Subgroups — #14580741

Order 2624.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050164	C₂	1	2	yes	maximal
2	#13050098	C₂	1	4	yes	maximal
3	C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₂	—	2	2	no	—
5	C₂×C₈×C₄₁	C₂²	1	2	yes	—
6	C₄²×C₄₁	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₄₁	—	2	2	no	—
8	C₄⋊Dic₄₁	C₂²	1	4	yes	—
9	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
10	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
11	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
12	C₂×Dic₄₁	—	2	4	no	—
13	Dic₈₂	—	4	4	no	—
14	C₈×C₄₁	Q₈	1	4	yes	complemented
15	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
16	C₄×C₄₁	D₈	1	2	yes	—
17	C₄×C₄₁	Q₁₆	1	2	yes	—
18	C₄×C₄₁	C₂×Q₈	1	2	yes	—
19	Dic₄₁	—	4	4	no	—
20	C₂×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
22	C₂×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
23	#500	—	41	1	no	maximal, Sylow
24	C₄₁	#500	1	1	yes	characteristic, complemented, Sylow
25	C₄⋊Q₈	—	41	2	no	—
26	#107	—	41	4	no	—
27	C₄×C₈	D₄₁	1	1	yes	characteristic
28	C₂×Q₈	—	82	2	no	—
29	C₂×C₈	C₂×D₄₁	1	2	yes	—
30	C₄⋊C₄	—	41	4	no	—
31	C₄⋊C₄	—	82	2	no	—
32	C₄²	C₂×D₄₁	1	1	yes	characteristic
33	Q₈	—	164	4	no	—
34	C₂×C₄	—	82	4	no	—
35	C₂×C₄	D₁₆₄	1	1	yes	characteristic
36	C₂×C₄	D₁₆₄	1	1	yes	characteristic
37	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
38	C₈	Dic₈₂	1	4	yes	complemented
39	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
40	C₄	—	164	4	no	—
41	C₄	D₃₂₈	1	2	yes	—
42	C₄	Dic₁₆₄	1	2	yes	—
43	C₄	C₂×Dic₈₂	1	2	yes	—
44	C₂	#13050164	1	1	yes	characteristic
45	C₂	C₂×D₃₂₈	1	1	yes	characteristic
46	C₂	C₂×Dic₁₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄×C₈×C₄₁	C₂
socle, prime radical	C₂²×C₄₁	C₂×D₄
hypercenter	C₄×C₈	D₄₁
hypocenter	C₄₁	#500
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₈×C₄₁	Q₈	semidirect
24. C₄₁	#500	semidirect
38. C₈	Dic₈₂	semidirect

Maximal subgroups

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

subgroup	index
#13050164	2
#13050098	2
C₄×C₈×C₄₁	2
#500	41