Subgroups — #65708

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#700	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#4038	C₂	1	1	yes	maximal, characteristic
4	#4004	C₂	1	2	yes	maximal, complemented
5	C₄⋊#170	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂×#170	C₂²	1	1	yes	characteristic
8	C₂×#170	C₂²	1	1	yes	characteristic
9	C₂×#170	C₂²	1	2	yes	—
10	#700	—	2	2	no	—
11	D₅⋊C₈	—	2	2	no	—
12	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
13	#685	C₄	1	2	yes	—
14	C₄⋊Dic₅	C₄	1	1	yes	characteristic
15	C₄×Dic₅	C₂²	1	2	yes	—
16	#424	—	5	1	no	maximal, Sylow
17	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₅	C₂×C₄	1	2	yes	—
21	C₂×Dic₅	C₂³	1	1	yes	characteristic
22	C₄×D₅	D₄	1	2	yes	—
23	C₄×D₅	Q₈	1	2	yes	—
24	#170	—	2	1	no	—
25	#170	—	2	1	no	—
26	#170	—	2	2	no	—
27	C₂×M₁₆	—	5	1	no	—
28	C₂²×C₈	—	5	1	no	—
29	#117	—	5	1	no	—
30	C₄⋊C₈	—	5	2	no	—
31	C₄⋊C₈	—	5	2	no	—
32	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₅	—	2	1	no	—
34	C₂×D₅	—	2	1	no	—
35	C₂×D₅	C₄⋊C₄	1	2	yes	—
36	C₄×C₅	—	2	2	no	—
37	C₄×C₅	C₄⋊C₄	1	2	yes	—
38	Dic₅	—	2	2	no	—
39	Dic₅	C₂×D₄	1	1	yes	characteristic
40	Dic₅	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄	—	5	1	no	—
42	M₁₆	—	10	2	no	—
43	C₂×C₈	—	5	1	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	5	2	no	—
46	C₂×C₈	—	10	2	no	—
47	C₄⋊C₄	—	5	1	no	—
48	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂²⋊C₄	—	5	2	no	—
50	C₄²	—	5	2	no	—
51	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₅	#131	1	1	yes	characteristic
53	C₂×C₅	#131	1	1	yes	characteristic
54	D₅	—	2	2	no	—
55	C₂³	—	5	1	no	—
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	2	no	—
58	C₂×C₄	—	5	2	no	—
59	C₂×C₄	—	5	2	no	—
60	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
61	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₈	—	10	1	no	—
63	C₈	—	10	1	no	—
64	C₈	—	10	2	no	—
65	C₅	#424	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	5	2	no	—
67	C₂²	—	10	1	no	—
68	C₂²	—	10	1	no	—
69	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	5	1	no	—
72	C₄	—	5	1	no	—
73	C₄	—	10	2	no	—
74	C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
75	C₂	—	10	2	no	—
76	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂	#4131	1	1	yes	characteristic
78	C₂	#4133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₅×C₄⋊C₄	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#424
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #4004	C₂	semidirect
5. C₄⋊#170	C₂	semidirect
65. C₅	#424	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
C₂×D₅⋊C₈	2
#4038	2
#4004	2
C₄⋊#170	2
#424	5