Subgroups — #14580715

Order 2624.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10620285	C₂	1	4	yes	maximal
2	#13050164	C₂	1	1	yes	maximal, characteristic
3	C₄×#66491	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₄₁	C₂²	1	2	yes	—
6	C₂×Dic₈₂	C₂²	1	2	yes	—
7	C₄⋊C₄×C₄₁	—	2	2	no	—
8	C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	#10620285	—	2	8	no	—
10	C₄⋊Dic₄₁	—	2	2	no	—
11	C₂×#66491	C₂²	1	2	yes	—
12	Q₈×C₄₁	—	2	4	no	—
13	C₂×C₄×C₄₁	—	2	2	no	—
14	C₂×C₄×C₄₁	D₄	1	2	yes	—
15	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
16	C₂×Dic₄₁	—	2	2	no	—
17	Dic₈₂	—	2	4	no	—
18	#66491	D₄	1	4	yes	—
19	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
20	C₄×C₄₁	—	4	2	no	—
21	C₄×C₄₁	Q₁₆	1	4	yes	complemented
22	C₄×C₄₁	C₂×D₄	1	2	yes	—
23	Dic₄₁	—	4	2	no	—
24	C₂×C₄₁	#127	1	1	yes	characteristic
25	C₂×C₄₁	C₂×Q₁₆	1	2	yes	—
26	#494	—	41	1	no	maximal, Sylow
27	C₄₁	#494	1	1	yes	characteristic, complemented, Sylow
28	C₂×Q₁₆	—	41	4	no	—
29	C₄⋊Q₈	—	41	1	no	—
30	C₄⋊Q₈	D₄₁	1	1	yes	characteristic
31	C₄×C₈	—	41	1	no	—
32	C₂×Q₈	—	41	2	no	—
33	C₂×Q₈	C₂×D₄₁	1	2	yes	—
34	Q₁₆	—	82	8	no	—
35	C₂×C₈	—	41	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	82	2	no	—
38	C₄²	C₂×D₄₁	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	82	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	82	2	no	—
43	C₂×C₄	C₂²⋊D₄₁	1	2	yes	—
44	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
45	C₈	—	41	4	no	—
46	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	164	2	no	—
49	C₄	#10620285	1	4	yes	complemented
50	C₄	D₄×D₄₁	1	2	yes	—
51	C₂	#13050160	1	1	yes	characteristic
52	C₂	C₂×#10620285	1	2	yes	—

	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	#494
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
21. C₄×C₄₁	Q₁₆	semidirect
27. C₄₁	#494	semidirect
49. C₄	#10620285	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10620285	2
#13050164	2
C₄×#66491	2
C₄⋊Q₈×C₄₁	2
#494	41