Subgroups — #34094595

Order 20032.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756519	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₅₀₄	C₂	1	2	yes	maximal, complemented
4	#23756442	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097561	—	2	1	no	—
7	C₂×Dic₆₂₆	—	2	1	no	—
8	Dic₁₂₅₂	—	2	1	no	—
9	Dic₁₂₅₂	C₂²	1	2	yes	—
10	Dic₁₂₅₂	C₂²	1	1	yes	characteristic
11	#19097550	—	2	1	no	—
12	D₂₅₀₄	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
14	C₁₆×C₃₁₃	C₂²	1	2	yes	—
15	Dic₆₂₆	—	2	1	no	—
16	Dic₆₂₆	—	2	2	no	—
17	Dic₆₂₆	—	4	1	no	—
18	D₁₂₅₂	—	2	1	no	—
19	C₂²⋊D₃₁₃	—	4	1	no	—
20	C₂×Dic₃₁₃	—	4	1	no	—
21	C₄×D₃₁₃	—	4	1	no	—
22	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
23	C₈×C₃₁₃	D₄	1	1	yes	characteristic
24	C₈×C₃₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₃₁₃	D₈	1	1	yes	characteristic
26	C₂×D₃₁₃	—	4	1	no	—
27	C₄×C₃₁₃	D₈	1	1	yes	characteristic
28	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
29	Dic₃₁₃	—	4	1	no	—
30	Dic₃₁₃	—	4	2	no	—
31	C₂×C₃₁₃	—	2	1	no	—
32	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
33	D₃₁₃	—	8	1	no	—
34	C₃₁₃	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	313	1	no	maximal, Sylow
36	#135	—	313	1	no	—
37	C₂×Q₁₆	—	313	1	no	—
38	Q₃₂	—	313	2	no	—
39	SD₃₂	—	313	2	no	—
40	M₃₂	D₃₁₃	1	1	yes	characteristic, complemented
41	#41	—	626	1	no	—
42	C₂×Q₈	—	626	1	no	—
43	Q₁₆	—	313	2	no	—
44	Q₁₆	—	313	1	no	—
45	Q₁₆	—	626	1	no	—
46	SD₁₆	—	626	1	no	—
47	D₈	—	313	1	no	—
48	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
49	C₁₆	C₂×D₃₁₃	1	2	yes	—
50	Q₈	—	626	1	no	—
51	Q₈	—	626	2	no	—
52	Q₈	—	1252	1	no	—
53	D₄	—	626	1	no	—
54	D₄	—	1252	1	no	—
55	C₂×C₄	—	1252	1	no	—
56	C₂×C₄	—	1252	1	no	—
57	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
58	C₈	D₁₂₅₂	1	1	yes	characteristic
59	C₈	C₂²×D₃₁₃	1	1	yes	characteristic
60	C₂²	—	1252	1	no	—
61	C₂²	D₂₅₀₄	1	1	yes	characteristic
62	C₄	—	1252	1	no	—
63	C₄	—	1252	2	no	—
64	C₄	D₂₅₀₄	1	1	yes	characteristic
65	C₄	C₂×D₁₂₅₂	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	2504	1	no	—
68	C₂	C₂×D₂₅₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₅₀₄
commutator, cosocle	C₈×C₃₁₃	C₂³
Frattini	C₈	C₂²×D₃₁₃
Fitting	M₃₂×C₃₁₃	C₂
socle	C₂×C₃₁₃	C₂×D₈
prime radical	#23756519	C₂
hypercenter	M₃₂	D₃₁₃
hypocenter	C₃₁₃	#510
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₂₅₂	C₂	semidirect
3. Dic₂₅₀₄	C₂	semidirect
4. #23756442	C₂	semidirect
5. M₃₂×C₃₁₃	C₂	semidirect
34. C₃₁₃	#510	semidirect
40. M₃₂	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
C₂×Dic₁₂₅₂	2
Dic₂₅₀₄	2
#23756442	2
M₃₂×C₃₁₃	2
#510	313