Subgroups — #22601169

Order 9088.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986722	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986709	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986368	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986359	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986354	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986307	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986300	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986285	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986215	C₂	1	1	yes	maximal, characteristic
10	#17986215	C₂	1	1	yes	maximal, characteristic
11	#17986205	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986112	C₂	1	1	yes	maximal, characteristic
13	#17986101	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11765854	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765854	C₂²	1	1	yes	characteristic
17	C₂×#11765854	C₂²	1	1	yes	characteristic
18	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
19	#14556128	C₂²	1	1	yes	characteristic
20	#14556128	C₂²	1	1	yes	characteristic
21	#14556123	—	2	1	no	—
22	#14556123	—	2	1	no	—
23	#14556120	C₂²	1	1	yes	characteristic
24	#14556120	C₂²	1	1	yes	characteristic
25	#14556116	C₂²	1	1	yes	characteristic
26	#14556116	C₂²	1	1	yes	characteristic
27	#14556106	—	2	1	no	—
28	#14556106	—	2	1	no	—
29	#14556104	—	2	1	no	—
30	#14556104	—	2	1	no	—
31	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
33	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
34	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
35	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
36	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
37	#14556086	—	2	1	no	—
38	C₄⋊C₄×D₇₁	—	2	1	no	—
39	#14556062	C₂²	1	1	yes	characteristic
40	#14556060	C₂²	1	1	yes	characteristic
41	#14556060	C₂²	1	1	yes	characteristic
42	#14556053	C₂²	1	1	yes	characteristic
43	#14556053	C₂²	1	1	yes	characteristic
44	#14556053	C₂²	1	1	yes	characteristic
45	#14556053	C₂²	1	1	yes	characteristic
46	#14556053	C₂²	1	1	yes	characteristic
47	#14556053	C₂²	1	1	yes	characteristic
48	#14556050	C₂²	1	1	yes	characteristic
49	#14556050	C₂²	1	1	yes	characteristic
50	#14556050	C₂²	1	1	yes	characteristic
51	#14556050	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
56	#97×C₇₁	C₂²	1	1	yes	characteristic
57	#97×C₇₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	#11765854	—	2	2	no	—
69	#11765854	—	2	2	no	—
70	D₅₆₈	—	2	2	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	—	2	1	no	—
83	Dic₇₁⋊C₄	—	2	1	no	—
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	—	2	1	no	—
94	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×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	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
102	C₄²×C₇₁	C₂³	1	1	yes	characteristic
103	C₄²×C₇₁	C₂³	1	1	yes	characteristic
104	C₄²×C₇₁	C₂³	1	1	yes	characteristic
105	C₂²×D₇₁	—	2	1	no	—
106	C₂²×D₇₁	—	2	1	no	—
107	Q₈×C₇₁	—	4	1	no	—
108	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	D₂₈₄	—	2	1	no	—
122	D₂₈₄	—	2	1	no	—
123	D₂₈₄	—	2	1	no	—
124	D₂₈₄	#41	1	2	yes	—
125	C₄×D₇₁	—	4	2	no	—
126	C₄×D₇₁	—	4	2	no	—
127	C₄×D₇₁	—	8	1	no	—
128	C₄×D₇₁	—	8	1	no	—
129	Dic₁₄₂	—	2	1	no	—
130	Dic₁₄₂	—	2	1	no	—
131	Dic₁₄₂	—	2	1	no	—
132	Dic₁₄₂	—	2	1	no	—
133	Dic₁₄₂	—	8	1	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₈×C₇₁	—	2	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₇₁	—	2	2	no	—
140	C₂×D₇₁	—	4	1	no	—
141	C₂×D₇₁	—	4	1	no	—
142	C₂×D₇₁	—	4	1	no	—
143	C₂×D₇₁	—	4	1	no	—
144	C₂×D₇₁	—	4	1	no	—
145	C₄×C₇₁	—	2	1	no	—
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	2	1	no	—
151	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
152	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₇₁	—	4	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	—	4	1	no	—
159	C₂×C₇₁	#548	1	1	yes	characteristic
160	C₂×C₇₁	#575	1	1	yes	characteristic
161	C₂×C₇₁	#577	1	1	yes	characteristic
162	D₇₁	—	4	2	no	—
163	D₇₁	—	8	1	no	—
164	#3376	—	71	1	no	maximal, Sylow
165	C₇₁	#3376	1	1	yes	characteristic, complemented, Sylow
166	#556	—	71	1	no	—
167	#554	—	71	1	no	—
168	#491	—	71	1	no	—
169	#489	—	71	1	no	—
170	#488	—	71	1	no	—
171	#479	—	71	1	no	—
172	#478	—	71	1	no	—
173	#475	—	71	1	no	—
174	#463	—	71	1	no	—
175	#463	—	71	1	no	—
176	#461	—	71	1	no	—
177	#446	D₇₁	1	1	yes	characteristic, complemented
178	#442	—	71	1	no	—
179	#440	—	71	1	no	—
180	C₄×SD₁₆	—	71	1	no	—
181	C₂×SD₁₆	—	71	1	no	—
182	C₂×SD₁₆	—	71	1	no	—
183	C₂×D₈	—	71	1	no	—
184	C₄⋊Q₈	—	71	1	no	—
185	C₄⋊Q₈	—	71	1	no	—
186	#126	—	142	1	no	—
187	#126	—	142	1	no	—
188	#125	—	71	1	no	—
189	#125	—	71	1	no	—
190	#124	—	71	1	no	—
191	#124	—	71	1	no	—
192	C₂²⋊Q₈	—	142	1	no	—
193	C₂²⋊Q₈	—	142	1	no	—
194	C₂²⋊Q₈	—	142	1	no	—
195	C₂²⋊Q₈	—	142	1	no	—
196	C₄×Q₈	—	71	1	no	—
197	C₄×Q₈	—	71	1	no	—
198	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
199	C₄×D₄	—	71	1	no	—
200	C₄×D₄	—	71	1	no	—
201	C₄×D₄	—	71	1	no	—
202	C₄×D₄	—	71	1	no	—
203	#117	—	142	1	no	—
204	C₂×C₄⋊C₄	—	142	1	no	—
205	#107	—	71	1	no	—
206	#106	—	71	1	no	—
207	#106	—	71	1	no	—
208	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
209	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
210	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
211	#103	—	71	1	no	—
212	#103	—	71	1	no	—
213	#103	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	—	71	1	no	—
217	#102	—	71	1	no	—
218	#102	—	71	1	no	—
219	#102	—	71	1	no	—
220	#102	—	71	1	no	—
221	#97	C₂×D₇₁	1	1	yes	characteristic
222	#97	C₂×D₇₁	1	1	yes	characteristic
223	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
224	C₂×Q₈	—	71	1	no	—
225	C₂×Q₈	—	71	1	no	—
226	C₂×Q₈	—	142	1	no	—
227	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
228	C₂×D₄	—	71	1	no	—
229	C₂×D₄	—	71	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	—	142	1	no	—
232	C₂²×C₄	—	142	1	no	—
233	C₂²×C₄	—	142	1	no	—
234	SD₁₆	—	142	2	no	—
235	SD₁₆	—	142	2	no	—
236	D₈	—	142	2	no	—
237	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
241	C₄⋊C₄	—	71	1	no	—
242	C₄⋊C₄	—	71	1	no	—
243	C₄⋊C₄	—	71	1	no	—
244	C₄⋊C₄	—	71	1	no	—
245	C₄⋊C₄	—	142	1	no	—
246	C₄⋊C₄	—	142	1	no	—
247	C₄⋊C₄	—	142	1	no	—
248	C₄⋊C₄	—	142	1	no	—
249	C₄⋊C₄	—	142	1	no	—
250	C₄⋊C₄	—	142	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
255	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
256	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
257	C₂²⋊C₄	—	142	1	no	—
258	C₂²⋊C₄	—	142	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₂²⋊C₄	—	142	1	no	—
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	C₄²	—	142	1	no	—
266	C₄²	—	142	1	no	—
267	C₄²	C₂²×D₇₁	1	1	yes	characteristic
268	C₄²	C₂²×D₇₁	1	1	yes	characteristic
269	C₄²	C₂²×D₇₁	1	1	yes	characteristic
270	C₂³	—	142	1	no	—
271	C₂³	—	142	1	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	568	1	no	—
274	Q₈	—	142	1	no	—
275	Q₈	—	142	1	no	—
276	Q₈	—	142	1	no	—
277	Q₈	—	142	1	no	—
278	D₄	—	71	2	no	—
279	D₄	—	142	1	no	—
280	D₄	—	142	1	no	—
281	D₄	—	142	1	no	—
282	C₂×C₄	—	284	2	no	—
283	C₂×C₄	—	284	2	no	—
284	C₂×C₄	—	568	1	no	—
285	C₂×C₄	—	568	1	no	—
286	C₂×C₄	—	142	1	no	—
287	C₂×C₄	—	142	1	no	—
288	C₂×C₄	—	142	1	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	—	142	1	no	—
291	C₂×C₄	—	142	1	no	—
292	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
293	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
294	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
295	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
296	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
297	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	284	1	no	—
304	C₂²	—	284	1	no	—
305	C₂²	—	284	1	no	—
306	C₂²	—	284	1	no	—
307	C₂²	—	284	1	no	—
308	C₂²	—	142	2	no	—
309	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	284	1	no	—
317	C₄	—	284	1	no	—
318	C₄	—	284	1	no	—
319	C₄	—	284	1	no	—
320	C₄	—	284	1	no	—
321	C₄	—	284	1	no	—
322	C₄	#41×D₇₁	1	1	yes	characteristic
323	C₄	#14556176	1	1	yes	characteristic
324	C₂	—	284	2	no	—
325	C₂	—	568	1	no	—
326	C₂	#17986661	1	1	yes	characteristic
327	C₂	#17986799	1	1	yes	characteristic
328	C₂	#17986811	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#446×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#446	D₇₁
hypocenter	C₇₁	#3376
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #17986722	C₂	semidirect
2. #17986709	C₂	semidirect
3. #17986368	C₂	semidirect
4. #17986359	C₂	semidirect
5. #17986354	C₂	semidirect
6. #17986307	C₂	semidirect
7. #17986300	C₂	semidirect
8. #17986285	C₂	semidirect
11. #17986205	C₂	semidirect
13. #17986101	C₂	semidirect
14. C₄×#11765854	C₂	semidirect
15. #446×C₇₁	C₂	semidirect
165. C₇₁	#3376	semidirect
177. #446	D₇₁	semidirect