Subgroups — #30874648

Order 17152.

732 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863524	C₂	1	2	yes	maximal
2	#22562686	C₂	1	1	yes	maximal, characteristic
3	#22560932	C₂	1	2	yes	maximal
4	#22560931	C₂	1	2	yes	maximal
5	#22560926	C₂	1	2	yes	maximal
6	#22560362	C₂	1	2	yes	maximal
7	#22560361	C₂	1	1	yes	maximal, characteristic
8	#22560068	C₂	1	2	yes	maximal
9	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#16863524	—	2	8	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
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×Dic₆₇⋊C₈	C₂²	1	2	yes	—
17	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
19	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
20	C₂×#14523953	C₂²	1	2	yes	—
21	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
22	#16863302	C₄	1	1	yes	characteristic
23	#16863302	C₄	1	1	yes	characteristic
24	#16863294	C₄	1	1	yes	characteristic
25	#16863294	C₄	1	1	yes	characteristic
26	#16863280	C₄	1	2	yes	—
27	#16863280	C₄	1	2	yes	—
28	C₄×Dic₆₇⋊C₄	C₂²	1	2	yes	—
29	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
30	#16863242	C₂²	1	2	yes	—
31	#16863150	C₂²	1	2	yes	—
32	#16863150	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₆₇	C₂²	1	2	yes	—
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	2	yes	—
38	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₄×Dic₁₃₄	D₄	1	4	yes	—
42	C₄×Dic₁₃₄	Q₈	1	4	yes	—
43	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
50	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
51	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
53	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
54	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	Dic₆₇⋊C₈	—	2	4	no	—
57	Dic₆₇⋊C₈	—	4	2	no	—
58	Dic₆₇⋊C₈	—	4	2	no	—
59	#14523953	—	2	4	no	—
60	C₈×Dic₆₇	—	2	4	no	—
61	C₆₇⋊#95	C₂×C₄	1	2	yes	—
62	C₆₇⋊#95	C₂×C₄	1	2	yes	—
63	C₆₇⋊#95	C₂×C₄	1	2	yes	—
64	C₆₇⋊#95	C₂×C₄	1	2	yes	—
65	C₆₇⋊#95	C₂×C₄	1	2	yes	—
66	C₆₇⋊#95	C₂×C₄	1	2	yes	—
67	C₆₇⋊#95	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	Q₈	1	4	yes	complemented
79	#97×C₆₇	—	2	4	no	—
80	C₂×Dic₁₃₄	—	4	1	no	—
81	C₂×Dic₁₃₄	—	4	1	no	—
82	C₂×Dic₁₃₄	C₄⋊C₄	1	4	yes	—
83	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	—	2	2	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₂×C₆₇⋊C₈	—	2	2	no	—
93	C₄⋊Dic₆₇	—	4	1	no	—
94	C₄⋊Dic₆₇	—	4	1	no	—
95	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
96	Dic₆₇⋊C₄	—	2	4	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	C₄⋊C₄	1	8	yes	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
102	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	2	1	no	—
118	C₄²×C₆₇	—	2	1	no	—
119	C₄²×C₆₇	—	4	1	no	—
120	C₄²×C₆₇	—	4	1	no	—
121	C₄²×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	C₂×D₄	1	2	yes	—
123	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
124	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
125	C₄²×C₆₇	#41	1	2	yes	—
126	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	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	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₆₇	#131	1	2	yes	—
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	2	2	no	—
154	C₂×Dic₆₇	—	2	2	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	2	2	no	—
157	C₂×Dic₆₇	—	2	2	no	—
158	C₂×Dic₆₇	—	2	2	no	—
159	C₂×Dic₆₇	—	2	2	no	—
160	C₂×Dic₆₇	—	2	2	no	—
161	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
162	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
163	Dic₁₃₄	—	4	4	no	—
164	C₈×C₆₇	—	2	4	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₆₇⋊C₈	—	4	2	no	—
168	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₂²×C₆₇	#520	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
172	C₂²×C₆₇	#568	1	1	yes	characteristic
173	C₂²×C₆₇	#568	1	1	yes	characteristic
174	C₂²×C₆₇	#568	1	1	yes	characteristic
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	2	no	—
180	C₄×C₆₇	—	4	1	no	—
181	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
182	C₄×C₆₇	#550	1	1	yes	characteristic
183	C₄×C₆₇	#557	1	1	yes	characteristic
184	C₄×C₆₇	Q₈²	1	1	yes	characteristic
185	Dic₆₇	—	2	4	no	—
186	Dic₆₇	—	2	4	no	—
187	Dic₆₇	—	4	2	no	—
188	Dic₆₇	—	4	2	no	—
189	#22723	—	67	1	no	maximal, Sylow
190	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₆₇	#2905	1	1	yes	characteristic
192	C₂×C₆₇	#2906	1	1	yes	characteristic
193	C₂×C₆₇	#2961	1	1	yes	characteristic
194	C₂×C₆₇	#2962	1	1	yes	characteristic
195	C₂×C₆₇	#2992	1	1	yes	characteristic
196	C₂×C₆₇	#2995	1	1	yes	characteristic
197	C₂×#446	—	67	2	no	—
198	#2295	—	67	1	no	—
199	#1940	—	67	2	no	—
200	#1940	—	67	2	no	—
201	#1939	—	67	2	no	—
202	#1843	—	67	2	no	—
203	#1842	—	67	1	no	—
204	#1768	—	67	2	no	—
205	#1767	D₆₇	1	1	yes	characteristic
206	C₆₇	#22723	1	1	yes	characteristic, complemented, Sylow
207	C₂×C₄×Q₈	—	67	1	no	—
208	#446	—	134	8	no	—
209	C₂×C₄⋊C₈	—	67	2	no	—
210	C₂×C₄⋊C₈	—	67	2	no	—
211	C₂×C₄⋊C₈	—	67	1	no	—
212	C₂×C₄⋊C₈	—	67	1	no	—
213	C₂×C₄⋊C₈	—	67	2	no	—
214	C₂×C₄⋊C₈	—	67	2	no	—
215	C₂×C₄⋊C₈	—	67	2	no	—
216	C₂×C₄⋊C₈	—	67	2	no	—
217	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
219	C₂×#97	—	67	2	no	—
220	C₂×#97	C₂×D₆₇	1	2	yes	—
221	C₂×C₄×C₈	—	67	2	no	—
222	#391	—	67	1	no	—
223	#391	—	67	1	no	—
224	#389	—	67	1	no	—
225	#389	—	67	1	no	—
226	#387	—	67	2	no	—
227	#387	—	67	2	no	—
228	C₄×C₄⋊C₄	—	67	1	no	—
229	C₄×C₄⋊C₄	—	67	2	no	—
230	#376	—	67	2	no	—
231	C₄³	C₂×D₆₇	1	1	yes	characteristic
232	#336	—	67	2	no	—
233	#336	—	67	2	no	—
234	#336	C₂×D₆₇	1	2	yes	—
235	C₂²×Q₈	—	67	1	no	—
236	C₂²×C₈	—	67	2	no	—
237	C₂²×C₈	—	67	2	no	—
238	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
239	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
240	C₄×Q₈	—	67	4	no	—
241	C₄×Q₈	—	67	4	no	—
242	C₂×C₄⋊C₄	—	67	2	no	—
243	C₂×C₄⋊C₄	—	67	1	no	—
244	C₂×C₄⋊C₄	—	67	1	no	—
245	C₂×C₄⋊C₄	—	67	2	no	—
246	C₂×C₄⋊C₄	—	67	1	no	—
247	C₂×C₄⋊C₄	—	67	2	no	—
248	C₂×C₄²	—	67	2	no	—
249	C₂×C₄²	—	67	2	no	—
250	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
253	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
254	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
255	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
256	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	268	1	no	—
259	C₄⋊C₈	—	268	1	no	—
260	C₄⋊C₈	—	268	2	no	—
261	C₄⋊C₈	—	268	2	no	—
262	C₄⋊C₈	—	134	4	no	—
263	C₄⋊C₈	—	134	4	no	—
264	C₄⋊C₈	—	134	4	no	—
265	C₄⋊C₈	—	134	4	no	—
266	C₄⋊C₈	Dic₁₃₄	1	4	yes	complemented
267	#97	—	2	4	no	—
268	#97	—	134	4	no	—
269	C₄×C₈	—	134	4	no	—
270	#95	—	67	2	no	—
271	#95	—	67	2	no	—
272	#95	—	67	2	no	—
273	#95	—	67	2	no	—
274	#95	—	67	2	no	—
275	#95	—	67	2	no	—
276	#95	—	67	2	no	—
277	C₂×Q₈	—	268	1	no	—
278	C₂×Q₈	—	268	1	no	—
279	C₂×Q₈	—	67	4	no	—
280	C₂²×C₄	—	67	2	no	—
281	C₂²×C₄	—	67	2	no	—
282	C₂²×C₄	—	67	2	no	—
283	C₂²×C₄	—	67	2	no	—
284	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	134	2	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	—	134	2	no	—
299	C₂×C₈	—	134	2	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
303	C₄⋊C₄	—	268	2	no	—
304	C₄⋊C₄	—	268	1	no	—
305	C₄⋊C₄	—	268	1	no	—
306	C₄⋊C₄	—	67	8	no	—
307	C₄⋊C₄	—	67	4	no	—
308	C₄⋊C₄	—	134	4	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	4	1	no	—
313	C₄²	—	4	1	no	—
314	C₄²	—	4	1	no	—
315	C₄²	—	67	4	no	—
316	C₄²	—	67	4	no	—
317	C₄²	—	134	2	no	—
318	C₄²	—	134	2	no	—
319	C₄²	D₄×D₆₇	1	2	yes	—
320	C₄²	Q₈×D₆₇	1	2	yes	—
321	C₄²	C₂×Dic₁₃₄	1	2	yes	—
322	C₄²	#11761628	1	2	yes	—
323	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
324	Q₈	—	268	4	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	67	4	no	—
339	C₂×C₄	—	67	4	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	—	134	2	no	—
342	C₂×C₄	—	134	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	—	134	2	no	—
345	C₂×C₄	—	134	2	no	—
346	C₂×C₄	—	134	2	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	2	no	—
349	C₂×C₄	—	134	2	no	—
350	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
351	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
352	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
353	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
354	C₂×C₄	#14524051	1	2	yes	—
355	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
358	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
359	C₂×C₄	#14524089	1	1	yes	characteristic
360	C₂×C₄	#14524090	1	1	yes	characteristic
361	C₈	—	2	4	no	—
362	C₈	—	4	2	no	—
363	C₈	—	268	2	no	—
364	C₈	—	268	2	no	—
365	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
366	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
367	C₂²	#16863876	1	1	yes	characteristic
368	C₂²	C₂×#14524051	1	1	yes	characteristic
369	C₂²	#16864162	1	1	yes	characteristic
370	C₂²	#16864162	1	1	yes	characteristic
371	C₂²	#16864162	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	2	no	—
377	C₄	—	4	1	no	—
378	C₄	—	268	2	no	—
379	C₄	—	268	2	no	—
380	C₄	—	134	4	no	—
381	C₄	—	134	4	no	—
382	C₄	D₄×Dic₁₃₄	1	1	yes	characteristic
383	C₄	#16864067	1	1	yes	characteristic
384	C₄	#16864124	1	1	yes	characteristic
385	C₄	Q₈×Dic₁₃₄	1	1	yes	characteristic
386	C₂	C₄⋊C₄×Dic₁₃₄	1	1	yes	characteristic
387	C₂	#22568771	1	1	yes	characteristic
388	C₂	#22568778	1	1	yes	characteristic
389	C₂	#22569149	1	1	yes	characteristic
390	C₂	#22569156	1	1	yes	characteristic
391	C₂	#22569350	1	1	yes	characteristic
392	C₂	#22569381	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1767	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1767	D₆₇
hypocenter	C₆₇	#22723
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1767

Complements

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

N	H ≅ G/N	type
78. C₄⋊C₈×C₆₇	Q₈	semidirect
206. C₆₇	#22723	semidirect
266. C₄⋊C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863524	2
#22562686	2
#22560932	2
#22560931	2
#22560926	2
#22560362	2
#22560361	2
#22560068	2
C₆₇×#1767	2
#22723	67