Subgroups — #34093963

Order 20032.

58 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	#23756429	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097561	—	2	1	no	—
5	Dic₁₂₅₂	C₄	1	1	yes	characteristic, complemented
6	#19097550	—	2	1	no	—
7	D₂₅₀₄	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₃₁₃	—	2	1	no	—
9	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₁₃	—	2	1	no	—
11	Dic₆₂₆	—	2	1	no	—
12	D₁₂₅₂	—	2	1	no	—
13	C₂²⋊D₃₁₃	—	4	1	no	—
14	C₂×Dic₃₁₃	—	4	1	no	—
15	C₄×D₃₁₃	—	4	1	no	—
16	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
17	C₈×C₃₁₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₃₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₃₁₃	D₈	1	1	yes	characteristic
20	C₂×D₃₁₃	—	4	1	no	—
21	C₄×C₃₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₁₃	SD₁₆	1	1	yes	characteristic
23	Dic₃₁₃	—	4	1	no	—
24	Dic₃₁₃	—	8	1	no	—
25	C₂×C₃₁₃	—	2	1	no	—
26	C₂×C₃₁₃	#102	1	1	yes	characteristic
27	D₃₁₃	—	8	1	no	—
28	C₃₁₃	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	313	1	no	maximal, Sylow
30	#135	—	313	1	no	—
31	M₃₂	D₃₁₃	1	1	yes	characteristic, complemented
32	#106	—	313	1	no	—
33	#41	—	626	1	no	—
34	Q₁₆	—	313	1	no	—
35	SD₁₆	—	626	1	no	—
36	D₈	—	313	1	no	—
37	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
38	C₄⋊C₄	—	626	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	626	1	no	—
41	D₄	—	626	1	no	—
42	D₄	—	1252	1	no	—
43	C₂×C₄	—	1252	1	no	—
44	C₂×C₄	—	1252	1	no	—
45	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
46	C₈	C₄×D₃₁₃	1	1	yes	characteristic
47	C₈	C₂²⋊D₃₁₃	1	1	yes	characteristic
48	C₂²	—	1252	1	no	—
49	C₂²	D₂₅₀₄	1	1	yes	characteristic
50	C₄	—	2504	1	no	—
51	C₄	—	1252	1	no	—
52	C₄	#19097538	1	1	yes	characteristic
53	C₄	#19097550	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	2504	1	no	—
56	C₂	#23756419	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23756419
commutator	C₈×C₃₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₁₃
Fitting	M₃₂×C₃₁₃	C₂
socle	C₂×C₃₁₃	#102
prime radical	#23756519	C₂
cosocle	C₂×C₈×C₃₁₃	C₂²
hypercenter	M₃₂	D₃₁₃
hypocenter	C₃₁₃	#360
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. #23756429	C₂	semidirect
3. M₃₂×C₃₁₃	C₂	semidirect
5. Dic₁₂₅₂	C₄	semidirect
7. D₂₅₀₄	C₄	semidirect
28. C₃₁₃	#360	semidirect
31. 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
#23756429	2
M₃₂×C₃₁₃	2
#360	313