Subgroups — #64864

Order 320.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4028	C₂	1	1	yes	maximal, characteristic
2	#4013	C₂	1	1	yes	maximal, characteristic, complemented
3	#4011	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
6	C₂×D₂₀	—	2	1	no	—
7	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
8	#704	—	2	1	no	—
9	C₅×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
10	#685	—	2	1	no	—
11	AΣL(1,𝔽₁₆)	—	5	1	no	maximal, Sylow
12	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×D₅	—	2	1	no	—
14	C₂²×D₅	—	4	2	no	—
15	C₂²×D₅	—	4	1	no	—
16	C₂²×D₅	D₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₅)	—	4	1	no	—
18	C₂×C₄×C₅	—	2	1	no	—
19	C₂²⋊D₅	—	4	1	no	—
20	C₂×Dic₅	D₄	1	1	yes	characteristic
21	D₂₀	—	4	2	no	—
22	#170	—	4	1	no	—
23	#120	—	5	1	no	—
24	#100	—	5	1	no	—
25	#99	—	5	1	no	—
26	C₂²×C₅	—	2	1	no	—
27	C₂²×C₅	—	4	1	no	—
28	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₅	—	2	2	no	—
30	C₂×D₅	—	2	1	no	—
31	C₂×D₅	—	4	1	no	—
32	C₂×D₅	—	4	2	no	—
33	C₂×D₅	—	4	1	no	—
34	C₂×D₅	—	4	1	no	—
35	C₂×D₅	—	4	2	no	—
36	AGL(1,𝔽₅)	—	8	1	no	—
37	C₄×C₅	—	4	1	no	—
38	Dic₅	—	2	1	no	—
39	C₂⁴	—	5	1	no	—
40	C₂×D₄	—	5	1	no	—
41	C₂×D₄	—	10	1	no	—
42	M₁₆	—	10	1	no	—
43	C₂²⋊C₄	—	10	1	no	—
44	C₂²⋊C₄	—	10	1	no	—
45	C₂²⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
46	C₂×C₅	—	2	1	no	—
47	C₂×C₅	—	4	1	no	—
48	C₂×C₅	#99	1	1	yes	characteristic
49	D₅	—	4	1	no	—
50	D₅	—	4	2	no	—
51	C₂³	—	5	1	no	—
52	C₂³	—	10	1	no	—
53	C₂³	—	20	1	no	—
54	C₂³	—	20	2	no	—
55	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	D₄	—	20	2	no	—
57	D₄	—	20	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	20	1	no	—
61	C₈	—	20	1	no	—
62	C₅	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	10	2	no	—
66	C₂²	—	10	1	no	—
67	C₂²	—	20	1	no	—
68	C₂²	—	20	1	no	—
69	C₂²	—	20	2	no	—
70	C₂²	—	20	2	no	—
71	C₂²	—	20	1	no	—
72	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₄	—	4	1	no	—
74	C₄	—	10	1	no	—
75	C₄	—	40	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	20	1	no	—
79	C₂	—	20	2	no	—
80	C₂	#3999	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3999
commutator	C₂³×C₅	C₂×C₄
Frattini	C₂³	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₂²⋊C₄	C₄
socle	C₂×C₅	#99
prime radical	#4028	C₂
cosocle	C₂×C₂²⋊D₅	C₂²
hypercenter	C₂²⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #4013	C₂	semidirect
3. #4011	C₂	semidirect
4. C₂³×D₅	C₄	semidirect
9. C₅×C₂²⋊C₄	C₄	semidirect
45. C₂²⋊C₄	AGL(1,𝔽₅)	semidirect
62. C₅	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#4028	2
#4013	2
#4011	2
AΣL(1,𝔽₁₆)	5