Subgroups — #65170

Order 320.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#676	C₂	1	1	yes	maximal, characteristic
2	#4038	C₂	1	1	yes	maximal, characteristic
3	D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	#3949	C₂	1	1	yes	maximal, characteristic, complemented
6	#3940	C₂	1	1	yes	maximal, characteristic, complemented
7	#3939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₅	—	2	1	no	—
9	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#685	—	2	1	no	—
14	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	#683	—	2	1	no	—
17	C₄×Dic₅	—	2	1	no	—
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#676	C₄	1	4	yes	complemented
20	#428	—	5	1	no	maximal, Sylow
21	C₂²×D₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	—	2	1	no	—
23	C₂×C₄×C₅	—	2	1	no	—
24	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×Dic₅	—	2	1	no	—
26	C₂×Dic₅	—	2	1	no	—
27	C₂×Dic₅	D₄	1	1	yes	characteristic
28	C₄×D₅	—	4	2	no	—
29	C₄×D₅	C₂×C₄	1	2	yes	—
30	C₄×D₅	Q₈	1	2	yes	—
31	C₅×C₈	C₂×C₄	1	2	yes	complemented
32	#168	C₂×C₄	1	2	yes	complemented
33	C₂×M₁₆	—	5	1	no	—
34	#117	—	5	1	no	—
35	C₂×C₄⋊C₄	—	5	1	no	—
36	#107	—	5	1	no	—
37	#107	D₅	1	1	yes	characteristic, complemented
38	#106	—	5	1	no	—
39	#106	—	5	1	no	—
40	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂×D₅	—	2	1	no	—
42	C₂×D₅	—	2	1	no	—
43	C₂×D₅	C₄⋊C₄	1	2	yes	—
44	C₄×C₅	—	4	1	no	—
45	C₄×C₅	—	4	1	no	—
46	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	Dic₅	—	4	1	no	—
49	Dic₅	—	4	1	no	—
50	Dic₅	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄	—	5	1	no	—
52	C₂²×C₄	—	10	1	no	—
53	M₁₆	—	5	4	no	—
54	C₂×C₈	—	5	1	no	—
55	C₂×C₈	C₂×D₅	1	1	yes	characteristic
56	C₄⋊C₄	—	5	1	no	—
57	C₄⋊C₄	—	5	1	no	—
58	C₄⋊C₄	—	10	1	no	—
59	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
60	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
61	C₂²⋊C₄	—	10	1	no	—
62	C₄²	—	10	1	no	—
63	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₅	#137	1	1	yes	characteristic
66	D₅	—	2	2	no	—
67	C₂³	—	5	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	5	1	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	—	10	1	no	—
74	C₂×C₄	—	10	1	no	—
75	C₂×C₄	—	20	2	no	—
76	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₈	—	5	2	no	—
78	C₈	C₄×D₅	1	2	yes	complemented
79	C₅	#428	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	5	2	no	—
81	C₂²	—	10	1	no	—
82	C₂²	—	10	1	no	—
83	C₂²	D₄×D₅	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	5	2	no	—
87	C₄	—	20	1	no	—
88	C₄	—	20	1	no	—
89	C₄	C₂×C₄×D₅	1	1	yes	characteristic
90	C₄	Q₈×D₅	1	1	yes	characteristic
91	C₂	—	10	2	no	—
92	C₂	D₅×C₄⋊C₄	1	1	yes	characteristic
93	C₂	#4057	1	1	yes	characteristic
94	C₂	#4064	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#107	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂²×D₅	D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#107	D₅
hypocenter	C₅	#428
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
4. C₅×#107	C₂	semidirect
5. #3949	C₂	semidirect
6. #3940	C₂	semidirect
7. #3939	C₂	semidirect
19. #676	C₄	semidirect
31. C₅×C₈	C₂×C₄	semidirect
32. #168	C₂×C₄	semidirect
37. #107	D₅	semidirect
78. C₈	C₄×D₅	semidirect
79. 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×#676	2
#4038	2
D₅×C₄⋊C₄	2
C₅×#107	2
#3949	2
#3940	2
#3939	2
#428	5