Subgroups — #36378142

Order 22592.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014984	C₂	1	1	yes	maximal, characteristic, complemented
2	#26014980	C₂	1	1	yes	maximal, characteristic
3	#26014917	C₂	1	2	yes	maximal, complemented
4	#26014914	C₂	1	2	yes	maximal
5	C₄×C₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇₀₆	C₂²	1	1	yes	characteristic
7	C₂×D₁₄₁₂	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₅₃	C₂²	1	2	yes	—
9	#19150716	—	2	2	no	—
10	#19150714	—	2	2	no	—
11	C₂×C₈×C₃₅₃	C₂²	1	2	yes	—
12	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₂²×D₃₅₃	—	2	1	no	—
14	Dic₇₀₆	—	2	2	no	—
15	D₁₄₁₂	—	2	2	no	—
16	C₂×Dic₃₅₃	—	2	1	no	—
17	C₂×Dic₃₅₃	—	2	2	no	—
18	C₂×C₄×C₃₅₃	D₄	1	2	yes	—
19	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
20	C₈×C₃₅₃	—	2	2	no	—
21	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
22	C₂×D₃₅₃	—	4	1	no	—
23	C₂×D₃₅₃	—	4	2	no	—
24	C₄×C₃₅₃	—	2	2	no	—
25	C₄×C₃₅₃	#41	1	2	yes	—
26	Dic₃₅₃	—	4	1	no	—
27	Dic₃₅₃	—	4	2	no	—
28	C₂×C₃₅₃	#124	1	1	yes	characteristic
29	C₂×C₃₅₃	#135	1	2	yes	—
30	D₃₅₃	—	8	1	no	—
31	C₃₅₃	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	353	1	no	maximal, Sylow
33	#125	—	353	1	no	—
34	#124	—	353	1	no	—
35	#103	—	353	2	no	—
36	#102	—	353	2	no	—
37	C₄×C₈	D₃₅₃	1	1	yes	characteristic, complemented
38	C₂×Q₈	—	353	1	no	—
39	C₂×D₄	—	353	1	no	—
40	C₂×C₈	C₂×D₃₅₃	1	2	yes	—
41	C₄⋊C₄	—	353	2	no	—
42	C₄⋊C₄	—	706	2	no	—
43	C₂²⋊C₄	—	706	2	no	—
44	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
45	C₂³	—	706	1	no	—
46	Q₈	—	706	2	no	—
47	D₄	—	706	2	no	—
48	C₂×C₄	—	706	1	no	—
49	C₂×C₄	—	706	2	no	—
50	C₂×C₄	D₁₄₁₂	1	2	yes	—
51	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₂²	—	1412	1	no	—
54	C₂²	—	1412	2	no	—
55	C₂²	C₂×D₁₄₁₂	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	1412	1	no	—
58	C₄	—	1412	2	no	—
59	C₄	#19150737	1	2	yes	—
60	C₂	—	2824	1	no	—
61	C₂	#26014980	1	1	yes	characteristic
62	C₂	#26015014	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₄₁₂
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	C₄×C₈×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₂×D₄
prime radical	C₂×D₁₄₁₂	C₂²
hypercenter	C₄×C₈	D₃₅₃
hypocenter	C₃₅₃	#488
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #26014984	C₂	semidirect
3. #26014917	C₂	semidirect
5. C₄×C₈×C₃₅₃	C₂	semidirect
31. C₃₅₃	#488	semidirect
37. C₄×C₈	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#26014984	2
#26014980	2
#26014917	2
#26014914	2
C₄×C₈×C₃₅₃	2
#488	353