Subgroups — #34094581

Order 20032.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756516	C₂	1	2	yes	maximal, complemented
2	Dic₂₅₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#23756441	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097560	—	2	2	no	—
7	Dic₁₂₅₂	C₂²	1	2	yes	—
8	#19097549	—	2	2	no	—
9	C₈×D₃₁₃	C₂²	1	1	yes	characteristic
10	#19097536	C₂²	1	1	yes	characteristic
11	D₈×C₃₁₃	C₂²	1	2	yes	—
12	C₁₆×C₃₁₃	C₂²	1	1	yes	characteristic
13	Dic₆₂₆	—	2	2	no	—
14	C₂²⋊D₃₁₃	—	4	2	no	—
15	C₂×Dic₃₁₃	—	4	2	no	—
16	C₄×D₃₁₃	D₄	1	1	yes	characteristic
17	#14578809	D₄	1	1	yes	characteristic
18	D₄×C₃₁₃	—	2	2	no	—
19	C₈×C₃₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₃₁₃	—	4	2	no	—
21	C₂×D₃₁₃	D₈	1	1	yes	characteristic
22	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
23	Dic₃₁₃	—	4	2	no	—
24	Dic₃₁₃	D₈	1	1	yes	characteristic
25	C₂×C₃₁₃	—	8	2	no	—
26	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
27	D₃₁₃	—	2	1	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	D₃₁₃	1	1	yes	characteristic, complemented
34	C₂×C₁₆	—	313	1	no	—
35	#41	—	626	2	no	—
36	Q₁₆	—	313	2	no	—
37	SD₁₆	—	626	2	no	—
38	D₈	C₂×D₃₁₃	1	2	yes	—
39	C₂×C₈	—	313	1	no	—
40	C₁₆	—	313	1	no	—
41	C₁₆	C₂×D₃₁₃	1	1	yes	characteristic
42	Q₈	—	626	2	no	—
43	D₄	—	2	2	no	—
44	D₄	—	1252	2	no	—
45	C₂×C₄	—	313	1	no	—
46	C₂×C₄	—	1252	2	no	—
47	C₈	—	313	1	no	—
48	C₈	C₂²×D₃₁₃	1	1	yes	characteristic
49	C₂²	—	4	2	no	—
50	C₂²	—	313	1	no	—
51	C₄	—	313	1	no	—
52	C₄	—	1252	2	no	—
53	C₄	D₄×D₃₁₃	1	1	yes	characteristic
54	C₂	—	8	2	no	—
55	C₂	—	626	1	no	—
56	C₂	D₈×D₃₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₁₃
commutator, cosocle	C₈×C₃₁₃	C₂³
Frattini	C₈	C₂²×D₃₁₃
Fitting	D₁₆×C₃₁₃	C₂
socle	C₂×C₃₁₃	C₂×D₈
hypercenter	D₁₆	D₃₁₃
hypocenter	C₃₁₃	#508
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #23756516	C₂	semidirect
2. Dic₂₅₀₄	C₂	semidirect
3. #23756441	C₂	semidirect
4. C₁₆×D₃₁₃	C₂	semidirect
5. D₁₆×C₃₁₃	C₂	semidirect
28. C₃₁₃	#508	semidirect
33. D₁₆	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756516	2
Dic₂₅₀₄	2
#23756441	2
C₁₆×D₃₁₃	2
D₁₆×C₃₁₃	2
#508	313