Subgroups — #34093964

Order 20032.

62 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	#23756437	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×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	—
7	#19097536	—	2	1	no	—
8	M₁₆×C₃₁₃	—	2	1	no	—
9	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
10	C₂²×D₃₁₃	—	4	1	no	—
11	D₁₂₅₂	—	2	2	no	—
12	D₁₂₅₂	—	4	1	no	—
13	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
14	C₈×C₃₁₃	—	4	1	no	—
15	C₈×C₃₁₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₃₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₃₁₃	SD₁₆	1	1	yes	characteristic
18	C₂×D₃₁₃	—	4	2	no	—
19	C₂×D₃₁₃	—	8	2	no	—
20	C₄×C₃₁₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₁₃	D₈	1	1	yes	characteristic
22	C₂×C₃₁₃	—	2	1	no	—
23	C₂×C₃₁₃	#102	1	1	yes	characteristic
24	D₃₁₃	—	8	2	no	—
25	C₃₁₃	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	313	1	no	maximal, Sylow
27	C₂×D₈	—	313	1	no	—
28	M₃₂	—	313	1	no	—
29	#108	D₃₁₃	1	1	yes	characteristic, complemented
30	C₂×D₄	—	626	1	no	—
31	D₈	—	313	2	no	—
32	D₈	—	626	1	no	—
33	M₁₆	—	2	1	no	—
34	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
35	C₁₆	—	626	1	no	—
36	C₂³	—	1252	1	no	—
37	D₄	—	626	2	no	—
38	D₄	—	1252	1	no	—
39	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
40	C₈	—	4	1	no	—
41	C₈	C₄×D₃₁₃	1	1	yes	characteristic
42	C₈	D₁₂₅₂	1	1	yes	characteristic
43	C₂²	—	2504	2	no	—
44	C₂²	—	1252	2	no	—
45	C₂²	#19097548	1	1	yes	characteristic
46	C₄	#19097538	1	1	yes	characteristic
47	C₄	#19097546	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	2504	2	no	—
50	C₂	#23756417	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23756417
commutator	C₈×C₃₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₁₃
Fitting	#108×C₃₁₃	C₂
socle	C₂×C₃₁₃	#102
prime radical	C₂×D₂₅₀₄	C₂
cosocle	C₂×C₈×C₃₁₃	C₂²
hypercenter	#108	D₃₁₃
hypocenter	C₃₁₃	#361
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #23756437	C₂	semidirect
3. #108×C₃₁₃	C₂	semidirect
25. C₃₁₃	#361	semidirect
29. #108	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
#23756437	2
#108×C₃₁₃	2
#361	313