Group #13107327 — subgroups

Subgroups — C₂×#11717252 #13107327

Order 1600.

1192 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 291 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	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87900	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87810	C₂	1	2	yes	maximal, complemented
4	#11717252	C₂	1	8	yes	maximal, direct product factor
5	C₂×#87809	C₂	1	2	yes	maximal, complemented
6	C₂×#87804	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₅²	—	2	1	no	—
8	C₂×C₅²×D₄	—	2	1	no	—
9	C₂×C₅²×D₄	—	2	2	no	—
10	C₂×C₅²×D₄	C₂²	1	2	yes	—
11	C₂×C₅²×D₄	C₂²	1	4	yes	complemented
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₅⋊D₅)	—	2	1	no	—
14	#87900	—	2	2	no	—
15	#87900	C₂²	1	4	yes	complemented
16	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×C₅⋊D₅	—	2	1	no	—
18	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
19	#87810	C₂²	1	8	yes	complemented
20	#87809	C₂²	1	8	yes	complemented
21	#87804	C₂²	1	4	yes	complemented
22	C₂×#6029	C₂²	1	2	yes	complemented
23	C₂×#4078	—	5	6	no	maximal
24	C₂³×C₅²	—	2	2	no	—
25	C₂³×C₅²	—	2	4	no	—
26	C₂³×C₅²	—	4	4	no	—
27	C₂³×C₅²	D₄	1	1	yes	characteristic
28	C₂²×C₅⋊D₅	—	2	1	no	—
29	C₅²×D₄	—	2	2	no	—
30	C₅²×D₄	—	2	4	no	—
31	C₅²×D₄	C₂³	1	4	yes	—
32	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
33	C₂×C₄×C₅²	D₄	1	2	yes	—
34	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅²	C₂³	1	2	yes	—
36	C₂²⋊(C₅⋊D₅)	—	2	4	no	—
37	C₂×#1039	—	2	1	no	—
38	C₄⋊(C₅⋊D₅)	—	2	1	no	—
39	C₄⋊(C₅⋊D₅)	C₂³	1	2	yes	complemented
40	C₄×C₅⋊D₅	—	2	4	no	—
41	#6045	—	2	1	no	—
42	#6045	C₂³	1	2	yes	complemented
43	#6029	C₂³	1	4	yes	complemented
44	C₂²×C₅×D₄	D₅	1	6	yes	complemented
45	C₂×#709	—	5	6	no	—
46	C₂×#687	—	5	12	no	—
47	#4078	—	5	48	no	—
48	C₂×#686	—	5	12	no	—
49	C₂×#681	—	5	6	no	—
50	C₂²×C₅²	—	2	2	no	—
51	C₂²×C₅²	—	2	4	no	—
52	C₂²×C₅²	—	4	8	no	—
53	C₂²×C₅²	—	4	4	no	—
54	C₂²×C₅²	C₂×D₄	1	2	yes	—
55	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅⋊D₅	—	2	2	no	—
57	C₂×C₅⋊D₅	—	4	2	no	—
58	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
59	C₄×C₅²	C₂×D₄	1	2	yes	—
60	C₄×C₅²	C₂⁴	1	1	yes	characteristic
61	#1039	—	2	2	no	—
62	C₂⁴×C₅	—	2	6	no	—
63	C₂×C₅×D₄	—	2	6	no	—
64	C₂×C₅×D₄	—	2	12	no	—
65	C₂×C₅×D₄	C₂×D₅	1	12	yes	—
66	C₂×C₅×D₄	C₂×D₅	1	24	yes	complemented
67	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
68	C₂×C₂²⋊D₅	—	10	6	no	—
69	#709	—	5	24	no	—
70	#709	—	10	12	no	—
71	C₂×D₂₀	—	5	6	no	—
72	C₂×C₄×D₅	—	10	6	no	—
73	C₂×Dic₁₀	—	5	6	no	—
74	#687	—	5	48	no	—
75	#686	—	5	48	no	—
76	#681	—	5	24	no	—
77	C₂×#168	—	5	12	no	—
78	C₂×AGL(1,ℤ/8)	—	25	1	no	Sylow
79	C₂×C₅²	—	2	2	no	—
80	C₂×C₅²	—	4	4	no	—
81	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	complemented
82	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
83	C₅⋊D₅	—	4	2	no	—
84	C₂³×C₅	—	2	12	no	—
85	C₂³×C₅	—	2	24	no	—
86	C₂³×C₅	—	4	24	no	—
87	C₂³×C₅	C₂²⋊D₅	1	6	yes	—
88	C₂²×D₅	—	10	6	no	—
89	C₅×D₄	—	2	12	no	—
90	C₅×D₄	—	2	24	no	—
91	C₅×D₄	C₂²×D₅	1	24	yes	—
92	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
93	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
94	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
95	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
96	C₂²⋊D₅	—	10	24	no	—
97	C₂×Dic₅	—	10	6	no	—
98	D₂₀	—	5	12	no	—
99	D₂₀	—	10	6	no	—
100	C₄×D₅	—	10	24	no	—
101	Dic₁₀	—	5	12	no	—
102	Dic₁₀	—	10	6	no	—
103	#168	—	5	24	no	—
104	C₂×#41	—	25	1	no	—
105	C₂²×D₄	C₅⋊D₅	1	1	yes	characteristic, complemented
106	AGL(1,ℤ/8)	—	25	8	no	—
107	C₂×SD₁₆	—	25	2	no	—
108	C₂×D₈	—	25	2	no	—
109	C₂×M₁₆	—	25	1	no	—
110	C₅²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₅	—	2	12	no	—
112	C₂²×C₅	—	2	24	no	—
113	C₂²×C₅	—	4	48	no	—
114	C₂²×C₅	—	4	24	no	—
115	C₂²×C₅	C₂×C₂²⋊D₅	1	12	yes	—
116	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
117	C₂×D₅	—	10	12	no	—
118	C₂×D₅	—	20	12	no	—
119	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
120	C₄×C₅	C₂×C₂²⋊D₅	1	12	yes	—
121	C₄×C₅	C₂³×D₅	1	6	yes	—
122	Dic₅	—	10	12	no	—
123	C₂⁴	—	2	1	no	—
124	#41	—	25	4	no	—
125	#41	—	50	2	no	—
126	C₂×Q₈	—	25	1	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	—	25	1	no	—
130	C₂×D₄	—	50	1	no	—
131	C₂×D₄	C₂×C₅⋊D₅	1	2	yes	—
132	C₂×D₄	C₂×C₅⋊D₅	1	4	yes	complemented
133	C₂²×C₄	—	50	1	no	—
134	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
135	SD₁₆	—	25	8	no	—
136	D₈	—	25	8	no	—
137	M₁₆	—	25	4	no	—
138	C₂×C₈	—	25	2	no	—
139	C₂×C₅	—	2	12	no	—
140	C₂×C₅	—	4	24	no	—
141	C₂×C₅	#4078	1	12	yes	complemented
142	C₂×C₅	C₂²×C₂²⋊D₅	1	6	yes	—
143	D₅	—	20	12	no	—
144	C₂³	—	2	4	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	4	4	no	—
147	C₂³	—	50	1	no	—
148	C₂³	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
149	Q₈	—	25	2	no	—
150	Q₈	—	50	1	no	—
151	D₄	—	2	2	no	—
152	D₄	—	2	4	no	—
153	D₄	—	25	2	no	—
154	D₄	—	50	4	no	—
155	D₄	—	50	1	no	—
156	D₄	C₂²×C₅⋊D₅	1	4	yes	—
157	C₂×C₄	—	50	1	no	—
158	C₂×C₄	—	50	4	no	—
159	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
160	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
162	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
163	C₈	—	25	4	no	—
164	C₅	C₂×#4078	1	6	yes	complemented
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	4	8	no	—
168	C₂²	—	4	4	no	—
169	C₂²	—	50	2	no	—
170	C₂²	—	100	2	no	—
171	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
172	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
173	C₄	—	50	2	no	—
174	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
175	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
176	C₄	C₂³×C₅⋊D₅	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	4	4	no	—
179	C₂	—	100	2	no	—
180	C₂	#11717252	1	2	yes	direct product factor
181	C₂	C₂²×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅²×D₄

C₂

yes

maximal, characteristic, complemented

C₂×#87900

C₂

yes

maximal, characteristic, complemented

C₂×#87810

C₂

yes

maximal, complemented

#11717252

C₂

yes

maximal, direct product factor

C₂×#87809

C₂

yes

maximal, complemented

C₂×#87804

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₅²

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

C₂²

yes

—

C₂×C₅²×D₄

C₂²

yes

complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂×C₂²⋊(C₅⋊D₅)

—

#87900

—

#87900

C₂²

yes

complemented

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic, complemented

C₂×C₄×C₅⋊D₅

—

C₂×#6045

C₂²

yes

characteristic, complemented

#87810

C₂²

yes

complemented

#87809

C₂²

yes

complemented

#87804

C₂²

yes

complemented

C₂×#6029

C₂²

yes

complemented

C₂×#4078

—

maximal

C₂³×C₅²

—

C₂³×C₅²

—

C₂³×C₅²

—

C₂³×C₅²

D₄

yes

characteristic

C₂²×C₅⋊D₅

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

C₂³

yes

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

C₂³

yes

complemented

C₄×C₅⋊D₅

—

#6045

—

#6045

C₂³

yes

complemented

#6029

C₂³

yes

complemented

C₂²×C₅×D₄

D₅

yes

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

—

C₄×C₅²

C₂⁴

yes

characteristic

#1039

—

C₂⁴×C₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

complemented

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂×C₂²⋊D₅

—

#709

—

#709

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

AGL(1,ℤ/8)

yes

complemented

C₂×C₅²

C₂²×D₄

yes

characteristic

C₅⋊D₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂²×D₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

100

C₄×D₅

—

101

Dic₁₀

—

102

Dic₁₀

—

103

#168

—

104

C₂×#41

—

105

C₂²×D₄

C₅⋊D₅

yes

characteristic, complemented

106

AGL(1,ℤ/8)

—

107

C₂×SD₁₆

—

108

C₂×D₈

—

109

C₂×M₁₆

—

110

C₅²

C₂×AGL(1,ℤ/8)

yes

characteristic, complemented, Sylow

111

C₂²×C₅

—

112

C₂²×C₅

—

113

C₂²×C₅

—

114

C₂²×C₅

—

115

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

116

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

117

C₂×D₅

—

118

C₂×D₅

—

119

C₄×C₅

C₂×C₂²⋊D₅

yes

—

120

C₄×C₅

C₂×C₂²⋊D₅

yes

—

121

C₄×C₅

C₂³×D₅

yes

—

122

Dic₅

—

123

C₂⁴

—

124

#41

—

125

#41

—

126

C₂×Q₈

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

C₂×C₅⋊D₅

yes

—

132

C₂×D₄

C₂×C₅⋊D₅

yes

complemented

133

C₂²×C₄

—

134

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

135

SD₁₆

—

136

D₈

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₅

—

140

C₂×C₅

—

141

C₂×C₅

#4078

yes

complemented

142

C₂×C₅

C₂²×C₂²⋊D₅

yes

—

143

D₅

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

C₂²⋊(C₅⋊D₅)

yes

characteristic

149

Q₈

—

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

C₂²×C₅⋊D₅

yes

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

160

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

161

C₂×C₄

C₂²×C₅⋊D₅

yes

—

162

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

163

C₈

—

164

C₅

C₂×#4078

yes

complemented

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

100

—

171

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

—

172

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

173

C₄

—

174

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

—

175

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

176

C₄

C₂³×C₅⋊D₅

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

—

100

—

180

C₂

#11717252

yes

direct product factor

181

C₂

C₂²×C₂²⋊(C₅⋊D₅)

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₄×C₅²	C₂⁴
Frattini	C₄	C₂³×C₅⋊D₅
Fitting	C₂²×C₅²×D₄	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂²×D₄	C₅⋊D₅
hypocenter	C₅²	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₄×C₅²

C₂⁴

Frattini

C₄

C₂³×C₅⋊D₅

Fitting

C₂²×C₅²×D₄

C₂

socle

C₂²×C₅²

C₂×D₄

hypercenter

C₂²×D₄

C₅⋊D₅

hypocenter

C₅²

C₂×AGL(1,ℤ/8)

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂²×C₅²×D₄	C₂	semidirect
2. C₂×#87900	C₂	semidirect
3. C₂×#87810	C₂	semidirect
4. #11717252	C₂	direct (N × H)
5. C₂×#87809	C₂	semidirect
6. C₂×#87804	C₂	semidirect
11. C₂×C₅²×D₄	C₂²	semidirect
15. #87900	C₂²	semidirect
16. C₂×C₄⋊(C₅⋊D₅)	C₂²	semidirect
18. C₂×#6045	C₂²	semidirect
19. #87810	C₂²	semidirect
20. #87809	C₂²	semidirect
21. #87804	C₂²	semidirect
22. C₂×#6029	C₂²	semidirect
39. C₄⋊(C₅⋊D₅)	C₂³	semidirect
42. #6045	C₂³	semidirect
43. #6029	C₂³	semidirect
44. C₂²×C₅×D₄	D₅	semidirect
66. C₂×C₅×D₄	C₂×D₅	semidirect
81. C₂×C₅²	AGL(1,ℤ/8)	semidirect
105. C₂²×D₄	C₅⋊D₅	semidirect
110. C₅²	C₂×AGL(1,ℤ/8)	semidirect
132. C₂×D₄	C₂×C₅⋊D₅	semidirect
141. C₂×C₅	#4078	semidirect
164. C₅	C₂×#4078	semidirect
180. C₂	#11717252	direct (N × H)

H ≅ G/N

type

1. C₂²×C₅²×D₄

C₂

semidirect

2. C₂×#87900

C₂

semidirect

3. C₂×#87810

C₂

semidirect

4. #11717252

C₂

direct (N × H)

5. C₂×#87809

C₂

semidirect

6. C₂×#87804

C₂

semidirect

11. C₂×C₅²×D₄

C₂²

semidirect

15. #87900

C₂²

semidirect

16. C₂×C₄⋊(C₅⋊D₅)

C₂²

semidirect

18. C₂×#6045

C₂²

semidirect

19. #87810

C₂²

semidirect

20. #87809

C₂²

semidirect

21. #87804

C₂²

semidirect

22. C₂×#6029

C₂²

semidirect

39. C₄⋊(C₅⋊D₅)

C₂³

semidirect

42. #6045

C₂³

semidirect

43. #6029

C₂³

semidirect

44. C₂²×C₅×D₄

D₅

semidirect

66. C₂×C₅×D₄

C₂×D₅

semidirect

81. C₂×C₅²

AGL(1,ℤ/8)

semidirect

105. C₂²×D₄

C₅⋊D₅

semidirect

110. C₅²

C₂×AGL(1,ℤ/8)

semidirect

132. C₂×D₄

C₂×C₅⋊D₅

semidirect

141. C₂×C₅

#4078

semidirect

164. C₅

C₂×#4078

semidirect

180. C₂

#11717252

direct (N × H)

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

subgroup	index
C₂²×C₅²×D₄	2
C₂×#87900	2
C₂×#87810	2
#11717252	2
C₂×#87809	2
C₂×#87804	2
C₂×#4078	5

subgroup

index

C₂²×C₅²×D₄

Subgroups — C₂×#11717252 #13107327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11717252 #13107327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11717252 #13107327