Subgroups — #34093954

Order 20032.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₀₄	C₂	1	1	yes	maximal, characteristic
2	#23756431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₂₅₂	—	2	1	no	—
5	D₂₅₀₄	—	2	1	no	—
6	D₂₅₀₄	C₄	1	2	yes	complemented
7	C₄⋊Dic₃₁₃	—	2	1	no	—
8	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
9	C₁₆×C₃₁₃	—	2	1	no	—
10	C₂²×D₃₁₃	—	4	1	no	—
11	D₁₂₅₂	—	2	2	no	—
12	D₁₂₅₂	—	4	1	no	—
13	C₂×Dic₃₁₃	—	4	1	no	—
14	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
15	C₈×C₃₁₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₃₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₃₁₃	D₈	1	1	yes	characteristic
18	C₂×D₃₁₃	—	4	2	no	—
19	C₂×D₃₁₃	—	8	1	no	—
20	C₂×D₃₁₃	—	8	1	no	—
21	C₄×C₃₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₁₃	SD₁₆	1	1	yes	characteristic
23	Dic₃₁₃	—	8	1	no	—
24	C₂×C₃₁₃	#102	1	1	yes	characteristic
25	C₂×C₃₁₃	D₁₆	1	1	yes	characteristic
26	C₂×C₃₁₃	SD₃₂	1	1	yes	characteristic
27	D₃₁₃	—	8	2	no	—
28	C₃₁₃	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	313	1	no	maximal, Sylow
30	C₂×D₈	—	313	1	no	—
31	C₂×C₁₆	D₃₁₃	1	1	yes	characteristic, complemented
32	#107	—	313	1	no	—
33	C₂×D₄	—	626	1	no	—
34	D₈	—	313	2	no	—
35	D₈	—	626	1	no	—
36	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
37	C₄⋊C₄	—	626	1	no	—
38	C₁₆	—	2	1	no	—
39	C₂³	—	1252	1	no	—
40	D₄	—	626	2	no	—
41	D₄	—	1252	1	no	—
42	C₂×C₄	—	1252	1	no	—
43	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
44	C₈	C₄×D₃₁₃	1	1	yes	characteristic
45	C₈	C₂²⋊D₃₁₃	1	1	yes	characteristic
46	C₂²	—	2504	1	no	—
47	C₂²	—	2504	1	no	—
48	C₂²	—	1252	2	no	—
49	C₂²	D₂₅₀₄	1	1	yes	characteristic
50	C₄	—	2504	1	no	—
51	C₄	#19097538	1	1	yes	characteristic
52	C₄	#19097550	1	1	yes	characteristic
53	C₂	—	2504	2	no	—
54	C₂	#23756419	1	1	yes	characteristic
55	C₂	D₅₀₀₈	1	1	yes	characteristic
56	C₂	#23756442	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₅₀₄
commutator	C₈×C₃₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₁₃
Fitting	C₂×C₁₆×C₃₁₃	C₂
socle	C₂²×C₃₁₃	D₈
prime radical	C₂×D₂₅₀₄	C₂
cosocle	C₂×C₈×C₃₁₃	C₂²
hypercenter	C₂×C₁₆	D₃₁₃
hypocenter	C₃₁₃	#357
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #23756431	C₂	semidirect
3. C₂×C₁₆×C₃₁₃	C₂	semidirect
6. D₂₅₀₄	C₄	semidirect
28. C₃₁₃	#357	semidirect
31. 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
C₂×D₂₅₀₄	2
#23756431	2
C₂×C₁₆×C₃₁₃	2
#357	313