Subgroups — #31885332

Order 18176.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1423	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₆₈	—	2	1	no	—
5	Dic₁₁₃₆	—	2	1	no	—
6	Dic₁₁₃₆	C₄	1	2	yes	complemented
7	C₇₁⋊C₆₄	—	2	1	no	—
8	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
9	#366×C₇₁	—	2	1	no	—
10	C₂×Dic₂₈₄	—	4	1	no	—
11	Dic₅₆₈	—	2	2	no	—
12	Dic₅₆₈	—	4	1	no	—
13	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
14	#107×C₇₁	—	4	1	no	—
15	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	D₄	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	8	1	no	—
18	Dic₂₈₄	—	4	2	no	—
19	Dic₂₈₄	—	8	1	no	—
20	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₇₁	—	8	1	no	—
22	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇₁	—	16	1	no	—
25	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
26	C₂×Dic₇₁	—	16	1	no	—
27	Dic₁₄₂	—	8	2	no	—
28	Dic₁₄₂	—	16	1	no	—
29	C₈×C₇₁	#102	1	1	yes	characteristic
30	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
31	C₂²×C₇₁	D₃₂	1	1	yes	characteristic
32	C₄×C₇₁	—	32	1	no	—
33	C₄×C₇₁	#357	1	1	yes	characteristic
34	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
35	Dic₇₁	—	16	2	no	—
36	#7538	—	71	1	no	maximal, Sylow
37	C₂×C₇₁	#1415	1	1	yes	characteristic
38	C₂×C₇₁	SD₁₂₈	1	1	yes	characteristic
39	C₂×C₇₁	Q₁₂₈	1	1	yes	characteristic
40	C₂×Q₆₄	—	71	1	no	—
41	C₂×C₆₄	—	71	1	no	—
42	#1423	D₇₁	1	1	yes	characteristic
43	C₇₁	#7538	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	142	1	no	—
45	Q₆₄	—	71	2	no	—
46	Q₆₄	—	142	1	no	—
47	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
48	#366	—	2	1	no	—
49	C₆₄	—	142	1	no	—
50	C₂×Q₁₆	—	284	1	no	—
51	Q₃₂	—	284	1	no	—
52	Q₃₂	—	142	2	no	—
53	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
54	#107	—	4	1	no	—
55	C₃₂	C₄×D₇₁	1	1	yes	characteristic
56	C₃₂	D₂₈₄	1	1	yes	characteristic
57	C₂×Q₈	—	568	1	no	—
58	Q₁₆	—	284	2	no	—
59	Q₁₆	—	568	1	no	—
60	C₂×C₈	#11765850	1	1	yes	characteristic
61	C₄⋊C₄	—	8	1	no	—
62	C₁₆	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
63	C₁₆	#11765852	1	1	yes	characteristic
64	Q₈	—	568	2	no	—
65	Q₈	—	1136	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	1136	1	no	—
68	C₂×C₄	#14556069	1	1	yes	characteristic
69	C₈	#14556048	1	1	yes	characteristic
70	C₈	#14556071	1	1	yes	characteristic
71	C₂²	#17985834	1	1	yes	characteristic
72	C₄	—	32	1	no	—
73	C₄	—	1136	2	no	—
74	C₄	#17985799	1	1	yes	characteristic
75	C₄	#17985836	1	1	yes	characteristic
76	C₂	#22586517	1	1	yes	characteristic
77	C₂	#22586555	1	1	yes	characteristic
78	C₂	#22586557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985834
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₇₁×#1423	C₂
socle, prime radical	C₂²×C₇₁	D₃₂
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	#1423	D₇₁
hypocenter	C₇₁	#7538
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
6. Dic₁₁₃₆	C₄	semidirect
43. C₇₁	#7538	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₁₃₆	2
C₂×C₇₁⋊C₆₄	2
C₇₁×#1423	2
#7538	71