Subgroups — #36378186

Order 22592.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014984	C₂	1	1	yes	maximal, characteristic
2	#26014925	C₂	1	2	yes	maximal
3	#26014923	C₂	1	2	yes	maximal
4	C₄×#15667725	C₂	1	1	yes	maximal, characteristic
5	#125×C₃₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667725	C₂²	1	1	yes	characteristic
7	C₂×#15667725	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₅₃	C₂²	1	2	yes	—
9	#19150716	—	2	2	no	—
10	C₄⋊C₄×C₃₅₃	—	2	2	no	—
11	C₄⋊C₄×C₃₅₃	C₂²	1	2	yes	—
12	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₅₃	—	2	2	no	—
14	#15667725	Q₈	1	2	yes	—
15	#15667725	Q₈	1	2	yes	—
16	C₂×C₄×C₃₅₃	—	2	2	no	—
17	C₂×C₄×C₃₅₃	D₄	1	2	yes	—
18	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₃₅₃	—	2	2	no	—
21	C₄×C₃₅₃	—	4	2	no	—
22	C₄×C₃₅₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₃₅₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₃₅₃	—	4	2	no	—
25	C₂×C₃₅₃	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₃₅₃	#135	1	2	yes	—
27	C₃₅₃	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	353	1	no	maximal, Sylow
29	#125	—	353	1	no	—
30	#125	D₃₅₃	1	1	yes	characteristic
31	#107	—	353	2	no	—
32	#106	—	353	2	no	—
33	C₄×C₈	—	353	1	no	—
34	C₂×C₈	—	353	1	no	—
35	C₂×C₈	—	353	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	353	2	no	—
38	C₄⋊C₄	—	706	2	no	—
39	C₄⋊C₄	C₂×D₃₅₃	1	2	yes	—
40	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	706	2	no	—
43	C₂×C₄	C₂²⋊D₃₅₃	1	2	yes	—
44	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
45	C₈	—	353	2	no	—
46	C₈	—	353	2	no	—
47	C₂²	C₂×C₂²⋊D₃₅₃	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	1412	2	no	—
51	C₄	Q₈×D₃₅₃	1	1	yes	characteristic
52	C₄	Q₈×D₃₅₃	1	1	yes	characteristic
53	C₂	#26014990	1	1	yes	characteristic
54	C₂	#26015012	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₅₃
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	#125×C₃₅₃	C₂
socle, prime radical	C₂²×C₃₅₃	C₂×D₄
hypercenter	#125	D₃₅₃
hypocenter	C₃₅₃	#499
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
27. C₃₅₃	#499	semidirect

Maximal subgroups

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

subgroup	index
#26014984	2
#26014925	2
#26014923	2
C₄×#15667725	2
#125×C₃₅₃	2
#499	353