Subgroups — #36377532

Order 22592.

64 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	C₄⋊#15667725	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₀₆	—	2	1	no	—
5	C₂×#15667725	—	2	1	no	—
6	C₄⋊Dic₃₅₃	—	2	1	no	—
7	C₄⋊Dic₃₅₃	C₄	1	2	yes	—
8	C₂×C₈×C₃₅₃	—	2	1	no	—
9	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
10	Dic₇₀₆	—	4	1	no	—
11	Dic₇₀₆	—	4	1	no	—
12	C₂×Dic₃₅₃	—	2	2	no	—
13	#15667725	—	4	1	no	—
14	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
17	C₈×C₃₅₃	—	4	1	no	—
18	C₂²×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₃₅₃	—	2	1	no	—
20	C₄×C₃₅₃	D₈	1	1	yes	characteristic
21	C₄×C₃₅₃	SD₁₆	1	1	yes	characteristic
22	C₄×C₃₅₃	SD₁₆	1	1	yes	characteristic
23	C₄×C₃₅₃	Q₁₆	1	1	yes	characteristic
24	Dic₃₅₃	—	4	2	no	—
25	C₂×C₃₅₃	#101	1	1	yes	characteristic
26	C₂×C₃₅₃	#102	1	1	yes	characteristic
27	C₂×C₃₅₃	#103	1	1	yes	characteristic
28	C₃₅₃	#332	1	1	yes	characteristic, complemented, Sylow
29	#332	—	353	1	no	maximal, Sylow
30	C₄⋊Q₈	—	353	1	no	—
31	C₄⋊C₈	—	353	1	no	—
32	C₄⋊C₈	D₃₅₃	1	1	yes	characteristic
33	C₂×Q₈	—	706	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	706	1	no	—
36	C₄⋊C₄	—	353	2	no	—
37	C₄⋊C₄	—	706	1	no	—
38	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
39	Q₈	—	1412	1	no	—
40	Q₈	—	1412	1	no	—
41	C₂×C₄	—	706	2	no	—
42	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
44	C₂×C₄	D₁₄₁₂	1	1	yes	characteristic
45	C₈	—	4	1	no	—
46	C₈	—	1412	1	no	—
47	C₂²	#19150714	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	1412	2	no	—
50	C₄	D₂₈₂₄	1	1	yes	characteristic
51	C₄	#19150725	1	1	yes	characteristic
52	C₄	#19150726	1	1	yes	characteristic
53	C₄	#19150727	1	1	yes	characteristic
54	C₂	#26014910	1	1	yes	characteristic
55	C₂	#26014914	1	1	yes	characteristic
56	C₂	#26014915	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19150714
commutator	C₂×C₄×C₃₅₃	C₂×C₄
Frattini	C₄²	C₂×D₃₅₃
Fitting	C₄⋊C₈×C₃₅₃	C₂
socle, prime radical	C₂²×C₃₅₃	C₂²⋊C₄
cosocle	C₄²×C₃₅₃	C₂²
hypercenter	C₄⋊C₈	D₃₅₃
hypocenter	C₃₅₃	#332
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
28. C₃₅₃	#332	semidirect

Maximal subgroups

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

subgroup	index
#26014992	2
C₄⋊#15667725	2
C₄⋊C₈×C₃₅₃	2
#332	353