Subgroups — C₂×C₅×S₅ #11925733

Order 1200.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×A₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×S₅	C₂	1	2	yes	maximal, direct product factor
3	C₅×A₅	C₂²	1	1	yes	characteristic, complemented
4	C₂×C₅×S₄	—	5	1	no	maximal
5	C₂×S₅	C₅	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₅×AGL(1,𝔽₅)	—	6	1	no	maximal
7	C₂²×C₅×D₃	—	10	1	no	maximal
8	C₂×C₅×A₄	—	5	1	no	—
9	C₅×S₄	—	5	2	no	—
10	C₂×A₅	C₂×C₅	1	1	yes	characteristic, complemented
11	S₅	C₂×C₅	1	2	yes	direct product factor
12	C₂×C₅×D₅	—	6	1	no	—
13	C₅×AGL(1,𝔽₅)	—	6	2	no	—
14	C₂×C₅×D₄	—	15	1	no	—
15	C₂²×C₃×C₅	—	10	1	no	—
16	C₂×C₅×D₃	—	10	1	no	—
17	C₂×C₅×D₃	—	10	2	no	—
18	C₂×C₅×D₃	—	10	1	no	—
19	C₂×C₅×D₃	—	10	2	no	—
20	C₅×A₄	—	5	1	no	—
21	A₅	C₂²×C₅	1	1	yes	characteristic, complemented
22	C₂×C₅²	—	6	1	no	—
23	C₅×D₅	—	6	1	no	—
24	C₅×D₅	—	6	1	no	—
25	C₂×S₄	—	5	1	no	Hall
26	C₂³×C₅	—	5	1	no	—
27	C₂³×C₅	—	15	1	no	—
28	C₂×AGL(1,𝔽₅)	—	6	1	no	—
29	C₅×D₄	—	15	2	no	—
30	C₅×D₄	—	15	2	no	—
31	C₂×C₄×C₅	—	15	1	no	—
32	C₂×C₃×C₅	—	10	2	no	—
33	C₂×C₃×C₅	—	10	1	no	—
34	C₅×D₃	—	10	2	no	—
35	C₅×D₃	—	10	1	no	—
36	C₅×D₃	—	10	1	no	—
37	C₅²	—	6	1	no	Sylow
38	C₂²×D₃	—	10	1	no	—
39	C₂×A₄	—	5	1	no	—
40	S₄	—	5	2	no	—
41	C₂²×C₅	—	5	1	no	—
42	C₂²×C₅	—	10	1	no	—
43	C₂²×C₅	—	15	2	no	—
44	C₂²×C₅	—	15	1	no	—
45	C₂²×C₅	—	15	1	no	—
46	C₂²×C₅	—	30	1	no	—
47	C₂×D₅	—	6	1	no	—
48	AGL(1,𝔽₅)	—	6	2	no	—
49	C₄×C₅	—	15	2	no	—
50	C₂×D₄	—	15	1	no	Sylow
51	C₃×C₅	—	10	1	no	—
52	C₂²×C₃	—	10	1	no	—
53	C₂×D₃	—	10	1	no	—
54	C₂×D₃	—	10	2	no	—
55	C₂×D₃	—	10	1	no	—
56	C₂×D₃	—	10	2	no	—
57	A₄	—	5	1	no	—
58	C₂×C₅	—	6	1	no	—
59	C₂×C₅	—	10	2	no	—
60	C₂×C₅	—	15	1	no	—
61	C₂×C₅	—	15	1	no	—
62	C₂×C₅	—	24	1	no	—
63	C₂×C₅	S₅	1	1	yes	characteristic, direct product factor
64	D₅	—	6	1	no	—
65	D₅	—	6	1	no	—
66	C₂³	—	5	1	no	—
67	C₂³	—	15	1	no	—
68	D₄	—	15	2	no	—
69	D₄	—	15	2	no	—
70	C₂×C₄	—	15	1	no	—
71	C₂×C₃	—	10	2	no	—
72	C₂×C₃	—	10	1	no	—
73	D₃	—	10	2	no	—
74	D₃	—	10	1	no	—
75	D₃	—	10	1	no	—
76	C₅	—	6	1	no	—
77	C₅	—	24	1	no	—
78	C₅	C₂×S₅	1	1	yes	characteristic, direct product factor
79	C₂²	—	5	1	no	—
80	C₂²	—	10	1	no	—
81	C₂²	—	15	2	no	—
82	C₂²	—	15	1	no	—
83	C₂²	—	15	1	no	—
84	C₂²	—	30	1	no	—
85	C₄	—	15	2	no	—
86	C₃	—	10	1	no	Sylow
87	C₂	—	10	2	no	—
88	C₂	—	15	1	no	—
89	C₂	—	15	1	no	—
90	C₂	C₅×S₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, Fitting, hypercenter	C₂×C₅	S₅
commutator, cosocle, hypocenter	A₅	C₂²×C₅
Frattini	C₁	G
socle	C₂×C₅×A₅	C₂
LCS	G ▷ A₅
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×A₅	C₂	semidirect
2. C₅×S₅	C₂	direct (N × H)
3. C₅×A₅	C₂²	semidirect
5. C₂×S₅	C₅	direct (N × H)
10. C₂×A₅	C₂×C₅	semidirect
11. S₅	C₂×C₅	direct (N × H)
21. A₅	C₂²×C₅	semidirect
63. C₂×C₅	S₅	direct (N × H)
78. C₅	C₂×S₅	direct (N × H)
90. C₂	C₅×S₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×A₅	2
C₅×S₅	2
C₂×C₅×S₄	5
C₂×S₅	5
C₂×C₅×AGL(1,𝔽₅)	6
C₂²×C₅×D₃	10

Subgroups — C2×C5×S5 #11925733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×S₅ #11925733