Subgroups — #22601295

Order 9088.

348 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986710	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986705	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986407	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986399	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986302	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986298	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986298	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986296	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986278	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986276	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986127	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986112	C₂	1	1	yes	maximal, characteristic
13	#17986110	C₂	1	1	yes	maximal, characteristic
14	#17986081	C₂	1	1	yes	maximal, characteristic
15	#501×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
17	#14556127	C₂²	1	1	yes	characteristic
18	#14556119	C₂²	1	1	yes	characteristic
19	#14556119	C₂²	1	1	yes	characteristic
20	#14556119	C₂²	1	1	yes	characteristic
21	#14556108	—	2	1	no	—
22	#14556108	—	2	1	no	—
23	#14556106	—	2	1	no	—
24	#14556106	—	2	1	no	—
25	#14556104	—	2	1	no	—
26	#14556104	—	2	1	no	—
27	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
28	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
29	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
30	#14556090	C₂²	1	1	yes	characteristic
31	#14556090	C₂²	1	1	yes	characteristic
32	#14556090	C₂²	1	1	yes	characteristic
33	#14556090	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₇₁	—	2	1	no	—
35	C₄⋊C₄×D₇₁	—	2	1	no	—
36	#14556061	C₂²	1	1	yes	characteristic
37	#14556061	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
44	#14556050	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	#14556048	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	#125×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#106×C₇₁	C₂²	1	1	yes	characteristic
57	#106×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
61	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
62	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₇₁	—	2	1	no	—
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	C₂×C₄×D₇₁	—	2	1	no	—
69	D₅₆₈	Q₈	1	4	yes	complemented
70	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₄⋊Dic₇₁	—	2	1	no	—
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	D₄	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	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	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
91	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₇₁	—	2	1	no	—
93	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₇₁	—	2	1	no	—
96	C₄⋊C₄×C₇₁	—	2	1	no	—
97	C₄⋊C₄×C₇₁	—	2	1	no	—
98	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₄²×C₇₁	C₂³	1	1	yes	characteristic
103	C₂²×D₇₁	—	2	1	no	—
104	C₂²×D₇₁	—	2	1	no	—
105	Q₈×C₇₁	—	8	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₇₁	—	2	1	no	—
114	C₂×Dic₇₁	—	2	1	no	—
115	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
119	D₂₈₄	—	2	1	no	—
120	D₂₈₄	—	2	1	no	—
121	D₂₈₄	C₂×Q₈	1	2	yes	—
122	D₂₈₄	C₂×Q₈	1	2	yes	—
123	C₄×D₇₁	—	4	2	no	—
124	C₄×D₇₁	—	4	2	no	—
125	C₄×D₇₁	—	8	1	no	—
126	C₄×D₇₁	—	8	1	no	—
127	C₄×D₇₁	—	8	1	no	—
128	C₄×D₇₁	—	8	1	no	—
129	Dic₁₄₂	—	4	1	no	—
130	Dic₁₄₂	—	8	1	no	—
131	C₈×C₇₁	—	2	1	no	—
132	C₈×C₇₁	C₂×Q₈	1	2	yes	—
133	C₇₁⋊C₈	—	2	1	no	—
134	C₇₁⋊C₈	—	2	1	no	—
135	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₇₁	—	2	2	no	—
137	C₂×D₇₁	—	2	2	no	—
138	C₂×D₇₁	—	4	1	no	—
139	C₂×D₇₁	—	4	1	no	—
140	C₂×D₇₁	—	4	1	no	—
141	C₂×D₇₁	—	4	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	2	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	—	4	1	no	—
148	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₇₁	—	2	1	no	—
151	Dic₇₁	—	2	1	no	—
152	Dic₇₁	—	2	1	no	—
153	Dic₇₁	—	2	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	—	4	1	no	—
156	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
157	C₂×C₇₁	#575	1	1	yes	characteristic
158	C₂×C₇₁	#576	1	1	yes	characteristic
159	D₇₁	—	4	2	no	—
160	D₇₁	—	4	2	no	—
161	#3389	—	71	1	no	maximal, Sylow
162	C₇₁	#3389	1	1	yes	characteristic, complemented, Sylow
163	#554	—	71	1	no	—
164	#554	—	71	1	no	—
165	#501	—	71	1	no	—
166	#501	D₇₁	1	1	yes	characteristic, complemented
167	#499	—	71	1	no	—
168	#478	—	71	1	no	—
169	#478	—	71	1	no	—
170	#478	—	71	1	no	—
171	#478	—	71	1	no	—
172	#474	—	71	1	no	—
173	#474	—	71	1	no	—
174	#446	—	71	1	no	—
175	#442	—	71	1	no	—
176	#442	—	71	1	no	—
177	C₄×D₈	—	71	1	no	—
178	C₂×D₈	—	71	1	no	—
179	C₄⋊Q₈	—	71	1	no	—
180	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
181	#125	—	71	1	no	—
182	#125	—	71	1	no	—
183	#125	—	71	1	no	—
184	#125	C₂×D₇₁	1	1	yes	characteristic
185	C₂²⋊Q₈	—	142	1	no	—
186	C₂²⋊Q₈	—	142	1	no	—
187	C₂²⋊Q₈	—	142	1	no	—
188	C₂²⋊Q₈	—	142	1	no	—
189	C₂²⋊Q₈	—	142	1	no	—
190	C₂²⋊Q₈	—	142	1	no	—
191	C₄×Q₈	—	71	1	no	—
192	C₄×D₄	—	71	1	no	—
193	C₄×D₄	—	71	1	no	—
194	C₄×D₄	—	71	1	no	—
195	C₄×D₄	—	71	1	no	—
196	C₄×D₄	—	71	1	no	—
197	C₄×D₄	—	71	1	no	—
198	C₂×C₄⋊C₄	—	142	1	no	—
199	C₂×C₄⋊C₄	—	142	1	no	—
200	#107	—	71	1	no	—
201	#107	—	71	1	no	—
202	#107	—	71	1	no	—
203	#107	C₂×D₇₁	1	1	yes	characteristic
204	#107	C₂×D₇₁	1	1	yes	characteristic
205	#106	—	71	1	no	—
206	#106	—	71	1	no	—
207	#106	C₂×D₇₁	1	1	yes	characteristic
208	#106	C₂×D₇₁	1	1	yes	characteristic
209	C₄⋊C₈	—	71	1	no	—
210	C₄⋊C₈	—	71	1	no	—
211	C₄⋊C₈	—	71	1	no	—
212	#102	—	71	1	no	—
213	#102	—	71	1	no	—
214	#102	—	71	1	no	—
215	#102	—	71	1	no	—
216	#102	—	71	1	no	—
217	#102	—	71	1	no	—
218	#97	—	71	1	no	—
219	#97	C₂×D₇₁	1	1	yes	characteristic
220	C₄×C₈	—	71	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	71	1	no	—
223	C₂×Q₈	—	142	1	no	—
224	C₂×D₄	—	71	1	no	—
225	C₂×D₄	—	71	1	no	—
226	C₂²×C₄	—	142	1	no	—
227	C₂²×C₄	—	142	1	no	—
228	C₂²×C₄	—	142	1	no	—
229	C₂²×C₄	—	142	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	—	142	1	no	—
232	D₈	—	71	4	no	—
233	C₂×C₈	—	71	1	no	—
234	C₂×C₈	—	71	1	no	—
235	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	71	1	no	—
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	C₂²×D₇₁	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
256	C₂²⋊C₄	—	142	1	no	—
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	C₄²	—	71	1	no	—
263	C₄²	—	71	1	no	—
264	C₄²	C₂²×D₇₁	1	1	yes	characteristic
265	C₂³	—	142	1	no	—
266	C₂³	—	142	1	no	—
267	Q₈	—	8	1	no	—
268	Q₈	—	284	1	no	—
269	Q₈	—	568	1	no	—
270	D₄	—	71	2	no	—
271	D₄	—	71	2	no	—
272	D₄	—	142	1	no	—
273	D₄	—	142	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	284	2	no	—
279	C₂×C₄	—	284	2	no	—
280	C₂×C₄	—	568	1	no	—
281	C₂×C₄	—	568	1	no	—
282	C₂×C₄	—	568	1	no	—
283	C₂×C₄	—	568	1	no	—
284	C₂×C₄	—	71	1	no	—
285	C₂×C₄	—	71	1	no	—
286	C₂×C₄	—	71	1	no	—
287	C₂×C₄	—	71	1	no	—
288	C₂×C₄	—	142	1	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
291	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	142	1	no	—
295	C₈	—	142	1	no	—
296	C₈	Q₈×D₇₁	1	2	yes	—
297	C₂²	—	284	1	no	—
298	C₂²	—	284	1	no	—
299	C₂²	—	284	1	no	—
300	C₂²	—	284	1	no	—
301	C₂²	—	142	2	no	—
302	C₂²	—	142	2	no	—
303	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	284	1	no	—
311	C₄	—	284	1	no	—
312	C₄	—	142	1	no	—
313	C₄	—	142	1	no	—
314	C₄	—	142	1	no	—
315	C₄	—	142	1	no	—
316	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
317	C₄	#14556176	1	1	yes	characteristic
318	C₂	—	284	2	no	—
319	C₂	—	284	2	no	—
320	C₂	#17986664	1	1	yes	characteristic
321	C₂	#17986798	1	1	yes	characteristic
322	C₂	#17986803	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#501×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#501	D₇₁
hypocenter	C₇₁	#3389
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #17986710	C₂	semidirect
2. #17986705	C₂	semidirect
3. #17986407	C₂	semidirect
4. #17986399	C₂	semidirect
5. #17986302	C₂	semidirect
6. #17986298	C₂	semidirect
7. #17986298	C₂	semidirect
8. #17986296	C₂	semidirect
9. #17986278	C₂	semidirect
10. #17986276	C₂	semidirect
11. #17986127	C₂	semidirect
15. #501×C₇₁	C₂	semidirect
69. D₅₆₈	Q₈	semidirect
162. C₇₁	#3389	semidirect
166. #501	D₇₁	semidirect