Subgroups — #22597349

Order 9088.

406 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11765865	C₂	1	1	yes	maximal, characteristic
2	#17986128	C₂	1	2	yes	maximal, complemented
3	#17986069	C₂	1	2	yes	maximal, complemented
4	#17986067	C₂	1	2	yes	maximal, complemented
5	(C₂²⋊D₇₁)⋊C₈	C₂	1	2	yes	maximal, complemented
6	#17986054	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986054	C₂	1	1	yes	maximal, characteristic, complemented
8	#17985951	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556039	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#432×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11765865	C₄	1	1	yes	characteristic
13	C₂×#11765848	C₂²	1	2	yes	—
14	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×D₇₁	C₂²	1	2	yes	—
16	C₄×Dic₁₄₂	C₄	1	1	yes	characteristic
17	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
18	C₄×D₂₈₄	C₄	1	1	yes	characteristic
19	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₇₁	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₇₁	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
24	#14556087	C₄	1	1	yes	characteristic
25	#14556085	C₄	1	1	yes	characteristic
26	#14556085	C₄	1	1	yes	characteristic
27	C₄²×D₇₁	C₂²	1	1	yes	characteristic
28	C₄²×D₇₁	C₂²	1	1	yes	characteristic
29	Dic₇₁⋊C₈	C₂²	1	2	yes	—
30	Dic₇₁⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₇₁)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₇₁)⋊C₈	C₂²	1	2	yes	—
33	#14556039	C₂²	1	1	yes	characteristic
34	#14556039	C₂²	1	2	yes	—
35	#14556039	C₂²	1	1	yes	characteristic
36	#14556038	C₂²	1	2	yes	—
37	C₈×Dic₇₁	C₂²	1	2	yes	—
38	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
41	#97×C₇₁	C₂²	1	1	yes	characteristic
42	#97×C₇₁	C₂²	1	1	yes	characteristic
43	#11765865	—	2	4	no	—
44	C₂×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
45	C₂×D₂₈₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
54	#11765848	—	2	4	no	—
55	C₂×C₇₁⋊C₈	—	2	4	no	—
56	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
57	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
58	C₈×D₇₁	—	2	4	no	—
59	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
61	Dic₇₁⋊C₄	C₂×C₄	1	1	yes	characteristic
62	Dic₇₁⋊C₄	C₂×C₄	1	1	yes	characteristic
63	Dic₇₁⋊C₄	C₂×C₄	1	1	yes	characteristic
64	Dic₇₁⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₇₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₇₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	(C₂×D₇₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
70	(C₂×D₇₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
76	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂³	1	2	yes	—
81	C₂×C₈×C₇₁	C₂³	1	2	yes	—
82	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₇₁	C₂³	1	1	yes	characteristic
85	C₄²×C₇₁	C₂³	1	1	yes	characteristic
86	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	—	2	2	no	—
90	C₂×C₄×C₇₁	—	2	2	no	—
91	C₂×C₄×C₇₁	—	4	1	no	—
92	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₇₁	—	2	2	no	—
99	C₂²⋊D₇₁	M₁₆	1	4	yes	complemented
100	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
106	D₂₈₄	—	2	2	no	—
107	C₄×D₇₁	—	2	2	no	—
108	C₄×D₇₁	—	2	2	no	—
109	C₄×D₇₁	—	2	2	no	—
110	C₄×D₇₁	—	2	2	no	—
111	C₄×D₇₁	—	4	1	no	—
112	C₄×D₇₁	—	4	1	no	—
113	Dic₁₄₂	—	2	2	no	—
114	C₈×C₇₁	—	2	2	no	—
115	C₈×C₇₁	—	2	2	no	—
116	C₇₁⋊C₈	—	2	2	no	—
117	C₇₁⋊C₈	#41	1	4	yes	complemented
118	C₂²×C₇₁	—	2	1	no	—
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
121	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₇₁	—	2	1	no	—
123	C₂×D₇₁	—	2	1	no	—
124	C₂×D₇₁	—	2	1	no	—
125	C₂×D₇₁	—	2	1	no	—
126	C₂×D₇₁	—	2	1	no	—
127	C₂×D₇₁	C₂×M₁₆	1	2	yes	—
128	C₄×C₇₁	—	2	1	no	—
129	C₄×C₇₁	—	2	1	no	—
130	C₄×C₇₁	—	2	1	no	—
131	C₄×C₇₁	—	2	1	no	—
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	C₂×#41	1	2	yes	—
134	Dic₇₁	—	2	1	no	—
135	Dic₇₁	—	2	1	no	—
136	Dic₇₁	—	2	1	no	—
137	Dic₇₁	—	2	1	no	—
138	Dic₇₁	—	2	1	no	—
139	Dic₇₁	C₂×M₁₆	1	2	yes	—
140	C₂×C₇₁	—	2	2	no	—
141	C₂×C₇₁	C₄×#41	1	1	yes	characteristic
142	C₂×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
143	C₂×C₇₁	#568	1	1	yes	characteristic
144	D₇₁	—	2	2	no	—
145	D₇₁	—	4	1	no	—
146	#2970	—	71	1	no	maximal, Sylow
147	C₇₁	#2970	1	1	yes	characteristic, complemented, Sylow
148	C₄×#41	—	71	1	no	—
149	#446	—	71	2	no	—
150	#435	—	71	2	no	—
151	#435	—	71	2	no	—
152	C₈×D₄	—	71	2	no	—
153	#432	—	71	1	no	—
154	#432	—	71	1	no	—
155	#432	D₇₁	1	1	yes	characteristic, complemented
156	C₄×M₁₆	—	71	1	no	—
157	C₂×#97	—	71	1	no	—
158	C₂×#97	—	71	1	no	—
159	C₂×#41	—	71	1	no	—
160	C₂×M₁₆	—	71	2	no	—
161	C₂²×C₈	—	71	2	no	—
162	C₂²×C₈	—	71	2	no	—
163	C₄×Q₈	—	71	1	no	—
164	C₄×Q₈	—	71	1	no	—
165	C₄×D₄	—	71	1	no	—
166	C₄×D₄	—	71	1	no	—
167	C₄×D₄	—	71	1	no	—
168	C₄×D₄	—	71	1	no	—
169	C₄×D₄	—	71	1	no	—
170	C₄×D₄	—	71	1	no	—
171	#117	—	71	1	no	—
172	#117	—	71	1	no	—
173	#117	—	71	1	no	—
174	C₂×C₄²	—	71	1	no	—
175	C₂×C₄²	—	71	1	no	—
176	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
177	C₄⋊C₈	—	71	2	no	—
178	C₄⋊C₈	—	71	2	no	—
179	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
180	C₂²⋊C₈	—	71	2	no	—
181	C₂²⋊C₈	—	71	2	no	—
182	C₂²⋊C₈	C₂×D₇₁	1	2	yes	—
183	#97	—	71	2	no	—
184	#97	—	71	1	no	—
185	#97	—	71	1	no	—
186	#97	—	71	2	no	—
187	#97	C₂×D₇₁	1	1	yes	characteristic
188	#97	C₂×D₇₁	1	1	yes	characteristic
189	C₄×C₈	—	71	2	no	—
190	#41	—	142	4	no	—
191	C₂×Q₈	—	71	1	no	—
192	C₂×D₄	—	71	1	no	—
193	C₂×D₄	—	71	1	no	—
194	C₂×D₄	—	71	1	no	—
195	C₂²×C₄	—	71	1	no	—
196	C₂²×C₄	—	71	1	no	—
197	C₂²×C₄	—	71	1	no	—
198	C₂²×C₄	—	71	1	no	—
199	C₂²×C₄	—	71	1	no	—
200	C₂²×C₄	—	71	1	no	—
201	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
202	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
204	M₁₆	—	142	4	no	—
205	C₂×C₈	—	71	2	no	—
206	C₂×C₈	—	71	2	no	—
207	C₂×C₈	—	142	4	no	—
208	C₂×C₈	—	142	4	no	—
209	C₂×C₈	C₂²×D₇₁	1	2	yes	—
210	C₂×C₈	C₂²×D₇₁	1	2	yes	—
211	C₄⋊C₄	—	71	1	no	—
212	C₄⋊C₄	—	71	1	no	—
213	C₄⋊C₄	—	71	1	no	—
214	C₄⋊C₄	—	71	1	no	—
215	C₄⋊C₄	—	71	1	no	—
216	C₄⋊C₄	—	71	1	no	—
217	C₂²⋊C₄	—	71	1	no	—
218	C₂²⋊C₄	—	71	1	no	—
219	C₂²⋊C₄	—	71	1	no	—
220	C₂²⋊C₄	—	71	1	no	—
221	C₂²⋊C₄	—	71	1	no	—
222	C₂²⋊C₄	—	71	1	no	—
223	C₄²	—	71	1	no	—
224	C₄²	—	71	1	no	—
225	C₄²	—	71	1	no	—
226	C₄²	—	71	1	no	—
227	C₄²	—	71	1	no	—
228	C₄²	—	71	1	no	—
229	C₄²	C₄×D₇₁	1	1	yes	characteristic
230	C₄²	C₄×D₇₁	1	1	yes	characteristic
231	C₄²	C₂²×D₇₁	1	1	yes	characteristic
232	C₄²	C₂²×D₇₁	1	1	yes	characteristic
233	C₂³	—	71	1	no	—
234	C₂³	—	71	1	no	—
235	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
236	Q₈	—	142	2	no	—
237	D₄	—	71	4	no	—
238	D₄	—	142	2	no	—
239	D₄	—	142	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	284	1	no	—
244	C₂×C₄	—	284	1	no	—
245	C₂×C₄	—	71	1	no	—
246	C₂×C₄	—	71	1	no	—
247	C₂×C₄	—	71	1	no	—
248	C₂×C₄	—	71	1	no	—
249	C₂×C₄	—	71	1	no	—
250	C₂×C₄	—	71	1	no	—
251	C₂×C₄	—	142	2	no	—
252	C₂×C₄	—	142	2	no	—
253	C₂×C₄	—	142	2	no	—
254	C₂×C₄	—	142	2	no	—
255	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	71	4	no	—
264	C₈	—	142	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	71	2	no	—
268	C₂²	—	142	1	no	—
269	C₂²	—	142	1	no	—
270	C₂²	—	142	1	no	—
271	C₂²	—	142	1	no	—
272	C₂²	—	142	1	no	—
273	C₂²	M₁₆×D₇₁	1	2	yes	—
274	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	71	2	no	—
281	C₄	—	142	1	no	—
282	C₄	—	142	1	no	—
283	C₄	—	142	1	no	—
284	C₄	—	142	1	no	—
285	C₄	—	142	1	no	—
286	C₄	#41×D₇₁	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	284	1	no	—
289	C₂	—	142	2	no	—
290	C₂	#17986467	1	1	yes	characteristic
291	C₂	C₂×M₁₆×D₇₁	1	1	yes	characteristic
292	C₂	#17986765	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	#432×C₇₁	C₂
hypercenter	#432	D₇₁
hypocenter	C₇₁	#2970
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #17986128	C₂	semidirect
3. #17986069	C₂	semidirect
4. #17986067	C₂	semidirect
5. (C₂²⋊D₇₁)⋊C₈	C₂	semidirect
6. #17986054	C₂	semidirect
7. #17986054	C₂	semidirect
8. #17985951	C₂	semidirect
9. C₂×#14556039	C₂	semidirect
10. #97×D₇₁	C₂	semidirect
11. #432×C₇₁	C₂	semidirect
99. C₂²⋊D₇₁	M₁₆	semidirect
117. C₇₁⋊C₈	#41	semidirect
147. C₇₁	#2970	semidirect
155. #432	D₇₁	semidirect