Subgroups — #22586409

Order 9088.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₁₆	—	2	1	no	—
5	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₇₁	—	2	1	no	—
8	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₇₁⋊C₈	—	2	1	no	—
10	C₂×C₇₁⋊C₈	C₈	1	1	yes	characteristic
11	C₂×C₇₁⋊C₈	C₈	1	1	yes	characteristic
12	C₇₁⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
14	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
15	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₇₁	—	4	1	no	—
17	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
20	C₈×C₇₁	—	2	1	no	—
21	C₈×C₇₁	D₈	1	1	yes	characteristic
22	C₈×C₇₁	Q₁₆	1	1	yes	characteristic
23	C₇₁⋊C₈	—	2	2	no	—
24	C₇₁⋊C₈	—	2	1	no	—
25	C₇₁⋊C₈	C₁₆	1	2	yes	complemented
26	C₂²×C₇₁	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₇₁	—	2	1	no	—
28	C₄×C₇₁	#107	1	1	yes	characteristic
29	C₄×C₇₁	#108	1	1	yes	characteristic
30	C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₇₁	M₃₂	1	1	yes	characteristic
32	C₂×C₇₁	#335	1	1	yes	characteristic
33	C₂×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₇₁	#364	1	1	yes	characteristic
35	#1370	—	71	1	no	maximal, Sylow
36	C₇₁	#1370	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊C₁₆	—	71	1	no	—
38	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
39	C₈²	—	71	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	142	1	no	—
42	C₄×C₈	—	71	1	no	—
43	C₄×C₈	—	71	1	no	—
44	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
45	C₂×C₈	—	71	1	no	—
46	C₂×C₈	—	71	1	no	—
47	C₂×C₈	—	142	1	no	—
48	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
50	C₄²	C₄×D₇₁	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	284	1	no	—
53	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
55	C₂×C₄	#11765848	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	71	2	no	—
58	C₈	—	142	2	no	—
59	C₈	—	142	1	no	—
60	C₈	#11765850	1	1	yes	characteristic
61	C₈	#11765855	1	1	yes	characteristic
62	C₂²	Dic₇₁⋊C₈	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#14556061	1	1	yes	characteristic
65	C₄	#14556063	1	1	yes	characteristic
66	C₄	C₁₆×D₇₁	1	1	yes	characteristic
67	C₄	#14556067	1	1	yes	characteristic
68	C₂	#17985751	1	1	yes	characteristic
69	C₂	Dic₇₁⋊C₁₆	1	1	yes	characteristic
70	C₂	#17985819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₇₁
Fitting	C₄⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₄⋊C₈
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#1370
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. C₇₁⋊C₈	C₁₆	semidirect
36. C₇₁	#1370	semidirect