Subgroups — C₅×#459 #65622

Order 320.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₈	C₂	1	2	yes	maximal, complemented
2	C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#102	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×D₄	—	2	1	no	—
7	C₂×C₅×D₄	—	2	1	no	—
8	C₂×C₅×D₄	C₂²	1	2	yes	—
9	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅	—	2	1	no	—
11	C₅×D₈	—	2	4	no	—
12	C₂×C₅×C₈	C₂²	1	2	yes	—
13	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₅×C₂²⋊C₄	—	2	1	no	—
15	C₄²×C₅	C₂²	1	1	yes	characteristic
16	#459	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₅	—	2	1	no	—
18	C₂³×C₅	—	2	2	no	—
19	C₅×D₄	—	2	2	no	—
20	C₅×D₄	—	2	1	no	—
21	C₅×D₄	—	2	2	no	—
22	C₅×D₄	—	4	2	no	—
23	C₅×D₄	—	4	2	no	—
24	C₅×D₄	D₄	1	2	yes	complemented
25	C₂×C₄×C₅	—	2	1	no	—
26	C₂×C₄×C₅	—	4	2	no	—
27	C₂×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₅×C₈	—	2	2	no	—
31	C₂×D₈	C₂×C₅	1	2	yes	complemented
32	#127	C₂×C₅	1	1	yes	characteristic, complemented
33	C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
34	C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
35	#102	C₂×C₅	1	2	yes	complemented
36	C₂²×C₅	—	2	2	no	—
37	C₂²×C₅	—	4	1	no	—
38	C₂²×C₅	—	4	1	no	—
39	C₂²×C₅	—	4	2	no	—
40	C₂²×C₅	—	4	2	no	—
41	C₂²×C₅	—	4	2	no	—
42	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄×C₅	—	2	1	no	—
44	C₄×C₅	—	4	1	no	—
45	C₄×C₅	D₈	1	2	yes	complemented
46	C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₄×C₅	#41	1	1	yes	characteristic
48	C₂×D₄	—	2	1	no	—
49	C₂×D₄	—	2	1	no	—
50	C₂×D₄	C₂²×C₅	1	2	yes	—
51	C₂×D₄	C₂²×C₅	1	1	yes	characteristic
52	C₂²×C₄	—	2	1	no	—
53	D₈	—	2	4	no	—
54	C₂×C₈	C₂²×C₅	1	2	yes	—
55	C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
56	C₂²⋊C₄	—	2	1	no	—
57	C₄²	C₂²×C₅	1	1	yes	characteristic
58	C₂×C₅	—	4	2	no	—
59	C₂×C₅	—	8	2	no	—
60	C₂×C₅	#121	1	1	yes	characteristic
61	C₂×C₅	C₂×D₈	1	1	yes	characteristic
62	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂³	—	2	1	no	—
64	C₂³	—	2	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	2	1	no	—
67	D₄	—	2	2	no	—
68	D₄	—	4	2	no	—
69	D₄	—	4	2	no	—
70	D₄	C₅×D₄	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₅×D₄	1	1	yes	characteristic
74	C₂×C₄	C₅×D₄	1	1	yes	characteristic
75	C₂×C₄	C₂³×C₅	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₅	#459	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂×C₅×D₄	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	C₅×D₈	1	2	yes	complemented
88	C₄	C₂×C₅×D₄	1	1	yes	characteristic
89	C₄	C₅×#41	1	1	yes	characteristic
90	C₂	—	4	2	no	—
91	C₂	—	8	2	no	—
92	C₂	C₅×#121	1	1	yes	characteristic
93	C₂	C₂×C₅×D₈	1	1	yes	characteristic
94	C₂	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₈	C₂	semidirect
2. C₅×#127	C₂	semidirect
3. C₄×C₅×D₄	C₂	semidirect
4. C₅×C₄⋊C₈	C₂	semidirect
5. C₅×#102	C₂	semidirect
16. #459	C₅	direct (N × H)
24. C₅×D₄	D₄	semidirect
31. C₂×D₈	C₂×C₅	semidirect
32. #127	C₂×C₅	semidirect
33. C₄×D₄	C₂×C₅	semidirect
34. C₄⋊C₈	C₂×C₅	semidirect
35. #102	C₂×C₅	semidirect
45. C₄×C₅	D₈	semidirect
70. D₄	C₅×D₄	semidirect
77. C₅	#459	direct (N × H)
87. C₄	C₅×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₈	2
C₅×#127	2
C₄×C₅×D₄	2
C₅×C₄⋊C₈	2
C₅×#102	2
#459	5

Subgroups — C5×#459 #65622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#459 #65622