Subgroups — #22586475

Order 9088.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17985833	C₂	1	1	yes	maximal, characteristic
2	#17985774	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×Dic₇₁	C₄	1	1	yes	characteristic
6	C₇₁⋊C₃₂	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	C₃₂×C₇₁	C₄	1	2	yes	complemented
9	C₂×C₇₁⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₇₁	C₈	1	1	yes	characteristic
11	C₇₁⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₇₁	—	2	1	no	—
17	C₈×C₇₁	C₄²	1	1	yes	characteristic
18	C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
19	C₇₁⋊C₈	—	2	1	no	—
20	C₂²×C₇₁	M₃₂	1	1	yes	characteristic
21	C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
22	C₄×C₇₁	M₃₂	1	1	yes	characteristic
23	Dic₇₁	—	4	1	no	—
24	C₂×C₇₁	—	2	1	no	—
25	C₂×C₇₁	#346	1	1	yes	characteristic
26	#1398	—	71	1	no	maximal, Sylow
27	C₇₁	#1398	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	71	1	no	—
29	M₆₄	D₇₁	1	1	yes	characteristic
30	#346	—	71	1	no	—
31	C₂×C₁₆	—	71	1	no	—
32	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
33	C₄×C₈	—	71	1	no	—
34	C₃₂	—	71	2	no	—
35	C₃₂	Dic₇₁	1	2	yes	complemented
36	C₂×C₈	—	71	1	no	—
37	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
38	C₄²	—	71	1	no	—
39	C₁₆	—	71	2	no	—
40	C₁₆	C₄×D₇₁	1	1	yes	characteristic
41	C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
42	C₂×C₄	—	142	1	no	—
43	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
44	C₈	—	142	1	no	—
45	C₈	C₄×Dic₇₁	1	1	yes	characteristic
46	C₈	C₈×D₇₁	1	1	yes	characteristic
47	C₂²	#14556067	1	1	yes	characteristic
48	C₄	—	284	1	no	—
49	C₄	C₈×Dic₇₁	1	1	yes	characteristic
50	C₄	#14556067	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#17985774	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₇₁
commutator	C₄×C₇₁	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₇₁
Fitting	M₆₄×C₇₁	C₂
socle	C₂×C₇₁	#346
hypercenter	M₆₄	D₇₁
hypocenter	C₇₁	#1398
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
6. C₇₁⋊C₃₂	C₄	semidirect
8. C₃₂×C₇₁	C₄	semidirect
27. C₇₁	#1398	semidirect
35. C₃₂	Dic₇₁	semidirect