Group #23771983 — subgroups

Subgroups — C₂×#19098603 #23771983

Order 10112.

536 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 269 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×#11929608	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579120	C₂	1	1	yes	maximal, characteristic
3	#19098603	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14579094	C₂	1	2	yes	maximal, complemented
5	C₂×#14579092	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#11929608	C₄	1	8	yes	complemented
8	C₂×#11929608	C₂²	1	4	yes	—
9	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
10	#14579120	—	2	2	no	—
11	#14579120	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₇₉	—	2	1	no	—
15	C₂×C₄×Dic₇₉	—	2	1	no	—
16	#14579094	C₂²	1	8	yes	complemented
17	#14579092	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₇₉	—	2	1	no	—
19	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
24	C₂²×Dic₇₉	—	2	1	no	—
25	C₂²×Dic₇₉	—	2	1	no	—
26	#11929608	C₂×C₄	1	16	yes	complemented
27	C₂×C₇₉⋊C₈	C₂×C₄	1	8	yes	complemented
28	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	C₂³	1	2	yes	—
32	C₄⋊Dic₇₉	C₂³	1	2	yes	—
33	C₂²⋊Dic₇₉	—	2	4	no	—
34	C₄×Dic₇₉	—	2	4	no	—
35	C₂⁴×C₇₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	—	2	4	no	—
37	C₂²×C₄×C₇₉	—	2	2	no	—
38	C₂²×C₄×C₇₉	—	4	4	no	—
39	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	D₄	1	2	yes	—
42	C₂²×C₄×C₇₉	Q₈	1	4	yes	—
43	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
45	C₄⋊C₄×C₇₉	—	2	4	no	—
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
48	C₂³×C₇₉	—	2	2	no	—
49	C₂³×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	C₄⋊C₄	1	4	yes	—
51	C₂³×C₇₉	C₂×D₄	1	2	yes	—
52	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	2	4	no	—
54	C₂×C₄×C₇₉	—	4	8	no	—
55	C₂×C₄×C₇₉	—	4	4	no	—
56	C₂×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
57	C₂×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₇₉	—	2	2	no	—
66	C₂×Dic₇₉	—	2	2	no	—
67	C₂×Dic₇₉	—	4	2	no	—
68	C₂×Dic₇₉	—	4	2	no	—
69	C₇₉⋊C₈	C₂²×C₄	1	8	yes	complemented
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	—	2	2	no	—
72	C₂²×C₇₉	—	2	8	no	—
73	C₂²×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
76	C₂²×C₇₉	#137	1	2	yes	—
77	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₇₉	—	4	4	no	—
79	C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
83	Dic₇₉	—	4	2	no	—
84	Dic₇₉	—	4	2	no	—
85	C₂×C₇₉	—	2	4	no	—
86	C₂×C₇₉	#428	1	4	yes	complemented
87	C₂×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
90	C₂×#428	—	79	1	no	maximal, Sylow
91	C₇₉	C₂×#428	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	—	79	1	no	—
93	C₂×#117	—	79	1	no	—
94	C₂²×C₄⋊C₄	D₇₉	1	1	yes	characteristic
95	#428	—	79	8	no	—
96	C₂×#107	—	79	2	no	—
97	C₂×#106	—	79	2	no	—
98	C₂³×C₄	—	2	1	no	—
99	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
100	C₂×M₁₆	—	79	4	no	—
101	C₂×M₁₆	—	79	8	no	—
102	C₂²×C₈	—	79	2	no	—
103	#117	—	79	4	no	—
104	#117	—	158	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	79	1	no	—
108	C₂×C₄⋊C₄	—	79	1	no	—
109	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
111	C₂×C₂²⋊C₄	—	158	1	no	—
112	C₂×C₄²	—	158	1	no	—
113	#107	—	79	8	no	—
114	#106	—	79	8	no	—
115	C₂⁴	C₂²⋊D₇₉	1	1	yes	characteristic
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	158	1	no	—
120	C₂²×C₄	—	158	1	no	—
121	C₂²×C₄	Dic₁₅₈	1	4	yes	—
122	C₂²×C₄	C₄×D₇₉	1	4	yes	—
123	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
126	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
127	M₁₆	—	79	16	no	—
128	C₂×C₈	—	79	8	no	—
129	C₂×C₈	—	79	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	79	2	no	—
133	C₄⋊C₄	—	79	2	no	—
134	C₄⋊C₄	—	158	1	no	—
135	C₄⋊C₄	—	158	1	no	—
136	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
137	C₂²⋊C₄	—	158	4	no	—
138	C₄²	—	158	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	Dic₇₉⋊C₄	1	4	yes	—
142	C₂³	C₂×C₂²⋊D₇₉	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	316	2	no	—
148	C₂×C₄	—	316	2	no	—
149	C₂×C₄	—	158	2	no	—
150	C₂×C₄	—	158	2	no	—
151	C₂×C₄	Dic₇₉⋊C₄	1	8	yes	—
152	C₂×C₄	Dic₇₉⋊C₄	1	4	yes	—
153	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₇₉	1	4	yes	—
155	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
158	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
159	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
160	C₈	—	79	8	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×Dic₇₉⋊C₄	1	4	yes	—
165	C₂²	C₂×Dic₇₉⋊C₄	1	2	yes	—
166	C₂²	#14579187	1	2	yes	—
167	C₂²	#14579192	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
169	C₄	—	4	4	no	—
170	C₄	—	316	2	no	—
171	C₄	—	316	2	no	—
172	C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
173	C₄	C₂×Dic₇₉⋊C₄	1	4	yes	—
174	C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
175	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	#19098603	1	4	yes	direct product factor
178	C₂	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
179	C₂	C₂×#14579187	1	1	yes	characteristic
180	C₂	C₂×#14579192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929608

C₂

yes

maximal, characteristic

C₂×#14579120

C₂

yes

maximal, characteristic

#19098603

C₂

yes

maximal, direct product factor

C₂×#14579094

C₂

yes

maximal, complemented

C₂×#14579092

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₄

yes

complemented

C₂×#11929608

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579120

—

#14579120

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₂²⋊Dic₇₉

—

C₂×C₄×Dic₇₉

—

#14579094

C₂²

yes

complemented

#14579092

C₂²

yes

complemented

C₂³×C₄×C₇₉

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

#11929608

C₂×C₄

yes

complemented

C₂×C₇₉⋊C₈

C₂×C₄

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂⁴×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

C₂²×C₄

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

—

C₂²×C₇₉

AGL(1,ℤ/8)

yes

—

C₂²×C₇₉

#137

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₄×C₇₉

C₂³×C₄

yes

characteristic

C₄×C₇₉

C₂²×Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#428

yes

complemented

C₂×C₇₉

C₂²×C₄⋊C₄

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#117

—

C₂²×C₄⋊C₄

D₇₉

yes

—

—

—

C₂³×C₄

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

#117

—

104

#117

—

158

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

110

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

111

C₂×C₂²⋊C₄

—

158

—

112

C₂×C₄²

—

158

—

113

#107

—

114

#106

—

115

C₂⁴

C₂²⋊D₇₉

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

158

—

120

C₂²×C₄

—

158

—

121

C₂²×C₄

Dic₁₅₈

yes

—

122

C₂²×C₄

C₄×D₇₉

yes

—

123

C₂²×C₄

C₂²⋊D₇₉

yes

—

124

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

125

C₂²×C₄

C₂²×D₇₉

yes

—

126

C₂²×C₄

C₂²×D₇₉

yes

characteristic

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

—

158

—

136

C₄⋊C₄

C₂²×D₇₉

yes

—

137

C₂²⋊C₄

—

158

—

138

C₄²

—

158

—

139

C₂³

—

140

C₂³

—

141

C₂³

Dic₇₉⋊C₄

yes

—

142

C₂³

C₂×C₂²⋊D₇₉

yes

—

143

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

316

—

148

C₂×C₄

—

316

—

149

C₂×C₄

—

158

—

150

C₂×C₄

—

158

—

151

C₂×C₄

Dic₇₉⋊C₄

yes

—

152

C₂×C₄

Dic₇₉⋊C₄

yes

—

153

C₂×C₄

C₂×C₄×D₇₉

yes

—

154

C₂×C₄

C₂×C₄×D₇₉

yes

—

155

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

156

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

157

C₂×C₄

C₂×Dic₁₅₈

yes

—

158

C₂×C₄

C₂×Dic₁₅₈

yes

—

159

C₂×C₄

C₂³×D₇₉

yes

characteristic

160

C₈

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂×Dic₇₉⋊C₄

yes

—

165

C₂²

C₂×Dic₇₉⋊C₄

yes

—

166

C₂²

#14579187

yes

—

167

C₂²

#14579192

yes

—

168

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

169

C₄

—

170

C₄

—

316

—

171

C₄

—

316

—

172

C₄

C₂×Dic₇₉⋊C₄

yes

—

173

C₄

C₂×Dic₇₉⋊C₄

yes

—

174

C₄

C₂²×C₄×D₇₉

yes

characteristic

175

C₄

C₂²×Dic₁₅₈

yes

characteristic

176

C₂

—

177

C₂

#19098603

yes

direct product factor

178

C₂

C₂²×Dic₇₉⋊C₄

yes

characteristic

179

C₂

C₂×#14579187

yes

characteristic

180

C₂

C₂×#14579192

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂²×C₄⋊C₄×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂⁴×C₇₉	D₄
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂×#428
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₄×C₇₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂²×C₄⋊C₄×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂⁴×C₇₉

D₄

cosocle

C₂×C₄×C₇₉

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₇₉

hypocenter

C₇₉

C₂×#428

LCS

G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #19098603	C₂	direct (N × H)
4. C₂×#14579094	C₂	semidirect
5. C₂×#14579092	C₂	semidirect
7. C₂×#11929608	C₄	semidirect
16. #14579094	C₂²	semidirect
17. #14579092	C₂²	semidirect
26. #11929608	C₂×C₄	semidirect
27. C₂×C₇₉⋊C₈	C₂×C₄	semidirect
69. C₇₉⋊C₈	C₂²×C₄	semidirect
86. C₂×C₇₉	#428	semidirect
91. C₇₉	C₂×#428	semidirect
177. C₂	#19098603	direct (N × H)

H ≅ G/N

type

3. #19098603

C₂

direct (N × H)

4. C₂×#14579094

C₂

semidirect

5. C₂×#14579092

C₂

semidirect

7. C₂×#11929608

C₄

semidirect

16. #14579094

C₂²

semidirect

17. #14579092

C₂²

semidirect

26. #11929608

C₂×C₄

semidirect

27. C₂×C₇₉⋊C₈

C₂×C₄

semidirect

69. C₇₉⋊C₈

C₂²×C₄

semidirect

86. C₂×C₇₉

#428

semidirect

91. C₇₉

C₂×#428

semidirect

177. C₂

#19098603

direct (N × H)

Subgroups — C₂×#19098603 #23771983

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098603 #23771983

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098603 #23771983