Subgroups — #30875731

Order 17152.

656 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561146	C₂	1	1	yes	maximal, characteristic
2	#22561146	C₂	1	1	yes	maximal, characteristic
3	#22560775	C₂	1	2	yes	maximal
4	#22560768	C₂	1	2	yes	maximal
5	#22560361	C₂	1	1	yes	maximal, characteristic
6	#22560329	C₂	1	2	yes	maximal
7	#22560078	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#22560066	C₂	1	2	yes	maximal
10	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#14523954	C₂²	1	2	yes	—
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
22	#16863151	C₂²	1	1	yes	characteristic
23	#16863151	C₂²	1	2	yes	—
24	#16863151	C₂²	1	2	yes	—
25	#16863151	C₂²	1	2	yes	—
26	#16863151	C₂²	1	1	yes	characteristic
27	#16863151	C₂²	1	2	yes	—
28	#16863151	C₂²	1	2	yes	—
29	#16863151	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	#391×C₆₇	C₄	1	1	yes	characteristic
32	#391×C₆₇	C₄	1	1	yes	characteristic
33	#389×C₆₇	C₄	1	1	yes	characteristic
34	#389×C₆₇	C₄	1	1	yes	characteristic
35	#387×C₆₇	C₄	1	2	yes	—
36	#387×C₆₇	C₄	1	2	yes	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	#376×C₆₇	C₂²	1	2	yes	—
40	C₄³×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
51	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
52	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
53	#14523954	—	4	2	no	—
54	C₄×C₆₇⋊C₈	—	2	2	no	—
55	C₄×C₆₇⋊C₈	—	2	2	no	—
56	C₄×C₆₇⋊C₈	—	4	2	no	—
57	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	#95×C₆₇	C₂×C₄	1	2	yes	—
76	#95×C₆₇	C₂×C₄	1	2	yes	—
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	#95×C₆₇	C₂×C₄	1	2	yes	—
79	#95×C₆₇	C₂×C₄	1	2	yes	—
80	#95×C₆₇	C₂×C₄	1	2	yes	—
81	#95×C₆₇	C₂×C₄	1	2	yes	—
82	C₂×C₆₇⋊C₈	—	2	1	no	—
83	C₂×C₆₇⋊C₈	—	2	1	no	—
84	C₂×C₆₇⋊C₈	—	2	1	no	—
85	C₂×C₆₇⋊C₈	—	2	1	no	—
86	C₂×C₆₇⋊C₈	—	2	1	no	—
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	2	no	—
93	C₂×C₆₇⋊C₈	—	2	2	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₂×C₆₇⋊C₈	—	2	2	no	—
96	C₂×C₆₇⋊C₈	—	2	1	no	—
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	4	1	no	—
130	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₄²×C₆₇	#41	1	2	yes	—
133	C₄²×C₆₇	#41	1	2	yes	—
134	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	#117	1	2	yes	—
160	C₂×C₄×C₆₇	#117	1	2	yes	—
161	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
162	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
163	C₂×C₄×C₆₇	#131	1	2	yes	—
164	C₂×C₄×C₆₇	#131	1	2	yes	—
165	C₂×C₄×C₆₇	#131	1	2	yes	—
166	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
172	C₆₇⋊C₈	—	2	2	no	—
173	C₆₇⋊C₈	—	2	2	no	—
174	C₆₇⋊C₈	—	4	1	no	—
175	C₆₇⋊C₈	—	4	1	no	—
176	C₆₇⋊C₈	—	4	2	no	—
177	C₆₇⋊C₈	—	4	2	no	—
178	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₂²×C₆₇	#520	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
183	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
184	C₂²×C₆₇	#568	1	1	yes	characteristic
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	2	no	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	—	4	1	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	#433	1	4	yes	—
195	C₄×C₆₇	#554	1	1	yes	characteristic
196	C₄×C₆₇	#556	1	1	yes	characteristic
197	C₄×C₆₇	#557	1	1	yes	characteristic
198	C₄×C₆₇	#559	1	1	yes	characteristic
199	#22822	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	#2355	1	1	yes	characteristic
201	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
202	C₂×C₆₇	#2925	1	1	yes	characteristic
203	C₂×C₆₇	#2960	1	1	yes	characteristic
204	C₂×C₆₇	#2962	1	1	yes	characteristic
205	C₂×C₆₇	#2993	1	1	yes	characteristic
206	C₂×C₆₇	#2994	1	1	yes	characteristic
207	#2295	D₆₇	1	1	yes	characteristic
208	#1984	—	67	1	no	—
209	#1984	—	67	1	no	—
210	#1917	—	67	2	no	—
211	#1916	—	67	2	no	—
212	#1842	—	67	1	no	—
213	#1832	—	67	2	no	—
214	#1770	—	67	1	no	—
215	C₈×C₄⋊C₄	—	67	1	no	—
216	#1768	—	67	2	no	—
217	C₄×C₄⋊C₈	—	67	1	no	—
218	C₆₇	#22822	1	1	yes	characteristic, complemented, Sylow
219	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
220	C₂×C₄⋊C₈	—	67	2	no	—
221	C₂×C₄⋊C₈	—	67	1	no	—
222	C₂×C₄⋊C₈	—	67	1	no	—
223	C₂×C₄⋊C₈	—	67	2	no	—
224	C₂×C₄⋊C₈	—	67	1	no	—
225	C₂×C₄⋊C₈	—	67	1	no	—
226	C₂×#97	—	67	2	no	—
227	C₂×C₄×C₈	—	67	2	no	—
228	C₂×C₄×C₈	—	67	1	no	—
229	C₂×C₄×C₈	—	67	1	no	—
230	#391	Dic₆₇	1	1	yes	characteristic
231	#391	Dic₆₇	1	1	yes	characteristic
232	#389	Dic₆₇	1	1	yes	characteristic
233	#389	Dic₆₇	1	1	yes	characteristic
234	#387	Dic₆₇	1	2	yes	—
235	#387	Dic₆₇	1	2	yes	—
236	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
237	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
238	#376	C₂×D₆₇	1	2	yes	—
239	C₄³	C₂×D₆₇	1	1	yes	characteristic
240	#336	—	67	2	no	—
241	#336	—	67	2	no	—
242	#336	—	67	2	no	—
243	#336	—	67	2	no	—
244	#336	—	67	1	no	—
245	#336	—	67	2	no	—
246	#336	—	67	1	no	—
247	#336	—	67	2	no	—
248	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
249	C₂²×C₈	—	67	1	no	—
250	C₂²×C₈	—	67	1	no	—
251	C₂²×C₈	—	67	1	no	—
252	C₂²×C₈	—	67	1	no	—
253	C₂²×C₈	—	67	2	no	—
254	C₂²×C₈	—	67	2	no	—
255	C₄×Q₈	—	2	2	no	—
256	C₄×Q₈	—	2	2	no	—
257	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
258	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
262	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
263	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
265	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
268	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
269	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
270	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
271	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
272	C₄⋊C₈	—	268	2	no	—
273	C₄⋊C₈	—	67	4	no	—
274	C₄⋊C₈	—	134	2	no	—
275	C₄⋊C₈	—	134	2	no	—
276	C₄⋊C₈	—	134	2	no	—
277	C₄⋊C₈	—	134	4	no	—
278	#97	—	268	2	no	—
279	C₄×C₈	—	268	2	no	—
280	C₄×C₈	—	134	2	no	—
281	C₄×C₈	—	134	2	no	—
282	#95	C₂×Dic₆₇	1	2	yes	—
283	#95	C₂×Dic₆₇	1	2	yes	—
284	#95	C₂×Dic₆₇	1	2	yes	—
285	#95	C₂×Dic₆₇	1	2	yes	—
286	#95	C₂×Dic₆₇	1	2	yes	—
287	#95	C₂×Dic₆₇	1	2	yes	—
288	#95	C₂×Dic₆₇	1	2	yes	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
293	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
294	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
295	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
297	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
303	C₂×C₈	—	67	2	no	—
304	C₂×C₈	—	67	2	no	—
305	C₂×C₈	—	134	1	no	—
306	C₂×C₈	—	134	1	no	—
307	C₂×C₈	—	134	2	no	—
308	C₂×C₈	—	134	2	no	—
309	C₂×C₈	—	134	1	no	—
310	C₂×C₈	—	134	1	no	—
311	C₂×C₈	—	134	2	no	—
312	C₂×C₈	—	134	2	no	—
313	C₂×C₈	—	134	2	no	—
314	C₂×C₈	—	134	2	no	—
315	C₂×C₈	—	134	1	no	—
316	C₂×C₈	—	134	1	no	—
317	C₂×C₈	—	134	1	no	—
318	C₂×C₈	—	134	1	no	—
319	C₂×C₈	—	134	1	no	—
320	C₂×C₈	—	134	1	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	1	no	—
337	C₄²	Q₈×D₆₇	1	2	yes	—
338	C₄²	#11761628	1	2	yes	—
339	C₄²	#11761628	1	2	yes	—
340	C₄²	#11761629	1	2	yes	—
341	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
342	Q₈	—	4	4	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	#14524002	1	2	yes	—
367	C₂×C₄	#14524002	1	2	yes	—
368	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
369	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
370	C₂×C₄	#14524050	1	2	yes	—
371	C₂×C₄	#14524050	1	2	yes	—
372	C₂×C₄	#14524050	1	2	yes	—
373	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
375	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
376	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
377	C₂×C₄	#14524089	1	1	yes	characteristic
378	C₂×C₄	#14524090	1	1	yes	characteristic
379	C₈	—	268	1	no	—
380	C₈	—	268	1	no	—
381	C₈	—	268	2	no	—
382	C₈	—	268	2	no	—
383	C₈	—	134	2	no	—
384	C₈	—	134	2	no	—
385	C₂²	C₂×#14524002	1	1	yes	characteristic
386	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
387	C₂²	#16863879	1	1	yes	characteristic
388	C₂²	C₂×#14524050	1	1	yes	characteristic
389	C₂²	C₂×#14524050	1	1	yes	characteristic
390	C₂²	C₂×#14524050	1	1	yes	characteristic
391	C₂²	#16864163	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	2	2	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	2	no	—
401	C₄	#16863457	1	4	yes	—
402	C₄	#16864106	1	1	yes	characteristic
403	C₄	#16864119	1	1	yes	characteristic
404	C₄	#16864124	1	1	yes	characteristic
405	C₄	#16864131	1	1	yes	characteristic
406	C₂	#22563166	1	1	yes	characteristic
407	C₂	C₂×#16863457	1	1	yes	characteristic
408	C₂	#22568875	1	1	yes	characteristic
409	C₂	#22569144	1	1	yes	characteristic
410	C₂	#22569158	1	1	yes	characteristic
411	C₂	#22569358	1	1	yes	characteristic
412	C₂	#22569369	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2295	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2295	D₆₇
hypocenter	C₆₇	#22822
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
52. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
218. C₆₇	#22822	semidirect

Maximal subgroups

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

subgroup	index
#22561146	2
#22560775	2
#22560768	2
#22560361	2
#22560329	2
#22560078	2
C₄⋊C₄×C₆₇⋊C₈	2
#22560066	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₆₇×#2295	2
#22822	67