Subgroups — #36378224

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015015	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#26014935	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅₆₄₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#26014933	C₂	1	1	yes	maximal, characteristic, complemented
6	#26014931	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#19150738	—	2	1	no	—
9	D₄×D₃₅₃	—	2	1	no	—
10	#19150724	—	2	1	no	—
11	D₂₈₂₄	C₂²	1	1	yes	characteristic
12	D₂₈₂₄	C₂²	1	1	yes	characteristic, complemented
13	#19150722	—	2	1	no	—
14	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
15	#19150712	C₂²	1	1	yes	characteristic, complemented
16	Q₁₆×C₃₅₃	C₂²	1	1	yes	characteristic, complemented
17	D₈×C₃₅₃	C₂²	1	1	yes	characteristic
18	C₁₆×C₃₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₃₅₃	—	4	1	no	—
20	D₁₄₁₂	—	2	1	no	—
21	D₁₄₁₂	—	2	1	no	—
22	D₁₄₁₂	—	4	1	no	—
23	C₂²⋊D₃₅₃	—	4	1	no	—
24	C₄×D₃₅₃	—	4	1	no	—
25	C₄×D₃₅₃	D₄	1	1	yes	characteristic
26	#15667725	D₄	1	1	yes	characteristic
27	Q₈×C₃₅₃	—	2	1	no	—
28	D₄×C₃₅₃	—	2	1	no	—
29	C₈×C₃₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₃₅₃	—	4	1	no	—
31	C₂×D₃₅₃	—	4	1	no	—
32	C₂×D₃₅₃	—	4	1	no	—
33	C₂×D₃₅₃	—	8	2	no	—
34	C₂×D₃₅₃	D₈	1	1	yes	characteristic
35	C₄×C₃₅₃	—	4	1	no	—
36	C₄×C₃₅₃	C₂×D₄	1	1	yes	characteristic
37	Dic₃₅₃	D₈	1	1	yes	characteristic
38	C₂×C₃₅₃	—	8	1	no	—
39	C₂×C₃₅₃	C₂×D₈	1	1	yes	characteristic
40	D₃₅₃	—	2	1	no	—
41	D₃₅₃	—	8	1	no	—
42	D₃₅₃	—	8	1	no	—
43	C₃₅₃	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	353	1	no	maximal, Sylow
45	#135	—	353	1	no	—
46	C₂×D₈	—	353	1	no	—
47	SD₃₂	—	353	1	no	—
48	SD₃₂	D₃₅₃	1	1	yes	characteristic, complemented
49	D₁₆	—	353	1	no	—
50	D₁₆	—	353	1	no	—
51	M₃₂	—	353	1	no	—
52	#41	—	706	1	no	—
53	C₂×D₄	—	706	1	no	—
54	Q₁₆	C₂×D₃₅₃	1	1	yes	characteristic, complemented
55	SD₁₆	—	706	1	no	—
56	D₈	—	353	1	no	—
57	D₈	—	353	1	no	—
58	D₈	—	706	1	no	—
59	D₈	C₂×D₃₅₃	1	1	yes	characteristic
60	C₂×C₈	—	353	1	no	—
61	C₁₆	—	353	1	no	—
62	C₁₆	C₂×D₃₅₃	1	1	yes	characteristic, complemented
63	C₂³	—	1412	1	no	—
64	Q₈	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	1412	1	no	—
67	D₄	—	1412	1	no	—
68	D₄	—	706	1	no	—
69	D₄	—	706	1	no	—
70	C₂×C₄	—	353	1	no	—
71	C₂×C₄	—	1412	1	no	—
72	C₈	—	353	1	no	—
73	C₈	C₂²×D₃₅₃	1	1	yes	characteristic
74	C₂²	—	4	1	no	—
75	C₂²	—	2824	2	no	—
76	C₂²	—	353	1	no	—
77	C₂²	—	1412	1	no	—
78	C₂²	—	1412	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	353	1	no	—
81	C₄	D₄×D₃₅₃	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	2824	1	no	—
84	C₂	—	2824	1	no	—
85	C₂	—	706	1	no	—
86	C₂	D₈×D₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₅₃
commutator, cosocle	C₈×C₃₅₃	C₂³
Frattini	C₈	C₂²×D₃₅₃
Fitting	SD₃₂×C₃₅₃	C₂
socle	C₂×C₃₅₃	C₂×D₈
hypercenter	SD₃₂	D₃₅₃
hypocenter	C₃₅₃	#509
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #26015015	C₂	semidirect
2. D₈×D₃₅₃	C₂	semidirect
3. #26014935	C₂	semidirect
4. D₅₆₄₈	C₂	semidirect
5. #26014933	C₂	semidirect
6. #26014931	C₂	semidirect
7. SD₃₂×C₃₅₃	C₂	semidirect
12. D₂₈₂₄	C₂²	semidirect
15. #19150712	C₂²	semidirect
16. Q₁₆×C₃₅₃	C₂²	semidirect
18. C₁₆×C₃₅₃	C₂²	semidirect
43. C₃₅₃	#509	semidirect
48. SD₃₂	D₃₅₃	semidirect
54. Q₁₆	C₂×D₃₅₃	semidirect
62. C₁₆	C₂×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
D₈×D₃₅₃	2
#26014935	2
D₅₆₄₈	2
#26014933	2
#26014931	2
SD₃₂×C₃₅₃	2
#509	353