Subgroups — #36378843

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015067	C₂	1	2	yes	maximal, complemented
2	#26015066	C₂	1	2	yes	maximal, complemented
3	#26015065	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic
5	#19150740	C₂²	1	4	yes	complemented
6	C₂×C₄×D₃₅₃	C₄	1	1	yes	characteristic
7	C₂×#15667725	C₂²	1	1	yes	characteristic
8	C₈×D₃₅₃	C₄	1	1	yes	characteristic
9	C₈×D₃₅₃	C₄	1	1	yes	characteristic
10	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
11	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₅₃	C₄	1	1	yes	characteristic
13	C₂²×D₃₅₃	C₈	1	1	yes	characteristic
14	C₂×Dic₃₅₃	C₈	1	1	yes	characteristic
15	C₄×D₃₅₃	C₈	1	1	yes	characteristic
16	C₄×D₃₅₃	C₈	1	1	yes	characteristic
17	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
19	#15667725	C₂×C₄	1	1	yes	characteristic
20	#15667725	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
22	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
23	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×D₃₅₃	—	2	2	no	—
26	C₂×D₃₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×D₃₅₃	C₂×C₈	1	1	yes	characteristic
28	C₄×C₃₅₃	C₂×C₈	1	1	yes	characteristic
29	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
30	Dic₃₅₃	C₂×C₈	1	1	yes	characteristic
31	Dic₃₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₃₅₃	—	2	1	no	—
33	C₂×C₃₅₃	C₂²×C₈	1	1	yes	characteristic
34	D₃₅₃	—	2	1	no	—
35	D₃₅₃	M₃₂	1	2	yes	complemented
36	C₃₅₃	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
37	C₂×M₃₂	—	353	1	no	maximal, Sylow
38	C₂²×C₈	—	353	1	no	—
39	M₃₂	—	353	2	no	—
40	M₃₂	—	353	2	no	—
41	C₂×C₁₆	—	353	2	no	—
42	C₂²×C₄	—	353	1	no	—
43	C₂×C₈	—	353	1	no	—
44	C₂×C₈	—	353	1	no	—
45	C₂×C₈	—	353	1	no	—
46	C₂×C₈	—	353	1	no	—
47	C₂×C₈	—	353	1	no	—
48	C₂×C₈	#13083443	1	1	yes	characteristic
49	C₁₆	—	353	4	no	—
50	C₂³	—	353	1	no	—
51	C₂×C₄	—	353	1	no	—
52	C₂×C₄	—	353	1	no	—
53	C₂×C₄	—	353	1	no	—
54	C₂×C₄	—	353	1	no	—
55	C₂×C₄	—	353	1	no	—
56	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
57	C₈	—	353	1	no	—
58	C₈	—	353	1	no	—
59	C₈	C₂×#13083443	1	1	yes	characteristic
60	C₈	C₂×#13083443	1	1	yes	characteristic
61	C₂²	—	353	1	no	—
62	C₂²	—	353	1	no	—
63	C₂²	—	706	2	no	—
64	C₂²	#19150745	1	1	yes	characteristic
65	C₄	—	353	1	no	—
66	C₄	—	353	1	no	—
67	C₄	#19150745	1	1	yes	characteristic
68	C₄	C₂²×#13083443	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	353	2	no	—
71	C₂	—	706	1	no	—
72	C₂	C₂×#19150745	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#13083443
commutator, socle	C₂×C₃₅₃	C₂²×C₈
Frattini	C₄	C₂²×#13083443
Fitting	C₂×C₈×C₃₅₃	C₄
prime radical	C₂²×D₃₅₃	C₈
cosocle	#15667725	C₂³
hypercenter	C₂×C₈	#13083443
hypocenter	C₃₅₃	C₂×M₃₂
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #26015067	C₂	semidirect
2. #26015066	C₂	semidirect
3. #26015065	C₂	semidirect
5. #19150740	C₂²	semidirect
35. D₃₅₃	M₃₂	semidirect
36. C₃₅₃	C₂×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#26015067	2
#26015066	2
#26015065	2
C₂×C₈×D₃₅₃	2
C₂×M₃₂	353