Subgroups — #36378220

Order 22592.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015015	C₂	1	2	yes	maximal, complemented
2	#26014935	C₂	1	2	yes	maximal, complemented
3	D₅₆₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150738	—	2	2	no	—
7	#19150724	—	2	2	no	—
8	D₂₈₂₄	C₂²	1	2	yes	—
9	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
10	#19150712	C₂²	1	1	yes	characteristic
11	Q₁₆×C₃₅₃	C₂²	1	2	yes	—
12	C₁₆×C₃₅₃	C₂²	1	1	yes	characteristic
13	D₁₄₁₂	—	2	2	no	—
14	D₁₄₁₂	—	4	2	no	—
15	C₄×D₃₅₃	—	4	2	no	—
16	C₄×D₃₅₃	D₄	1	1	yes	characteristic
17	#15667725	D₄	1	1	yes	characteristic
18	Q₈×C₃₅₃	—	2	2	no	—
19	C₈×C₃₅₃	C₂³	1	1	yes	characteristic
20	C₂×D₃₅₃	—	4	2	no	—
21	C₂×D₃₅₃	D₈	1	1	yes	characteristic
22	C₄×C₃₅₃	—	4	2	no	—
23	C₄×C₃₅₃	C₂×D₄	1	1	yes	characteristic
24	Dic₃₅₃	D₈	1	1	yes	characteristic
25	C₂×C₃₅₃	C₂×D₈	1	1	yes	characteristic
26	D₃₅₃	—	2	1	no	—
27	D₃₅₃	—	8	2	no	—
28	C₃₅₃	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	353	1	no	maximal, Sylow
30	#135	—	353	2	no	—
31	Q₃₂	D₃₅₃	1	1	yes	characteristic, complemented
32	SD₃₂	—	353	2	no	—
33	D₁₆	—	353	1	no	—
34	C₂×C₁₆	—	353	1	no	—
35	#41	—	706	2	no	—
36	Q₁₆	C₂×D₃₅₃	1	2	yes	—
37	SD₁₆	—	706	2	no	—
38	D₈	—	353	2	no	—
39	C₂×C₈	—	353	1	no	—
40	C₁₆	—	353	1	no	—
41	C₁₆	C₂×D₃₅₃	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	1412	2	no	—
44	D₄	—	706	2	no	—
45	C₂×C₄	—	353	1	no	—
46	C₂×C₄	—	1412	2	no	—
47	C₈	—	353	1	no	—
48	C₈	C₂²×D₃₅₃	1	1	yes	characteristic
49	C₂²	—	353	1	no	—
50	C₂²	—	1412	2	no	—
51	C₄	—	4	2	no	—
52	C₄	—	353	1	no	—
53	C₄	D₄×D₃₅₃	1	1	yes	characteristic
54	C₂	—	2824	2	no	—
55	C₂	—	706	1	no	—
56	C₂	D₈×D₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₅₃
commutator, cosocle	C₈×C₃₅₃	C₂³
Frattini	C₈	C₂²×D₃₅₃
Fitting	Q₃₂×C₃₅₃	C₂
socle	C₂×C₃₅₃	C₂×D₈
hypercenter	Q₃₂	D₃₅₃
hypocenter	C₃₅₃	#508
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #26015015	C₂	semidirect
2. #26014935	C₂	semidirect
3. D₅₆₄₈	C₂	semidirect
4. C₁₆×D₃₅₃	C₂	semidirect
5. Q₃₂×C₃₅₃	C₂	semidirect
28. C₃₅₃	#508	semidirect
31. Q₃₂	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#26015015	2
#26014935	2
D₅₆₄₈	2
C₁₆×D₃₅₃	2
Q₃₂×C₃₅₃	2
#508	353