Subgroups — #36378230

Order 22592.

82 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	C₂×Dic₁₄₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₈₂₄	C₂	1	2	yes	maximal, complemented
4	#26014937	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150737	—	2	1	no	—
7	C₂×Dic₇₀₆	—	2	1	no	—
8	Dic₁₄₁₂	—	2	1	no	—
9	Dic₁₄₁₂	C₂²	1	2	yes	—
10	Dic₁₄₁₂	C₂²	1	1	yes	characteristic
11	#19150726	—	2	1	no	—
12	D₂₈₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
14	C₁₆×C₃₅₃	C₂²	1	2	yes	—
15	Dic₇₀₆	—	2	1	no	—
16	Dic₇₀₆	—	2	2	no	—
17	Dic₇₀₆	—	4	1	no	—
18	D₁₄₁₂	—	2	1	no	—
19	C₂²⋊D₃₅₃	—	4	1	no	—
20	C₂×Dic₃₅₃	—	4	1	no	—
21	C₄×D₃₅₃	—	4	1	no	—
22	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
23	C₈×C₃₅₃	D₄	1	1	yes	characteristic
24	C₈×C₃₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₃₅₃	D₈	1	1	yes	characteristic
26	C₂×D₃₅₃	—	4	1	no	—
27	C₄×C₃₅₃	D₈	1	1	yes	characteristic
28	C₄×C₃₅₃	C₂×D₄	1	1	yes	characteristic
29	Dic₃₅₃	—	4	1	no	—
30	Dic₃₅₃	—	4	2	no	—
31	C₂×C₃₅₃	—	2	1	no	—
32	C₂×C₃₅₃	C₂×D₈	1	1	yes	characteristic
33	D₃₅₃	—	8	1	no	—
34	C₃₅₃	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	353	1	no	maximal, Sylow
36	#135	—	353	1	no	—
37	C₂×Q₁₆	—	353	1	no	—
38	Q₃₂	—	353	2	no	—
39	SD₃₂	—	353	2	no	—
40	M₃₂	D₃₅₃	1	1	yes	characteristic, complemented
41	#41	—	706	1	no	—
42	C₂×Q₈	—	706	1	no	—
43	Q₁₆	—	353	2	no	—
44	Q₁₆	—	353	1	no	—
45	Q₁₆	—	706	1	no	—
46	SD₁₆	—	706	1	no	—
47	D₈	—	353	1	no	—
48	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
49	C₁₆	C₂×D₃₅₃	1	2	yes	—
50	Q₈	—	1412	1	no	—
51	Q₈	—	706	1	no	—
52	Q₈	—	706	2	no	—
53	D₄	—	1412	1	no	—
54	D₄	—	706	1	no	—
55	C₂×C₄	—	1412	1	no	—
56	C₂×C₄	—	1412	1	no	—
57	C₂×C₄	D₁₄₁₂	1	1	yes	characteristic
58	C₈	D₁₄₁₂	1	1	yes	characteristic
59	C₈	C₂²×D₃₅₃	1	1	yes	characteristic
60	C₂²	—	1412	1	no	—
61	C₂²	D₂₈₂₄	1	1	yes	characteristic
62	C₄	—	1412	1	no	—
63	C₄	—	1412	2	no	—
64	C₄	D₂₈₂₄	1	1	yes	characteristic
65	C₄	C₂×D₁₄₁₂	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	2824	1	no	—
68	C₂	C₂×D₂₈₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₈₂₄
commutator, cosocle	C₈×C₃₅₃	C₂³
Frattini	C₈	C₂²×D₃₅₃
Fitting	M₃₂×C₃₅₃	C₂
socle	C₂×C₃₅₃	C₂×D₈
prime radical	#26015014	C₂
hypercenter	M₃₂	D₃₅₃
hypocenter	C₃₅₃	#510
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₄₁₂	C₂	semidirect
3. Dic₂₈₂₄	C₂	semidirect
4. #26014937	C₂	semidirect
5. M₃₂×C₃₅₃	C₂	semidirect
34. C₃₅₃	#510	semidirect
40. M₃₂	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
C₂×Dic₁₄₁₂	2
Dic₂₈₂₄	2
#26014937	2
M₃₂×C₃₅₃	2
#510	353