Subgroups — #36378699

Order 22592.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015065	C₂	1	1	yes	maximal, characteristic, complemented
2	#26015061	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13083443	—	2	1	no	—
5	#19150740	—	2	1	no	—
6	D₄×D₃₅₃	—	2	1	no	—
7	D₂₈₂₄	C₄	1	1	yes	characteristic, complemented
8	#19150722	—	2	1	no	—
9	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
10	D₈×C₃₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#13083443	—	4	1	no	—
12	C₂²×D₃₅₃	—	4	1	no	—
13	D₁₄₁₂	—	2	1	no	—
14	C₂²⋊D₃₅₃	—	4	1	no	—
15	C₄×D₃₅₃	D₄	1	1	yes	characteristic
16	#15667725	D₄	1	1	yes	characteristic
17	D₄×C₃₅₃	—	2	1	no	—
18	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃₅₃	—	4	1	no	—
20	C₂×D₃₅₃	—	4	1	no	—
21	C₂×D₃₅₃	—	8	1	no	—
22	C₂×D₃₅₃	—	8	1	no	—
23	C₂×D₃₅₃	D₈	1	1	yes	characteristic
24	#13083443	—	8	1	no	—
25	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₃₅₃	SD₁₆	1	1	yes	characteristic
27	C₂×C₃₅₃	—	8	1	no	—
28	C₂×C₃₅₃	#102	1	1	yes	characteristic
29	D₃₅₃	—	8	1	no	—
30	D₃₅₃	D₁₆	1	1	yes	characteristic
31	D₃₅₃	SD₃₂	1	1	yes	characteristic
32	C₃₅₃	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	353	1	no	maximal, Sylow
34	C₂×D₈	—	353	1	no	—
35	C₂×C₁₆	—	353	1	no	—
36	#107	—	353	1	no	—
37	C₂×D₄	—	706	1	no	—
38	D₈	—	353	1	no	—
39	D₈	—	706	1	no	—
40	D₈	#13083443	1	1	yes	characteristic, complemented
41	C₂×C₈	—	353	1	no	—
42	C₄⋊C₄	—	706	1	no	—
43	C₁₆	—	706	1	no	—
44	C₂³	—	1412	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	1412	1	no	—
47	D₄	—	706	1	no	—
48	C₂×C₄	—	353	1	no	—
49	C₂×C₄	—	1412	1	no	—
50	C₈	—	353	1	no	—
51	C₈	C₂×#13083443	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	2824	1	no	—
54	C₂²	—	2824	1	no	—
55	C₂²	—	353	1	no	—
56	C₂²	—	1412	1	no	—
57	C₄	—	2824	1	no	—
58	C₄	—	353	1	no	—
59	C₄	C₂²⋊#13083443	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	2824	1	no	—
62	C₂	—	353	1	no	—
63	C₂	—	353	1	no	—
64	C₂	#26015054	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26015054
commutator	C₈×C₃₅₃	C₂×C₄
Frattini	C₈	C₂×#13083443
Fitting	D₈×C₃₅₃	C₄
socle	C₂×C₃₅₃	#102
prime radical	D₈×D₃₅₃	C₂
cosocle	C₈×D₃₅₃	C₂²
hypercenter	D₈	#13083443
hypocenter	C₃₅₃	#357
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #26015065	C₂	semidirect
2. #26015061	C₂	semidirect
7. D₂₈₂₄	C₄	semidirect
10. D₈×C₃₅₃	C₄	semidirect
32. C₃₅₃	#357	semidirect
40. D₈	#13083443	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26015065	2
#26015061	2
D₈×D₃₅₃	2
#357	353