Group #13100564 — subgroups

Subgroups — #13100564

Order 1600.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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	#11717195	C₂	1	1	yes	maximal, characteristic
2	C₅×#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87776	C₂	1	2	yes	maximal, complemented
4	C₂×#87774	C₂	1	2	yes	maximal
5	C₅×#3935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
7	C₂×#6045	C₂²	1	1	yes	characteristic
8	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	#87808	—	2	2	no	—
11	C₅×C₄⋊Dic₅	—	2	2	no	—
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	#87776	—	2	4	no	—
14	#87774	—	2	4	no	—
15	#65375	—	5	1	no	maximal
16	#65004	—	5	1	no	maximal
17	C₂²×C₅⋊D₅	—	2	1	no	—
18	C₂×C₄×C₅²	D₄	1	2	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	—	2	1	no	—
21	C₄⋊(C₅⋊D₅)	—	2	2	no	—
22	#6045	—	2	2	no	—
23	C₂×C₅×Dic₅	—	2	2	no	—
24	C₅×Dic₁₀	—	2	2	no	—
25	C₅×Dic₁₀	—	2	2	no	—
26	C₅×#168	D₄	1	4	yes	—
27	C₅×C₄⋊Q₈	—	5	1	no	—
28	C₂×#689	—	5	2	no	—
29	C₂×#688	—	5	2	no	—
30	C₂×Dic₂₀	—	5	2	no	—
31	C₂×#677	—	5	2	no	—
32	#4021	—	5	1	no	—
33	#4021	—	5	1	no	—
34	#4021	—	10	2	no	—
35	#4015	D₅	1	1	yes	characteristic, complemented
36	C₅×#97	—	5	1	no	—
37	#3935	D₅	1	1	yes	characteristic, complemented
38	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₂×C₅⋊D₅	—	4	1	no	—
40	C₂×C₅⋊D₅	—	4	2	no	—
41	C₄×C₅²	—	2	2	no	—
42	C₄×C₅²	C₂×D₄	1	2	yes	—
43	#1039	—	4	1	no	—
44	C₅×Dic₅	—	4	2	no	—
45	C₂×C₅×Q₈	—	5	2	no	—
46	C₂×D₂₀	—	5	1	no	—
47	C₂×D₂₀	—	5	1	no	—
48	C₂×D₂₀	—	10	2	no	—
49	C₂×Dic₁₀	—	5	1	no	—
50	C₂×Dic₁₀	—	5	1	no	—
51	C₂×Dic₁₀	—	10	2	no	—
52	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
53	C₂×C₅×C₈	—	5	2	no	—
54	C₅×C₄⋊C₄	—	10	2	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
57	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
58	#689	—	10	4	no	—
59	#688	—	10	4	no	—
60	#685	—	10	8	no	—
61	#685	—	10	2	no	—
62	#685	—	10	2	no	—
63	C₄⋊Dic₅	—	2	2	no	—
64	C₂×#168	C₂×D₅	1	2	yes	—
65	Dic₂₀	—	10	4	no	—
66	#677	—	10	4	no	—
67	#497	—	25	1	no	Sylow
68	C₂×C₅²	#127	1	1	yes	characteristic
69	C₂×C₅²	#137	1	2	yes	—
70	C₅⋊D₅	—	8	1	no	—
71	C₂²×D₅	—	10	4	no	—
72	C₂²×D₅	—	10	1	no	—
73	C₂²×D₅	—	10	1	no	—
74	C₅×Q₈	—	10	2	no	—
75	C₅×Q₈	—	10	2	no	—
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	10	2	no	—
79	C₂×C₄×C₅	D₂₀	1	2	yes	—
80	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
81	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
83	C₂×Dic₅	—	2	2	no	—
84	C₂×Dic₅	—	10	4	no	—
85	C₂×Dic₅	—	10	1	no	—
86	C₂×Dic₅	—	10	1	no	—
87	D₂₀	—	10	2	no	—
88	D₂₀	—	10	2	no	—
89	D₂₀	—	20	4	no	—
90	Dic₁₀	—	2	2	no	—
91	Dic₁₀	—	2	2	no	—
92	Dic₁₀	—	10	2	no	—
93	Dic₁₀	—	10	2	no	—
94	Dic₁₀	—	20	4	no	—
95	C₅×C₈	—	5	4	no	—
96	#168	D₂₀	1	4	yes	—
97	C₂×Q₁₆	—	25	2	no	—
98	C₂×SD₁₆	—	25	2	no	—
99	C₄⋊Q₈	—	5	1	no	—
100	#124	—	25	1	no	—
101	#97	—	5	1	no	—
102	C₅²	#497	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
105	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
106	C₂×D₅	—	20	4	no	—
107	C₂×D₅	—	20	8	no	—
108	C₂×D₅	—	20	1	no	—
109	C₂×D₅	—	20	2	no	—
110	C₂×D₅	—	20	1	no	—
111	C₂×D₅	—	20	2	no	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	4	4	no	—
116	C₄×C₅	—	20	2	no	—
117	C₄×C₅	C₂×D₂₀	1	2	yes	—
118	C₄×C₅	D₄×D₅	1	2	yes	—
119	Dic₅	—	4	2	no	—
120	Dic₅	—	20	4	no	—
121	Dic₅	—	20	1	no	—
122	Dic₅	—	20	1	no	—
123	C₂×Q₈	—	5	2	no	—
124	C₂×Q₈	—	25	1	no	—
125	C₂×D₄	—	25	1	no	—
126	Q₁₆	—	50	4	no	—
127	SD₁₆	—	50	4	no	—
128	C₂×C₈	—	5	2	no	—
129	C₄⋊C₄	—	10	2	no	—
130	C₂²⋊C₄	—	50	2	no	—
131	C₄²	D₅²	1	1	yes	characteristic
132	C₂×C₅	—	2	4	no	—
133	C₂×C₅	—	2	2	no	—
134	C₂×C₅	#4020	1	1	yes	characteristic
135	C₂×C₅	#4055	1	2	yes	—
136	C₂×C₅	#4086	1	1	yes	characteristic
137	C₂×C₅	#4088	1	2	yes	—
138	D₅	—	40	4	no	—
139	D₅	—	40	1	no	—
140	D₅	—	40	1	no	—
141	C₂³	—	50	1	no	—
142	Q₈	—	10	2	no	—
143	Q₈	—	10	2	no	—
144	Q₈	—	50	2	no	—
145	D₄	—	50	2	no	—
146	C₂×C₄	—	10	2	no	—
147	C₂×C₄	—	50	1	no	—
148	C₂×C₄	#6038	1	2	yes	—
149	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
150	C₈	—	5	4	no	—
151	C₅	—	2	2	no	—
152	C₅	#65004	1	1	yes	characteristic, complemented
153	C₅	#65375	1	1	yes	characteristic, complemented
154	C₂²	—	100	2	no	—
155	C₂²	—	100	1	no	—
156	C₂²	C₂×#6038	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	20	2	no	—
159	C₄	—	100	1	no	—
160	C₄	D₅×D₂₀	1	2	yes	—
161	C₂	—	200	1	no	—
162	C₂	#11716951	1	2	yes	—
163	C₂	#11717029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717195

C₂

yes

maximal, characteristic

C₅×#4015

C₂

yes

maximal, characteristic, complemented

C₂×#87776

C₂

yes

maximal, complemented

C₂×#87774

C₂

yes

maximal

C₅×#3935

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₂×C₅×Dic₁₀

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

#87808

—

C₅×C₄⋊Dic₅

—

C₂×C₅×#168

C₂²

yes

—

—

—

—

—

C₂²×C₅⋊D₅

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

#6045

—

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

—

C₅×#168

D₄

yes

—

C₅×C₄⋊Q₈

—

C₂×#689

—

C₂×#688

—

C₂×Dic₂₀

—

—

—

—

—

D₅

yes

characteristic, complemented

C₅×#97

—

#3935

D₅

yes

characteristic, complemented

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

characteristic

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

C₄⋊Dic₅

—

C₂×#168

C₂×D₅

yes

—

Dic₂₀

—

—

—

C₂×C₅²

yes

C₂×C₅²

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

D₂₀

yes

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

100

#124

—

101

#97

—

102

C₅²

#497

yes

characteristic, complemented, Sylow

103

C₂²×C₅

—

104

C₂²×C₅

C₂×D₂₀

yes

characteristic

105

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

106

C₂×D₅

—

107

C₂×D₅

—

108

C₂×D₅

—

109

C₂×D₅

—

110

C₂×D₅

—

111

C₂×D₅

—

112

C₄×C₅

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

C₂×D₂₀

yes

—

118

C₄×C₅

D₄×D₅

yes

—

119

Dic₅

—

120

Dic₅

—

121

Dic₅

—

122

Dic₅

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

Q₁₆

—

127

SD₁₆

—

128

C₂×C₈

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

D₅²

yes

characteristic

132

C₂×C₅

—

133

C₂×C₅

—

134

C₂×C₅

#4020

yes

characteristic

135

C₂×C₅

#4055

yes

—

136

C₂×C₅

#4086

yes

characteristic

137

C₂×C₅

#4088

yes

—

138

D₅

—

139

D₅

—

140

D₅

—

141

C₂³

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#6038

yes

—

149

C₂×C₄

C₂×D₅²

yes

characteristic

150

C₈

—

151

C₅

—

152

C₅

#65004

yes

characteristic, complemented

153

C₅

#65375

yes

characteristic, complemented

154

C₂²

—

100

—

155

C₂²

—

100

—

156

C₂²

C₂×#6038

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

100

—

160

C₄

D₅×D₂₀

yes

—

161

C₂

—

200

—

162

C₂

#11716951

yes

—

163

C₂

#11717029

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6038
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₄²×C₅²	C₂²
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₄²	D₅²
hypocenter	C₅²	#497
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6038

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₄²×C₅²

C₂²

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₄⋊(C₅⋊D₅)

C₂²

hypercenter

C₄²

D₅²

hypocenter

C₅²

#497

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₅×#4015	C₂	semidirect
3. C₂×#87776	C₂	semidirect
5. C₅×#3935	C₂	semidirect
35. #4015	D₅	semidirect
37. #3935	D₅	semidirect
102. C₅²	#497	semidirect
152. C₅	#65004	semidirect
153. C₅	#65375	semidirect

H ≅ G/N

type

2. C₅×#4015

C₂

semidirect

3. C₂×#87776

C₂

semidirect

5. C₅×#3935

C₂

semidirect

35. #4015

D₅

semidirect

37. #3935

D₅

semidirect

102. C₅²

#497

semidirect

152. C₅

#65004

semidirect

153. C₅

#65375

semidirect

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

subgroup	index
#11717195	2
C₅×#4015	2
C₂×#87776	2
C₂×#87774	2
C₅×#3935	2
#65375	5
#65004	5

subgroup

index

Subgroups — #13100564

Conjugacy classes of subgroups

Complements

Maximal subgroups