Subgroups — #21440607

Order 7712.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₈₂	C₂	1	1	yes	maximal, characteristic
2	#16838749	C₂	1	2	yes	maximal
3	C₄×Dic₂₄₁	C₂	1	2	yes	maximal
4	C₄×Dic₂₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
6	Dic₄₈₂	C₄	1	4	yes	complemented
7	C₂×Dic₂₄₁	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₄₁	C₂²	1	2	yes	—
9	C₂×Dic₂₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₂₄₁	C₂²	1	2	yes	—
11	C₂×C₄×C₂₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₂₄₁	C₂³	1	1	yes	characteristic
13	C₄×C₂₄₁	—	2	2	no	—
14	C₄×C₂₄₁	C₂×C₄	1	2	yes	—
15	Dic₂₄₁	—	2	1	no	—
16	Dic₂₄₁	C₂×C₄	1	4	yes	—
17	Dic₂₄₁	Q₈	1	2	yes	complemented
18	C₂×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₂₄₁	#41	1	1	yes	characteristic
21	C₂₄₁	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
22	C₄×Q₈	—	241	1	no	maximal, Sylow
23	C₂×Q₈	—	241	1	no	—
24	C₄⋊C₄	—	241	2	no	—
25	C₄⋊C₄	D₂₄₁	1	1	yes	characteristic
26	C₄²	—	241	1	no	—
27	C₄²	—	241	2	no	—
28	Q₈	—	241	4	no	—
29	C₂×C₄	—	241	1	no	—
30	C₂×C₄	—	241	1	no	—
31	C₂×C₄	—	241	2	no	—
32	C₂×C₄	C₂×D₂₄₁	1	2	yes	—
33	C₂×C₄	C₂×D₂₄₁	1	1	yes	characteristic
34	C₂²	C₂²×D₂₄₁	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	482	1	no	—
37	C₄	—	241	4	no	—
38	C₄	—	241	2	no	—
39	C₄	C₄×D₂₄₁	1	2	yes	—
40	C₂	C₂×C₄×D₂₄₁	1	1	yes	characteristic
41	C₂	Q₈×D₂₄₁	1	1	yes	characteristic
42	C₂	#16838769	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₄₁
commutator	C₂×C₂₄₁	C₂²×C₄
Fitting	C₄⋊C₄×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₂³
hypercenter	C₄⋊C₄	D₂₄₁
hypocenter	C₂₄₁	C₄×Q₈
LCS	G ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. Dic₄₈₂	C₄	semidirect
17. Dic₂₄₁	Q₈	semidirect
21. C₂₄₁	C₄×Q₈	semidirect