Subgroups — #22586529

Order 9088.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#17985833	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₆₈	—	2	1	no	—
5	D₁₁₃₆	—	2	1	no	—
6	D₁₁₃₆	C₄	1	2	yes	—
7	C₇₁⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	#108×C₇₁	—	2	1	no	—
10	C₂×D₂₈₄	—	4	1	no	—
11	D₅₆₈	—	2	2	no	—
12	D₅₆₈	—	4	1	no	—
13	M₁₆×C₇₁	—	4	1	no	—
14	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
15	C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₇₁	D₄	1	1	yes	characteristic
17	C₂²×D₇₁	—	8	1	no	—
18	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
19	D₂₈₄	—	4	2	no	—
20	D₂₈₄	—	8	1	no	—
21	C₈×C₇₁	—	8	1	no	—
22	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₇₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₇₁	SD₃₂	1	1	yes	characteristic
25	C₂×D₇₁	—	8	2	no	—
26	C₂×D₇₁	—	16	2	no	—
27	C₄×C₇₁	#102	1	1	yes	characteristic
28	C₄×C₇₁	D₁₆	1	1	yes	characteristic
29	C₂×C₇₁	—	2	1	no	—
30	C₂×C₇₁	#357	1	1	yes	characteristic
31	D₇₁	—	16	2	no	—
32	#1419	—	71	1	no	maximal, Sylow
33	C₇₁	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	71	1	no	—
35	M₆₄	—	71	1	no	—
36	#368	D₇₁	1	1	yes	characteristic, complemented
37	C₂×D₈	—	142	1	no	—
38	D₁₆	—	71	2	no	—
39	D₁₆	—	142	1	no	—
40	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
41	#108	—	2	1	no	—
42	C₃₂	—	142	1	no	—
43	C₂×D₄	—	284	1	no	—
44	D₈	—	284	1	no	—
45	D₈	—	142	2	no	—
46	M₁₆	—	4	1	no	—
47	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
48	C₁₆	C₄×D₇₁	1	1	yes	characteristic
49	C₁₆	D₂₈₄	1	1	yes	characteristic
50	C₂³	—	568	1	no	—
51	D₄	—	284	2	no	—
52	D₄	—	568	1	no	—
53	C₂×C₄	#11765850	1	1	yes	characteristic
54	C₈	—	8	1	no	—
55	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
56	C₈	#11765852	1	1	yes	characteristic
57	C₂²	—	568	2	no	—
58	C₂²	—	1136	2	no	—
59	C₂²	#14556071	1	1	yes	characteristic
60	C₄	#14556048	1	1	yes	characteristic
61	C₄	#14556069	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	1136	2	no	—
64	C₂	#17985799	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17985799
commutator	C₁₆×C₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₇₁
Fitting	#368×C₇₁	C₂
socle	C₂×C₇₁	#357
hypercenter	#368	D₇₁
hypocenter	C₇₁	#1419
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #17985833	C₂	semidirect
3. #368×C₇₁	C₂	semidirect
33. C₇₁	#1419	semidirect
36. #368	D₇₁	semidirect