Group #13102919 — subgroups

Subgroups — #13102919

Order 1600.

520 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₅²×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87811	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#87809	C₂	1	1	yes	maximal, characteristic
6	#11717195	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6029	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	C₅²×C₂²⋊C₄	—	2	2	no	—
13	C₄²×C₅²	C₂²	1	1	yes	characteristic
14	#87812	—	2	2	no	—
15	#87811	—	2	2	no	—
16	#87810	—	2	2	no	—
17	#87809	—	2	2	no	—
18	#87808	—	2	2	no	—
19	C₂×#6029	C₂²	1	1	yes	characteristic
20	C₂×#6029	C₂²	1	1	yes	characteristic
21	#65348	—	5	6	no	maximal
22	C₂³×C₅²	—	2	1	no	—
23	C₂²×C₅⋊D₅	—	2	1	no	—
24	C₅²×Q₈	—	2	1	no	—
25	C₅²×Q₈	—	2	1	no	—
26	C₅²×D₄	—	2	1	no	—
27	C₅²×D₄	—	2	1	no	—
28	C₂×C₄×C₅²	—	2	1	no	—
29	C₂×C₄×C₅²	D₄	1	2	yes	—
30	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
31	C₂×#1039	—	2	1	no	—
32	C₄⋊(C₅⋊D₅)	—	2	1	no	—
33	C₄⋊(C₅⋊D₅)	—	2	1	no	—
34	#6045	—	2	1	no	—
35	#6045	—	2	1	no	—
36	#6029	D₄	1	2	yes	—
37	#6029	D₄	1	2	yes	—
38	C₅×#124	D₅	1	6	yes	complemented
39	C₂×#689	—	5	6	no	—
40	C₂×#688	—	5	6	no	—
41	C₂×#687	—	5	6	no	—
42	C₂×#686	—	5	6	no	—
43	#4021	—	5	6	no	—
44	C₄×#168	—	5	6	no	—
45	C₂²×C₅²	—	4	1	no	—
46	C₂²×C₅²	—	4	2	no	—
47	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₅⋊D₅	—	4	1	no	—
49	C₂×C₅⋊D₅	—	4	2	no	—
50	C₄×C₅²	—	2	2	no	—
51	C₄×C₅²	—	4	1	no	—
52	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
53	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
54	#1039	—	4	1	no	—
55	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
56	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
57	C₂×D₂₀	—	5	6	no	—
58	C₂×Dic₁₀	—	5	6	no	—
59	C₅×C₂²⋊C₄	—	2	12	no	—
60	C₄²×C₅	C₂×D₅	1	6	yes	—
61	#689	—	10	12	no	—
62	#688	—	10	12	no	—
63	#687	—	10	12	no	—
64	#686	—	10	12	no	—
65	#685	—	10	12	no	—
66	C₂×#168	—	5	6	no	—
67	C₂×#168	—	5	6	no	—
68	#495	—	25	1	no	Sylow
69	C₂×C₅²	—	8	1	no	—
70	C₂×C₅²	#127	1	1	yes	characteristic
71	C₂×C₅²	#135	1	2	yes	—
72	C₅⋊D₅	—	8	1	no	—
73	C₂³×C₅	—	2	6	no	—
74	C₂²×D₅	—	10	6	no	—
75	C₅×Q₈	—	2	6	no	—
76	C₅×Q₈	—	2	6	no	—
77	C₅×D₄	—	2	6	no	—
78	C₅×D₄	—	2	6	no	—
79	C₂×C₄×C₅	—	2	6	no	—
80	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
81	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
82	C₂×Dic₅	—	10	6	no	—
83	D₂₀	—	10	6	no	—
84	D₂₀	—	10	6	no	—
85	Dic₁₀	—	10	6	no	—
86	Dic₁₀	—	10	6	no	—
87	#168	—	5	12	no	—
88	#168	—	5	12	no	—
89	C₂×Q₁₆	—	25	1	no	—
90	C₂×SD₁₆	—	25	1	no	—
91	C₂×SD₁₆	—	25	1	no	—
92	C₂×D₈	—	25	1	no	—
93	#124	—	25	1	no	—
94	#124	C₅⋊D₅	1	1	yes	characteristic, complemented
95	C₄×C₈	—	25	1	no	—
96	C₅²	#495	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₅	—	4	12	no	—
98	C₂²×C₅	—	4	6	no	—
99	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
100	C₂×D₅	—	20	12	no	—
101	C₂×D₅	—	20	6	no	—
102	C₄×C₅	—	2	12	no	—
103	C₄×C₅	—	4	6	no	—
104	C₄×C₅	D₄×D₅	1	6	yes	—
105	C₄×C₅	D₄×D₅	1	6	yes	—
106	Dic₅	—	20	6	no	—
107	C₂×Q₈	—	25	1	no	—
108	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
109	C₂×D₄	—	25	1	no	—
110	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
111	Q₁₆	—	50	2	no	—
112	SD₁₆	—	50	2	no	—
113	SD₁₆	—	50	2	no	—
114	D₈	—	50	2	no	—
115	C₂×C₈	—	25	1	no	—
116	C₂×C₈	—	25	1	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	50	2	no	—
119	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
120	C₂×C₅	—	8	6	no	—
121	C₂×C₅	#4086	1	6	yes	—
122	C₂×C₅	#4096	1	12	yes	—
123	D₅	—	40	6	no	—
124	C₂³	—	2	1	no	—
125	C₂³	—	50	1	no	—
126	Q₈	—	2	1	no	—
127	Q₈	—	2	1	no	—
128	Q₈	—	50	1	no	—
129	Q₈	—	50	1	no	—
130	D₄	—	2	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	50	1	no	—
133	D₄	—	50	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	50	1	no	—
136	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
137	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
138	C₈	—	25	2	no	—
139	C₈	—	25	2	no	—
140	C₅	#65348	1	6	yes	complemented
141	C₂²	—	4	1	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	100	1	no	—
144	C₂²	—	100	2	no	—
145	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	4	1	no	—
148	C₄	—	100	1	no	—
149	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
150	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	—	200	1	no	—
153	C₂	#11717260	1	1	yes	characteristic
154	C₂	#11717270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#124

C₂

yes

maximal, characteristic, complemented

C₂×#87812

C₂

yes

maximal, characteristic, complemented

C₂×#87811

C₂

yes

maximal, characteristic, complemented

C₂×#87810

C₂

yes

maximal, characteristic, complemented

C₂×#87809

C₂

yes

maximal, characteristic

#11717195

C₂

yes

maximal, characteristic, complemented

C₄×#6029

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×C₂²⋊C₄

—

C₄²×C₅²

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#65348

—

maximal

C₂³×C₅²

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

#6045

—

#6045

—

#6029

D₄

yes

—

#6029

D₄

yes

—

C₅×#124

D₅

yes

—

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

#127

yes

characteristic

C₂×C₅²

#135

yes

—

C₅⋊D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

Dic₁₀

—

#168

—

#168

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#124

—

#124

C₅⋊D₅

yes

characteristic, complemented

C₄×C₈

—

C₅²

#495

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

100

C₂×D₅

—

101

C₂×D₅

—

102

C₄×C₅

—

103

C₄×C₅

—

104

C₄×C₅

D₄×D₅

yes

—

105

C₄×C₅

D₄×D₅

yes

—

106

Dic₅

—

107

C₂×Q₈

—

108

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

111

Q₁₆

—

112

SD₁₆

—

113

SD₁₆

—

114

D₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₂×C₅⋊D₅

yes

characteristic

120

C₂×C₅

—

121

C₂×C₅

#4086

yes

—

122

C₂×C₅

#4096

yes

—

123

D₅

—

124

C₂³

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

137

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

138

C₈

—

139

C₈

—

140

C₅

#65348

yes

complemented

141

C₂²

—

142

C₂²

—

143

C₂²

—

100

—

144

C₂²

—

100

—

145

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

100

—

149

C₄

D₄×C₅⋊D₅

yes

characteristic

150

C₄

D₄×C₅⋊D₅

yes

characteristic

151

C₂

—

152

C₂

—

200

—

153

C₂

#11717260

yes

characteristic

154

C₂

#11717270

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#124	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×#87809	C₂
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	#495
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#124

C₂

socle

C₂²×C₅²

C₂×D₄

C₂

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
1. C₅²×#124	C₂	semidirect
2. C₂×#87812	C₂	semidirect
3. C₂×#87811	C₂	semidirect
4. C₂×#87810	C₂	semidirect
6. #11717195	C₂	semidirect
7. C₄×#6029	C₂	semidirect
38. C₅×#124	D₅	semidirect
94. #124	C₅⋊D₅	semidirect
96. C₅²	#495	semidirect
140. C₅	#65348	semidirect

H ≅ G/N

type

1. C₅²×#124

C₂

semidirect

2. C₂×#87812

C₂

semidirect

3. C₂×#87811

C₂

semidirect

4. C₂×#87810

C₂

semidirect

6. #11717195

C₂

semidirect

7. C₄×#6029

C₂

semidirect

38. C₅×#124

D₅

semidirect

94. #124

C₅⋊D₅

semidirect

96. C₅²

#495

semidirect

140. C₅

#65348

semidirect

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

subgroup	index
C₅²×#124	2
C₂×#87812	2
C₂×#87811	2
C₂×#87810	2
C₂×#87809	2
#11717195	2
C₄×#6029	2
#65348	5

subgroup

index

C₅²×#124

Subgroups — #13102919

Conjugacy classes of subgroups

Complements

Maximal subgroups