Subgroups — #46830578

Order 34624.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772628	C₂	1	1	yes	maximal, characteristic
2	#31772561	C₂	1	4	yes	maximal
3	C₄×#16864513	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₅₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₀₈₂	—	2	1	no	—
6	C₂×#16864513	C₂²	1	2	yes	—
7	C₄⋊Dic₅₄₁	—	2	1	no	—
8	C₄⋊Dic₅₄₁	C₂²	1	2	yes	—
9	C₂×Q₈×C₅₄₁	—	2	1	no	—
10	C₄⋊C₄×C₅₄₁	—	2	1	no	—
11	C₄⋊C₄×C₅₄₁	C₂²	1	2	yes	—
12	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
13	Dic₁₀₈₂	—	4	2	no	—
14	C₂×Dic₅₄₁	—	2	2	no	—
15	#16864513	Q₈	1	4	yes	complemented
16	Q₈×C₅₄₁	—	4	2	no	—
17	C₂×C₄×C₅₄₁	—	2	2	no	—
18	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
21	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
22	C₄×C₅₄₁	—	4	2	no	—
23	C₄×C₅₄₁	D₈	1	2	yes	—
24	C₄×C₅₄₁	Q₁₆	1	2	yes	—
25	C₄×C₅₄₁	C₂×Q₈	1	2	yes	—
26	Dic₅₄₁	—	4	2	no	—
27	C₂×C₅₄₁	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₅₄₁	C₂×D₈	1	1	yes	characteristic
29	C₂×C₅₄₁	C₂×Q₁₆	1	1	yes	characteristic
30	C₅₄₁	#500	1	1	yes	characteristic, complemented, Sylow
31	#500	—	541	1	no	maximal, Sylow
32	C₄⋊Q₈	—	541	1	no	—
33	C₄⋊Q₈	D₅₄₁	1	1	yes	characteristic
34	#107	—	541	4	no	—
35	C₄×C₈	—	541	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	1082	1	no	—
38	C₂×C₈	—	541	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	541	2	no	—
41	C₄⋊C₄	—	1082	1	no	—
42	C₄⋊C₄	C₂×D₅₄₁	1	2	yes	—
43	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	2164	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	1082	2	no	—
48	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
51	C₈	—	541	4	no	—
52	C₂²	C₂×C₂²⋊D₅₄₁	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	2164	2	no	—
55	C₄	#22575821	1	2	yes	—
56	C₄	#22575826	1	2	yes	—
57	C₄	Q₈×D₅₄₁	1	2	yes	—
58	C₂	#31772626	1	1	yes	characteristic
59	C₂	C₂×#22575821	1	1	yes	characteristic
60	C₂	C₂×#22575826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₄₁
commutator, cosocle	C₂×C₄×C₅₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	C₄⋊Q₈×C₅₄₁	C₂
socle, prime radical	C₂²×C₅₄₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₄₁
hypocenter	C₅₄₁	#500
LCS	G ▷ C₂×C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
15. #16864513	Q₈	semidirect
30. C₅₄₁	#500	semidirect

Maximal subgroups

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

subgroup	index
#31772628	2
#31772561	2
C₄×#16864513	2
C₄⋊Q₈×C₅₄₁	2
#500	541