Subgroups — #65253

Order 320.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#4072	C₂	1	1	yes	maximal, characteristic
3	C₂×#681	C₂	1	1	yes	maximal, characteristic
4	#3940	C₂	1	2	yes	maximal, complemented
5	#3939	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅	—	2	1	no	—
7	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₅×C₄⋊C₄	—	2	1	no	—
9	C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²⋊Dic₅	—	2	1	no	—
11	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
13	C₄×Dic₅	—	2	1	no	—
14	#681	C₄	1	4	yes	complemented
15	C₂×#168	C₂²	1	2	yes	—
16	#428	—	5	1	no	maximal, Sylow
17	C₂³×C₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	—	2	2	no	—
19	C₂×C₄×C₅	—	4	2	no	—
20	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	Q₈	1	2	yes	—
23	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×Dic₅	—	2	1	no	—
25	C₂×Dic₅	—	2	1	no	—
26	#168	C₂×C₄	1	4	yes	complemented
27	C₂×M₁₆	—	5	1	no	—
28	#117	—	5	1	no	—
29	C₂×C₄⋊C₄	D₅	1	1	yes	characteristic
30	#107	—	5	2	no	—
31	#106	—	5	2	no	—
32	C₂²×C₅	—	2	1	no	—
33	C₂²×C₅	—	2	1	no	—
34	C₂²×C₅	C₄⋊C₄	1	2	yes	—
35	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
36	C₄×C₅	—	4	2	no	—
37	C₄×C₅	C₄⋊C₄	1	2	yes	—
38	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
40	Dic₅	—	4	1	no	—
41	Dic₅	—	4	1	no	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
44	M₁₆	—	5	4	no	—
45	C₂×C₈	—	5	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	5	1	no	—
48	C₄⋊C₄	—	5	1	no	—
49	C₄⋊C₄	C₂×D₅	1	2	yes	—
50	C₂²⋊C₄	—	10	1	no	—
51	C₄²	—	10	1	no	—
52	C₂×C₅	—	2	2	no	—
53	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₅	#137	1	1	yes	characteristic
56	C₂³	C₂²⋊D₅	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	10	1	no	—
60	C₂×C₄	—	10	1	no	—
61	C₂×C₄	Dic₁₀	1	2	yes	—
62	C₂×C₄	C₄×D₅	1	2	yes	—
63	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
65	C₈	—	5	4	no	—
66	C₅	#428	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	#683	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	20	1	no	—
73	C₄	—	20	1	no	—
74	C₄	#683	1	2	yes	—
75	C₄	C₂×Dic₁₀	1	1	yes	characteristic
76	C₄	C₂×C₄×D₅	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×#683	1	1	yes	characteristic
79	C₂	#4078	1	1	yes	characteristic
80	C₂	#4088	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂×C₅×C₄⋊C₄	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×C₄⋊C₄	D₅
hypocenter	C₅	#428
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #3940	C₂	semidirect
5. #3939	C₂	semidirect
14. #681	C₄	semidirect
26. #168	C₂×C₄	semidirect
66. C₅	#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊C₄	2
#4072	2
C₂×#681	2
#3940	2
#3939	2
#428	5