Subgroups — #34094584

Order 20032.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756519	C₂	1	2	yes	maximal, complemented
2	Dic₂₅₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#23756442	C₂	1	2	yes	maximal, complemented
4	D₅₀₀₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097561	—	2	2	no	—
7	Dic₁₂₅₂	C₂²	1	2	yes	—
8	#19097550	—	2	2	no	—
9	D₂₅₀₄	C₂²	1	2	yes	—
10	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
11	C₁₆×C₃₁₃	C₂²	1	1	yes	characteristic
12	C₁₆×C₃₁₃	C₂²	1	1	yes	characteristic
13	Dic₆₂₆	—	2	2	no	—
14	D₁₂₅₂	—	2	2	no	—
15	C₂²⋊D₃₁₃	—	4	2	no	—
16	C₄×D₃₁₃	—	4	2	no	—
17	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
18	C₈×C₃₁₃	D₄	1	1	yes	characteristic
19	C₈×C₃₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₃₁₃	D₈	1	1	yes	characteristic
21	C₂×D₃₁₃	—	4	2	no	—
22	C₄×C₃₁₃	D₈	1	1	yes	characteristic
23	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
24	Dic₃₁₃	—	4	2	no	—
25	C₂×C₃₁₃	—	2	1	no	—
26	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
27	D₃₁₃	—	8	2	no	—
28	C₃₁₃	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	313	1	no	maximal, Sylow
30	#135	—	313	2	no	—
31	Q₃₂	—	313	1	no	—
32	SD₃₂	—	313	2	no	—
33	D₁₆	—	313	1	no	—
34	C₂×C₁₆	D₃₁₃	1	1	yes	characteristic, complemented
35	#41	—	626	2	no	—
36	Q₁₆	—	313	2	no	—
37	SD₁₆	—	626	2	no	—
38	D₈	—	313	2	no	—
39	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
40	C₁₆	C₂×D₃₁₃	1	1	yes	characteristic
41	C₁₆	C₂×D₃₁₃	1	1	yes	characteristic
42	Q₈	—	626	2	no	—
43	D₄	—	626	2	no	—
44	D₄	—	1252	2	no	—
45	C₂×C₄	—	1252	2	no	—
46	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
47	C₈	D₁₂₅₂	1	1	yes	characteristic
48	C₈	C₂²×D₃₁₃	1	1	yes	characteristic
49	C₂²	—	1252	2	no	—
50	C₂²	D₂₅₀₄	1	1	yes	characteristic
51	C₄	—	1252	2	no	—
52	C₄	D₂₅₀₄	1	1	yes	characteristic
53	C₄	C₂×D₁₂₅₂	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	2504	2	no	—
56	C₂	C₂×D₂₅₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₅₀₄
commutator, cosocle	C₈×C₃₁₃	C₂³
Frattini	C₈	C₂²×D₃₁₃
Fitting	C₂×C₁₆×C₃₁₃	C₂
socle	C₂×C₃₁₃	C₂×D₈
hypercenter	C₂×C₁₆	D₃₁₃
hypocenter	C₃₁₃	#508
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #23756519	C₂	semidirect
2. Dic₂₅₀₄	C₂	semidirect
3. #23756442	C₂	semidirect
4. D₅₀₀₈	C₂	semidirect
5. C₂×C₁₆×C₃₁₃	C₂	semidirect
28. C₃₁₃	#508	semidirect
34. C₂×C₁₆	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756519	2
Dic₂₅₀₄	2
#23756442	2
D₅₀₀₈	2
C₂×C₁₆×C₃₁₃	2
#508	313