Subgroups — #65682

Order 320.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#700	C₂	1	2	yes	maximal
2	C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	#4004	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₅	C₄	1	2	yes	—
5	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₂×#170	C₂²	1	4	yes	—
7	#700	—	2	4	no	—
8	C₄²×C₅	C₄	1	1	yes	characteristic
9	C₄×Dic₅	C₄	1	1	yes	characteristic
10	C₄×Dic₅	C₂²	1	2	yes	—
11	#423	—	5	1	no	maximal, Sylow
12	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₅	C₂×C₄	1	2	yes	—
16	C₂×Dic₅	C₂³	1	1	yes	characteristic
17	C₄×D₅	—	2	4	no	—
18	C₄×D₅	D₄	1	2	yes	—
19	C₄×D₅	Q₈	1	2	yes	—
20	#170	—	2	4	no	—
21	C₂×M₁₆	—	5	2	no	—
22	C₂×C₄²	—	5	1	no	—
23	C₄⋊C₈	—	5	4	no	—
24	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
25	C₂×D₅	—	2	2	no	—
26	C₂×D₅	C₄⋊C₄	1	2	yes	—
27	C₄×C₅	—	2	2	no	—
28	C₄×C₅	C₄⋊C₄	1	2	yes	—
29	Dic₅	M₁₆	1	4	yes	complemented
30	Dic₅	C₂×D₄	1	1	yes	characteristic
31	Dic₅	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	5	2	no	—
34	M₁₆	—	10	4	no	—
35	C₂×C₈	—	5	4	no	—
36	C₄²	—	5	2	no	—
37	C₄²	—	5	1	no	—
38	C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅	C₂×M₁₆	1	2	yes	—
41	D₅	—	2	2	no	—
42	C₂³	—	5	1	no	—
43	C₂×C₄	—	5	1	no	—
44	C₂×C₄	—	5	2	no	—
45	C₂×C₄	—	5	2	no	—
46	C₂×C₄	—	5	2	no	—
47	C₂×C₄	—	10	4	no	—
48	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
49	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₈	—	10	4	no	—
51	C₅	#423	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	5	2	no	—
53	C₂²	—	10	2	no	—
54	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	5	4	no	—
57	C₄	—	5	1	no	—
58	C₄	—	5	1	no	—
59	C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
60	C₂	—	10	2	no	—
61	C₂	D₅⋊M₁₆	1	2	yes	—
62	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₄²×C₅	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄²	AGL(1,𝔽₅)
hypocenter	C₅	#423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
3. #4004	C₂	semidirect
29. Dic₅	M₁₆	semidirect
51. C₅	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#700	2
C₄²×D₅	2
#4004	2
#423	5