Subgroups — #36378682

Order 22592.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150740	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic
3	#19150740	—	2	2	no	—
4	C₂×C₄×D₃₅₃	C₄	1	1	yes	characteristic
5	C₂×#15667725	C₂²	1	1	yes	characteristic
6	C₈×D₃₅₃	—	2	2	no	—
7	C₂×C₈×C₃₅₃	C₄	1	1	yes	characteristic
8	C₂²×D₃₅₃	C₈	1	1	yes	characteristic
9	C₂×Dic₃₅₃	C₈	1	1	yes	characteristic
10	C₄×D₃₅₃	—	2	1	no	—
11	C₄×D₃₅₃	—	2	1	no	—
12	#15667725	D₄	1	2	yes	—
13	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
14	C₈×C₃₅₃	—	2	1	no	—
15	C₂²×C₃₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×D₃₅₃	—	2	1	no	—
17	C₂×D₃₅₃	—	2	1	no	—
18	C₂×D₃₅₃	C₁₆	1	2	yes	complemented
19	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₃₅₃	M₁₆	1	1	yes	characteristic
21	Dic₃₅₃	—	2	1	no	—
22	C₂×C₃₅₃	C₂²⋊C₈	1	1	yes	characteristic
23	C₂×C₃₅₃	C₂×C₁₆	1	1	yes	characteristic
24	C₂×C₃₅₃	M₃₂	1	1	yes	characteristic
25	D₃₅₃	—	2	2	no	—
26	C₃₅₃	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂²⋊C₁₆	—	353	1	no	maximal, Sylow
28	C₂²×C₈	—	353	1	no	—
29	C₂×C₁₆	—	353	2	no	—
30	C₂²×C₄	—	353	1	no	—
31	C₂×C₈	—	353	1	no	—
32	C₂×C₈	—	706	2	no	—
33	C₂×C₈	#13083443	1	1	yes	characteristic
34	C₁₆	—	706	2	no	—
35	C₂³	—	353	1	no	—
36	C₂×C₄	—	353	1	no	—
37	C₂×C₄	—	706	1	no	—
38	C₂×C₄	—	706	1	no	—
39	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	—	353	2	no	—
42	C₂²	—	353	2	no	—
43	C₂²	—	706	1	no	—
44	C₂²	—	706	1	no	—
45	C₂²	#19150745	1	1	yes	characteristic
46	C₄	—	706	1	no	—
47	C₄	C₂²⋊#13083443	1	1	yes	characteristic
48	C₄	#19150747	1	1	yes	characteristic
49	C₂	—	706	2	no	—
50	C₂	#26015049	1	1	yes	characteristic
51	C₂	#26015065	1	1	yes	characteristic
52	C₂	#26015067	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#13083443
commutator	C₂×C₃₅₃	C₂×C₁₆
Fitting	C₂×C₈×C₃₅₃	C₄
socle	C₂²×C₃₅₃	C₂×C₈
prime radical	C₂²×D₃₅₃	C₈
cosocle	C₂×#15667725	C₂²
hypercenter	C₂×C₈	#13083443
hypocenter	C₃₅₃	C₂²⋊C₁₆
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19150740	C₂	semidirect
18. C₂×D₃₅₃	C₁₆	semidirect
26. C₃₅₃	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150740	2
C₂×C₈×D₃₅₃	2
C₂²⋊C₁₆	353