Subgroups — #46830541

Order 34624.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772625	C₂	1	1	yes	maximal, characteristic
2	#31772548	C₂	1	4	yes	maximal, complemented
3	#31772539	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₁₆₄	—	2	2	no	—
6	C₂×D₂₁₆₄	C₂²	1	2	yes	—
7	C₂×#16864513	C₂²	1	2	yes	—
8	C₄⋊C₄×C₅₄₁	—	2	2	no	—
9	C₄⋊C₄×C₅₄₁	C₂²	1	2	yes	—
10	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
11	C₂²×D₅₄₁	—	2	2	no	—
12	D₂₁₆₄	—	2	4	no	—
13	D₂₁₆₄	—	4	4	no	—
14	#16864513	—	2	2	no	—
15	C₂×C₄×C₅₄₁	—	2	2	no	—
16	C₂×C₄×C₅₄₁	D₄	1	2	yes	—
17	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
18	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
19	C₂×D₅₄₁	—	4	2	no	—
20	C₂×D₅₄₁	—	4	4	no	—
21	C₄×C₅₄₁	—	2	2	no	—
22	C₄×C₅₄₁	—	4	2	no	—
23	C₄×C₅₄₁	#41	1	2	yes	—
24	C₂×C₅₄₁	#124	1	1	yes	characteristic
25	C₂×C₅₄₁	AGL(1,ℤ/8)	1	2	yes	—
26	D₅₄₁	—	8	2	no	—
27	C₅₄₁	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	541	1	no	maximal, Sylow
29	#127	—	541	1	no	—
30	#125	D₅₄₁	1	1	yes	characteristic, complemented
31	#102	—	541	4	no	—
32	#97	—	541	1	no	—
33	C₂×D₄	—	541	2	no	—
34	C₂×D₄	—	1082	2	no	—
35	C₂×C₈	—	541	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₅₄₁	1	2	yes	—
38	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
39	C₂³	—	1082	2	no	—
40	D₄	—	1082	4	no	—
41	D₄	—	2164	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₅₄₁	1	2	yes	—
44	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
45	C₈	—	1082	2	no	—
46	C₂²	—	2164	2	no	—
47	C₂²	—	2164	4	no	—
48	C₂²	C₂×C₂²⋊D₅₄₁	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#22575837	1	2	yes	—
52	C₂	—	4328	2	no	—
53	C₂	#31772617	1	1	yes	characteristic
54	C₂	#31772653	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₄₁
commutator, cosocle	C₂×C₄×C₅₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	#125×C₅₄₁	C₂
socle	C₂²×C₅₄₁	C₂×D₄
prime radical	#31772625	C₂
hypercenter	#125	D₅₄₁
hypocenter	C₅₄₁	#490
LCS	G ▷ C₂×C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #31772548	C₂	semidirect
3. #31772539	C₂	semidirect
4. #125×C₅₄₁	C₂	semidirect
27. C₅₄₁	#490	semidirect
30. #125	D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#31772625	2
#31772548	2
#31772539	2
#125×C₅₄₁	2
#490	541