Subgroups — #72244608

Order 3280.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄₁⋊C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#4223⋊C₅	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄₁⋊C₅	C₄	1	1	yes	characteristic, complemented
4	C₂×D₄₁⋊C₅	—	2	2	no	—
5	C₂×D₄₁⋊C₅	C₄	1	1	yes	characteristic, complemented
6	C₂×D₄₁⋊C₅	C₂²	1	1	yes	characteristic
7	#4223⋊C₅	—	2	2	no	—
8	C₂²⋊#4223	C₅	1	1	yes	maximal, characteristic, complemented, Hall
9	C₂×C₄₁⋊C₅	—	2	1	no	—
10	C₂×C₄₁⋊C₅	C₂×C₄	1	1	yes	characteristic
11	D₄₁⋊C₅	—	2	1	no	—
12	D₄₁⋊C₅	D₄	1	2	yes	—
13	C₂²×D₄₁	C₂×C₅	1	1	yes	characteristic
14	C₂×#4223	C₂×C₅	1	2	yes	complemented
15	C₄₁⋊C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₄₁	C₄×C₅	1	1	yes	characteristic, complemented
17	C₂×D₄₁	—	2	2	no	—
18	C₂×D₄₁	C₄×C₅	1	1	yes	characteristic, complemented
19	C₂×D₄₁	C₂²×C₅	1	1	yes	characteristic
20	#4223	—	2	2	no	—
21	C₂×C₄₁	—	2	1	no	—
22	C₂×C₄₁	C₂×C₄×C₅	1	1	yes	characteristic
23	D₄₁	—	2	1	no	—
24	D₄₁	C₅×D₄	1	2	yes	—
25	C₅×C₂²⋊C₄	—	41	1	no	maximal, Hall
26	C₄₁	C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₅	—	41	1	no	—
28	C₂×C₄×C₅	—	41	2	no	—
29	C₂²×C₅	—	41	1	no	—
30	C₂²×C₅	—	41	1	no	—
31	C₂²×C₅	—	82	2	no	—
32	C₂²×C₅	—	164	1	no	—
33	C₄×C₅	—	82	2	no	—
34	C₂²⋊C₄	—	41	1	no	Sylow
35	C₂×C₅	—	41	2	no	—
36	C₂×C₅	—	82	1	no	—
37	C₂×C₅	—	164	1	no	—
38	C₂×C₅	—	164	1	no	—
39	C₂³	—	41	1	no	—
40	C₂×C₄	—	41	2	no	—
41	C₅	—	164	1	no	Sylow
42	C₂²	—	41	1	no	—
43	C₂²	—	41	1	no	—
44	C₂²	—	82	2	no	—
45	C₂²	#4223⋊C₅	1	1	yes	characteristic, complemented
46	C₄	—	82	2	no	—
47	C₂	—	2	1	no	—
48	C₂	—	41	2	no	—
49	C₂	—	82	1	no	—
50	C₂	C₂×#4223⋊C₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×#4223⋊C₅
commutator, socle	C₂×C₄₁	C₂×C₄×C₅
Fitting	C₂²×C₄₁	C₄×C₅
prime radical	C₂²×D₄₁⋊C₅	C₂
cosocle	C₂×D₄₁	C₂²×C₅
hypercenter	C₂²	#4223⋊C₅
hypocenter	C₄₁	C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×#4223⋊C₅	C₂	semidirect
3. C₂²×C₄₁⋊C₅	C₄	semidirect
5. C₂×D₄₁⋊C₅	C₄	semidirect
8. C₂²⋊#4223	C₅	semidirect
14. C₂×#4223	C₂×C₅	semidirect
15. C₄₁⋊C₅	C₂²⋊C₄	semidirect
16. C₂²×C₄₁	C₄×C₅	semidirect
18. C₂×D₄₁	C₄×C₅	semidirect
26. C₄₁	C₅×C₂²⋊C₄	semidirect
45. C₂²	#4223⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄₁⋊C₅	2
C₂×#4223⋊C₅	2
C₂²⋊#4223	5
C₅×C₂²⋊C₄	41