Subgroups — #65774

Order 320.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#704	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#170	C₂	1	2	yes	maximal, complemented
4	C₄⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₄	C₄	1	1	yes	characteristic
7	C₂²×Dic₅	C₂²	1	2	yes	—
8	#704	—	2	4	no	—
9	C₂×#170	C₂²	1	2	yes	—
10	C₂×#170	C₂²	1	1	yes	characteristic
11	C₂×#170	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₅	C₄	1	2	yes	—
13	C₄⋊Dic₅	C₄	1	1	yes	characteristic
14	C₄×Dic₅	C₂²	1	1	yes	characteristic
15	#436	—	5	1	no	maximal, Sylow
16	C₂³×C₅	C₂×C₄	1	2	yes	—
17	C₅×D₄	—	2	2	no	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	—	2	2	no	—
20	C₂×Dic₅	—	2	2	no	—
21	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅	C₂×C₄	1	2	yes	—
23	C₂×Dic₅	C₂³	1	1	yes	characteristic
24	#170	—	2	2	no	—
25	#170	—	2	1	no	—
26	#170	D₄	1	2	yes	complemented
27	C₂×M₁₆	—	5	2	no	—
28	C₄×D₄	—	5	1	no	—
29	C₄⋊C₈	—	5	1	no	—
30	C₂²⋊C₈	—	5	2	no	—
31	C₄×C₈	—	5	1	no	—
32	C₂²×C₅	—	2	2	no	—
33	C₂²×C₅	—	2	2	no	—
34	C₂²×C₅	—	2	2	no	—
35	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅	M₁₆	1	2	yes	complemented
37	Dic₅	—	2	2	no	—
38	Dic₅	—	2	1	no	—
39	Dic₅	C₂×D₄	1	1	yes	characteristic
40	Dic₅	#41	1	1	yes	characteristic
41	C₂×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂²×C₄	—	5	2	no	—
43	M₁₆	—	10	4	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	5	1	no	—
46	C₂×C₈	—	5	2	no	—
47	C₄⋊C₄	—	5	1	no	—
48	C₂²⋊C₄	—	5	2	no	—
49	C₄²	—	5	1	no	—
50	C₂×C₅	—	4	2	no	—
51	C₂×C₅	C₄×D₄	1	1	yes	characteristic
52	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₅	#131	1	1	yes	characteristic
54	C₂³	C₂×AGL(1,𝔽₅)	1	2	yes	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	—	5	2	no	—
59	C₂×C₄	—	10	2	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₈	—	5	2	no	—
63	C₈	—	10	2	no	—
64	C₈	—	10	1	no	—
65	C₅	#436	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₄	—	5	1	no	—
71	C₄	—	5	1	no	—
72	C₄	—	10	1	no	—
73	C₄	—	10	2	no	—
74	C₄	#704	1	2	yes	complemented
75	C₂	—	4	2	no	—
76	C₂	#4131	1	1	yes	characteristic
77	C₂	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂	C₂×#704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting, prime radical	C₂×C₅×D₄	C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂×D₄	AGL(1,𝔽₅)
hypocenter	C₅	#436
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#704	C₂	semidirect
3. C₂²⋊#170	C₂	semidirect
4. C₄⋊#170	C₂	semidirect
5. C₄×#170	C₂	semidirect
26. #170	D₄	semidirect
36. C₄×C₅	M₁₆	semidirect
65. C₅	#436	semidirect
74. C₄	#704	semidirect

Maximal subgroups

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

subgroup	index
C₂×#704	2
D₄×Dic₅	2
C₂²⋊#170	2
C₄⋊#170	2
C₄×#170	2
#436	5