Subgroups — #36377914

Order 22592.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₀₆	C₂	1	1	yes	maximal, characteristic
2	#26014921	C₂	1	2	yes	maximal
3	#26014902	C₂	1	2	yes	maximal
4	C₄×#15667725	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₀₆	C₄	1	1	yes	characteristic
7	C₂×#15667725	C₂²	1	1	yes	characteristic
8	C₂×#15667725	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₃₅₃	C₄	1	1	yes	characteristic
10	#19150716	C₄	1	2	yes	—
11	C₄×Dic₃₅₃	C₂²	1	2	yes	—
12	C₂×C₈×C₃₅₃	C₂²	1	2	yes	—
13	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
14	Dic₇₀₆	—	2	2	no	—
15	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
17	#15667725	—	2	1	no	—
18	#15667725	Q₈	1	2	yes	complemented
19	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
22	C₈×C₃₅₃	—	2	2	no	—
23	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₅₃	—	2	1	no	—
25	C₄×C₃₅₃	M₁₆	1	2	yes	—
26	C₄×C₃₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₃₅₃	#41	1	1	yes	characteristic
28	Dic₃₅₃	—	2	2	no	—
29	Dic₃₅₃	—	2	2	no	—
30	C₂×C₃₅₃	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₃₅₃	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₃₅₃	#131	1	1	yes	characteristic
33	C₃₅₃	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	353	1	no	maximal, Sylow
35	C₄×Q₈	—	353	1	no	—
36	C₄⋊C₈	—	353	2	no	—
37	C₄⋊C₈	D₃₅₃	1	1	yes	characteristic
38	#97	—	353	2	no	—
39	C₄×C₈	—	353	1	no	—
40	C₂×Q₈	—	353	1	no	—
41	C₂×C₈	—	353	1	no	—
42	C₂×C₈	—	353	1	no	—
43	C₂×C₈	C₂×D₃₅₃	1	2	yes	—
44	C₄⋊C₄	—	353	2	no	—
45	C₄⋊C₄	—	353	1	no	—
46	C₄²	—	353	2	no	—
47	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
48	Q₈	—	706	2	no	—
49	C₂×C₄	—	353	2	no	—
50	C₂×C₄	—	353	2	no	—
51	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
52	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	353	2	no	—
56	C₈	—	706	1	no	—
57	C₂²	C₂×C₄×D₃₅₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	706	2	no	—
60	C₄	—	706	2	no	—
61	C₄	#19150720	1	2	yes	—
62	C₄	Q₈×D₃₅₃	1	1	yes	characteristic
63	C₄	#19150736	1	1	yes	characteristic
64	C₂	#26014957	1	1	yes	characteristic
65	C₂	C₂×#19150720	1	1	yes	characteristic
66	C₂	#26015000	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₅₃
commutator, socle, prime radical	C₂²×C₃₅₃	C₂²×C₄
Fitting	C₄⋊C₈×C₃₅₃	C₂
cosocle	C₂×C₄×C₃₅₃	C₂³
hypercenter	C₄⋊C₈	D₃₅₃
hypocenter	C₃₅₃	#446
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
18. #15667725	Q₈	semidirect
33. C₃₅₃	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₇₀₆	2
#26014921	2
#26014902	2
C₄×#15667725	2
C₄⋊C₈×C₃₅₃	2
#446	353