Group #13105015 — subgroups

Subgroups — #13105015

Order 1600.

560 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#4016	C₂	1	2	yes	maximal, complemented
3	#11717018	C₂	1	2	yes	maximal
4	#11717001	C₂	1	4	yes	maximal
5	#11716991	C₂	1	2	yes	maximal, complemented
6	#11716989	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
8	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
9	C₂×#6036	C₂²	1	4	yes	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₄×#1039	C₂²	1	2	yes	—
12	C₄×#1039	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊Dic₅	C₂²	1	4	yes	—
14	C₅×#683	C₂²	1	8	yes	—
15	#87781	C₂²	1	4	yes	—
16	#87780	C₂²	1	8	yes	—
17	D₅×#125	—	5	2	no	maximal
18	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	2	yes	—
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₂×#1039	C₂³	1	2	yes	—
22	C₂×#1039	C₂³	1	1	yes	characteristic
23	C₄×C₅⋊D₅	—	2	4	no	—
24	C₄×C₅⋊D₅	Q₈	1	4	yes	—
25	C₂×C₅×Dic₅	C₂³	1	8	yes	—
26	#6036	—	2	8	no	—
27	C₅×#125	—	5	2	no	—
28	D₅×C₄⋊C₄	—	5	4	no	—
29	D₅×C₄⋊C₄	—	5	8	no	—
30	#4036	—	5	4	no	—
31	#4035	—	5	8	no	—
32	C₄²×D₅	—	5	2	no	—
33	C₄²×D₅	—	10	2	no	—
34	#4016	D₅	1	2	yes	complemented
35	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
36	C₂×C₅⋊D₅	C₂×Q₈	1	2	yes	—
37	C₂×C₅⋊D₅	#41	1	4	yes	—
38	C₄×C₅²	—	2	2	no	—
39	C₄×C₅²	C₂×Q₈	1	2	yes	—
40	#1039	—	2	2	no	—
41	#1039	C₂×Q₈	1	2	yes	—
42	C₅×Dic₅	—	2	8	no	—
43	C₂×C₄×D₅	—	5	2	no	—
44	C₂×C₄×D₅	—	5	4	no	—
45	C₂×C₄×D₅	—	5	8	no	—
46	C₂×C₄×D₅	—	10	2	no	—
47	C₂×C₄×D₅	—	10	4	no	—
48	C₅×C₄⋊C₄	—	5	8	no	—
49	C₅×C₄⋊C₄	—	5	4	no	—
50	C₄²×C₅	—	2	2	no	—
51	C₄²×C₅	C₂×D₅	1	2	yes	—
52	C₄⋊Dic₅	—	5	8	no	—
53	C₄⋊Dic₅	C₂×D₅	1	4	yes	—
54	#683	—	5	8	no	—
55	#683	—	5	8	no	—
56	#683	C₂×D₅	1	8	yes	—
57	C₄×Dic₅	—	5	4	no	—
58	C₄×Dic₅	—	5	2	no	—
59	C₄×Dic₅	—	10	4	no	—
60	C₄×Dic₅	—	10	2	no	—
61	C₂×#125	—	25	1	no	Sylow
62	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₅²	C₂×#41	1	2	yes	—
64	C₅⋊D₅	#125	1	4	yes	complemented
65	C₂²×D₅	—	5	2	no	—
66	C₂²×D₅	—	10	2	no	—
67	C₂×C₄×C₅	—	2	4	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	5	8	no	—
70	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
71	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
72	C₂×Dic₅	—	5	2	no	—
73	C₂×Dic₅	—	5	4	no	—
74	C₂×Dic₅	—	10	2	no	—
75	C₂×Dic₅	—	10	4	no	—
76	C₂×Dic₅	C₂²×D₅	1	8	yes	—
77	C₄×D₅	—	5	8	no	—
78	C₄×D₅	—	10	8	no	—
79	C₄×D₅	—	10	16	no	—
80	C₄×D₅	—	10	8	no	—
81	C₄×D₅	—	10	16	no	—
82	#125	—	5	2	no	—
83	#125	—	25	4	no	—
84	#125	—	25	2	no	—
85	C₂×C₄⋊C₄	—	25	4	no	—
86	C₂×C₄⋊C₄	—	25	2	no	—
87	C₂×C₄²	—	25	1	no	—
88	C₅²	C₂×#125	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₅	—	2	2	no	—
90	C₂²×C₅	C₂³×D₅	1	2	yes	—
91	C₂×D₅	—	5	4	no	—
92	C₂×D₅	—	5	8	no	—
93	C₂×D₅	—	10	4	no	—
94	C₂×D₅	—	10	8	no	—
95	C₄×C₅	—	2	8	no	—
96	C₄×C₅	—	2	4	no	—
97	C₄×C₅	—	2	4	no	—
98	C₄×C₅	—	10	8	no	—
99	C₄×C₅	Q₈×D₅	1	4	yes	—
100	Dic₅	—	2	8	no	—
101	Dic₅	—	5	4	no	—
102	Dic₅	—	10	4	no	—
103	Dic₅	—	10	8	no	—
104	Dic₅	—	10	4	no	—
105	C₂²×C₄	—	25	4	no	—
106	C₂²×C₄	—	25	1	no	—
107	C₂²×C₄	—	25	2	no	—
108	C₄⋊C₄	—	5	8	no	—
109	C₄⋊C₄	—	5	4	no	—
110	C₄⋊C₄	—	25	4	no	—
111	C₄⋊C₄	—	25	8	no	—
112	C₄²	—	25	2	no	—
113	C₄²	—	25	1	no	—
114	C₄²	D₅²	1	1	yes	characteristic
115	C₂×C₅	—	2	2	no	—
116	C₂×C₅	—	2	4	no	—
117	C₂×C₅	C₂×Q₈×D₅	1	2	yes	—
118	C₂×C₅	D₅×#41	1	4	yes	—
119	D₅	—	5	8	no	—
120	D₅	—	10	8	no	—
121	C₂³	—	25	1	no	—
122	C₂×C₄	—	5	8	no	—
123	C₂×C₄	—	25	1	no	—
124	C₂×C₄	—	25	4	no	—
125	C₂×C₄	—	25	2	no	—
126	C₂×C₄	—	50	4	no	—
127	C₂×C₄	—	50	8	no	—
128	C₂×C₄	C₂×D₅²	1	2	yes	—
129	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
130	C₅	—	2	2	no	—
131	C₅	D₅×#125	1	2	yes	complemented
132	C₂²	—	25	2	no	—
133	C₂²	—	25	4	no	—
134	C₂²	C₂²×D₅²	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	10	8	no	—
137	C₄	—	25	2	no	—
138	C₄	—	50	2	no	—
139	C₄	#87872	1	2	yes	—
140	C₂	—	25	4	no	—
141	C₂	C₂×#87872	1	1	yes	characteristic
142	C₂	#11717471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₅⋊D₅

C₂

yes

maximal, characteristic

C₅×#4016

C₂

yes

maximal, complemented

#11717018

C₂

yes

maximal

#11717001

C₂

yes

maximal

#11716991

C₂

yes

maximal, complemented

#11716989

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×#6036

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

C₂²

yes

—

#87781

C₂²

yes

—

#87780

C₂²

yes

—

D₅×#125

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

characteristic

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

Q₈

yes

—

C₂×C₅×Dic₅

C₂³

yes

—

#6036

—

C₅×#125

—

D₅×C₄⋊C₄

—

D₅×C₄⋊C₄

—

#4036

—

#4035

—

C₄²×D₅

—

C₄²×D₅

—

#4016

D₅

yes

complemented

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

C₂×Q₈

yes

—

C₂×C₅⋊D₅

#41

yes

—

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

#1039

—

#1039

C₂×Q₈

yes

—

C₅×Dic₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

—

#683

C₂×D₅

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#125

—

Sylow

C₂×C₅²

C₂²×Q₈

yes

characteristic

C₂×C₅²

C₂×#41

yes

—

C₅⋊D₅

#125

yes

complemented

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂²×D₅

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₅²

C₂×#125

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

Q₈×D₅

yes

—

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

Dic₅

—

104

Dic₅

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

D₅²

yes

characteristic

115

C₂×C₅

—

116

C₂×C₅

—

117

C₂×C₅

C₂×Q₈×D₅

yes

—

118

C₂×C₅

D₅×#41

yes

—

119

D₅

—

120

D₅

—

121

C₂³

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×D₅²

yes

—

129

C₂×C₄

C₂×D₅²

yes

characteristic

130

C₅

—

131

C₅

D₅×#125

yes

complemented

132

C₂²

—

133

C₂²

—

134

C₂²

C₂²×D₅²

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#87872

yes

—

140

C₂

—

141

C₂

C₂×#87872

yes

characteristic

142

C₂

#11717471

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
prime radical	C₂²×C₅⋊D₅	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	C₂×#125
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×D₅²

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₄²×C₅²

C₂²

prime radical

C₂²×C₅⋊D₅

C₂³

hypercenter

C₄²

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₅×#4016	C₂	semidirect
5. #11716991	C₂	semidirect
6. #11716989	C₂	semidirect
34. #4016	D₅	semidirect
64. C₅⋊D₅	#125	semidirect
88. C₅²	C₂×#125	semidirect
131. C₅	D₅×#125	semidirect

H ≅ G/N

type

2. C₅×#4016

C₂

semidirect

5. #11716991

C₂

semidirect

6. #11716989

C₂

semidirect

34. #4016

D₅

semidirect

64. C₅⋊D₅

#125

semidirect

88. C₅²

C₂×#125

semidirect

131. C₅

D₅×#125

semidirect

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

subgroup	index
C₄²×C₅⋊D₅	2
C₅×#4016	2
#11717018	2
#11717001	2
#11716991	2
#11716989	2
D₅×#125	5

subgroup

index

C₄²×C₅⋊D₅

Subgroups — #13105015

Conjugacy classes of subgroups

Complements

Maximal subgroups