Subgroups — #22587949

Order 9088.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986404	C₂	1	1	yes	maximal, characteristic
2	#17986351	C₂	1	2	yes	maximal
3	#17986349	C₂	1	2	yes	maximal
4	C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#14556128	C₂²	1	2	yes	—
7	#14556062	—	2	2	no	—
8	#14556052	—	2	4	no	—
9	#14556051	—	2	4	no	—
10	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
13	#107×C₇₁	—	2	2	no	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₂	—	2	2	no	—
16	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
17	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
18	C₄⋊Dic₇₁	—	2	2	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×Q₈×C₇₁	—	2	2	no	—
21	C₂×C₈×C₇₁	D₄	1	2	yes	—
22	C₄⋊C₄×C₇₁	—	2	2	no	—
23	C₄⋊C₄×C₇₁	—	2	2	no	—
24	C₄²×C₇₁	C₂³	1	1	yes	characteristic
25	Q₈×C₇₁	—	4	4	no	—
26	C₂×C₄×C₇₁	—	4	2	no	—
27	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₇₁	—	4	2	no	—
31	Dic₁₄₂	—	4	4	no	—
32	C₈×C₇₁	Q₁₆	1	4	yes	—
33	C₇₁⋊C₈	—	2	2	no	—
34	C₇₁⋊C₈	—	2	2	no	—
35	C₂²×C₇₁	#127	1	1	yes	characteristic
36	C₄×C₇₁	—	8	2	no	—
37	C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
38	C₄×C₇₁	#135	1	2	yes	—
39	C₄×C₇₁	#135	1	2	yes	—
40	Dic₇₁	—	8	2	no	—
41	C₂×C₇₁	#494	1	1	yes	characteristic
42	C₂×C₇₁	#495	1	1	yes	characteristic
43	C₂×C₇₁	#495	1	1	yes	characteristic
44	#1710	—	71	1	no	maximal, Sylow
45	C₇₁	#1710	1	1	yes	characteristic, complemented, Sylow
46	#500	—	71	1	no	—
47	#500	D₇₁	1	1	yes	characteristic
48	#487	—	71	2	no	—
49	#487	—	71	2	no	—
50	C₈²	—	71	1	no	—
51	C₄⋊Q₈	—	71	2	no	—
52	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
53	#107	—	2	2	no	—
54	#107	—	142	2	no	—
55	#103	—	142	4	no	—
56	#103	—	142	4	no	—
57	C₄×C₈	—	71	1	no	—
58	C₄×C₈	—	71	1	no	—
59	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	142	2	no	—
62	C₂×C₈	—	71	2	no	—
63	C₂×C₈	—	71	2	no	—
64	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	142	2	no	—
68	C₄⋊C₄	—	142	2	no	—
69	C₄²	C₂²×D₇₁	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	284	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	284	2	no	—
74	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
75	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
77	C₈	—	142	2	no	—
78	C₈	—	142	2	no	—
79	C₈	#11765855	1	4	yes	—
80	C₂²	#14556124	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	568	2	no	—
83	C₄	C₂×#11765855	1	2	yes	—
84	C₄	#14556147	1	2	yes	—
85	C₄	#14556151	1	2	yes	—
86	C₂	#17986378	1	1	yes	characteristic
87	C₂	#17986382	1	1	yes	characteristic
88	C₂	#17986384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14556124
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#500×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#500	D₇₁
hypocenter	C₇₁	#1710
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
45. C₇₁	#1710	semidirect