Group #35157488 — subgroups

Subgroups — C₂²×#19098931 #35157488

Order 20224.

2142 conjugacy classes of subgroups; only the 747 normal subgroups are shown.

normal subgroups only

This group has 2142 conjugacy classes of subgroups — too many to draw. Only its 747 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×#14579153	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098931	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14579085	C₂	1	4	yes	maximal
4	C₂²×#14579072	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂×#14579153	—	2	6	no	—
8	C₂×#14579153	C₂²	1	12	yes	—
9	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	C₂²×Dic₇₉⋊C₄	—	2	2	no	—
11	#19098931	C₂²	1	64	yes	direct product factor
12	C₂×#14579085	C₂²	1	48	yes	—
13	C₂×#14579072	C₂²	1	12	yes	—
14	C₂³×Q₈×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₇₉	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₇₉	C₂²	1	12	yes	—
17	C₂²×C₄⋊C₄×C₇₉	—	2	2	no	—
18	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	C₂³×Dic₇₉	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	—	2	8	no	—
21	C₂²×C₇₉⋊C₈	C₂³	1	12	yes	—
22	#14579153	—	2	24	no	—
23	#14579153	C₂³	1	16	yes	—
24	C₂×C₄⋊Dic₇₉	—	2	6	no	—
25	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
26	C₂×Dic₇₉⋊C₄	—	2	24	no	—
27	#14579085	C₂³	1	64	yes	—
28	#14579072	C₂³	1	16	yes	—
29	C₂²×Q₈×C₇₉	—	2	6	no	—
30	C₂²×Q₈×C₇₉	—	2	16	no	—
31	C₂²×Q₈×C₇₉	C₂³	1	12	yes	—
32	C₂³×C₄×C₇₉	—	2	2	no	—
33	C₂³×C₄×C₇₉	D₄	1	2	yes	—
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₇₉	—	2	24	no	—
36	C₄⋊Q₈×C₇₉	C₂³	1	16	yes	—
37	C₂×C₄⋊C₄×C₇₉	—	2	24	no	—
38	C₂×C₄²×C₇₉	D₄	1	6	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
40	C₂²×Dic₇₉	—	2	12	no	—
41	C₂²×Dic₇₉	—	4	8	no	—
42	C₂×C₇₉⋊C₈	—	2	24	no	—
43	C₂×C₇₉⋊C₈	C₂⁴	1	8	yes	—
44	C₄⋊Dic₇₉	—	2	12	no	—
45	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
46	Dic₇₉⋊C₄	—	2	32	no	—
47	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₇₉	—	2	12	no	—
49	C₂×Q₈×C₇₉	—	2	48	no	—
50	C₂×Q₈×C₇₉	—	4	24	no	—
51	C₂×Q₈×C₇₉	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₇₉	—	2	8	no	—
53	C₂²×C₄×C₇₉	—	2	12	no	—
54	C₂²×C₄×C₇₉	—	4	8	no	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₇₉	#41	1	8	yes	—
58	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
59	C₄⋊C₄×C₇₉	—	2	32	no	—
60	C₄²×C₇₉	C₂×D₄	1	12	yes	—
61	C₄²×C₇₉	C₂⁴	1	4	yes	—
62	C₂³×C₇₉	#124	1	4	yes	—
63	C₂³×C₇₉	#137	1	8	yes	—
64	C₂³×C₇₉	C₂²×D₄	1	3	yes	—
65	Q₈×C₇₉	—	2	16	no	—
66	Q₈×C₇₉	—	4	24	no	—
67	C₂×C₄×C₇₉	—	2	24	no	—
68	C₂×C₄×C₇₉	—	2	8	no	—
69	C₂×C₄×C₇₉	—	4	24	no	—
70	C₂×C₄×C₇₉	#124	1	12	yes	—
71	C₂×C₄×C₇₉	C₂²×D₄	1	8	yes	—
72	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
73	C₂×C₄×C₇₉	C₂×#41	1	12	yes	—
74	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
75	C₂×Dic₇₉	—	2	8	no	—
76	C₂×Dic₇₉	—	4	24	no	—
77	C₇₉⋊C₈	—	2	8	no	—
78	C₂²×C₇₉	#491	1	16	yes	complemented
79	C₂²×C₇₉	C₂×#124	1	6	yes	—
80	C₂²×C₇₉	C₂×#137	1	12	yes	—
81	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
82	C₄×C₇₉	—	2	8	no	—
83	C₄×C₇₉	—	4	8	no	—
84	C₄×C₇₉	C₂×#124	1	6	yes	—
85	C₄×C₇₉	C₂²×#41	1	2	yes	—
86	Dic₇₉	—	4	8	no	—
87	C₂²×#491	—	79	1	no	maximal, Sylow
88	C₂×C₇₉	C₂×#491	1	12	yes	complemented
89	C₂×C₇₉	C₂²×#124	1	1	yes	characteristic
90	C₂×C₇₉	C₂²×#137	1	2	yes	—
91	C₂²×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
92	C₂²×#125	—	79	1	no	—
93	C₂×#491	—	79	24	no	—
94	C₂²×#103	—	79	4	no	—
95	C₂²×#97	—	79	1	no	—
96	C₇₉	C₂²×#491	1	1	yes	characteristic, complemented, Sylow
97	C₂³×Q₈	C₂×D₇₉	1	2	yes	—
98	C₂³×C₈	—	79	2	no	—
99	C₂×C₄⋊Q₈	—	2	6	no	—
100	C₂×C₄⋊Q₈	C₂×D₇₉	1	12	yes	—
101	C₂×#125	—	79	12	no	—
102	C₂×#125	—	158	6	no	—
103	C₂²×C₄⋊C₄	—	2	2	no	—
104	C₂²×C₄⋊C₄	—	79	2	no	—
105	C₂²×C₄⋊C₄	—	158	2	no	—
106	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
107	#491	—	79	64	no	—
108	C₂×#103	—	79	48	no	—
109	C₂×#97	—	79	12	no	—
110	C₂²×Q₈	—	2	6	no	—
111	C₂²×Q₈	—	2	16	no	—
112	C₂²×Q₈	C₂²×D₇₉	1	12	yes	—
113	C₂³×C₄	—	2	2	no	—
114	C₂³×C₄	—	158	2	no	—
115	C₂³×C₄	C₂²⋊D₇₉	1	2	yes	—
116	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
117	C₂²×C₈	—	79	12	no	—
118	C₂²×C₈	—	158	8	no	—
119	C₄⋊Q₈	—	2	24	no	—
120	C₄⋊Q₈	C₂²×D₇₉	1	16	yes	—
121	#125	—	79	16	no	—
122	#125	—	158	24	no	—
123	C₂×C₄⋊C₄	—	2	24	no	—
124	C₂×C₄⋊C₄	—	79	12	no	—
125	C₂×C₄⋊C₄	—	158	6	no	—
126	C₂×C₄⋊C₄	—	158	24	no	—
127	C₂×C₄²	C₂²⋊D₇₉	1	6	yes	—
128	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
129	#103	—	79	64	no	—
130	#97	—	79	16	no	—
131	C₂⁴	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
132	C₂×Q₈	—	2	12	no	—
133	C₂×Q₈	—	2	48	no	—
134	C₂×Q₈	—	4	24	no	—
135	C₂×Q₈	C₂³×D₇₉	1	8	yes	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	12	no	—
138	C₂²×C₄	—	4	8	no	—
139	C₂²×C₄	—	316	8	no	—
140	C₂²×C₄	—	158	12	no	—
141	C₂²×C₄	C₂×C₂²⋊D₇₉	1	12	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
143	C₂²×C₄	#11929623	1	8	yes	—
144	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
145	C₂×C₈	—	79	8	no	—
146	C₂×C₈	—	158	24	no	—
147	C₄⋊C₄	—	2	32	no	—
148	C₄⋊C₄	—	79	8	no	—
149	C₄⋊C₄	—	158	32	no	—
150	C₄⋊C₄	—	158	12	no	—
151	C₄²	C₂×C₂²⋊D₇₉	1	12	yes	—
152	C₄²	C₂³×D₇₉	1	4	yes	—
153	C₂³	#14579148	1	4	yes	—
154	C₂³	#14579192	1	8	yes	—
155	C₂³	C₂²×C₂²⋊D₇₉	1	3	yes	—
156	Q₈	—	2	16	no	—
157	Q₈	—	4	24	no	—
158	C₂×C₄	—	2	24	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	24	no	—
161	C₂×C₄	—	316	24	no	—
162	C₂×C₄	—	158	8	no	—
163	C₂×C₄	#14579148	1	12	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₇₉	1	8	yes	—
165	C₂×C₄	C₂²×C₂²⋊D₇₉	1	3	yes	—
166	C₂×C₄	C₂×#11929623	1	12	yes	—
167	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
168	C₈	—	158	8	no	—
169	C₂²	#19098931	1	16	yes	direct product factor
170	C₂²	C₂×#14579148	1	6	yes	—
171	C₂²	C₂×#14579192	1	12	yes	—
172	C₂²	C₂³×C₂²⋊D₇₉	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	4	8	no	—
175	C₄	—	316	8	no	—
176	C₄	C₂×#14579148	1	6	yes	—
177	C₄	C₂²×#11929623	1	2	yes	—
178	C₂	C₂×#19098931	1	12	yes	direct product factor
179	C₂	C₂²×#14579148	1	1	yes	characteristic
180	C₂	C₂²×#14579192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14579153

C₂

yes

maximal, characteristic

C₂×#19098931

C₂

yes

maximal, direct product factor

C₂²×#14579085

C₂

yes

maximal

C₂²×#14579072

C₂

yes

maximal, characteristic

C₂²×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂³×C₇₉⋊C₈

C₂²

yes

—

C₂×#14579153

—

C₂×#14579153

C₂²

yes

—

C₂²×C₄⋊Dic₇₉

C₂²

yes

—

C₂²×Dic₇₉⋊C₄

—

#19098931

C₂²

yes

direct product factor

C₂×#14579085

C₂²

yes

—

C₂×#14579072

C₂²

yes

—

C₂³×Q₈×C₇₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₇₉

—

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

—

C₂²×C₄⋊C₄×C₇₉

—

C₂²×C₄²×C₇₉

C₂²

yes

characteristic

C₂³×Dic₇₉

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

C₂³

yes

—

#14579153

—

#14579153

C₂³

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂³

yes

—

C₂×Dic₇₉⋊C₄

—

#14579085

C₂³

yes

—

#14579072

C₂³

yes

—

C₂²×Q₈×C₇₉

—

C₂²×Q₈×C₇₉

—

C₂²×Q₈×C₇₉

C₂³

yes

—

C₂³×C₄×C₇₉

—

C₂³×C₄×C₇₉

D₄

yes

—

C₂³×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂³

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂⁴

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂⁴

yes

—

Dic₇₉⋊C₄

—

C₂⁴×C₇₉

C₂×D₄

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂⁴

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

#41

yes

—

C₂²×C₄×C₇₉

C₂⁴

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂⁴

yes

—

C₂³×C₇₉

#124

yes

—

C₂³×C₇₉

#137

yes

—

C₂³×C₇₉

C₂²×D₄

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

—

C₂×C₄×C₇₉

C₂×#41

yes

—

C₂×C₄×C₇₉

C₂⁵

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#491

yes

complemented

C₂²×C₇₉

C₂×#124

yes

—

C₂²×C₇₉

C₂×#137

yes

—

C₂²×C₇₉

C₂³×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#124

yes

—

C₄×C₇₉

C₂²×#41

yes

—

Dic₇₉

—

C₂²×#491

—

maximal, Sylow

C₂×C₇₉

C₂×#491

yes

complemented

C₂×C₇₉

C₂²×#124

yes

characteristic

C₂×C₇₉

C₂²×#137

yes

—

C₂²×C₄⋊Q₈

D₇₉

yes

characteristic

C₂²×#125

—

C₂×#491

—

C₂²×#103

—

C₂²×#97

—

C₇₉

C₂²×#491

yes

characteristic, complemented, Sylow

C₂³×Q₈

C₂×D₇₉

yes

—

C₂³×C₈

—

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

C₂×D₇₉

yes

—

101

C₂×#125

—

102

C₂×#125

—

158

—

103

C₂²×C₄⋊C₄

—

104

C₂²×C₄⋊C₄

—

105

C₂²×C₄⋊C₄

—

158

—

106

C₂²×C₄²

C₂×D₇₉

yes

characteristic

107

#491

—

108

C₂×#103

—

109

C₂×#97

—

110

C₂²×Q₈

—

111

C₂²×Q₈

—

112

C₂²×Q₈

C₂²×D₇₉

yes

—

113

C₂³×C₄

—

114

C₂³×C₄

—

158

—

115

C₂³×C₄

C₂²⋊D₇₉

yes

—

116

C₂³×C₄

C₂²×D₇₉

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

—

158

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

C₂²×D₇₉

yes

—

121

#125

—

122

#125

—

158

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

158

—

126

C₂×C₄⋊C₄

—

158

—

127

C₂×C₄²

C₂²⋊D₇₉

yes

—

128

C₂×C₄²

C₂²×D₇₉

yes

—

129

#103

—

130

#97

—

131

C₂⁴

C₂×C₂²⋊D₇₉

yes

characteristic

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

C₂³×D₇₉

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

316

—

140

C₂²×C₄

—

158

—

141

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

143

C₂²×C₄

#11929623

yes

—

144

C₂²×C₄

C₂³×D₇₉

yes

—

145

C₂×C₈

—

146

C₂×C₈

—

158

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

158

—

150

C₄⋊C₄

—

158

—

151

C₄²

C₂×C₂²⋊D₇₉

yes

—

152

C₄²

C₂³×D₇₉

yes

—

153

C₂³

#14579148

yes

—

154

C₂³

#14579192

yes

—

155

C₂³

C₂²×C₂²⋊D₇₉

yes

—

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

158

—

163

C₂×C₄

#14579148

yes

—

164

C₂×C₄

C₂²×C₂²⋊D₇₉

yes

—

165

C₂×C₄

C₂²×C₂²⋊D₇₉

yes

—

166

C₂×C₄

C₂×#11929623

yes

—

167

C₂×C₄

C₂⁴×D₇₉

yes

characteristic

168

C₈

—

158

—

169

C₂²

#19098931

yes

direct product factor

170

C₂²

C₂×#14579148

yes

—

171

C₂²

C₂×#14579192

yes

—

172

C₂²

C₂³×C₂²⋊D₇₉

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

316

—

176

C₄

C₂×#14579148

yes

—

177

C₄

C₂²×#11929623

yes

—

178

C₂

C₂×#19098931

yes

direct product factor

179

C₂

C₂²×#14579148

yes

characteristic

180

C₂

C₂²×#14579192

yes

—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂²×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	C₂²×#491
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

subgroup

G/H

center

C₂⁴

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁵

Frattini

C₂×C₄

C₂⁴×D₇₉

Fitting

C₂²×C₄⋊Q₈×C₇₉

C₂

socle, prime radical

C₂⁴×C₇₉

C₂×D₄

hypercenter

C₂²×C₄⋊Q₈

D₇₉

hypocenter

C₇₉

C₂²×#491

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#19098931	C₂	direct (N × H)
11. #19098931	C₂²	direct (N × H)
78. C₂²×C₇₉	#491	semidirect
88. C₂×C₇₉	C₂×#491	semidirect
96. C₇₉	C₂²×#491	semidirect
169. C₂²	#19098931	direct (N × H)
178. C₂	C₂×#19098931	direct (N × H)

H ≅ G/N

type

2. C₂×#19098931

C₂

direct (N × H)

11. #19098931

C₂²

direct (N × H)

78. C₂²×C₇₉

#491

semidirect

88. C₂×C₇₉

C₂×#491

semidirect

96. C₇₉

C₂²×#491

semidirect

169. C₂²

#19098931

direct (N × H)

178. C₂

C₂×#19098931

direct (N × H)

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

subgroup	index
C₂²×#14579153	2
C₂×#19098931	2
C₂²×#14579085	2
C₂²×#14579072	2
C₂²×C₄⋊Q₈×C₇₉	2
C₂²×#491	79

subgroup

index

C₂²×C₄⋊Q₈×C₇₉

C₂²×#491

Subgroups — C₂²×#19098931 #35157488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×#19098931 #35157488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19098931 #35157488