Subgroups — #64918

Order 320.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#170	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
4	C₂²×Dic₅	C₄	1	1	yes	characteristic
5	C₂²×Dic₅	C₂²	1	1	yes	characteristic
6	C₂×#170	—	2	2	no	—
7	C₂×#170	—	2	2	no	—
8	C₂×#170	C₄	1	4	yes	complemented
9	C₄×Dic₅	—	2	2	no	—
10	#336	—	5	1	no	maximal, Sylow
11	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅	—	2	1	no	—
13	C₂×C₄×C₅	—	2	1	no	—
14	C₂×C₄×C₅	C₈	1	2	yes	complemented
15	C₂×Dic₅	—	2	1	no	—
16	C₂×Dic₅	—	2	1	no	—
17	C₂×Dic₅	C₈	1	2	yes	complemented
18	C₂×Dic₅	C₂×C₄	1	2	yes	—
19	C₂×Dic₅	D₄	1	1	yes	characteristic
20	C₂×Dic₅	D₄	1	2	yes	—
21	C₂×Dic₅	Q₈	1	1	yes	characteristic
22	#170	—	2	4	no	—
23	C₂²×C₈	—	5	2	no	—
24	C₂×C₄²	—	5	1	no	—
25	C₂²×C₅	C₄²	1	1	yes	characteristic
26	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₅	M₁₆	1	1	yes	characteristic
31	C₂²×C₅	M₁₆	1	1	yes	characteristic
32	C₄×C₅	—	2	2	no	—
33	Dic₅	—	2	2	no	—
34	Dic₅	C₂²⋊C₄	1	2	yes	—
35	Dic₅	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄	—	5	1	no	—
37	C₂²×C₄	—	5	1	no	—
38	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₂×C₈	—	5	4	no	—
40	C₂×C₈	—	10	2	no	—
41	C₂×C₈	—	10	2	no	—
42	C₄²	—	10	2	no	—
43	C₂×C₅	#95	1	1	yes	characteristic
44	C₂×C₅	C₄×C₈	1	1	yes	characteristic
45	C₂×C₅	#97	1	1	yes	characteristic
46	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₅	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₅	C₄⋊C₈	1	1	yes	characteristic
50	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	5	2	no	—
54	C₂×C₄	—	5	1	no	—
55	C₂×C₄	—	5	1	no	—
56	C₂×C₄	—	5	2	no	—
57	C₂×C₄	—	5	2	no	—
58	C₂×C₄	—	10	1	no	—
59	C₂×C₄	—	10	1	no	—
60	C₂×C₄	#170	1	2	yes	complemented
61	C₈	—	10	4	no	—
62	C₅	#336	1	1	yes	characteristic, complemented, Sylow
63	C₂²	D₅⋊C₈	1	1	yes	characteristic
64	C₂²	#700	1	1	yes	characteristic
65	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₂²	C₂×#170	1	1	yes	characteristic
68	C₂²	#704	1	1	yes	characteristic
69	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	5	2	no	—
72	C₄	—	5	2	no	—
73	C₄	—	10	2	no	—
74	C₂	C₄×#170	1	1	yes	characteristic
75	C₂	C₄⋊#170	1	1	yes	characteristic
76	C₂	#4002	1	1	yes	characteristic
77	C₂	#4003	1	1	yes	characteristic
78	C₂	#4004	1	1	yes	characteristic
79	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
80	C₂	C₂²⋊#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₄×C₈
Fitting	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂²
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#336
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
8. C₂×#170	C₄	semidirect
14. C₂×C₄×C₅	C₈	semidirect
17. C₂×Dic₅	C₈	semidirect
60. C₂×C₄	#170	semidirect
62. C₅	#336	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#170	2
C₂×C₄×Dic₅	2
#336	5