Subgroups — #34094592

Order 20032.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₅₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#23756443	C₂	1	2	yes	maximal, complemented
3	#23756441	C₂	1	2	yes	maximal, complemented
4	#23756437	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₃₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₂₆	—	2	1	no	—
7	Dic₁₂₅₂	—	2	1	no	—
8	Dic₁₂₅₂	C₂²	1	2	yes	—
9	#19097536	C₂²	1	2	yes	—
10	#41×C₃₁₃	—	2	1	no	—
11	Q₁₆×C₃₁₃	C₂²	1	1	yes	characteristic
12	SD₁₆×C₃₁₃	—	2	1	no	—
13	D₈×C₃₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
15	Dic₆₂₆	—	2	2	no	—
16	Dic₆₂₆	—	4	1	no	—
17	C₂×Dic₃₁₃	—	4	1	no	—
18	Q₈×C₃₁₃	—	2	1	no	—
19	D₄×C₃₁₃	—	2	1	no	—
20	D₄×C₃₁₃	—	4	1	no	—
21	C₂×C₄×C₃₁₃	—	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	—	4	1	no	—
26	C₂²×C₃₁₃	D₈	1	1	yes	characteristic
27	C₄×C₃₁₃	—	4	1	no	—
28	C₄×C₃₁₃	D₈	1	1	yes	characteristic
29	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
30	Dic₃₁₃	—	4	2	no	—
31	C₂×C₃₁₃	—	2	1	no	—
32	C₂×C₃₁₃	—	8	1	no	—
33	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
34	C₃₁₃	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	313	1	no	maximal, Sylow
36	#135	D₃₁₃	1	1	yes	characteristic
37	C₂×Q₁₆	—	313	1	no	—
38	Q₃₂	—	313	2	no	—
39	SD₃₂	—	313	2	no	—
40	M₃₂	—	313	1	no	—
41	#41	—	2	1	no	—
42	C₂×Q₈	—	626	1	no	—
43	Q₁₆	—	313	2	no	—
44	Q₁₆	—	626	1	no	—
45	Q₁₆	C₂×D₃₁₃	1	1	yes	characteristic
46	SD₁₆	—	2	1	no	—
47	D₈	C₂×D₃₁₃	1	1	yes	characteristic
48	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
49	C₁₆	—	313	2	no	—
50	Q₈	—	2	1	no	—
51	Q₈	—	626	2	no	—
52	Q₈	—	1252	1	no	—
53	D₄	—	2	1	no	—
54	D₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	1252	1	no	—
57	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
58	C₈	C₂²⋊D₃₁₃	1	1	yes	characteristic
59	C₈	C₂²×D₃₁₃	1	1	yes	characteristic
60	C₂²	—	4	1	no	—
61	C₂²	#19097546	1	1	yes	characteristic
62	C₄	—	4	1	no	—
63	C₄	—	1252	2	no	—
64	C₄	#19097546	1	1	yes	characteristic
65	C₄	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	8	1	no	—
68	C₂	C₂×#19097546	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19097546
commutator, cosocle	C₈×C₃₁₃	C₂³
Frattini	C₈	C₂²×D₃₁₃
Fitting, prime radical	#135×C₃₁₃	C₂
socle	C₂×C₃₁₃	C₂×D₈
hypercenter	#135	D₃₁₃
hypocenter	C₃₁₃	#510
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₂₅₂	C₂	semidirect
2. #23756443	C₂	semidirect
3. #23756441	C₂	semidirect
4. #23756437	C₂	semidirect
34. C₃₁₃	#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₂₅₂	2
#23756443	2
#23756441	2
#23756437	2
#135×C₃₁₃	2
#510	313