Subgroups — #34094518

Order 20032.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756494	C₂	1	1	yes	maximal, characteristic
2	#23756417	C₂	1	4	yes	maximal, complemented
3	#23756408	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₂₅₂	—	2	2	no	—
6	C₂×D₁₂₅₂	C₂²	1	2	yes	—
7	C₂×#14578809	C₂²	1	2	yes	—
8	C₄⋊C₄×C₃₁₃	—	2	2	no	—
9	C₄⋊C₄×C₃₁₃	C₂²	1	2	yes	—
10	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
11	C₂²×D₃₁₃	—	2	2	no	—
12	D₁₂₅₂	—	2	4	no	—
13	D₁₂₅₂	—	4	4	no	—
14	#14578809	—	2	2	no	—
15	C₂×C₄×C₃₁₃	—	2	2	no	—
16	C₂×C₄×C₃₁₃	D₄	1	2	yes	—
17	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
18	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
19	C₂×D₃₁₃	—	4	2	no	—
20	C₂×D₃₁₃	—	4	4	no	—
21	C₄×C₃₁₃	—	2	2	no	—
22	C₄×C₃₁₃	—	4	2	no	—
23	C₄×C₃₁₃	#41	1	2	yes	—
24	C₂×C₃₁₃	#124	1	1	yes	characteristic
25	C₂×C₃₁₃	AGL(1,ℤ/8)	1	2	yes	—
26	D₃₁₃	—	8	2	no	—
27	C₃₁₃	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	313	1	no	maximal, Sylow
29	#127	—	313	1	no	—
30	#125	D₃₁₃	1	1	yes	characteristic, complemented
31	#102	—	313	4	no	—
32	#97	—	313	1	no	—
33	C₂×D₄	—	313	2	no	—
34	C₂×D₄	—	626	2	no	—
35	C₂×C₈	—	313	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₃₁₃	1	2	yes	—
38	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
39	C₂³	—	626	2	no	—
40	D₄	—	626	4	no	—
41	D₄	—	1252	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₃₁₃	1	2	yes	—
44	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
45	C₈	—	626	2	no	—
46	C₂²	—	1252	2	no	—
47	C₂²	—	1252	4	no	—
48	C₂²	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#19097562	1	2	yes	—
52	C₂	—	2504	2	no	—
53	C₂	#23756486	1	1	yes	characteristic
54	C₂	#23756522	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁₃
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	#125×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
prime radical	#23756494	C₂
hypercenter	#125	D₃₁₃
hypocenter	C₃₁₃	#490
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #23756417	C₂	semidirect
3. #23756408	C₂	semidirect
4. #125×C₃₁₃	C₂	semidirect
27. C₃₁₃	#490	semidirect
30. #125	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756494	2
#23756417	2
#23756408	2
#125×C₃₁₃	2
#490	313