Subgroups — #22586528

Order 9088.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986431	C₂	1	1	yes	maximal, characteristic
2	#17985827	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14556150	—	2	1	no	—
5	Dic₅₆₈	C₄	1	1	yes	characteristic, complemented
6	#14556073	—	2	1	no	—
7	D₁₁₃₆	C₄	1	1	yes	characteristic, complemented
8	#14556062	—	2	1	no	—
9	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₇₁	—	2	1	no	—
11	#11765865	—	4	1	no	—
12	Dic₂₈₄	—	2	1	no	—
13	#11765854	—	4	1	no	—
14	D₅₆₈	—	2	1	no	—
15	C₄⋊Dic₇₁	—	4	1	no	—
16	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
20	C₂²⋊D₇₁	—	8	1	no	—
21	C₂×Dic₇₁	—	8	1	no	—
22	D₂₈₄	—	4	1	no	—
23	C₄×D₇₁	—	8	1	no	—
24	Dic₁₄₂	—	4	1	no	—
25	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₇₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₇₁	D₁₆	1	1	yes	characteristic
28	C₂×D₇₁	—	8	1	no	—
29	C₄×C₇₁	#102	1	1	yes	characteristic
30	C₄×C₇₁	SD₃₂	1	1	yes	characteristic
31	Dic₇₁	—	8	1	no	—
32	Dic₇₁	—	16	1	no	—
33	C₂×C₇₁	—	2	1	no	—
34	C₂×C₇₁	#357	1	1	yes	characteristic
35	D₇₁	—	16	1	no	—
36	#1418	—	71	1	no	maximal, Sylow
37	C₇₁	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	—	71	1	no	—
39	M₆₄	D₇₁	1	1	yes	characteristic, complemented
40	#367	—	71	1	no	—
41	#135	—	142	1	no	—
42	Q₃₂	—	71	1	no	—
43	SD₃₂	—	142	1	no	—
44	D₁₆	—	71	1	no	—
45	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
46	#107	—	142	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	284	1	no	—
49	Q₁₆	—	142	1	no	—
50	SD₁₆	—	284	1	no	—
51	D₈	—	142	1	no	—
52	C₂×C₈	D₂₈₄	1	1	yes	characteristic
53	C₄⋊C₄	—	284	1	no	—
54	C₁₆	C₄×D₇₁	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
56	Q₈	—	284	1	no	—
57	D₄	—	284	1	no	—
58	D₄	—	568	1	no	—
59	C₂×C₄	—	568	1	no	—
60	C₂×C₄	—	568	1	no	—
61	C₂×C₄	D₅₆₈	1	1	yes	characteristic
62	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
63	C₈	#11765854	1	1	yes	characteristic
64	C₂²	—	568	1	no	—
65	C₂²	D₁₁₃₆	1	1	yes	characteristic
66	C₄	—	568	1	no	—
67	C₄	—	1136	1	no	—
68	C₄	#14556050	1	1	yes	characteristic
69	C₄	#14556073	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1136	1	no	—
72	C₂	#17985801	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17985801
commutator	C₁₆×C₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₇₁
Fitting	M₆₄×C₇₁	C₂
socle	C₂×C₇₁	#357
hypercenter	M₆₄	D₇₁
hypocenter	C₇₁	#1418
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #17985827	C₂	semidirect
3. M₆₄×C₇₁	C₂	semidirect
5. Dic₅₆₈	C₄	semidirect
7. D₁₁₃₆	C₄	semidirect
37. C₇₁	#1418	semidirect
39. M₆₄	D₇₁	semidirect