Subgroups — C₆₇×#21776 #30701027

Order 17152.

1520 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3484	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2883	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1913	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1889	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1828	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1781	C₂	1	2	yes	maximal, complemented
7	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
9	#546×C₆₇	—	2	4	no	—
10	D₄²×C₆₇	—	2	4	no	—
11	#535×C₆₇	C₄	1	2	yes	complemented
12	#534×C₆₇	C₄	1	2	yes	complemented
13	#534×C₆₇	C₂²	1	2	yes	—
14	#525×C₆₇	C₄	1	2	yes	complemented
15	C₂×#120×C₆₇	C₄	1	2	yes	complemented
16	C₂×#120×C₆₇	C₂²	1	2	yes	—
17	#518×C₆₇	C₂²	1	1	yes	characteristic
18	#518×C₆₇	C₂²	1	4	yes	complemented
19	#410×C₆₇	—	2	4	no	—
20	C₂×#99×C₆₇	—	2	4	no	—
21	C₂×#99×C₆₇	C₂²	1	4	yes	—
22	C₂×#99×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#99×C₆₇	C₂²	1	4	yes	complemented
24	#379×C₆₇	C₂²	1	4	yes	complemented
25	#342×C₆₇	C₂²	1	8	yes	complemented
26	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
28	H(5,𝔽₂)×C₆₇	—	4	4	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×#41×C₆₇	D₄	1	4	yes	—
31	C₂²×D₄×C₆₇	—	2	2	no	—
32	C₂²×D₄×C₆₇	—	2	2	no	—
33	C₂²×D₄×C₆₇	—	2	4	no	—
34	C₂²×D₄×C₆₇	—	4	4	no	—
35	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂²×D₄×C₆₇	D₄	1	4	yes	—
39	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
43	#127×C₆₇	—	2	2	no	—
44	#124×C₆₇	—	2	2	no	—
45	#123×C₆₇	—	2	2	no	—
46	#123×C₆₇	—	2	2	no	—
47	#123×C₆₇	C₂×C₄	1	4	yes	complemented
48	C₂²⋊Q₈×C₆₇	—	2	2	no	—
49	#121×C₆₇	—	2	2	no	—
50	#121×C₆₇	—	2	4	no	—
51	#121×C₆₇	—	2	4	no	—
52	#121×C₆₇	—	2	4	no	—
53	#120×C₆₇	—	2	2	no	—
54	#120×C₆₇	—	2	2	no	—
55	#120×C₆₇	—	2	4	no	—
56	#120×C₆₇	—	2	4	no	—
57	#120×C₆₇	C₂×C₄	1	4	yes	complemented
58	#120×C₆₇	C₂×C₄	1	4	yes	complemented
59	#120×C₆₇	C₂×C₄	1	4	yes	complemented
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	4	8	no	—
63	#117×C₆₇	—	2	4	no	—
64	#117×C₆₇	C₂×C₄	1	2	yes	complemented
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	complemented
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
72	#99×C₆₇	—	2	8	no	—
73	#99×C₆₇	—	2	8	no	—
74	#99×C₆₇	—	4	4	no	—
75	C₂⁴×C₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	4	2	no	—
79	C₂⁴×C₆₇	—	8	2	no	—
80	C₂⁴×C₆₇	—	8	2	no	—
81	C₂⁴×C₆₇	C₂²⋊C₄	1	2	yes	complemented
82	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
87	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
88	#41×C₆₇	—	4	4	no	—
89	#41×C₆₇	—	4	8	no	—
90	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	complemented
91	C₂×D₄×C₆₇	—	2	8	no	—
92	C₂×D₄×C₆₇	—	2	4	no	—
93	C₂×D₄×C₆₇	—	2	4	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	4	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	4	no	—
103	C₂×D₄×C₆₇	—	4	8	no	—
104	C₂×D₄×C₆₇	—	4	16	no	—
105	C₂×D₄×C₆₇	—	8	4	no	—
106	C₂×D₄×C₆₇	—	8	4	no	—
107	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	complemented
108	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	complemented
109	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	8	no	—
116	C₂²×C₄×C₆₇	—	4	2	no	—
117	C₂²×C₄×C₆₇	—	4	4	no	—
118	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	complemented
119	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	complemented
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	4	no	—
125	C₄⋊C₄×C₆₇	—	4	4	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	2	4	no	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₂²⋊C₄×C₆₇	—	2	4	no	—
130	C₂²⋊C₄×C₆₇	—	2	4	no	—
131	C₂²⋊C₄×C₆₇	—	2	4	no	—
132	C₂²⋊C₄×C₆₇	—	2	8	no	—
133	C₂²⋊C₄×C₆₇	—	2	8	no	—
134	C₂²⋊C₄×C₆₇	—	4	4	no	—
135	C₂²⋊C₄×C₆₇	—	8	8	no	—
136	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	4	4	no	—
139	C₂³×C₆₇	—	2	1	no	—
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	8	no	—
150	C₂³×C₆₇	—	4	4	no	—
151	C₂³×C₆₇	—	4	4	no	—
152	C₂³×C₆₇	—	4	4	no	—
153	C₂³×C₆₇	—	4	4	no	—
154	C₂³×C₆₇	—	8	4	no	—
155	C₂³×C₆₇	—	8	1	no	—
156	C₂³×C₆₇	—	8	1	no	—
157	C₂³×C₆₇	—	8	2	no	—
158	C₂³×C₆₇	—	8	1	no	—
159	C₂³×C₆₇	—	8	1	no	—
160	C₂³×C₆₇	—	8	4	no	—
161	C₂³×C₆₇	—	8	2	no	—
162	C₂³×C₆₇	—	8	2	no	—
163	C₂³×C₆₇	—	8	2	no	—
164	C₂³×C₆₇	—	8	4	no	—
165	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
167	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
168	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
169	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
170	Q₈×C₆₇	—	4	2	no	—
171	D₄×C₆₇	—	2	8	no	—
172	D₄×C₆₇	—	2	8	no	—
173	D₄×C₆₇	—	2	8	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	4	no	—
176	D₄×C₆₇	—	4	4	no	—
177	D₄×C₆₇	—	4	4	no	—
178	D₄×C₆₇	—	4	8	no	—
179	D₄×C₆₇	—	4	4	no	—
180	D₄×C₆₇	—	8	4	no	—
181	D₄×C₆₇	—	8	4	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	4	4	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	4	no	—
190	C₂×C₄×C₆₇	—	4	8	no	—
191	C₂×C₄×C₆₇	—	8	2	no	—
192	C₂×C₄×C₆₇	—	8	2	no	—
193	C₂×C₄×C₆₇	—	8	8	no	—
194	C₂×C₄×C₆₇	—	16	4	no	—
195	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	4	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	1	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	4	1	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	4	4	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	—	4	1	no	—
213	C₂²×C₆₇	—	4	1	no	—
214	C₂²×C₆₇	—	8	4	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	—	8	8	no	—
217	C₂²×C₆₇	—	8	2	no	—
218	C₂²×C₆₇	—	8	8	no	—
219	C₂²×C₆₇	—	16	2	no	—
220	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₆₇	#518	1	2	yes	—
222	C₂²×C₆₇	#534	1	4	yes	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	4	4	no	—
226	C₄×C₆₇	—	4	1	no	—
227	C₄×C₆₇	—	8	8	no	—
228	#21776	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	2	1	no	—
230	C₂×C₆₇	—	2	4	no	—
231	C₂×C₆₇	—	2	2	no	—
232	C₂×C₆₇	—	4	2	no	—
233	C₂×C₆₇	—	4	2	no	—
234	C₂×C₆₇	—	4	4	no	—
235	C₂×C₆₇	—	8	1	no	—
236	C₂×C₆₇	—	8	1	no	—
237	C₂×C₆₇	#2308	1	1	yes	characteristic
238	#3484	C₂×C₆₇	1	1	yes	characteristic
239	#2883	C₂×C₆₇	1	2	yes	complemented
240	#1913	C₂×C₆₇	1	4	yes	complemented
241	#1889	C₂×C₆₇	1	4	yes	complemented
242	#1828	C₂×C₆₇	1	2	yes	complemented
243	#1781	C₂×C₆₇	1	2	yes	complemented
244	C₆₇	#21776	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
246	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
247	#546	—	2	4	no	—
248	D₄²	—	2	4	no	—
249	#535	C₄×C₆₇	1	2	yes	complemented
250	#534	C₄×C₆₇	1	2	yes	complemented
251	#534	C₂²×C₆₇	1	2	yes	—
252	#525	C₄×C₆₇	1	2	yes	complemented
253	C₂×#120	C₄×C₆₇	1	2	yes	complemented
254	C₂×#120	C₂²×C₆₇	1	2	yes	—
255	#518	C₂²×C₆₇	1	1	yes	characteristic
256	#518	C₂²×C₆₇	1	4	yes	complemented
257	#410	—	2	4	no	—
258	C₂×#99	—	2	4	no	—
259	C₂×#99	C₂²×C₆₇	1	4	yes	—
260	C₂×#99	C₂²×C₆₇	1	4	yes	complemented
261	C₂×#99	C₂²×C₆₇	1	4	yes	complemented
262	#379	C₂²×C₆₇	1	4	yes	complemented
263	#342	C₂²×C₆₇	1	8	yes	complemented
264	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
265	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
266	H(5,𝔽₂)	—	4	4	no	—
267	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
268	C₂×#41	D₄×C₆₇	1	4	yes	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	—	2	4	no	—
272	C₂²×D₄	—	4	4	no	—
273	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
274	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
275	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
276	C₂²×D₄	D₄×C₆₇	1	4	yes	—
277	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
279	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
280	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
281	#127	—	2	2	no	—
282	#124	—	2	2	no	—
283	#123	—	2	2	no	—
284	#123	—	2	2	no	—
285	#123	C₂×C₄×C₆₇	1	4	yes	complemented
286	C₂²⋊Q₈	—	2	2	no	—
287	#121	—	2	2	no	—
288	#121	—	2	4	no	—
289	#121	—	2	4	no	—
290	#121	—	2	4	no	—
291	#120	—	2	2	no	—
292	#120	—	2	2	no	—
293	#120	—	2	4	no	—
294	#120	—	2	4	no	—
295	#120	C₂×C₄×C₆₇	1	4	yes	complemented
296	#120	C₂×C₄×C₆₇	1	4	yes	complemented
297	#120	C₂×C₄×C₆₇	1	4	yes	complemented
298	C₄×D₄	—	2	4	no	—
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	4	8	no	—
301	#117	—	2	4	no	—
302	#117	C₂×C₄×C₆₇	1	2	yes	complemented
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	4	no	—
306	C₂×C₂²⋊C₄	—	4	4	no	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	complemented
308	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
310	#99	—	2	8	no	—
311	#99	—	2	8	no	—
312	#99	—	4	4	no	—
313	C₂⁴	—	2	2	no	—
314	C₂⁴	—	2	2	no	—
315	C₂⁴	—	2	4	no	—
316	C₂⁴	—	4	2	no	—
317	C₂⁴	—	8	2	no	—
318	C₂⁴	—	8	2	no	—
319	C₂⁴	C₂²⋊C₄×C₆₇	1	2	yes	complemented
320	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
321	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
323	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
324	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
325	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
326	#41	—	4	4	no	—
327	#41	—	4	8	no	—
328	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	complemented
329	C₂×D₄	—	2	8	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	4	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	4	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	4	no	—
341	C₂×D₄	—	4	8	no	—
342	C₂×D₄	—	4	16	no	—
343	C₂×D₄	—	8	4	no	—
344	C₂×D₄	—	8	4	no	—
345	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	complemented
346	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	complemented
347	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
348	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
349	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	—	4	4	no	—
356	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	complemented
357	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	complemented
358	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	2	2	no	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	4	no	—
368	C₂²⋊C₄	—	2	4	no	—
369	C₂²⋊C₄	—	2	4	no	—
370	C₂²⋊C₄	—	2	8	no	—
371	C₂²⋊C₄	—	2	8	no	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₂²⋊C₄	—	8	8	no	—
374	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
375	C₄²	—	2	2	no	—
376	C₄²	—	4	4	no	—
377	C₂³	—	2	1	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	8	no	—
388	C₂³	—	4	4	no	—
389	C₂³	—	4	4	no	—
390	C₂³	—	4	4	no	—
391	C₂³	—	4	4	no	—
392	C₂³	—	8	4	no	—
393	C₂³	—	8	1	no	—
394	C₂³	—	8	1	no	—
395	C₂³	—	8	2	no	—
396	C₂³	—	8	1	no	—
397	C₂³	—	8	1	no	—
398	C₂³	—	8	4	no	—
399	C₂³	—	8	2	no	—
400	C₂³	—	8	2	no	—
401	C₂³	—	8	2	no	—
402	C₂³	—	8	4	no	—
403	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
404	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
405	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
406	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
407	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
408	Q₈	—	4	2	no	—
409	D₄	—	2	8	no	—
410	D₄	—	2	8	no	—
411	D₄	—	2	8	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	4	no	—
414	D₄	—	4	4	no	—
415	D₄	—	4	4	no	—
416	D₄	—	4	8	no	—
417	D₄	—	4	4	no	—
418	D₄	—	8	4	no	—
419	D₄	—	8	4	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	8	no	—
429	C₂×C₄	—	8	2	no	—
430	C₂×C₄	—	8	2	no	—
431	C₂×C₄	—	8	8	no	—
432	C₂×C₄	—	16	4	no	—
433	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
434	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	8	4	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	8	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	8	no	—
457	C₂²	—	16	2	no	—
458	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
459	C₂²	#518×C₆₇	1	2	yes	—
460	C₂²	#534×C₆₇	1	4	yes	—
461	C₄	—	2	2	no	—
462	C₄	—	2	4	no	—
463	C₄	—	4	4	no	—
464	C₄	—	4	1	no	—
465	C₄	—	8	8	no	—
466	C₂	—	2	1	no	—
467	C₂	—	2	4	no	—
468	C₂	—	2	2	no	—
469	C₂	—	4	2	no	—
470	C₂	—	4	2	no	—
471	C₂	—	4	4	no	—
472	C₂	—	8	1	no	—
473	C₂	—	8	1	no	—
474	C₂	C₆₇×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2308
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₆₇×#3484	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2883	C₂	semidirect
3. C₆₇×#1913	C₂	semidirect
4. C₆₇×#1889	C₂	semidirect
5. C₆₇×#1828	C₂	semidirect
6. C₆₇×#1781	C₂	semidirect
11. #535×C₆₇	C₄	semidirect
12. #534×C₆₇	C₄	semidirect
14. #525×C₆₇	C₄	semidirect
15. C₂×#120×C₆₇	C₄	semidirect
18. #518×C₆₇	C₂²	semidirect
22. C₂×#99×C₆₇	C₂²	semidirect
23. C₂×#99×C₆₇	C₂²	semidirect
24. #379×C₆₇	C₂²	semidirect
25. #342×C₆₇	C₂²	semidirect
26. C₂⁵×C₆₇	C₂×C₄	semidirect
36. C₂²×D₄×C₆₇	C₂×C₄	semidirect
37. C₂²×D₄×C₆₇	C₂×C₄	semidirect
42. C₂³×C₄×C₆₇	C₂×C₄	semidirect
47. #123×C₆₇	C₂×C₄	semidirect
57. #120×C₆₇	C₂×C₄	semidirect
58. #120×C₆₇	C₂×C₄	semidirect
59. #120×C₆₇	C₂×C₄	semidirect
64. #117×C₆₇	C₂×C₄	semidirect
69. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
81. C₂⁴×C₆₇	C₂²⋊C₄	semidirect
90. C₂×Q₈×C₆₇	C₂²⋊C₄	semidirect
107. C₂×D₄×C₆₇	C₂²⋊C₄	semidirect
108. C₂×D₄×C₆₇	C₂²⋊C₄	semidirect
118. C₂²×C₄×C₆₇	C₂²⋊C₄	semidirect
119. C₂²×C₄×C₆₇	C₂²⋊C₄	semidirect
228. #21776	C₆₇	direct (N × H)
239. #2883	C₂×C₆₇	semidirect
240. #1913	C₂×C₆₇	semidirect
241. #1889	C₂×C₆₇	semidirect
242. #1828	C₂×C₆₇	semidirect
243. #1781	C₂×C₆₇	semidirect
244. C₆₇	#21776	direct (N × H)
249. #535	C₄×C₆₇	semidirect
250. #534	C₄×C₆₇	semidirect
252. #525	C₄×C₆₇	semidirect
253. C₂×#120	C₄×C₆₇	semidirect
256. #518	C₂²×C₆₇	semidirect
260. C₂×#99	C₂²×C₆₇	semidirect
261. C₂×#99	C₂²×C₆₇	semidirect
262. #379	C₂²×C₆₇	semidirect
263. #342	C₂²×C₆₇	semidirect
264. C₂⁵	C₂×C₄×C₆₇	semidirect
274. C₂²×D₄	C₂×C₄×C₆₇	semidirect
275. C₂²×D₄	C₂×C₄×C₆₇	semidirect
280. C₂³×C₄	C₂×C₄×C₆₇	semidirect
285. #123	C₂×C₄×C₆₇	semidirect
295. #120	C₂×C₄×C₆₇	semidirect
296. #120	C₂×C₄×C₆₇	semidirect
297. #120	C₂×C₄×C₆₇	semidirect
302. #117	C₂×C₄×C₆₇	semidirect
307. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
319. C₂⁴	C₂²⋊C₄×C₆₇	semidirect
328. C₂×Q₈	C₂²⋊C₄×C₆₇	semidirect
345. C₂×D₄	C₂²⋊C₄×C₆₇	semidirect
346. C₂×D₄	C₂²⋊C₄×C₆₇	semidirect
356. C₂²×C₄	C₂²⋊C₄×C₆₇	semidirect
357. C₂²×C₄	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3484	2
C₆₇×#2883	2
C₆₇×#1913	2
C₆₇×#1889	2
C₆₇×#1828	2
C₆₇×#1781	2
#21776	67

Subgroups — C67×#21776 #30701027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21776 #30701027