Group #24987515 — subgroups

Subgroups — C₄×#14582641 #24987515

Order 11008.

532 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₄×#13063236	C₂	1	2	yes	maximal
2	C₂×#14582641	C₂	1	2	yes	maximal
3	#19132476	C₂	1	2	yes	maximal
4	C₄×#125×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063236	—	2	2	no	—
6	C₂×#13063236	C₂²	1	4	yes	—
7	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
8	#14582641	C₄	1	8	yes	direct product factor
9	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
10	C₄₃×#389	C₄	1	2	yes	—
11	C₄₃×#387	—	2	2	no	—
12	C₄₃×#382	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
15	C₄³×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₈	—	2	4	no	—
17	#13063236	—	2	8	no	—
18	#13063236	—	2	4	no	—
19	#13063236	C₂×C₄	1	8	yes	—
20	C₄×C₄₃⋊C₈	—	2	8	no	—
21	#125×C₄₃	—	2	2	no	—
22	#125×C₄₃	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₃	—	2	2	no	—
28	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₃	D₄	1	2	yes	—
31	C₂×C₄²×C₄₃	D₄	1	2	yes	—
32	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	#95×C₄₃	—	2	2	no	—
34	#95×C₄₃	—	2	2	no	—
35	#95×C₄₃	—	2	2	no	—
36	C₂²×C₄×C₄₃	—	2	2	no	—
37	C₂²×C₄×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₄₃	—	2	2	no	—
44	C₄⋊C₄×C₄₃	—	2	8	no	—
45	C₄⋊C₄×C₄₃	—	4	2	no	—
46	C₄⋊C₄×C₄₃	—	4	4	no	—
47	C₄⋊C₄×C₄₃	C₄²	1	4	yes	—
48	C₄²×C₄₃	—	2	2	no	—
49	C₄²×C₄₃	—	2	4	no	—
50	C₄²×C₄₃	—	2	4	no	—
51	C₄²×C₄₃	—	4	1	no	—
52	C₄²×C₄₃	—	4	4	no	—
53	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
54	C₄²×C₄₃	#41	1	2	yes	—
55	C₂×C₄₃⋊C₈	—	2	8	no	—
56	C₂×C₄₃⋊C₈	—	2	8	no	—
57	C₂×C₄₃⋊C₈	—	2	8	no	—
58	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	2	1	no	—
60	C₂×C₄×C₄₃	—	2	1	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	2	no	—
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	4	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	4	2	no	—
69	C₂×C₄×C₄₃	—	4	2	no	—
70	C₂×C₄×C₄₃	—	4	2	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	#101	1	2	yes	—
74	C₂×C₄×C₄₃	#104	1	4	yes	—
75	C₂×C₄×C₄₃	#104	1	4	yes	—
76	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
79	C₄₃⋊C₈	—	2	16	no	—
80	C₄×#330	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
83	C₂²×C₄₃	#413	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×#104	1	2	yes	—
85	C₂²×C₄₃	C₂×#104	1	2	yes	—
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	—	2	4	no	—
89	C₄×C₄₃	—	2	4	no	—
90	C₄×C₄₃	—	4	4	no	—
91	C₄×C₄₃	—	4	1	no	—
92	C₄×C₄₃	—	8	2	no	—
93	C₄×C₄₃	#330	1	4	yes	complemented
94	C₄×#125	D₄₃	1	1	yes	characteristic
95	C₄×#97	—	43	2	no	—
96	C₂×#330	—	43	2	no	—
97	#1295	—	43	2	no	—
98	C₂×C₄₃	C₂×#330	1	1	yes	characteristic
99	C₂×C₄₃	#1524	1	2	yes	—
100	C₂×C₄₃	#1526	1	1	yes	characteristic
101	C₂×C₄₃	C₄×#101	1	1	yes	characteristic
102	C₂×C₄₃	C₄×#104	1	2	yes	—
103	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
104	C₂×#97	—	43	4	no	—
105	C₂×#97	—	86	2	no	—
106	C₂×C₄×C₈	—	86	2	no	—
107	#389	Dic₄₃	1	2	yes	—
108	#387	—	2	2	no	—
109	#382	C₂×D₄₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	2	2	no	—
111	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
112	C₄³	C₂×D₄₃	1	1	yes	characteristic
113	#330	—	43	8	no	—
114	C₄₃	C₄×#330	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	86	4	no	—
116	#125	—	2	2	no	—
117	#125	C₄×D₄₃	1	4	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
128	#97	—	43	8	no	—
129	#97	—	86	8	no	—
130	#97	—	86	4	no	—
131	C₄×C₈	—	86	8	no	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
139	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
142	C₂×C₈	—	86	8	no	—
143	C₂×C₈	—	86	8	no	—
144	C₂×C₈	—	86	8	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	C₄×Dic₄₃	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	4	no	—
153	C₄²	—	4	1	no	—
154	C₄²	—	4	4	no	—
155	C₄²	C₂×C₄×D₄₃	1	2	yes	—
156	C₄²	#10621898	1	2	yes	—
157	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	#13063244	1	2	yes	—
173	C₂×C₄	#13063252	1	4	yes	—
174	C₂×C₄	#13063252	1	4	yes	—
175	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
176	C₂×C₄	#13063284	1	1	yes	characteristic
177	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
178	C₈	—	86	16	no	—
179	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
180	C₂²	C₂×#13063244	1	1	yes	characteristic
181	C₂²	#14582883	1	1	yes	characteristic
182	C₂²	C₂×#13063252	1	2	yes	—
183	C₂²	C₂×#13063252	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	4	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	#14582641	1	4	yes	direct product factor
192	C₂	C₂×#14582641	1	1	yes	characteristic
193	C₂	#19133183	1	2	yes	—
194	C₂	#19133190	1	1	yes	characteristic
195	C₂	C₄×#13063244	1	1	yes	characteristic
196	C₂	C₄×#13063252	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063236

C₂

yes

maximal

C₂×#14582641

C₂

yes

maximal

#19132476

C₂

yes

maximal

C₄×#125×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063236

—

C₂×#13063236

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

—

#14582641

C₄

yes

direct product factor

C₂×#125×C₄₃

C₂²

yes

characteristic

C₄₃×#389

C₄

yes

—

C₄₃×#387

—

C₄₃×#382

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₄₃

—

C₄×C₄⋊C₄×C₄₃

C₄

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

#13063236

—

#13063236

—

#13063236

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

—

#125×C₄₃

—

#125×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

#95×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₄²

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#101

yes

—

C₂×C₄×C₄₃

#104

yes

—

C₂×C₄×C₄₃

#104

yes

—

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

#117

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₄₃⋊C₈

—

C₄×#330

—

maximal, Sylow

C₂²×C₄₃

C₄×C₂²⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

C₂×#104

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#330

yes

complemented

C₄×#125

D₄₃

yes

characteristic

C₄×#97

—

C₂×#330

—

#1295

—

C₂×C₄₃

C₂×#330

yes

characteristic

C₂×C₄₃

#1524

yes

—

100

C₂×C₄₃

#1526

yes

characteristic

101

C₂×C₄₃

C₄×#101

yes

characteristic

102

C₂×C₄₃

C₄×#104

yes

—

103

C₂×#125

C₂×D₄₃

yes

characteristic

104

C₂×#97

—

105

C₂×#97

—

106

C₂×C₄×C₈

—

107

#389

Dic₄₃

yes

—

108

#387

—

109

#382

C₂×D₄₃

yes

characteristic

110

C₄×C₄⋊C₄

—

111

C₄×C₄⋊C₄

Dic₄₃

yes

—

112

C₄³

C₂×D₄₃

yes

characteristic

113

#330

—

114

C₄₃

C₄×#330

yes

characteristic, complemented, Sylow

115

C₂²×C₈

—

116

#125

—

117

#125

C₄×D₄₃

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

121

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

124

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

125

C₂×C₄²

C₂²⋊D₄₃

yes

—

126

C₂×C₄²

C₂²⋊D₄₃

yes

—

127

C₂×C₄²

C₂²×D₄₃

yes

characteristic

128

#97

—

129

#97

—

130

#97

—

131

C₄×C₈

—

132

#95

—

133

#95

—

134

#95

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

138

C₂²×C₄

C₂²⋊Dic₄₃

yes

—

139

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

140

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

C₄×Dic₄₃

yes

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

C₂×C₄×D₄₃

yes

—

156

C₄²

#10621898

yes

—

157

C₂³

C₂×C₂²⋊Dic₄₃

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#13063244

yes

—

173

C₂×C₄

#13063252

yes

—

174

C₂×C₄

#13063252

yes

—

175

C₂×C₄

C₂×C₄×Dic₄₃

yes

characteristic

176

C₂×C₄

#13063284

yes

characteristic

177

C₂×C₄

C₄×C₂²⋊D₄₃

yes

—

178

C₈

—

179

C₂²

C₄×C₂²⋊Dic₄₃

yes

characteristic

180

C₂²

C₂×#13063244

yes

characteristic

181

C₂²

#14582883

yes

characteristic

182

C₂²

C₂×#13063252

yes

—

183

C₂²

C₂×#13063252

yes

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#14582641

yes

direct product factor

192

C₂

C₂×#14582641

yes

characteristic

193

C₂

#19133183

yes

—

194

C₂

#19133190

yes

characteristic

195

C₂

C₄×#13063244

yes

characteristic

196

C₂

C₄×#13063252

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#125×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#125	D₄₃
hypocenter	C₄₃	C₄×#330
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₄₃

commutator

C₂×C₄×C₄₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄×#125×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
8. #14582641	C₄	direct (N × H)
93. C₄×C₄₃	#330	semidirect
114. C₄₃	C₄×#330	semidirect
191. C₄	#14582641	direct (N × H)

H ≅ G/N

type

8. #14582641

C₄

direct (N × H)

93. C₄×C₄₃

#330

semidirect

114. C₄₃

C₄×#330

semidirect

191. C₄

#14582641

direct (N × H)

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

subgroup	index
C₄×#13063236	2
C₂×#14582641	2
#19132476	2
C₄×#125×C₄₃	2
C₄×#330	43

subgroup

index

C₄×#125×C₄₃

Subgroups — C₄×#14582641 #24987515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#14582641 #24987515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582641 #24987515