Subgroups — #22599528

Order 9088.

368 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986546	C₂	1	1	yes	maximal, characteristic
2	#125×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986407	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986406	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986362	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986362	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986319	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986316	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986314	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986311	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986300	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986299	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986296	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986055	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765865	—	2	1	no	—
17	#14556127	C₂²	1	1	yes	characteristic
18	#14556124	C₂²	1	1	yes	characteristic
19	#14556120	C₂²	1	1	yes	characteristic
20	#14556119	C₂²	1	1	yes	characteristic
21	#14556119	C₂²	1	1	yes	characteristic
22	#14556118	—	2	1	no	—
23	#14556118	—	2	1	no	—
24	#14556114	—	2	1	no	—
25	#14556103	C₂²	1	1	yes	characteristic
26	#14556101	—	2	1	no	—
27	#14556099	C₂²	1	1	yes	characteristic
28	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₇₁	—	2	1	no	—
30	C₄⋊C₄×D₇₁	—	2	1	no	—
31	C₄⋊C₄×D₇₁	—	2	1	no	—
32	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
34	C₄²×D₇₁	C₂²	1	1	yes	characteristic
35	#14556062	C₂²	1	1	yes	characteristic
36	#14556061	C₂²	1	1	yes	characteristic
37	#14556061	C₂²	1	1	yes	characteristic
38	#14556060	C₂²	1	1	yes	characteristic
39	#14556059	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#14556050	C₂²	1	1	yes	characteristic
43	#14556050	C₂²	1	1	yes	characteristic
44	#14556049	C₂²	1	1	yes	characteristic
45	#14556049	C₂²	1	1	yes	characteristic
46	#14556048	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	#14556038	C₂²	1	1	yes	characteristic
52	#125×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#106×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	#11765865	—	4	4	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×D₂₈₄	—	2	1	no	—
62	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₇₁	—	2	1	no	—
64	C₂×C₂²⋊D₇₁	—	2	1	no	—
65	C₂×C₂²⋊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	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
71	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	—	2	1	no	—
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	Dic₇₁⋊C₄	—	2	1	no	—
85	Dic₇₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	—	2	1	no	—
89	C₄×Dic₇₁	D₄	1	1	yes	characteristic
90	C₄×Dic₇₁	D₄	1	1	yes	characteristic
91	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	—	2	1	no	—
94	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
102	C₄²×C₇₁	C₂³	1	1	yes	characteristic
103	C₂³×C₇₁	—	2	1	no	—
104	C₂²×D₇₁	—	2	1	no	—
105	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
106	D₄×C₇₁	—	2	1	no	—
107	D₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	4	2	no	—
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	C₂²⋊D₇₁	—	4	2	no	—
116	C₂²⋊D₇₁	—	4	2	no	—
117	C₂²⋊D₇₁	—	4	2	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	—	2	1	no	—
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	—	4	2	no	—
127	C₄×D₇₁	—	2	2	no	—
128	C₄×D₇₁	—	2	1	no	—
129	C₄×D₇₁	—	2	1	no	—
130	C₄×D₇₁	—	4	1	no	—
131	C₄×D₇₁	—	4	2	no	—
132	C₄×D₇₁	—	4	2	no	—
133	C₄×D₇₁	—	4	2	no	—
134	Dic₁₄₂	—	4	2	no	—
135	C₈×C₇₁	—	2	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₇₁⋊C₈	—	2	1	no	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₂²×C₇₁	—	4	1	no	—
140	C₂²×C₇₁	—	4	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₇₁	—	2	1	no	—
144	C₂×D₇₁	—	2	1	no	—
145	C₂×D₇₁	—	4	1	no	—
146	C₂×D₇₁	—	4	1	no	—
147	C₂×D₇₁	—	4	1	no	—
148	C₂×D₇₁	AGL(1,ℤ/8)	1	2	yes	—
149	C₄×C₇₁	—	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	2₋¹⁺⁴	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	C₂×C₇₁	—	8	1	no	—
163	C₂×C₇₁	#536	1	1	yes	characteristic
164	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₇₁	#575	1	1	yes	characteristic
166	D₇₁	—	2	2	no	—
167	D₇₁	—	8	1	no	—
168	#3234	—	71	1	no	maximal, Sylow
169	C₇₁	#3234	1	1	yes	characteristic, complemented, Sylow
170	#532	—	71	1	no	—
171	C₂×#125	—	71	1	no	—
172	#501	—	71	1	no	—
173	#501	—	71	1	no	—
174	#490	—	71	1	no	—
175	#490	—	71	1	no	—
176	#481	—	71	1	no	—
177	#481	—	71	1	no	—
178	#480	—	71	1	no	—
179	#480	—	71	1	no	—
180	#478	—	71	1	no	—
181	#478	—	71	1	no	—
182	#478	—	71	1	no	—
183	#478	D₇₁	1	1	yes	characteristic, complemented
184	#432	—	71	1	no	—
185	C₂×#41	—	142	1	no	—
186	C₄⋊Q₈	—	71	1	no	—
187	#127	—	71	1	no	—
188	#125	—	71	1	no	—
189	#125	—	71	1	no	—
190	#125	—	71	1	no	—
191	#125	—	142	1	no	—
192	#125	—	142	1	no	—
193	#125	C₂×D₇₁	1	1	yes	characteristic
194	#124	—	142	1	no	—
195	#121	—	71	1	no	—
196	#121	—	71	1	no	—
197	#121	—	142	1	no	—
198	C₄×D₄	—	71	1	no	—
199	C₄×D₄	—	142	1	no	—
200	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	71	1	no	—
202	C₂×C₄⋊C₄	—	71	1	no	—
203	C₂×C₄⋊C₄	—	142	1	no	—
204	C₂×C₄⋊C₄	—	142	1	no	—
205	C₂×C₄²	—	71	1	no	—
206	#107	—	71	1	no	—
207	#107	—	71	1	no	—
208	#107	—	71	1	no	—
209	#107	C₂×D₇₁	1	1	yes	characteristic
210	#106	—	71	1	no	—
211	#106	—	71	1	no	—
212	#106	—	71	1	no	—
213	#106	C₂×D₇₁	1	1	yes	characteristic
214	C₄⋊C₈	—	71	1	no	—
215	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
216	#102	—	71	1	no	—
217	#102	—	71	1	no	—
218	#102	—	71	1	no	—
219	#102	—	71	1	no	—
220	#102	—	71	1	no	—
221	#102	—	71	1	no	—
222	#102	C₂×D₇₁	1	1	yes	characteristic
223	#102	C₂×D₇₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	71	1	no	—
225	C₂²⋊C₈	—	71	1	no	—
226	#97	—	71	1	no	—
227	#97	—	71	1	no	—
228	#41	—	284	4	no	—
229	C₂×Q₈	—	142	1	no	—
230	C₂×D₄	—	71	1	no	—
231	C₂×D₄	—	142	1	no	—
232	C₂×D₄	—	142	1	no	—
233	C₂×D₄	—	142	1	no	—
234	C₂×D₄	—	142	1	no	—
235	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	71	1	no	—
238	C₂²×C₄	—	71	1	no	—
239	C₂²×C₄	—	71	1	no	—
240	C₂²×C₄	—	142	1	no	—
241	C₂²×C₄	—	142	1	no	—
242	C₂²×C₄	—	142	1	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	—	142	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	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
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	—	142	1	no	—
266	C₂²⋊C₄	—	142	1	no	—
267	C₂²⋊C₄	—	142	1	no	—
268	C₄²	—	71	1	no	—
269	C₄²	—	71	1	no	—
270	C₄²	—	71	1	no	—
271	C₄²	C₂²×D₇₁	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	71	1	no	—
274	C₂³	—	142	1	no	—
275	Q₈	—	284	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	284	2	no	—
279	D₄	—	284	2	no	—
280	D₄	—	284	2	no	—
281	D₄	—	284	2	no	—
282	D₄	—	142	1	no	—
283	D₄	—	142	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	284	2	no	—
289	C₂×C₄	—	284	1	no	—
290	C₂×C₄	—	284	2	no	—
291	C₂×C₄	—	284	2	no	—
292	C₂×C₄	—	71	1	no	—
293	C₂×C₄	—	71	1	no	—
294	C₂×C₄	—	71	1	no	—
295	C₂×C₄	—	142	2	no	—
296	C₂×C₄	—	142	1	no	—
297	C₂×C₄	—	142	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
302	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	142	1	no	—
307	C₈	—	142	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	284	1	no	—
312	C₂²	—	284	1	no	—
313	C₂²	—	284	1	no	—
314	C₂²	—	71	2	no	—
315	C₂²	—	142	1	no	—
316	C₂²	—	142	1	no	—
317	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	284	1	no	—
324	C₄	—	284	1	no	—
325	C₄	—	284	1	no	—
326	C₄	—	142	1	no	—
327	C₄	—	142	1	no	—
328	C₄	—	142	1	no	—
329	C₄	#14556175	1	1	yes	characteristic
330	C₄	#14556176	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	—	568	1	no	—
333	C₂	—	142	2	no	—
334	C₂	#17986572	1	1	yes	characteristic
335	C₂	AGL(1,ℤ/8)×D₇₁	1	1	yes	characteristic
336	C₂	#17986796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#478×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#478	D₇₁
hypocenter	C₇₁	#3234
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #125×D₇₁	C₂	semidirect
3. #17986407	C₂	semidirect
4. #17986406	C₂	semidirect
5. #17986362	C₂	semidirect
6. #17986362	C₂	semidirect
7. #17986319	C₂	semidirect
8. #17986316	C₂	semidirect
9. #17986314	C₂	semidirect
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14. #17986055	C₂	semidirect
15. #478×C₇₁	C₂	semidirect
169. C₇₁	#3234	semidirect
183. #478	D₇₁	semidirect