#123	C₂×C₄×C₆₇	1	4	yes	—
530	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
531	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
532	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
533	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
534	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
535	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
536	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
537	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
538	#121	C₂×C₄×C₆₇	1	2	yes	—
539	#121	C₂×C₄×C₆₇	1	2	yes	—
540	#121	C₂×C₄×C₆₇	1	2	yes	—
541	#121	C₂×C₄×C₆₇	1	2	yes	—
542	#121	C₂×C₄×C₆₇	1	2	yes	—
543	#121	C₂×C₄×C₆₇	1	2	yes	—
544	#121	C₂×C₄×C₆₇	1	2	yes	—
545	#121	C₂×C₄×C₆₇	1	2	yes	—
546	#120	C₂×C₄×C₆₇	1	4	yes	—
547	#120	C₂×C₄×C₆₇	1	4	yes	—
548	#120	C₂×C₄×C₆₇	1	4	yes	—
549	#120	C₂×C₄×C₆₇	1	4	yes	—
550	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
551	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
552	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
553	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
554	C₄×D₄	C₂³×C₆₇	1	4	yes	—
555	C₄×D₄	C₂³×C₆₇	1	4	yes	—
556	C₄×D₄	C₂³×C₆₇	1	4	yes	—
557	C₄×D₄	C₂³×C₆₇	1	4	yes	—
558	#117	C₂×C₄×C₆₇	1	2	yes	—
559	#117	C₂×C₄×C₆₇	1	2	yes	—
560	#117	C₂³×C₆₇	1	2	yes	—
561	#117	C₂³×C₆₇	1	2	yes	—
562	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
563	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
564	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
565	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
566	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
567	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
568	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
569	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
570	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
571	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
572	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
573	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
574	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
575	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
576	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
577	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
578	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
579	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
580	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
581	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
582	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
583	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
584	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
585	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
586	#97	C₂³×C₆₇	1	8	yes	complemented
587	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
588	C₂⁴	—	4	2	no	—
589	C₂⁴	—	4	2	no	—
590	C₂⁴	—	4	2	no	—
591	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
592	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
593	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
594	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
595	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
596	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
597	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
598	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
599	#41	—	2	8	no	—
600	#41	—	4	4	no	—
601	#41	—	4	4	no	—
602	#41	—	4	4	no	—
603	C₂×Q₈	—	4	2	no	—
604	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
605	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
606	C₂×D₄	—	4	2	no	—
607	C₂×D₄	—	4	4	no	—
608	C₂×D₄	—	4	4	no	—
609	C₂×D₄	—	4	4	no	—
610	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
611	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
612	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
613	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
614	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
615	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
616	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
617	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
618	C₂²×C₄	—	2	4	no	—
619	C₂²×C₄	—	2	4	no	—
620	C₂²×C₄	—	2	4	no	—
621	C₂²×C₄	—	4	2	no	—
622	C₂²×C₄	—	4	2	no	—
623	C₂²×C₄	—	4	2	no	—
624	C₂²×C₄	—	4	2	no	—
625	C₂²×C₄	—	4	2	no	—
626	C₂²×C₄	—	4	2	no	—
627	C₂²×C₄	—	4	2	no	—
628	C₂²×C₄	—	4	2	no	—
629	C₂²×C₄	—	4	4	no	—
630	C₂²×C₄	—	4	2	no	—
631	C₂²×C₄	—	4	2	no	—
632	C₂²×C₄	—	4	2	no	—
633	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
634	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
635	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
636	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
637	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
638	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
639	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
640	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
641	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
642	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
643	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
644	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
645	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
646	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
647	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
648	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
649	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
650	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
651	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
652	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
653	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
654	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
655	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
656	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
657	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
658	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
659	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
660	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
661	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
662	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
663	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
664	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
665	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
666	M₁₆	—	2	8	no	—
667	M₁₆	—	4	2	no	—
668	M₁₆	—	4	4	no	—
669	M₁₆	—	4	2	no	—
670	M₁₆	—	4	4	no	—
671	M₁₆	—	4	8	no	—
672	C₂×C₈	—	2	8	no	—
673	C₂×C₈	—	4	4	no	—
674	C₂×C₈	—	4	8	no	—
675	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
676	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
677	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
678	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
679	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
680	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
681	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
682	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
683	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
684	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
685	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
686	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
687	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
688	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
689	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
690	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
691	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
692	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
693	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
694	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
695	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
696	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
697	C₄²	C₂⁴×C₆₇	1	4	yes	—
698	C₂³	—	2	2	no	—
699	C₂³	—	2	2	no	—
700	C₂³	—	2	2	no	—
701	C₂³	—	2	2	no	—
702	C₂³	—	2	2	no	—
703	C₂³	—	2	2	no	—
704	C₂³	—	2	2	no	—
705	C₂³	—	2	2	no	—
706	C₂³	—	2	2	no	—
707	C₂³	—	2	4	no	—
708	C₂³	—	4	1	no	—
709	C₂³	—	4	1	no	—
710	C₂³	—	4	1	no	—
711	C₂³	—	4	1	no	—
712	C₂³	—	4	1	no	—
713	C₂³	—	4	1	no	—
714	C₂³	—	4	1	no	—
715	C₂³	—	4	1	no	—
716	C₂³	—	4	1	no	—
717	C₂³	—	4	1	no	—
718	C₂³	—	4	1	no	—
719	C₂³	—	4	1	no	—
720	C₂³	—	4	2	no	—
721	C₂³	—	4	2	no	—
722	C₂³	—	4	16	no	—
723	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
724	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
725	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
726	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
727	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
728	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
729	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
730	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
731	Q₈	—	2	4	no	—
732	Q₈	—	4	2	no	—
733	D₄	—	2	8	no	—
734	D₄	—	2	8	no	—
735	D₄	—	2	4	no	—
736	D₄	—	4	2	no	—
737	D₄	—	4	4	no	—
738	D₄	—	4	4	no	—
739	D₄	—	4	4	no	—
740	D₄	—	4	4	no	—
741	C₂×C₄	—	2	2	no	—
742	C₂×C₄	—	2	2	no	—
743	C₂×C₄	—	2	2	no	—
744	C₂×C₄	—	2	4	no	—
745	C₂×C₄	—	2	2	no	—
746	C₂×C₄	—	2	2	no	—
747	C₂×C₄	—	2	2	no	—
748	C₂×C₄	—	2	2	no	—
749	C₂×C₄	—	2	2	no	—
750	C₂×C₄	—	2	2	no	—
751	C₂×C₄	—	2	2	no	—
752	C₂×C₄	—	2	8	no	—
753	C₂×C₄	—	2	2	no	—
754	C₂×C₄	—	2	2	no	—
755	C₂×C₄	—	2	2	no	—
756	C₂×C₄	—	2	2	no	—
757	C₂×C₄	—	2	2	no	—
758	C₂×C₄	—	4	1	no	—
759	C₂×C₄	—	4	1	no	—
760	C₂×C₄	—	4	1	no	—
761	C₂×C₄	—	4	1	no	—
762	C₂×C₄	—	4	1	no	—
763	C₂×C₄	—	4	1	no	—
764	C₂×C₄	—	4	1	no	—
765	C₂×C₄	—	4	1	no	—
766	C₂×C₄	—	4	1	no	—
767	C₂×C₄	—	4	1	no	—
768	C₂×C₄	—	4	1	no	—
769	C₂×C₄	—	4	1	no	—
770	C₂×C₄	—	4	2	no	—
771	C₂×C₄	—	4	4	no	—
772	C₂×C₄	—	4	2	no	—
773	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
774	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
775	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
776	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
777	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
778	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
779	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
780	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
781	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
782	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
783	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
784	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
785	C₈	—	2	4	no	—
786	C₈	—	2	4	no	—
787	C₈	—	2	8	no	—
788	C₂²	—	2	1	no	—
789	C₂²	—	2	1	no	—
790	C₂²	—	2	1	no	—
791	C₂²	—	2	1	no	—
792	C₂²	—	2	1	no	—
793	C₂²	—	2	1	no	—
794	C₂²	—	2	4	no	—
795	C₂²	—	2	1	no	—
796	C₂²	—	2	1	no	—
797	C₂²	—	2	1	no	—
798	C₂²	—	2	4	no	—
799	C₂²	—	2	1	no	—
800	C₂²	—	2	1	no	—
801	C₂²	—	2	1	no	—
802	C₂²	—	2	1	no	—
803	C₂²	—	2	1	no	—
804	C₂²	—	2	1	no	—
805	C₂²	—	2	1	no	—
806	C₂²	—	2	1	no	—
807	C₂²	—	2	1	no	—
808	C₂²	—	2	4	no	—
809	C₂²	—	4	4	no	—
810	C₂²	—	4	4	no	—
811	C₂²	—	4	4	no	—
812	C₂²	—	4	4	no	—
813	C₂²	—	4	4	no	—
814	C₂²	—	4	4	no	—
815	C₂²	—	4	4	no	—
816	C₂²	—	8	4	no	—
817	C₂²	#518×C₆₇	1	2	yes	—
818	C₂²	#568×C₆₇	1	2	yes	—
819	C₂²	#568×C₆₇	1	2	yes	—
820	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
821	C₄	—	2	4	no	—
822	C₄	—	2	4	no	—
823	C₄	—	2	1	no	—
824	C₄	—	2	1	no	—
825	C₄	—	2	1	no	—
826	C₄	—	2	1	no	—
827	C₄	—	2	1	no	—
828	C₄	—	2	1	no	—
829	C₄	—	2	4	no	—
830	C₄	#518×C₆₇	1	2	yes	—
831	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
832	C₂	—	2	2	no	—
833	C₂	—	2	2	no	—
834	C₂	—	2	2	no	—
835	C₂	—	4	1	no	—
836	C₂	—	4	1	no	—
837	C₂	—	4	1	no	—
838	C₂	—	4	1	no	—
839	C₂	—	4	4	no	—
840	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
841	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
842	C₂	C₂×#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₂×#525×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2978	C₂	semidirect
3. C₆₇×#2976	C₂	semidirect
4. C₆₇×#2975	C₂	semidirect
5. C₆₇×#2974	C₂	semidirect
6. C₆₇×#2973	C₂	semidirect
7. C₆₇×#2933	C₂	semidirect
8. C₆₇×#2930	C₂	semidirect
9. C₆₇×#2924	C₂	semidirect
10. C₆₇×#2880	C₂	semidirect
11. C₂×#407×C₆₇	C₂	semidirect
12. C₂×#407×C₆₇	C₂	semidirect
15. C₂×#131×C₆₇	C₂²	semidirect
16. C₂²×M₁₆×C₆₇	C₂²	semidirect
17. C₂²×M₁₆×C₆₇	C₂²	semidirect
35. #436×C₆₇	C₂²	semidirect
36. #436×C₆₇	C₂²	semidirect
37. #436×C₆₇	C₂²	semidirect
38. #435×C₆₇	C₂²	semidirect
39. #435×C₆₇	C₂²	semidirect
40. #435×C₆₇	C₂²	semidirect
41. #435×C₆₇	C₂²	semidirect
42. C₈×D₄×C₆₇	C₂²	semidirect
43. #433×C₆₇	C₂²	semidirect
44. #432×C₆₇	C₂²	semidirect
45. #431×C₆₇	C₂²	semidirect
46. #424×C₆₇	C₂²	semidirect
47. #408×C₆₇	C₂²	semidirect
48. #408×C₆₇	C₂²	semidirect
54. #407×C₆₇	C₂²	semidirect
55. #407×C₆₇	C₂²	semidirect
56. #407×C₆₇	C₂²	semidirect
57. #407×C₆₇	C₂²	semidirect
58. #407×C₆₇	C₂²	semidirect
61. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
62. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
63. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
64. #405×C₆₇	C₂²	semidirect
100. C₂×M₁₆×C₆₇	C₂³	semidirect
103. C₂²×C₈×C₆₇	C₂³	semidirect
150. C₄⋊C₈×C₆₇	C₂³	semidirect
151. C₄⋊C₈×C₆₇	C₂³	semidirect
162. C₂²⋊C₈×C₆₇	C₂³	semidirect
163. C₂²⋊C₈×C₆₇	C₂³	semidirect
164. #97×C₆₇	C₂³	semidirect
165. C₄×C₈×C₆₇	C₂³	semidirect
410. #60234	C₆₇	direct (N × H)
423. #2978	C₂×C₆₇	semidirect
424. #2976	C₂×C₆₇	semidirect
425. #2975	C₂×C₆₇	semidirect
426. #2974	C₂×C₆₇	semidirect
427. #2973	C₂×C₆₇	semidirect
428. #2933	C₂×C₆₇	semidirect
429. #2930	C₂×C₆₇	semidirect
430. #2924	C₂×C₆₇	semidirect
431. #2880	C₂×C₆₇	semidirect
432. C₂×#407	C₂×C₆₇	semidirect
433. C₂×#407	C₂×C₆₇	semidirect
434. C₆₇	#60234	direct (N × H)
437. C₂×#131	C₂²×C₆₇	semidirect
438. C₂²×M₁₆	C₂²×C₆₇	semidirect
439. C₂²×M₁₆	C₂²×C₆₇	semidirect
457. #436	C₂²×C₆₇	semidirect
458. #436	C₂²×C₆₇	semidirect
459. #436	C₂²×C₆₇	semidirect
460. #435	C₂²×C₆₇	semidirect
461. #435	C₂²×C₆₇	semidirect
462. #435	C₂²×C₆₇	semidirect
463. #435	C₂²×C₆₇	semidirect
464. C₈×D₄	C₂²×C₆₇	semidirect
465. #433	C₂²×C₆₇	semidirect
466. #432	C₂²×C₆₇	semidirect
467. #431	C₂²×C₆₇	semidirect
468. #424	C₂²×C₆₇	semidirect
469. #408	C₂²×C₆₇	semidirect
470. #408	C₂²×C₆₇	semidirect
476. #407	C₂²×C₆₇	semidirect
477. #407	C₂²×C₆₇	semidirect
478. #407	C₂²×C₆₇	semidirect
479. #407	C₂²×C₆₇	semidirect
480. #407	C₂²×C₆₇	semidirect
483. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
484. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
485. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
486. #405	C₂²×C₆₇	semidirect
522. C₂×M₁₆	C₂³×C₆₇	semidirect
525. C₂²×C₈	C₂³×C₆₇	semidirect
572. C₄⋊C₈	C₂³×C₆₇	semidirect
573. C₄⋊C₈	C₂³×C₆₇	semidirect
584. C₂²⋊C₈	C₂³×C₆₇	semidirect
585. C₂²⋊C₈	C₂³×C₆₇	semidirect
586. #97	C₂³×C₆₇	semidirect
587. C₄×C₈	C₂³×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#525×C₆₇	2
C₆₇×#2978	2
C₆₇×#2976	2
C₆₇×#2975	2
C₆₇×#2974	2
C₆₇×#2973	2
C₆₇×#2933	2
C₆₇×#2930	2
C₆₇×#2924	2
C₆₇×#2880	2
C₂×#407×C₆₇	2
#60234	67

Subgroups — C67×#60234 #30739485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60234 #30739485