Subgroups — #36378906

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015089	C₂	1	2	yes	maximal, complemented
2	#26015088	C₂	1	2	yes	maximal, complemented
3	#26015087	C₂	1	2	yes	maximal, complemented
4	C₂×#19150745	C₂	1	1	yes	maximal, characteristic
5	#19150749	C₂²	1	4	yes	complemented
6	#19150745	C₄	1	1	yes	characteristic
7	#19150745	C₄	1	1	yes	characteristic
8	#19150745	C₂²	1	1	yes	characteristic
9	#19150745	C₂²	1	1	yes	characteristic
10	C₂×#15667732	C₄	1	1	yes	characteristic
11	C₂×#15667732	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃₅₃	C₄	1	1	yes	characteristic
13	#15667732	C₂×C₄	1	1	yes	characteristic
14	#15667732	C₂×C₄	1	1	yes	characteristic
15	#15667732	C₂×C₄	1	1	yes	characteristic
16	#15667732	C₂³	1	1	yes	characteristic
17	C₂²×D₃₅₃	C₈	1	1	yes	characteristic
18	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₅₃	C₈	1	1	yes	characteristic
20	C₄×D₃₅₃	C₈	1	1	yes	characteristic
21	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₅₃	C₈	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	C₂×C₈	1	1	yes	characteristic
30	Dic₃₅₃	C₂×C₈	1	1	yes	characteristic
31	Dic₃₅₃	C₂²×C₄	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	—	353	1	no	—
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	#15667734	1	1	yes	characteristic
57	C₈	—	353	1	no	—
58	C₈	—	353	1	no	—
59	C₈	—	353	1	no	—
60	C₈	—	353	1	no	—
61	C₂²	—	353	1	no	—
62	C₂²	—	353	1	no	—
63	C₂²	—	706	2	no	—
64	C₂²	C₂×#15667734	1	1	yes	characteristic
65	C₄	—	353	1	no	—
66	C₄	—	353	1	no	—
67	C₄	C₂×#15667734	1	1	yes	characteristic
68	C₄	C₂×#15667734	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	353	2	no	—
71	C₂	—	706	1	no	—
72	C₂	C₂²×#15667734	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#15667734
commutator, socle	C₂×C₃₅₃	C₂²×C₈
Frattini	C₂	C₂²×#15667734
Fitting	C₂×C₄×C₃₅₃	C₈
prime radical	C₂²×D₃₅₃	C₈
cosocle	#15667732	C₂³
hypercenter	C₂×C₄	#15667734
hypocenter	C₃₅₃	C₂×M₃₂
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. #26015089	C₂	semidirect
2. #26015088	C₂	semidirect
3. #26015087	C₂	semidirect
5. #19150749	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
#26015089	2
#26015088	2
#26015087	2
C₂×#19150745	2
C₂×M₃₂	353