Subgroups — #36377596

Order 22592.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015014	C₂	1	1	yes	maximal, characteristic
2	#26014932	C₂	1	1	yes	maximal, characteristic, complemented
3	#106×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19150737	—	2	1	no	—
5	Dic₁₄₁₂	C₄	1	1	yes	characteristic, complemented
6	#19150726	—	2	1	no	—
7	D₂₈₂₄	C₄	1	1	yes	characteristic, complemented
8	#19150712	—	2	1	no	—
9	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₅₃	—	2	1	no	—
11	Dic₇₀₆	—	2	1	no	—
12	D₁₄₁₂	—	2	1	no	—
13	C₂²⋊D₃₅₃	—	4	1	no	—
14	C₄×D₃₅₃	—	4	1	no	—
15	C₂×C₄×C₃₅₃	—	4	1	no	—
16	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
17	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₃₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₃₅₃	D₈	1	1	yes	characteristic
20	C₂×D₃₅₃	—	4	1	no	—
21	C₄×C₃₅₃	—	8	1	no	—
22	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₃₅₃	SD₁₆	1	1	yes	characteristic
24	Dic₃₅₃	—	4	1	no	—
25	C₂×C₃₅₃	—	2	1	no	—
26	C₂×C₃₅₃	#102	1	1	yes	characteristic
27	D₃₅₃	—	8	1	no	—
28	C₃₅₃	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	353	1	no	maximal, Sylow
30	#135	—	353	1	no	—
31	M₃₂	—	353	1	no	—
32	#106	D₃₅₃	1	1	yes	characteristic, complemented
33	#41	—	706	1	no	—
34	Q₁₆	—	353	1	no	—
35	SD₁₆	—	706	1	no	—
36	D₈	—	353	1	no	—
37	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
38	C₄⋊C₄	—	2	1	no	—
39	C₁₆	—	706	1	no	—
40	Q₈	—	706	1	no	—
41	D₄	—	1412	1	no	—
42	D₄	—	706	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	1412	1	no	—
45	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
46	C₈	C₄×D₃₅₃	1	1	yes	characteristic
47	C₈	D₁₄₁₂	1	1	yes	characteristic
48	C₂²	—	1412	1	no	—
49	C₂²	#19150722	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	—	1412	1	no	—
52	C₄	#19150714	1	1	yes	characteristic
53	C₄	#19150724	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	2824	1	no	—
56	C₂	#26014912	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26014912
commutator	C₈×C₃₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₅₃
Fitting	#106×C₃₅₃	C₂
socle	C₂×C₃₅₃	#102
prime radical	#26015014	C₂
cosocle	C₂×C₈×C₃₅₃	C₂²
hypercenter	#106	D₃₅₃
hypocenter	C₃₅₃	#360
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #26014932	C₂	semidirect
3. #106×C₃₅₃	C₂	semidirect
5. Dic₁₄₁₂	C₄	semidirect
7. D₂₈₂₄	C₄	semidirect
28. C₃₅₃	#360	semidirect
32. #106	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#26015014	2
#26014932	2
#106×C₃₅₃	2
#360	353