Subgroups — #31885342

Order 18176.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586551	C₂	1	1	yes	maximal, characteristic, complemented
2	#22586544	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985829	—	2	1	no	—
5	C₇₁⋊C₆₄	—	2	1	no	—
6	C₂×D₁₆×C₇₁	—	2	1	no	—
7	D₃₂×C₇₁	—	2	1	no	—
8	D₃₂×C₇₁	C₄	1	2	yes	—
9	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	#14556064	—	4	1	no	—
11	C₂×D₈×C₇₁	—	4	1	no	—
12	D₁₆×C₇₁	—	2	2	no	—
13	D₁₆×C₇₁	—	4	1	no	—
14	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
15	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	D₄	1	1	yes	characteristic
17	#11765849	—	8	1	no	—
18	C₂×D₄×C₇₁	—	8	1	no	—
19	D₈×C₇₁	—	4	2	no	—
20	D₈×C₇₁	—	8	1	no	—
21	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
22	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
24	C₂³×C₇₁	—	16	1	no	—
25	D₄×C₇₁	—	8	2	no	—
26	D₄×C₇₁	—	16	1	no	—
27	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
28	C₈×C₇₁	#102	1	1	yes	characteristic
29	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
30	C₇₁⋊C₈	—	16	1	no	—
31	C₂²×C₇₁	—	16	2	no	—
32	C₂²×C₇₁	—	32	2	no	—
33	C₂²×C₇₁	SD₆₄	1	1	yes	characteristic
34	C₄×C₇₁	#357	1	1	yes	characteristic
35	C₄×C₇₁	D₃₂	1	1	yes	characteristic
36	#7541	—	71	1	no	maximal, Sylow
37	C₂×C₇₁	—	2	1	no	—
38	C₂×C₇₁	—	32	2	no	—
39	C₂×C₇₁	#1415	1	1	yes	characteristic
40	C₂×D₃₂	D₇₁	1	1	yes	characteristic
41	M₁₂₈	—	71	1	no	—
42	#1425	—	71	1	no	—
43	C₇₁	#7541	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₁₆	—	2	1	no	—
45	D₃₂	—	2	1	no	—
46	D₃₂	Dic₇₁	1	2	yes	—
47	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
48	#368	—	142	1	no	—
49	C₆₄	—	142	1	no	—
50	C₂×D₈	—	4	1	no	—
51	D₁₆	—	2	2	no	—
52	D₁₆	—	4	1	no	—
53	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
54	#108	—	284	1	no	—
55	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
57	C₂×D₄	—	8	1	no	—
58	D₈	—	4	2	no	—
59	D₈	—	8	1	no	—
60	M₁₆	—	568	1	no	—
61	C₂×C₈	#11765850	1	1	yes	characteristic
62	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
63	C₁₆	#11765853	1	1	yes	characteristic
64	C₂³	—	16	1	no	—
65	D₄	—	8	2	no	—
66	D₄	—	16	1	no	—
67	C₂×C₄	#14556069	1	1	yes	characteristic
68	C₈	—	1136	1	no	—
69	C₈	#14556049	1	1	yes	characteristic
70	C₈	#14556072	1	1	yes	characteristic
71	C₂²	—	16	2	no	—
72	C₂²	—	32	2	no	—
73	C₂²	#17985837	1	1	yes	characteristic
74	C₄	#17985800	1	1	yes	characteristic
75	C₄	#17985834	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	32	2	no	—
78	C₂	#22586518	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22586518
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting, prime radical	C₂×D₃₂×C₇₁	C₂
socle	C₂×C₇₁	#1415
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	C₂×D₃₂	D₇₁
hypocenter	C₇₁	#7541
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×D₃₂

Complements

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

N	H ≅ G/N	type
1. #22586551	C₂	semidirect
2. #22586544	C₂	semidirect
43. C₇₁	#7541	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22586551	2
#22586544	2
C₂×D₃₂×C₇₁	2
#7541	71