Subgroups — #36378197

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014992	C₂	1	1	yes	maximal, characteristic
2	#26014925	C₂	1	2	yes	maximal
3	#26014923	C₂	1	2	yes	maximal
4	#26014903	C₂	1	1	yes	maximal, characteristic
5	#125×C₃₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₀₆	—	2	1	no	—
7	C₂×#15667725	C₂²	1	2	yes	—
8	C₄⋊Dic₃₅₃	—	2	1	no	—
9	C₄⋊Dic₃₅₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₅₃	—	2	1	no	—
11	C₄⋊C₄×C₃₅₃	—	2	1	no	—
12	C₄⋊C₄×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₅₃	C₂²	1	1	yes	characteristic
14	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
15	Dic₇₀₆	—	4	2	no	—
16	C₂×Dic₃₅₃	—	2	2	no	—
17	#15667725	Q₈	1	4	yes	complemented
18	C₂×C₄×C₃₅₃	—	2	1	no	—
19	C₂×C₄×C₃₅₃	—	2	1	no	—
20	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₃₅₃	—	2	1	no	—
25	C₄×C₃₅₃	—	2	1	no	—
26	C₄×C₃₅₃	—	4	1	no	—
27	C₄×C₃₅₃	—	4	1	no	—
28	C₄×C₃₅₃	C₂×Q₈	1	2	yes	—
29	Dic₃₅₃	—	4	2	no	—
30	C₂×C₃₅₃	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₃₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₃₅₃	#137	1	1	yes	characteristic
33	C₃₅₃	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	353	1	no	maximal, Sylow
35	C₄⋊Q₈	—	353	1	no	—
36	#125	D₃₅₃	1	1	yes	characteristic
37	#107	—	353	2	no	—
38	#106	—	353	2	no	—
39	#97	—	353	1	no	—
40	C₂×Q₈	—	706	1	no	—
41	C₂×C₈	—	353	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	353	2	no	—
45	C₄⋊C₄	—	706	1	no	—
46	C₄⋊C₄	C₂×D₃₅₃	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₃₅₃	1	1	yes	characteristic
48	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
49	Q₈	—	1412	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	706	2	no	—
53	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
56	C₈	—	353	4	no	—
57	C₂²	C₂×C₂²⋊D₃₅₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	1412	2	no	—
63	C₄	Q₈×D₃₅₃	1	2	yes	—
64	C₂	#26014990	1	1	yes	characteristic
65	C₂	#26015017	1	1	yes	characteristic
66	C₂	#26015022	1	1	yes	characteristic

	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	#501
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
17. #15667725	Q₈	semidirect
33. C₃₅₃	#501	semidirect

Maximal subgroups

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

subgroup	index
#26014992	2
#26014925	2
#26014923	2
#26014903	2
#125×C₃₅₃	2
#501	353