Subgroups — #65347

Order 320.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#688	C₂	1	2	yes	maximal, complemented
3	C₂×#686	C₂	1	2	yes	maximal, complemented
4	#4020	C₂	1	1	yes	maximal, characteristic, complemented
5	#3935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂×D₂₀	—	2	2	no	—
9	C₂×D₂₀	C₂²	1	2	yes	—
10	C₅×C₂²⋊C₄	—	2	2	no	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	#688	—	2	4	no	—
13	#686	—	2	4	no	—
14	C₂×#168	C₂²	1	2	yes	—
15	#496	—	5	1	no	maximal, Sylow
16	C₂³×C₅	—	2	1	no	—
17	C₂²×D₅	—	2	2	no	—
18	C₅×Q₈	—	2	2	no	—
19	C₅×D₄	—	2	2	no	—
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	D₄	1	2	yes	—
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	D₂₀	—	2	2	no	—
24	D₂₀	—	2	2	no	—
25	D₂₀	—	4	4	no	—
26	#168	D₄	1	4	yes	complemented
27	C₂×SD₁₆	—	5	2	no	—
28	C₂×D₈	—	5	2	no	—
29	#127	—	5	1	no	—
30	#124	D₅	1	1	yes	characteristic, complemented
31	#97	—	5	1	no	—
32	C₂²×C₅	—	4	1	no	—
33	C₂²×C₅	—	4	2	no	—
34	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂×D₅	—	4	2	no	—
36	C₂×D₅	—	4	4	no	—
37	C₄×C₅	—	2	2	no	—
38	C₄×C₅	—	4	1	no	—
39	C₄×C₅	C₂×D₄	1	2	yes	—
40	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
41	C₂×D₄	—	5	2	no	—
42	C₂×D₄	—	10	2	no	—
43	C₂×D₄	C₂×D₅	1	1	yes	characteristic
44	SD₁₆	—	10	4	no	—
45	D₈	—	10	4	no	—
46	C₂×C₈	—	5	2	no	—
47	C₂²⋊C₄	—	2	2	no	—
48	C₄²	C₂×D₅	1	1	yes	characteristic
49	C₂×C₅	—	8	1	no	—
50	C₂×C₅	#127	1	1	yes	characteristic
51	C₂×C₅	AGL(1,ℤ/8)	1	2	yes	—
52	D₅	—	8	2	no	—
53	C₂³	—	2	1	no	—
54	C₂³	—	10	2	no	—
55	Q₈	—	2	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	10	2	no	—
58	D₄	—	10	2	no	—
59	D₄	—	20	4	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	C₂²⋊D₅	1	2	yes	—
62	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
63	C₈	—	5	4	no	—
64	C₅	#496	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	20	2	no	—
68	C₂²	—	20	4	no	—
69	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₅	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	40	2	no	—
75	C₂	#4086	1	1	yes	characteristic
76	C₂	#4095	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#124	C₂
socle	C₂²×C₅	C₂×D₄
hypercenter	#124	D₅
hypocenter	C₅	#496
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₅×#124	C₂	semidirect
2. C₂×#688	C₂	semidirect
3. C₂×#686	C₂	semidirect
4. #4020	C₂	semidirect
5. #3935	C₂	semidirect
26. #168	D₄	semidirect
30. #124	D₅	semidirect
64. C₅	#496	semidirect

Maximal subgroups

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

subgroup	index
C₅×#124	2
C₂×#688	2
C₂×#686	2
#4020	2
#3935	2
#496	5