Subgroups — #22600831

Order 9088.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986707	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986701	C₂	1	1	yes	maximal, characteristic
3	#17986406	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986339	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986332	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986315	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986311	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986308	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986277	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986242	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986100	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986070	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986038	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14556062	C₂	1	1	yes	maximal, characteristic, complemented
15	#465×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765848	C₂²	1	1	yes	characteristic
17	#14556127	C₂²	1	1	yes	characteristic
18	#14556121	—	2	1	no	—
19	#14556119	C₂²	1	1	yes	characteristic
20	#14556118	—	2	1	no	—
21	#14556112	—	2	1	no	—
22	#14556108	—	2	1	no	—
23	#14556105	—	2	1	no	—
24	#14556104	C₂²	1	1	yes	characteristic
25	#14556100	—	2	1	no	—
26	#14556099	C₂²	1	1	yes	characteristic
27	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
28	D₄×Dic₇₁	—	2	1	no	—
29	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
32	#14556086	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
36	#14556062	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556062	C₂²	1	1	yes	characteristic
39	#14556062	C₂²	1	1	yes	characteristic
40	#14556061	C₂²	1	1	yes	characteristic
41	#14556061	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	#14556059	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	#14556052	C₂²	1	1	yes	characteristic
46	#14556052	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#14556038	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#121×C₇₁	C₂²	1	1	yes	characteristic
56	#106×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×C₂²⋊D₇₁	—	2	1	no	—
61	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
62	C₂²×Dic₇₁	—	2	1	no	—
63	C₂²×Dic₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
67	#11765848	—	2	2	no	—
68	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₄⋊Dic₇₁	—	2	1	no	—
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
89	C₄×Dic₇₁	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₇₁	—	2	1	no	—
94	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
96	SD₁₆×C₇₁	—	2	2	no	—
97	C₂×C₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
104	C₂²⋊C₄×C₇₁	—	2	1	no	—
105	C₂³×C₇₁	—	2	1	no	—
106	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₇₁	—	2	1	no	—
109	Q₈×C₇₁	—	2	1	no	—
110	D₄×C₇₁	—	2	1	no	—
111	D₄×C₇₁	—	2	1	no	—
112	D₄×C₇₁	—	4	2	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₇₁	—	2	2	no	—
121	C₂²⋊D₇₁	—	8	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	8	1	no	—
127	C₂×Dic₇₁	—	8	1	no	—
128	C₂×Dic₇₁	—	8	1	no	—
129	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
132	C₄×D₇₁	—	2	1	no	—
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	8	1	no	—
135	Dic₁₄₂	—	8	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	#41	1	2	yes	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	—	2	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	—	4	1	no	—
144	C₂²×C₇₁	—	4	1	no	—
145	C₂²×C₇₁	#137	1	2	yes	—
146	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₇₁	—	2	1	no	—
148	C₂×D₇₁	—	2	1	no	—
149	C₂×D₇₁	—	2	1	no	—
150	C₄×C₇₁	—	2	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	—	4	1	no	—
162	Dic₇₁	—	4	1	no	—
163	C₂×C₇₁	—	2	2	no	—
164	C₂×C₇₁	—	8	1	no	—
165	C₂×C₇₁	#547	1	1	yes	characteristic
166	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
167	C₂×C₇₁	#576	1	1	yes	characteristic
168	D₇₁	—	4	1	no	—
169	#3346	—	71	1	no	maximal, Sylow
170	C₇₁	#3346	1	1	yes	characteristic, complemented, Sylow
171	#554	—	71	1	no	—
172	#553	—	71	1	no	—
173	#501	—	71	1	no	—
174	#485	—	71	1	no	—
175	#484	—	71	1	no	—
176	#481	—	71	1	no	—
177	#480	—	71	1	no	—
178	#479	—	71	1	no	—
179	#474	—	71	1	no	—
180	#468	—	71	1	no	—
181	#465	D₇₁	1	1	yes	characteristic, complemented
182	#440	—	71	1	no	—
183	#435	—	71	1	no	—
184	#428	—	71	1	no	—
185	C₂×#107	—	71	1	no	—
186	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
187	C₂×M₁₆	—	71	1	no	—
188	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
189	C₄⋊Q₈	—	71	1	no	—
190	#126	—	142	1	no	—
191	#125	—	71	1	no	—
192	#125	—	142	1	no	—
193	#123	—	142	1	no	—
194	C₂²⋊Q₈	—	71	1	no	—
195	C₂²⋊Q₈	—	142	1	no	—
196	C₂²⋊Q₈	—	142	1	no	—
197	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
198	#121	—	71	1	no	—
199	#121	—	142	1	no	—
200	#121	C₂×D₇₁	1	1	yes	characteristic
201	C₄×Q₈	—	71	1	no	—
202	C₄×D₄	—	71	1	no	—
203	C₄×D₄	—	71	1	no	—
204	C₄×D₄	—	142	1	no	—
205	C₄×D₄	—	142	1	no	—
206	#117	—	71	1	no	—
207	C₂×C₄⋊C₄	—	71	1	no	—
208	C₂×C₄⋊C₄	—	71	1	no	—
209	C₂×C₄⋊C₄	—	71	1	no	—
210	#107	—	71	1	no	—
211	#107	—	71	1	no	—
212	#107	—	71	1	no	—
213	#107	—	71	1	no	—
214	#107	—	71	1	no	—
215	#107	—	71	1	no	—
216	#106	—	71	1	no	—
217	#106	—	71	1	no	—
218	#106	C₂×D₇₁	1	1	yes	characteristic
219	C₄⋊C₈	—	71	1	no	—
220	#103	—	71	1	no	—
221	#103	—	71	1	no	—
222	#103	C₂×D₇₁	1	1	yes	characteristic
223	#102	—	71	1	no	—
224	#102	—	71	1	no	—
225	#102	C₂×D₇₁	1	1	yes	characteristic
226	C₂²⋊C₈	—	71	1	no	—
227	C₂²⋊C₈	—	71	1	no	—
228	#97	—	71	1	no	—
229	C₂×Q₈	—	142	1	no	—
230	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	71	1	no	—
233	C₂×D₄	—	142	1	no	—
234	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
235	C₂²×C₄	—	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	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
241	SD₁₆	—	2	2	no	—
242	M₁₆	—	142	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	71	1	no	—
245	C₂×C₈	—	71	1	no	—
246	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
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	—	71	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	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₄⋊C₄	—	142	1	no	—
262	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	71	1	no	—
267	C₂²⋊C₄	—	71	1	no	—
268	C₂²⋊C₄	—	142	1	no	—
269	C₂²⋊C₄	—	142	1	no	—
270	C₂²⋊C₄	—	142	1	no	—
271	C₂²⋊C₄	—	142	1	no	—
272	C₄²	—	71	1	no	—
273	C₄²	—	142	1	no	—
274	C₄²	—	142	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	71	1	no	—
277	C₂³	D₄×D₇₁	1	1	yes	characteristic
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	568	1	no	—
281	D₄	—	2	1	no	—
282	D₄	—	2	1	no	—
283	D₄	—	4	2	no	—
284	D₄	—	568	1	no	—
285	D₄	—	142	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	568	1	no	—
292	C₂×C₄	—	568	1	no	—
293	C₂×C₄	—	568	1	no	—
294	C₂×C₄	—	568	1	no	—
295	C₂×C₄	—	71	1	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	—	142	1	no	—
302	C₂×C₄	—	142	1	no	—
303	C₂×C₄	—	142	1	no	—
304	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	142	1	no	—
308	C₈	—	142	1	no	—
309	C₈	#11765864	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	142	1	no	—
316	C₂²	—	142	1	no	—
317	C₂²	—	142	1	no	—
318	C₂²	#14556157	1	2	yes	—
319	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	284	1	no	—
325	C₄	—	284	1	no	—
326	C₄	—	284	1	no	—
327	C₄	—	284	1	no	—
328	C₄	—	142	1	no	—
329	C₄	—	142	1	no	—
330	C₄	—	142	1	no	—
331	C₄	C₂×#11765864	1	1	yes	characteristic
332	C₄	#14556175	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	—	8	1	no	—
335	C₂	—	284	1	no	—
336	C₂	#17986649	1	1	yes	characteristic
337	C₂	C₂×#14556157	1	1	yes	characteristic
338	C₂	#17986801	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#465×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#465	D₇₁
hypocenter	C₇₁	#3346
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #17986707	C₂	semidirect
3. #17986406	C₂	semidirect
4. #17986339	C₂	semidirect
5. #17986332	C₂	semidirect
6. #17986315	C₂	semidirect
7. #17986311	C₂	semidirect
8. #17986308	C₂	semidirect
9. #17986277	C₂	semidirect
10. #17986242	C₂	semidirect
11. #17986100	C₂	semidirect
12. #17986070	C₂	semidirect
13. #17986038	C₂	semidirect
14. C₂×#14556062	C₂	semidirect
15. #465×C₇₁	C₂	semidirect
170. C₇₁	#3346	semidirect
181. #465	D₇₁	semidirect