Group #72277324 — subgroups

Subgroups — #72277324

Order 2400.

186 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#6732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊#6732	C₂	1	2	yes	maximal, complemented
3	C₅×SL(2,𝔽₅)	C₂²	1	1	yes	characteristic, complemented
4	C₅⋊#994	—	5	1	no	maximal
5	#91883	—	5	1	no	maximal
6	#87874	—	6	1	no	maximal
7	C₅×#41⋊C₃	—	5	1	no	—
8	#6769	—	10	1	no	maximal
9	C₅⋊(#41⋊C₃)	—	5	2	no	—
10	#6732	—	5	2	no	—
11	#6732	D₅	1	1	yes	characteristic, complemented
12	C₄⋊(C₅⋊D₅)	—	6	1	no	—
13	C₄×C₅×D₅	—	6	1	no	—
14	C₅×Dic₁₀	—	6	1	no	—
15	#6038	—	6	2	no	—
16	#6036	—	6	2	no	—
17	#4149	—	5	1	no	—
18	D₆₀	—	10	1	no	—
19	C₄×C₅×D₃	—	10	1	no	—
20	C₃×Dic₁₀	—	10	1	no	—
21	C₅×SL(2,𝔽₃)	—	5	1	no	—
22	#1203	—	10	2	no	—
23	#1200	—	10	2	no	—
24	SL(2,𝔽₅)	C₂×D₅	1	1	yes	characteristic, complemented
25	C₂×C₅⋊D₅	—	6	2	no	—
26	C₂×C₅×D₅	—	6	1	no	—
27	C₄×C₅²	—	6	1	no	—
28	C₅×Dic₅	—	6	1	no	—
29	C₅×Dic₅	—	6	2	no	—
30	#994	—	25	1	no	Hall
31	C₅×#41	—	5	1	no	—
32	#713	—	5	2	no	—
33	#713	—	30	1	no	—
34	D₄×D₅	—	15	2	no	—
35	#709	—	15	1	no	—
36	C₂×D₂₀	—	15	1	no	—
37	C₂×D₁₅	—	10	2	no	—
38	C₂×C₅×D₃	—	10	1	no	—
39	C₃×C₄×C₅	—	10	1	no	—
40	C₃×Dic₅	—	10	2	no	—
41	C₅×Dic₃	—	10	1	no	—
42	C₂×C₅²	—	6	1	no	—
43	C₅⋊D₅	—	12	2	no	—
44	C₅×D₅	—	12	1	no	—
45	#239	—	50	1	no	—
46	#41⋊C₃	—	5	1	no	—
47	#41⋊C₃	—	25	2	no	—
48	C₂²×D₅	—	15	2	no	—
49	C₅×Q₈	—	5	1	no	—
50	C₅×Q₈	—	30	1	no	—
51	C₅×D₄	—	15	1	no	—
52	C₂×C₄×C₅	—	15	1	no	—
53	C₂²⋊D₅	—	15	2	no	—
54	D₂₀	—	15	2	no	—
55	D₂₀	—	15	1	no	—
56	D₂₀	—	30	1	no	—
57	D₂₀	—	30	2	no	—
58	D₂₀	—	60	2	no	—
59	C₄×D₅	—	6	1	no	—
60	C₄×D₅	—	15	2	no	—
61	C₄×D₅	—	30	2	no	—
62	Dic₁₀	A₅	1	1	yes	maximal normal, characteristic
63	H(5,𝔽₂)	—	25	1	no	Sylow
64	C₂×C₃×C₅	—	10	1	no	—
65	D₁₅	—	20	2	no	—
66	C₅×D₃	—	20	1	no	—
67	C₅²	—	6	1	no	Sylow
68	C₃×Q₈	—	50	1	no	—
69	D₁₂	—	50	1	no	—
70	D₁₂	—	50	2	no	—
71	C₄×D₃	—	10	1	no	—
72	C₄×D₃	—	50	2	no	—
73	SL(2,𝔽₃)	—	5	1	no	—
74	C₂²×C₅	—	15	1	no	—
75	C₂×D₅	—	6	1	no	—
76	C₂×D₅	—	15	2	no	—
77	C₂×D₅	—	30	2	no	—
78	C₂×D₅	—	60	2	no	—
79	C₂×D₅	—	60	4	no	—
80	C₄×C₅	—	6	1	no	—
81	C₄×C₅	—	12	2	no	—
82	C₄×C₅	—	15	1	no	—
83	C₄×C₅	—	30	2	no	—
84	C₄×C₅	C₂×A₅	1	1	yes	characteristic
85	Dic₅	—	6	1	no	—
86	Dic₅	C₂×A₅	1	2	yes	—
87	#41	—	5	1	no	—
88	#41	—	25	2	no	—
89	#41	—	75	1	no	—
90	C₂×D₄	—	75	1	no	—
91	C₂×D₄	—	75	2	no	—
92	C₃×C₅	—	10	1	no	—
93	C₂×D₃	—	10	1	no	—
94	C₂×D₃	—	50	2	no	—
95	C₃×C₄	—	10	1	no	—
96	C₃×C₄	—	50	2	no	—
97	Dic₃	—	10	1	no	—
98	C₂×C₅	—	6	1	no	—
99	C₂×C₅	—	12	2	no	—
100	C₂×C₅	—	30	1	no	—
101	C₂×C₅	C₂²×A₅	1	1	yes	characteristic
102	D₅	—	12	1	no	—
103	D₅	—	30	2	no	—
104	D₅	—	60	2	no	—
105	D₅	—	120	4	no	—
106	C₂³	—	75	2	no	—
107	Q₈	—	5	1	no	—
108	Q₈	—	5	1	no	—
109	D₄	—	15	1	no	—
110	D₄	—	75	2	no	—
111	D₄	—	75	2	no	—
112	D₄	—	75	1	no	—
113	C₂×C₄	—	15	1	no	—
114	C₂×C₄	—	75	2	no	—
115	C₂×C₃	—	10	1	no	—
116	D₃	—	20	1	no	—
117	D₃	—	100	2	no	—
118	C₅	—	6	1	no	—
119	C₅	—	12	2	no	—
120	C₅	#91883	1	1	yes	characteristic, complemented
121	C₂²	—	15	1	no	—
122	C₂²	—	300	2	no	—
123	C₂²	—	75	2	no	—
124	C₄	—	5	2	no	—
125	C₄	—	15	1	no	—
126	C₄	D₅×A₅	1	1	yes	characteristic
127	C₃	—	10	1	no	Sylow
128	C₂	—	30	1	no	—
129	C₂	—	150	2	no	—
130	C₂	C₂×D₅×A₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#6732

C₂

yes

maximal, characteristic, complemented

C₅⋊#6732

C₂

yes

maximal, complemented

C₅×SL(2,𝔽₅)

C₂²

yes

characteristic, complemented

—

—

—

C₅×#41⋊C₃

—

#6769

—

maximal

C₅⋊(#41⋊C₃)

—

#6732

—

#6732

D₅

yes

characteristic, complemented

C₄⋊(C₅⋊D₅)

—

C₄×C₅×D₅

—

C₅×Dic₁₀

—

#6038

—

#6036

—

#4149

—

D₆₀

—

C₄×C₅×D₃

—

C₃×Dic₁₀

—

C₅×SL(2,𝔽₃)

—

#1203

—

#1200

—

SL(2,𝔽₅)

C₂×D₅

yes

characteristic, complemented

C₂×C₅⋊D₅

—

C₂×C₅×D₅

—

C₄×C₅²

—

C₅×Dic₅

—

C₅×Dic₅

—

#994

—

Hall

C₅×#41

—

#713

—

#713

—

D₄×D₅

—

#709

—

C₂×D₂₀

—

C₂×D₁₅

—

C₂×C₅×D₃

—

C₃×C₄×C₅

—

C₃×Dic₅

—

C₅×Dic₃

—

C₂×C₅²

—

C₅⋊D₅

—

C₅×D₅

—

#239

—

#41⋊C₃

—

#41⋊C₃

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂²⋊D₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

A₅

yes

maximal normal, characteristic

H(5,𝔽₂)

—

Sylow

C₂×C₃×C₅

—

D₁₅

—

C₅×D₃

—

C₅²

—

Sylow

C₃×Q₈

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

SL(2,𝔽₃)

—

C₂²×C₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×A₅

yes

characteristic

Dic₅

—

Dic₅

C₂×A₅

yes

—

#41

—

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₃×C₅

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

C₂×C₅

—

C₂×C₅

—

100

C₂×C₅

—

101

C₂×C₅

C₂²×A₅

yes

characteristic

102

D₅

—

103

D₅

—

104

D₅

—

105

D₅

—

120

—

106

C₂³

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₃

—

116

D₃

—

117

D₃

—

100

—

118

C₅

—

119

C₅

—

120

C₅

#91883

yes

characteristic, complemented

121

C₂²

—

122

C₂²

—

300

—

123

C₂²

—

124

C₄

—

125

C₄

—

126

C₄

D₅×A₅

yes

characteristic

127

C₃

—

Sylow

128

C₂

—

129

C₂

—

150

—

130

C₂

C₂×D₅×A₅

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×A₅
commutator, hypocenter	C₅×SL(2,𝔽₅)	C₂²
Fitting	C₄×C₅	C₂×A₅
socle, cosocle	C₂×C₅	C₂²×A₅
hypercenter	C₄	D₅×A₅
LCS	G ▷ C₅×SL(2,𝔽₅)
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center, Frattini

C₂

C₂×D₅×A₅

commutator, hypocenter

C₅×SL(2,𝔽₅)

C₂²

Fitting

C₄×C₅

C₂×A₅

socle, cosocle

C₂×C₅

C₂²×A₅

hypercenter

C₄

D₅×A₅

LCS

G ▷ C₅×SL(2,𝔽₅)

UCS

C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₅×#6732	C₂	semidirect
2. C₅⋊#6732	C₂	semidirect
3. C₅×SL(2,𝔽₅)	C₂²	semidirect
11. #6732	D₅	semidirect
24. SL(2,𝔽₅)	C₂×D₅	semidirect
120. C₅	#91883	semidirect

H ≅ G/N

type

1. C₅×#6732

C₂

semidirect

2. C₅⋊#6732

C₂

semidirect

3. C₅×SL(2,𝔽₅)

C₂²

semidirect

11. #6732

D₅

semidirect

24. SL(2,𝔽₅)

C₂×D₅

semidirect

120. C₅

#91883

semidirect

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

subgroup	index
C₅×#6732	2
C₅⋊#6732	2
C₅⋊#994	5
#91883	5
#87874	6
#6769	10

subgroup

index

Subgroups — #72277324

Conjugacy classes of subgroups

Complements

Maximal subgroups