Subgroups — #64875

Order 320.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×#700	C₂	1	1	yes	maximal, characteristic
3	#4038	C₂	1	1	yes	maximal, characteristic
4	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
5	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₂×#170	—	2	1	no	—
7	C₄×AGL(1,𝔽₅)	—	2	1	no	—
8	C₄×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
9	#700	—	2	1	no	—
10	#700	C₄	1	2	yes	complemented
11	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
12	#685	—	2	1	no	—
13	C₄⋊Dic₅	C₄	1	1	yes	characteristic, complemented
14	C₄×Dic₅	—	2	1	no	—
15	#339	—	5	1	no	maximal, Sylow
16	C₂²×D₅	D₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₅)	—	2	2	no	—
18	C₂×AGL(1,𝔽₅)	—	2	2	no	—
19	C₂×AGL(1,𝔽₅)	—	2	2	no	—
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅	—	2	1	no	—
23	C₂×Dic₅	D₄	1	1	yes	characteristic
24	C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₄×D₅	C₂×C₄	1	1	yes	characteristic
26	C₄×D₅	D₄	1	1	yes	characteristic
27	C₄×D₅	Q₈	1	1	yes	characteristic
28	#170	—	2	2	no	—
29	C₂×M₁₆	—	5	1	no	—
30	#117	—	5	1	no	—
31	C₂×C₄²	—	5	1	no	—
32	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×D₅	—	2	1	no	—
34	C₂×D₅	—	2	1	no	—
35	C₂×D₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
37	AGL(1,𝔽₅)	—	2	4	no	—
38	C₄×C₅	—	4	1	no	—
39	C₄×C₅	C₄²	1	1	yes	characteristic
40	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
41	Dic₅	—	4	1	no	—
42	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
43	Dic₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄	—	5	1	no	—
45	C₂²×C₄	—	10	1	no	—
46	M₁₆	—	5	2	no	—
47	M₁₆	—	10	1	no	—
48	C₂×C₈	—	10	1	no	—
49	C₄⋊C₄	—	5	1	no	—
50	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
51	C₂²⋊C₄	—	10	1	no	—
52	C₄²	—	5	2	no	—
53	C₄²	—	10	1	no	—
54	C₄²	—	10	1	no	—
55	C₂×C₅	#95	1	1	yes	characteristic
56	C₂×C₅	#104	1	1	yes	characteristic
57	C₂×C₅	#104	1	1	yes	characteristic
58	D₅	—	2	1	no	—
59	D₅	—	2	1	no	—
60	C₂³	—	5	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	5	1	no	—
67	C₂×C₄	—	10	2	no	—
68	C₂×C₄	—	10	2	no	—
69	C₂×C₄	—	10	2	no	—
70	C₂×C₄	—	10	1	no	—
71	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₈	—	10	2	no	—
73	C₅	#339	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	5	1	no	—
75	C₂²	—	5	1	no	—
76	C₂²	—	10	1	no	—
77	C₂²	—	10	1	no	—
78	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	—	5	1	no	—
81	C₄	—	5	1	no	—
82	C₄	—	10	4	no	—
83	C₄	—	20	1	no	—
84	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₂	—	10	1	no	—
87	C₂	—	10	1	no	—
88	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
89	C₂	#4008	1	1	yes	characteristic
90	C₂	#4010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊C₄	C₄
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#339
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. C₄×AGL(1,𝔽₅)	C₄	semidirect
10. #700	C₄	semidirect
11. C₅×C₄⋊C₄	C₄	semidirect
13. C₄⋊Dic₅	C₄	semidirect
50. C₄⋊C₄	AGL(1,𝔽₅)	semidirect
73. C₅	#339	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×AGL(1,𝔽₅)	2
C₂×#700	2
#4038	2
#339	5