Subgroups — #14580739

Order 2624.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050164	C₂	1	1	yes	maximal, characteristic
2	#13050097	C₂	1	4	yes	maximal
3	C₄×#66491	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₄₁	—	2	1	no	—
6	C₂×Dic₈₂	—	2	1	no	—
7	C₄⋊C₄×C₄₁	—	2	1	no	—
8	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
9	C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₁	—	2	1	no	—
11	C₄⋊Dic₄₁	C₂²	1	2	yes	—
12	C₂×#66491	C₂²	1	2	yes	—
13	Q₈×C₄₁	—	4	2	no	—
14	C₂×C₄×C₄₁	—	2	2	no	—
15	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
18	C₂×Dic₄₁	—	2	2	no	—
19	Dic₈₂	—	4	2	no	—
20	#66491	Q₈	1	4	yes	complemented
21	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
22	C₄×C₄₁	—	4	2	no	—
23	C₄×C₄₁	D₈	1	2	yes	—
24	C₄×C₄₁	Q₁₆	1	2	yes	—
25	C₄×C₄₁	C₂×Q₈	1	2	yes	—
26	Dic₄₁	—	4	2	no	—
27	C₂×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
29	C₂×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
30	#500	—	41	1	no	maximal, Sylow
31	C₄₁	#500	1	1	yes	characteristic, complemented, Sylow
32	C₄⋊Q₈	—	41	1	no	—
33	C₄⋊Q₈	D₄₁	1	1	yes	characteristic
34	#107	—	41	4	no	—
35	C₄×C₈	—	41	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	82	1	no	—
38	C₂×C₈	—	41	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	41	2	no	—
41	C₄⋊C₄	—	82	1	no	—
42	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
43	C₄²	C₂×D₄₁	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	164	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	82	2	no	—
48	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
51	C₈	—	41	4	no	—
52	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	164	2	no	—
55	C₄	#10620282	1	2	yes	—
56	C₄	#10620285	1	2	yes	—
57	C₄	Q₈×D₄₁	1	2	yes	—
58	C₂	#13050162	1	1	yes	characteristic
59	C₂	C₂×#10620282	1	1	yes	characteristic
60	C₂	C₂×#10620285	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄⋊Q₈×C₄₁	C₂
socle, prime radical	C₂²×C₄₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₄₁
hypocenter	C₄₁	#500
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
20. #66491	Q₈	semidirect
31. C₄₁	#500	semidirect

Maximal subgroups

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

subgroup	index
#13050164	2
#13050097	2
C₄×#66491	2
C₄⋊Q₈×C₄₁	2
#500	41