Subgroups — C₆₇×#60312 #30739563

Order 17152.

1902 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3460	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2994	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2988	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2979	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2976	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2969	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2965	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2935	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2929	C₂	1	4	yes	maximal, complemented
11	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	#544×C₆₇	C₄	1	1	yes	characteristic
15	#542×C₆₇	C₄	1	4	yes	—
16	#540×C₆₇	C₄	1	4	yes	—
17	#538×C₆₇	C₄	1	4	yes	—
18	#537×C₆₇	C₄	1	1	yes	characteristic
19	#535×C₆₇	C₄	1	2	yes	—
20	#532×C₆₇	C₂²	1	4	yes	—
21	#529×C₆₇	C₂²	1	4	yes	—
22	#525×C₆₇	C₂²	1	4	yes	—
23	C₄×#41×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	#446×C₆₇	C₂²	1	2	yes	complemented
26	#446×C₆₇	C₂²	1	2	yes	complemented
27	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
28	#436×C₆₇	C₂²	1	4	yes	—
29	#436×C₆₇	C₂²	1	4	yes	—
30	#436×C₆₇	C₂²	1	2	yes	complemented
31	#436×C₆₇	C₂²	1	2	yes	complemented
32	#436×C₆₇	C₂²	1	8	yes	complemented
33	#435×C₆₇	C₂²	1	4	yes	—
34	#435×C₆₇	C₂²	1	4	yes	—
35	#435×C₆₇	C₂²	1	8	yes	complemented
36	#435×C₆₇	C₂²	1	8	yes	complemented
37	C₈×D₄×C₆₇	C₂²	1	4	yes	—
38	C₈×D₄×C₆₇	C₂²	1	4	yes	—
39	C₈×D₄×C₆₇	C₂²	1	8	yes	—
40	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
41	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
42	#433×C₆₇	C₂²	1	8	yes	complemented
43	#431×C₆₇	C₂²	1	2	yes	—
44	#431×C₆₇	C₂²	1	4	yes	complemented
45	#431×C₆₇	C₂²	1	4	yes	complemented
46	#424×C₆₇	C₂²	1	4	yes	complemented
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
49	#408×C₆₇	C₂²	1	4	yes	complemented
50	#408×C₆₇	C₂²	1	8	yes	complemented
51	#407×C₆₇	C₂²	1	4	yes	—
52	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
53	#405×C₆₇	C₂²	1	4	yes	complemented
54	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
55	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×#41×C₆₇	C₂³	1	4	yes	—
58	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	#131×C₆₇	—	4	8	no	—
61	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
62	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
63	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
64	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
65	C₂²×C₈×C₆₇	—	4	4	no	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
73	#127×C₆₇	C₂×C₄	1	4	yes	—
74	#126×C₆₇	C₂×C₄	1	8	yes	—
75	#125×C₆₇	C₂×C₄	1	4	yes	—
76	#124×C₆₇	C₂×C₄	1	4	yes	—
77	#124×C₆₇	C₂×C₄	1	8	yes	—
78	#123×C₆₇	C₂×C₄	1	8	yes	—
79	#123×C₆₇	C₂×C₄	1	8	yes	—
80	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
81	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
82	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
83	#121×C₆₇	C₂×C₄	1	4	yes	—
84	#121×C₆₇	C₂×C₄	1	4	yes	—
85	#121×C₆₇	C₂×C₄	1	4	yes	—
86	#121×C₆₇	C₂×C₄	1	4	yes	—
87	#121×C₆₇	C₂×C₄	1	4	yes	—
88	#121×C₆₇	C₂×C₄	1	8	yes	—
89	#120×C₆₇	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
92	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
93	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
94	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₆₇	C₂³	1	4	yes	—
96	C₄×D₄×C₆₇	C₂³	1	4	yes	—
97	C₄×D₄×C₆₇	C₂³	1	2	yes	—
98	C₄×D₄×C₆₇	C₂³	1	8	yes	—
99	#117×C₆₇	C₂×C₄	1	2	yes	—
100	#117×C₆₇	C₂×C₄	1	4	yes	—
101	#117×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
106	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
107	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
108	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
110	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
111	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
112	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
114	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
115	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
116	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
117	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
118	#97×C₆₇	C₂³	1	4	yes	—
119	#97×C₆₇	C₂³	1	4	yes	—
120	C₄×C₈×C₆₇	C₂³	1	2	yes	—
121	C₄×C₈×C₆₇	C₂³	1	2	yes	—
122	C₄×C₈×C₆₇	C₂³	1	4	yes	—
123	C₄×C₈×C₆₇	C₂³	1	8	yes	—
124	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
125	#41×C₆₇	—	2	8	no	—
126	#41×C₆₇	—	4	4	no	—
127	#41×C₆₇	—	4	4	no	—
128	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
133	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
135	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₆₇	—	4	2	no	—
138	C₂²×C₄×C₆₇	—	4	2	no	—
139	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	M₁₆×C₆₇	—	2	8	no	—
149	M₁₆×C₆₇	—	4	4	no	—
150	M₁₆×C₆₇	—	4	4	no	—
151	M₁₆×C₆₇	—	4	4	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	8	no	—
154	C₂×C₈×C₆₇	—	4	2	no	—
155	C₂×C₈×C₆₇	—	4	4	no	—
156	C₂×C₈×C₆₇	—	4	1	no	—
157	C₂×C₈×C₆₇	—	4	8	no	—
158	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
159	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
163	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
164	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
168	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
170	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
174	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
175	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
176	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
177	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
178	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
179	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
180	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
181	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
182	C₄²×C₆₇	C₂⁴	1	2	yes	—
183	C₄²×C₆₇	C₂⁴	1	4	yes	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	2	no	—
187	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
188	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
189	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
190	Q₈×C₆₇	—	2	4	no	—
191	Q₈×C₆₇	—	4	4	no	—
192	D₄×C₆₇	—	2	8	no	—
193	D₄×C₆₇	—	2	8	no	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	4	8	no	—
196	D₄×C₆₇	—	4	4	no	—
197	D₄×C₆₇	—	4	8	no	—
198	C₂×C₄×C₆₇	—	2	8	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	#131	1	4	yes	complemented
210	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
211	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
213	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
214	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
215	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
216	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
217	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
218	C₈×C₆₇	—	2	1	no	—
219	C₈×C₆₇	—	2	4	no	—
220	C₈×C₆₇	—	2	4	no	—
221	C₈×C₆₇	—	2	2	no	—
222	C₈×C₆₇	—	2	4	no	—
223	C₈×C₆₇	H(5,𝔽₂)	1	2	yes	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	4	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	2	4	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	4	no	—
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	4	8	no	—
233	C₂²×C₆₇	C₂×#131	1	2	yes	—
234	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	—	2	4	no	—
237	C₄×C₆₇	—	2	4	no	—
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	C₂×#131	1	4	yes	—
242	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₄×C₆₇	#585	1	1	yes	characteristic
244	#60312	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂×C₆₇	—	2	2	no	—
246	C₂×C₆₇	—	4	4	no	—
247	C₂×C₆₇	—	4	4	no	—
248	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
250	C₂×C₆₇	#3573	1	1	yes	characteristic
251	#3460	C₂×C₆₇	1	1	yes	characteristic
252	#2994	C₂×C₆₇	1	4	yes	complemented
253	#2988	C₂×C₆₇	1	2	yes	complemented
254	#2979	C₂×C₆₇	1	4	yes	complemented
255	#2976	C₂×C₆₇	1	4	yes	complemented
256	#2975	C₂×C₆₇	1	1	yes	characteristic, complemented
257	#2969	C₂×C₆₇	1	4	yes	complemented
258	#2965	C₂×C₆₇	1	2	yes	complemented
259	#2935	C₂×C₆₇	1	4	yes	complemented
260	#2929	C₂×C₆₇	1	4	yes	complemented
261	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₆₇	#60312	1	1	yes	characteristic, direct product factor, Sylow
263	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
264	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
265	#544	C₄×C₆₇	1	1	yes	characteristic
266	#542	C₄×C₆₇	1	4	yes	—
267	#540	C₄×C₆₇	1	4	yes	—
268	#538	C₄×C₆₇	1	4	yes	—
269	#537	C₄×C₆₇	1	1	yes	characteristic
270	#535	C₄×C₆₇	1	2	yes	—
271	#532	C₂²×C₆₇	1	4	yes	—
272	#529	C₂²×C₆₇	1	4	yes	—
273	#525	C₂²×C₆₇	1	4	yes	—
274	C₄×#41	C₂²×C₆₇	1	2	yes	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
276	#446	C₂²×C₆₇	1	2	yes	complemented
277	#446	C₂²×C₆₇	1	2	yes	complemented
278	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
279	#436	C₂²×C₆₇	1	4	yes	—
280	#436	C₂²×C₆₇	1	4	yes	—
281	#436	C₂²×C₆₇	1	2	yes	complemented
282	#436	C₂²×C₆₇	1	2	yes	complemented
283	#436	C₂²×C₆₇	1	8	yes	complemented
284	#435	C₂²×C₆₇	1	4	yes	—
285	#435	C₂²×C₆₇	1	4	yes	—
286	#435	C₂²×C₆₇	1	8	yes	complemented
287	#435	C₂²×C₆₇	1	8	yes	complemented
288	C₈×D₄	C₂²×C₆₇	1	4	yes	—
289	C₈×D₄	C₂²×C₆₇	1	4	yes	—
290	C₈×D₄	C₂²×C₆₇	1	8	yes	—
291	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
292	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
293	#433	C₂²×C₆₇	1	8	yes	complemented
294	#431	C₂²×C₆₇	1	2	yes	—
295	#431	C₂²×C₆₇	1	4	yes	complemented
296	#431	C₂²×C₆₇	1	4	yes	complemented
297	#424	C₂²×C₆₇	1	4	yes	complemented
298	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
300	#408	C₂²×C₆₇	1	4	yes	complemented
301	#408	C₂²×C₆₇	1	8	yes	complemented
302	#407	C₂²×C₆₇	1	4	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
304	#405	C₂²×C₆₇	1	4	yes	complemented
305	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
306	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
308	C₂×#41	C₂³×C₆₇	1	4	yes	—
309	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
311	#131	—	4	8	no	—
312	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
313	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
314	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
315	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
316	C₂²×C₈	—	4	4	no	—
317	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
319	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
320	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
321	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
322	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
323	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
324	#127	C₂×C₄×C₆₇	1	4	yes	—
325	#126	C₂×C₄×C₆₇	1	8	yes	—
326	#125	C₂×C₄×C₆₇	1	4	yes	—
327	#124	C₂×C₄×C₆₇	1	4	yes	—
328	#124	C₂×C₄×C₆₇	1	8	yes	—
329	#123	C₂×C₄×C₆₇	1	8	yes	—
330	#123	C₂×C₄×C₆₇	1	8	yes	—
331	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
332	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
333	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
334	#121	C₂×C₄×C₆₇	1	4	yes	—
335	#121	C₂×C₄×C₆₇	1	4	yes	—
336	#121	C₂×C₄×C₆₇	1	4	yes	—
337	#121	C₂×C₄×C₆₇	1	4	yes	—
338	#121	C₂×C₄×C₆₇	1	4	yes	—
339	#121	C₂×C₄×C₆₇	1	8	yes	—
340	#120	C₂×C₄×C₆₇	1	8	yes	—
341	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
342	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
343	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
344	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
345	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
346	C₄×D₄	C₂³×C₆₇	1	4	yes	—
347	C₄×D₄	C₂³×C₆₇	1	4	yes	—
348	C₄×D₄	C₂³×C₆₇	1	2	yes	—
349	C₄×D₄	C₂³×C₆₇	1	8	yes	—
350	#117	C₂×C₄×C₆₇	1	2	yes	—
351	#117	C₂×C₄×C₆₇	1	4	yes	—
352	#117	C₂³×C₆₇	1	4	yes	—
353	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
355	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
357	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
358	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
359	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
360	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
361	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
362	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
363	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
364	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
365	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
366	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
367	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
368	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
369	#97	C₂³×C₆₇	1	4	yes	—
370	#97	C₂³×C₆₇	1	4	yes	—
371	C₄×C₈	C₂³×C₆₇	1	2	yes	—
372	C₄×C₈	C₂³×C₆₇	1	2	yes	—
373	C₄×C₈	C₂³×C₆₇	1	4	yes	—
374	C₄×C₈	C₂³×C₆₇	1	8	yes	—
375	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
376	#41	—	2	8	no	—
377	#41	—	4	4	no	—
378	#41	—	4	4	no	—
379	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
380	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
383	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
384	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
385	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
386	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
387	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
391	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
393	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
399	M₁₆	—	2	8	no	—
400	M₁₆	—	4	4	no	—
401	M₁₆	—	4	4	no	—
402	M₁₆	—	4	4	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	8	no	—
405	C₂×C₈	—	4	2	no	—
406	C₂×C₈	—	4	4	no	—
407	C₂×C₈	—	4	8	no	—
408	C₂×C₈	—	4	1	no	—
409	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
410	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
411	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
412	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
413	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
414	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
416	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
417	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
418	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
419	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
420	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
421	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
422	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
423	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
424	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
425	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
426	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
427	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
428	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
429	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
430	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
431	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
432	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
433	C₄²	C₂⁴×C₆₇	1	2	yes	—
434	C₄²	C₂⁴×C₆₇	1	4	yes	—
435	C₂³	—	2	4	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
439	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
440	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
441	Q₈	—	2	4	no	—
442	Q₈	—	4	4	no	—
443	D₄	—	2	8	no	—
444	D₄	—	2	8	no	—
445	D₄	—	2	4	no	—
446	D₄	—	4	4	no	—
447	D₄	—	4	8	no	—
448	D₄	—	4	8	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	#131×C₆₇	1	4	yes	complemented
461	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
464	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
465	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
466	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
467	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
468	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
469	C₈	—	2	1	no	—
470	C₈	—	2	4	no	—
471	C₈	—	2	4	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	4	no	—
474	C₈	H(5,𝔽₂)×C₆₇	1	2	yes	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	4	8	no	—
484	C₂²	C₂×#131×C₆₇	1	2	yes	—
485	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
486	C₄	—	2	4	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	4	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	4	no	—
491	C₄	—	2	4	no	—
492	C₄	C₂×#131×C₆₇	1	4	yes	—
493	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
494	C₄	#585×C₆₇	1	1	yes	characteristic
495	C₂	—	2	2	no	—
496	C₂	—	4	4	no	—
497	C₂	—	4	4	no	—
498	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
499	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
500	C₂	C₆₇×#3573	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	C₆₇×#3460	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×#2994	C₂	semidirect
3. C₆₇×#2988	C₂	semidirect
4. C₆₇×#2979	C₂	semidirect
5. C₆₇×#2976	C₂	semidirect
6. C₆₇×#2975	C₂	semidirect
7. C₆₇×#2969	C₂	semidirect
8. C₆₇×#2965	C₂	semidirect
9. C₆₇×#2935	C₂	semidirect
10. C₆₇×#2929	C₂	semidirect
11. C₂×C₈×D₄×C₆₇	C₂	semidirect
12. C₂×#131×C₆₇	C₂²	semidirect
25. #446×C₆₇	C₂²	semidirect
26. #446×C₆₇	C₂²	semidirect
27. C₈×Q₈×C₆₇	C₂²	semidirect
30. #436×C₆₇	C₂²	semidirect
31. #436×C₆₇	C₂²	semidirect
32. #436×C₆₇	C₂²	semidirect
35. #435×C₆₇	C₂²	semidirect
36. #435×C₆₇	C₂²	semidirect
40. C₈×D₄×C₆₇	C₂²	semidirect
41. C₈×D₄×C₆₇	C₂²	semidirect
42. #433×C₆₇	C₂²	semidirect
44. #431×C₆₇	C₂²	semidirect
45. #431×C₆₇	C₂²	semidirect
46. #424×C₆₇	C₂²	semidirect
48. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
49. #408×C₆₇	C₂²	semidirect
50. #408×C₆₇	C₂²	semidirect
53. #405×C₆₇	C₂²	semidirect
54. C₄×M₁₆×C₆₇	C₂²	semidirect
209. C₂×C₄×C₆₇	#131	semidirect
244. #60312	C₆₇	direct (N × H)
252. #2994	C₂×C₆₇	semidirect
253. #2988	C₂×C₆₇	semidirect
254. #2979	C₂×C₆₇	semidirect
255. #2976	C₂×C₆₇	semidirect
256. #2975	C₂×C₆₇	semidirect
257. #2969	C₂×C₆₇	semidirect
258. #2965	C₂×C₆₇	semidirect
259. #2935	C₂×C₆₇	semidirect
260. #2929	C₂×C₆₇	semidirect
261. C₂×C₈×D₄	C₂×C₆₇	semidirect
262. C₆₇	#60312	direct (N × H)
263. C₂×#131	C₂²×C₆₇	semidirect
276. #446	C₂²×C₆₇	semidirect
277. #446	C₂²×C₆₇	semidirect
278. C₈×Q₈	C₂²×C₆₇	semidirect
281. #436	C₂²×C₆₇	semidirect
282. #436	C₂²×C₆₇	semidirect
283. #436	C₂²×C₆₇	semidirect
286. #435	C₂²×C₆₇	semidirect
287. #435	C₂²×C₆₇	semidirect
291. C₈×D₄	C₂²×C₆₇	semidirect
292. C₈×D₄	C₂²×C₆₇	semidirect
293. #433	C₂²×C₆₇	semidirect
295. #431	C₂²×C₆₇	semidirect
296. #431	C₂²×C₆₇	semidirect
297. #424	C₂²×C₆₇	semidirect
299. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
300. #408	C₂²×C₆₇	semidirect
301. #408	C₂²×C₆₇	semidirect
304. #405	C₂²×C₆₇	semidirect
305. C₄×M₁₆	C₂²×C₆₇	semidirect
460. C₂×C₄	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3460	2
C₆₇×#2994	2
C₆₇×#2988	2
C₆₇×#2979	2
C₆₇×#2976	2
C₆₇×#2975	2
C₆₇×#2969	2
C₆₇×#2965	2
C₆₇×#2935	2
C₆₇×#2929	2
C₂×C₈×D₄×C₆₇	2
#60312	67

Subgroups — C67×#60312 #30739563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60312 #30739563