Subgroups — #22601061

Order 9088.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986721	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986664	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986361	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986360	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986348	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986278	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986217	C₂	1	1	yes	maximal, characteristic
9	#17986215	C₂	1	1	yes	maximal, characteristic
10	#17986203	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986128	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986098	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986097	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986081	C₂	1	1	yes	maximal, characteristic
15	#477×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₇₁	—	2	1	no	—
17	C₂×#11765854	C₂²	1	1	yes	characteristic
18	C₂×#11765852	C₂²	1	1	yes	characteristic
19	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556126	C₂²	1	1	yes	characteristic
23	#14556123	—	2	1	no	—
24	#14556122	—	2	1	no	—
25	#14556117	C₂²	1	1	yes	characteristic
26	#14556116	C₂²	1	1	yes	characteristic
27	#14556106	—	2	1	no	—
28	#14556106	—	2	1	no	—
29	#14556106	—	2	1	no	—
30	#14556104	—	2	1	no	—
31	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
32	#14556093	C₂²	1	1	yes	characteristic
33	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
34	#14556090	C₂²	1	1	yes	characteristic
35	#14556090	C₂²	1	1	yes	characteristic
36	#14556090	C₂²	1	1	yes	characteristic
37	#14556085	—	2	1	no	—
38	#14556061	C₂²	1	1	yes	characteristic
39	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
41	#14556053	C₂²	1	1	yes	characteristic
42	#14556052	C₂²	1	1	yes	characteristic
43	#14556051	C₂²	1	1	yes	characteristic
44	#14556051	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556048	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	#14556039	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#103×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
62	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
63	Q₈×D₇₁	—	4	4	no	—
64	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
65	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	#11765854	—	2	2	no	—
71	#11765852	—	2	2	no	—
72	D₅₆₈	—	2	2	no	—
73	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	Dic₇₁⋊C₄	—	2	1	no	—
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
83	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	—	2	1	no	—
89	(C₂×D₇₁)⋊C₄	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	—	2	1	no	—
91	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	C₄×Dic₇₁	—	2	1	no	—
93	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
94	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₇₁	—	2	1	no	—
96	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	—	2	1	no	—
100	C₄⋊C₄×C₇₁	—	2	1	no	—
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₄²×C₇₁	—	2	1	no	—
105	C₄²×C₇₁	C₂³	1	1	yes	characteristic
106	C₂²×D₇₁	—	2	1	no	—
107	C₂²×D₇₁	—	2	1	no	—
108	Q₈×C₇₁	—	2	1	no	—
109	Q₈×C₇₁	—	2	1	no	—
110	Q₈×C₇₁	—	4	2	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
124	D₂₈₄	—	2	1	no	—
125	D₂₈₄	—	2	1	no	—
126	D₂₈₄	—	2	1	no	—
127	D₂₈₄	#41	1	2	yes	—
128	C₄×D₇₁	—	4	2	no	—
129	C₄×D₇₁	—	4	2	no	—
130	C₄×D₇₁	—	4	2	no	—
131	C₄×D₇₁	—	4	2	no	—
132	Dic₁₄₂	—	2	2	no	—
133	Dic₁₄₂	—	2	1	no	—
134	Dic₁₄₂	—	2	1	no	—
135	Dic₁₄₂	—	4	2	no	—
136	Dic₁₄₂	—	4	2	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₇₁	—	2	2	no	—
143	C₂×D₇₁	—	4	1	no	—
144	C₂×D₇₁	—	4	1	no	—
145	C₂×D₇₁	—	4	1	no	—
146	C₂×D₇₁	—	4	1	no	—
147	C₂×D₇₁	—	4	1	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	1	no	—
152	C₄×C₇₁	—	4	1	no	—
153	C₄×C₇₁	—	4	1	no	—
154	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
155	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	2	1	no	—
158	Dic₇₁	—	2	1	no	—
159	Dic₇₁	—	4	1	no	—
160	Dic₇₁	—	4	1	no	—
161	Dic₇₁	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₇₁	#548	1	1	yes	characteristic
163	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×C₇₁	#578	1	1	yes	characteristic
165	D₇₁	—	4	2	no	—
166	D₇₁	—	8	1	no	—
167	#3367	—	71	1	no	maximal, Sylow
168	C₇₁	#3367	1	1	yes	characteristic, complemented, Sylow
169	#556	—	71	1	no	—
170	D₄×Q₈	—	71	1	no	—
171	#489	—	71	1	no	—
172	#489	—	71	1	no	—
173	#487	—	71	1	no	—
174	#477	—	71	1	no	—
175	#477	D₇₁	1	1	yes	characteristic, complemented
176	#474	—	71	1	no	—
177	#463	—	71	1	no	—
178	#463	—	71	1	no	—
179	#461	—	71	1	no	—
180	#446	—	71	1	no	—
181	#440	—	71	1	no	—
182	#440	—	71	1	no	—
183	C₄×D₈	—	71	1	no	—
184	C₂²×Q₈	—	142	1	no	—
185	C₂×SD₁₆	—	71	1	no	—
186	C₂×SD₁₆	—	71	1	no	—
187	C₂×D₈	—	71	1	no	—
188	C₄⋊Q₈	—	71	1	no	—
189	C₄⋊Q₈	—	71	1	no	—
190	C₄⋊Q₈	—	71	1	no	—
191	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
192	#126	—	142	1	no	—
193	#126	—	142	1	no	—
194	#124	—	71	1	no	—
195	#124	—	71	1	no	—
196	C₂²⋊Q₈	—	142	1	no	—
197	C₂²⋊Q₈	—	142	1	no	—
198	C₂²⋊Q₈	—	142	1	no	—
199	C₂²⋊Q₈	—	142	1	no	—
200	C₄×Q₈	—	71	1	no	—
201	C₄×Q₈	—	71	1	no	—
202	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
203	C₄×D₄	—	71	1	no	—
204	C₄×D₄	—	71	1	no	—
205	C₄×D₄	—	71	1	no	—
206	C₄×D₄	—	71	1	no	—
207	#117	—	142	1	no	—
208	#107	—	71	1	no	—
209	#107	C₂×D₇₁	1	1	yes	characteristic
210	#107	C₂×D₇₁	1	1	yes	characteristic
211	C₄⋊C₈	—	71	1	no	—
212	C₄⋊C₈	—	71	1	no	—
213	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	C₂×D₇₁	1	1	yes	characteristic
219	#103	C₂×D₇₁	1	1	yes	characteristic
220	#102	—	71	1	no	—
221	#102	—	71	1	no	—
222	#102	—	71	1	no	—
223	#102	—	71	1	no	—
224	#97	—	71	1	no	—
225	#97	—	71	1	no	—
226	C₄×C₈	—	71	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	284	4	no	—
229	C₂×Q₈	—	71	1	no	—
230	C₂×Q₈	—	71	1	no	—
231	C₂×Q₈	—	142	1	no	—
232	C₂×Q₈	—	142	1	no	—
233	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
234	C₂×D₄	—	71	1	no	—
235	C₂×D₄	—	71	1	no	—
236	C₂²×C₄	—	142	1	no	—
237	C₂²×C₄	—	142	1	no	—
238	C₂²×C₄	—	142	1	no	—
239	C₂²×C₄	—	142	1	no	—
240	SD₁₆	—	142	2	no	—
241	SD₁₆	—	142	2	no	—
242	D₈	—	142	2	no	—
243	C₂×C₈	—	71	1	no	—
244	C₂×C₈	—	71	1	no	—
245	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	71	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	71	1	no	—
252	C₄⋊C₄	—	71	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	—	142	1	no	—
255	C₄⋊C₄	—	142	1	no	—
256	C₄⋊C₄	—	142	1	no	—
257	C₄⋊C₄	—	142	1	no	—
258	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	142	1	no	—
262	C₂²⋊C₄	—	142	1	no	—
263	C₂²⋊C₄	—	142	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₂²⋊C₄	—	142	1	no	—
266	C₂²⋊C₄	—	142	1	no	—
267	C₂²⋊C₄	—	142	1	no	—
268	C₂²⋊C₄	—	142	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	71	1	no	—
271	C₄²	—	71	1	no	—
272	C₄²	—	142	1	no	—
273	C₄²	C₂²×D₇₁	1	1	yes	characteristic
274	C₂³	—	142	1	no	—
275	C₂³	—	142	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	284	2	no	—
280	Q₈	—	284	2	no	—
281	Q₈	—	142	2	no	—
282	Q₈	—	142	1	no	—
283	Q₈	—	142	1	no	—
284	D₄	—	71	2	no	—
285	D₄	—	142	1	no	—
286	D₄	—	142	1	no	—
287	D₄	—	142	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	284	2	no	—
293	C₂×C₄	—	284	2	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	284	2	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	—	142	1	no	—
302	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
303	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	142	1	no	—
308	C₈	—	142	1	no	—
309	C₂²	—	284	1	no	—
310	C₂²	—	284	1	no	—
311	C₂²	—	284	1	no	—
312	C₂²	—	284	1	no	—
313	C₂²	—	284	1	no	—
314	C₂²	—	142	2	no	—
315	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	284	1	no	—
323	C₄	—	284	1	no	—
324	C₄	—	71	2	no	—
325	C₄	—	142	1	no	—
326	C₄	—	142	1	no	—
327	C₄	—	142	1	no	—
328	C₄	#41×D₇₁	1	1	yes	characteristic
329	C₄	#14556176	1	1	yes	characteristic
330	C₂	—	284	2	no	—
331	C₂	—	568	1	no	—
332	C₂	#17986659	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₇₁	1	1	yes	characteristic
334	C₂	#17986812	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#477×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#477	D₇₁
hypocenter	C₇₁	#3367
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #17986721	C₂	semidirect
2. #17986664	C₂	semidirect
3. #17986361	C₂	semidirect
4. #17986360	C₂	semidirect
5. #17986348	C₂	semidirect
6. #17986293	C₂	semidirect
7. #17986278	C₂	semidirect
10. #17986203	C₂	semidirect
11. #17986128	C₂	semidirect
12. #17986098	C₂	semidirect
13. #17986097	C₂	semidirect
15. #477×C₇₁	C₂	semidirect
168. C₇₁	#3367	semidirect
175. #477	D₇₁	semidirect