Subgroups — #64920

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4084	C₂	1	1	yes	maximal, characteristic
2	(C₂×AGL(1,𝔽₅))⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4003	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
5	C₂×C₅×D₄	C₄	1	1	yes	characteristic, complemented
6	C₂×C₂²⋊D₅	—	2	1	no	—
7	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂×#170	—	2	1	no	—
9	C₂²⋊Dic₅	—	2	1	no	—
10	C₄⋊Dic₅	C₄	1	1	yes	characteristic, complemented
11	#327	—	5	1	no	maximal, Sylow
12	C₂³×C₅	—	2	1	no	—
13	C₂²×D₅	D₄	1	1	yes	characteristic
14	C₂×AGL(1,𝔽₅)	—	2	2	no	—
15	C₂×AGL(1,𝔽₅)	—	4	1	no	—
16	C₂×AGL(1,𝔽₅)	—	4	1	no	—
17	C₅×D₄	—	4	1	no	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²⋊D₅	—	4	1	no	—
20	C₂²⋊D₅	—	4	1	no	—
21	C₂×Dic₅	—	2	1	no	—
22	C₂×Dic₅	D₄	1	1	yes	characteristic
23	C₄×D₅	—	4	1	no	—
24	#170	—	4	1	no	—
25	#121	—	5	1	no	—
26	C₂²⋊C₈	—	5	1	no	—
27	#95	—	5	1	no	—
28	C₂²×C₅	—	4	1	no	—
29	C₂²×C₅	—	4	1	no	—
30	C₂²×C₅	—	4	1	no	—
31	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₅	—	2	1	no	—
33	C₂×D₅	—	2	1	no	—
34	C₂×D₅	D₈	1	1	yes	characteristic
35	C₂×D₅	SD₁₆	1	1	yes	characteristic
36	AGL(1,𝔽₅)	—	4	2	no	—
37	C₄×C₅	—	2	1	no	—
38	Dic₅	—	2	1	no	—
39	Dic₅	—	4	1	no	—
40	C₂×D₄	—	10	1	no	—
41	C₂×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
42	C₂²×C₄	—	5	1	no	—
43	C₂²×C₄	—	10	1	no	—
44	C₂×C₈	—	10	1	no	—
45	C₄⋊C₄	—	5	1	no	—
46	C₂²⋊C₄	—	10	1	no	—
47	C₂×C₅	—	8	1	no	—
48	C₂×C₅	#99	1	1	yes	characteristic
49	C₂×C₅	#102	1	1	yes	characteristic
50	C₂×C₅	#104	1	1	yes	characteristic
51	D₅	—	2	1	no	—
52	D₅	—	2	1	no	—
53	C₂³	—	2	1	no	—
54	C₂³	—	5	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	20	1	no	—
57	D₄	—	20	1	no	—
58	C₂×C₄	—	5	1	no	—
59	C₂×C₄	—	10	2	no	—
60	C₂×C₄	—	10	1	no	—
61	C₂×C₄	—	20	1	no	—
62	C₂×C₄	—	20	1	no	—
63	C₂×C₄	—	20	1	no	—
64	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₈	—	20	1	no	—
66	C₅	#327	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	5	1	no	—
71	C₂²	—	5	1	no	—
72	C₂²	—	10	1	no	—
73	C₂²	—	10	1	no	—
74	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	10	1	no	—
77	C₄	—	20	2	no	—
78	C₄	—	20	1	no	—
79	C₂	—	8	1	no	—
80	C₂	—	10	1	no	—
81	C₂	—	10	1	no	—
82	C₂	#4007	1	1	yes	characteristic
83	C₂	#4008	1	1	yes	characteristic
84	C₂	#4011	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×D₄	C₄
socle	C₂²×C₅	C₂²⋊C₄
prime radical	#4084	C₂
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₂×D₄	AGL(1,𝔽₅)
hypocenter	C₅	#327
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. (C₂×AGL(1,𝔽₅))⋊C₄	C₂	semidirect
3. #4003	C₂	semidirect
5. C₂×C₅×D₄	C₄	semidirect
10. C₄⋊Dic₅	C₄	semidirect
41. C₂×D₄	AGL(1,𝔽₅)	semidirect
66. C₅	#327	semidirect

Maximal subgroups

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

subgroup	index
#4084	2
(C₂×AGL(1,𝔽₅))⋊C₄	2
#4003	2
#327	5