Subgroups — #22586523

Order 9088.

72 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	C₂×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14556150	—	2	1	no	—
5	Dic₅₆₈	C₄	1	1	yes	characteristic
6	#14556073	—	2	1	no	—
7	D₁₁₃₆	C₄	1	1	yes	characteristic
8	C₇₁⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	#108×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	M₁₆×C₇₁	—	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	D₂₈₄	—	4	1	no	—
22	C₄×D₇₁	—	8	1	no	—
23	Dic₁₄₂	—	4	1	no	—
24	C₈×C₇₁	—	8	1	no	—
25	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₇₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₇₁	SD₃₂	1	1	yes	characteristic
28	C₂×D₇₁	—	8	1	no	—
29	C₄×C₇₁	#102	1	1	yes	characteristic
30	C₄×C₇₁	D₁₆	1	1	yes	characteristic
31	Dic₇₁	—	8	1	no	—
32	C₂×C₇₁	—	2	1	no	—
33	C₂×C₇₁	#357	1	1	yes	characteristic
34	D₇₁	—	16	1	no	—
35	#1417	—	71	1	no	maximal, Sylow
36	C₇₁	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	71	1	no	—
38	C₂×C₃₂	—	71	1	no	—
39	#368	D₇₁	1	1	yes	characteristic, complemented
40	#135	—	142	1	no	—
41	Q₃₂	—	71	1	no	—
42	SD₃₂	—	142	1	no	—
43	D₁₆	—	71	1	no	—
44	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	142	1	no	—
47	#41	—	284	1	no	—
48	Q₁₆	—	142	1	no	—
49	SD₁₆	—	284	1	no	—
50	D₈	—	142	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
53	C₁₆	C₄×D₇₁	1	1	yes	characteristic
54	C₁₆	D₂₈₄	1	1	yes	characteristic
55	Q₈	—	284	1	no	—
56	D₄	—	284	1	no	—
57	D₄	—	568	1	no	—
58	C₂×C₄	—	568	1	no	—
59	C₂×C₄	#11765850	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
62	C₈	#11765852	1	1	yes	characteristic
63	C₂²	—	568	1	no	—
64	C₂²	#14556071	1	1	yes	characteristic
65	C₄	—	568	1	no	—
66	C₄	#14556048	1	1	yes	characteristic
67	C₄	#14556069	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1136	1	no	—
70	C₂	#17985799	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14556069
commutator	C₁₆×C₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₇₁
Fitting	#368×C₇₁	C₂
socle	C₂×C₇₁	#357
hypercenter	#368	D₇₁
hypocenter	C₇₁	#1417
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₇₁⋊C₃₂	C₂	semidirect
3. #368×C₇₁	C₂	semidirect
36. C₇₁	#1417	semidirect
39. #368	D₇₁	semidirect