Group #24995691 — subgroups

Subgroups — #24995691

Order 11008.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#19142993	C₂	1	1	yes	maximal, characteristic
2	#19134712	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582646	C₂	1	4	yes	maximal
4	C₄₃×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063325	—	2	1	no	—
6	C₂×#13063325	—	2	1	no	—
7	C₂×#13063325	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	#14582769	—	2	2	no	—
10	#14582769	C₄	1	4	yes	—
11	#14582654	—	2	2	no	—
12	#14582646	—	2	8	no	—
13	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
14	C₂×#97×C₄₃	—	2	1	no	—
15	C₄³×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#336	—	2	1	no	—
17	C₂²×Dic₈₆	—	2	2	no	—
18	C₂²×C₄₃⋊C₈	—	2	2	no	—
19	#13063325	—	2	4	no	—
20	#13063325	—	2	4	no	—
21	#13063325	—	4	4	no	—
22	#13063325	—	4	2	no	—
23	#13063325	—	4	2	no	—
24	C₂×C₄⋊Dic₄₃	—	2	4	no	—
25	C₂×C₄⋊Dic₄₃	—	2	4	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
28	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
29	C₂²×C₈×C₄₃	—	2	2	no	—
30	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₃	D₄	1	2	yes	—
32	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₃	—	2	2	no	—
36	C₄⋊C₈×C₄₃	D₄	1	4	yes	—
37	#97×C₄₃	—	4	2	no	—
38	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₃	—	2	4	no	—
44	C₂×Dic₈₆	—	4	4	no	—
45	C₂×Dic₈₆	—	4	2	no	—
46	C₂×Dic₈₆	—	4	2	no	—
47	C₂×Dic₈₆	—	4	4	no	—
48	C₂×C₈×C₄₃	—	2	4	no	—
49	C₂×C₈×C₄₃	—	4	2	no	—
50	C₂×C₈×C₄₃	—	4	2	no	—
51	C₄²×C₄₃	—	2	4	no	—
52	C₄²×C₄₃	—	2	2	no	—
53	C₄²×C₄₃	—	2	4	no	—
54	C₄²×C₄₃	D₈	1	2	yes	—
55	C₄²×C₄₃	SD₁₆	1	2	yes	—
56	C₄²×C₄₃	C₂×D₄	1	2	yes	—
57	C₄²×C₄₃	#41	1	2	yes	—
58	C₄⋊Dic₄₃	—	2	8	no	—
59	C₄⋊Dic₄₃	—	4	8	no	—
60	C₄⋊Dic₄₃	—	4	8	no	—
61	C₄⋊Dic₄₃	—	4	2	no	—
62	C₄⋊Dic₄₃	—	4	2	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	C₂×C₄₃⋊C₈	—	4	4	no	—
65	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	#101	1	4	yes	—
72	C₂×C₄×C₄₃	#102	1	4	yes	—
73	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₄₃	#121	1	2	yes	—
75	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₄₃	#137	1	4	yes	—
81	C₂×Dic₄₃	—	4	4	no	—
82	C₂×Dic₄₃	—	4	4	no	—
83	C₂×Dic₄₃	—	4	4	no	—
84	Dic₈₆	—	8	4	no	—
85	Dic₈₆	—	8	4	no	—
86	C₈×C₄₃	—	4	4	no	—
87	C₄₃⋊C₈	—	8	2	no	—
88	#9705	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	#390	1	1	yes	characteristic
90	C₂²×C₄₃	C₂×#101	1	2	yes	—
91	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
92	C₂²×C₄₃	#418	1	1	yes	characteristic
93	C₂²×C₄₃	#419	1	2	yes	—
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	—	2	4	no	—
96	C₄×C₄₃	—	2	2	no	—
97	C₄×C₄₃	#459	1	2	yes	—
98	C₄×C₄₃	#460	1	2	yes	—
99	C₄×C₄₃	#491	1	2	yes	—
100	C₄×C₄₃	#497	1	2	yes	—
101	Dic₄₃	—	8	4	no	—
102	#2868	—	43	1	no	—
103	#1831	—	43	1	no	—
104	#1767	D₄₃	1	1	yes	characteristic
105	C₂×#332	—	43	4	no	—
106	C₂×C₄₃	#1546	1	2	yes	—
107	C₂×C₄₃	#1556	1	2	yes	—
108	C₂×C₄₃	#1960	1	1	yes	characteristic
109	C₂×C₄₃	#1966	1	1	yes	characteristic
110	C₂×C₄₃	#1982	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	43	2	no	—
112	C₂×C₄⋊Q₈	—	86	1	no	—
113	C₂×C₄⋊Q₈	—	86	1	no	—
114	C₂×C₄⋊C₈	—	43	2	no	—
115	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
116	C₂×#97	—	2	1	no	—
117	#384	—	43	4	no	—
118	#384	—	86	2	no	—
119	C₄³	C₂×D₄₃	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#336	—	86	2	no	—
122	#332	—	86	8	no	—
123	C₄₃	#9705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	86	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	86	2	no	—
127	C₄⋊Q₈	—	86	4	no	—
128	C₄⋊Q₈	—	86	4	no	—
129	C₄⋊Q₈	—	172	4	no	—
130	C₄⋊Q₈	—	172	2	no	—
131	C₄⋊Q₈	—	172	2	no	—
132	C₂×C₄⋊C₄	—	43	4	no	—
133	C₂×C₄⋊C₄	—	86	4	no	—
134	C₂×C₄⋊C₄	—	86	4	no	—
135	C₂×C₄⋊C₄	—	86	2	no	—
136	C₂×C₄²	C₄×D₄₃	1	2	yes	—
137	C₂×C₄²	D₁₇₂	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	86	4	no	—
143	C₄⋊C₈	D₁₇₂	1	4	yes	—
144	#97	—	4	2	no	—
145	C₂×Q₈	—	172	4	no	—
146	C₂×Q₈	—	172	2	no	—
147	C₂×Q₈	—	172	2	no	—
148	C₂×Q₈	—	172	4	no	—
149	C₂²×C₄	—	86	4	no	—
150	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
151	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	172	2	no	—
159	C₂×C₈	—	172	4	no	—
160	C₄⋊C₄	—	86	8	no	—
161	C₄⋊C₄	—	172	8	no	—
162	C₄⋊C₄	—	172	8	no	—
163	C₄⋊C₄	—	172	2	no	—
164	C₄⋊C₄	—	172	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	#10621882	1	2	yes	—
169	C₄²	#10621884	1	2	yes	—
170	C₄²	C₂×D₁₇₂	1	2	yes	—
171	C₄²	#10621898	1	2	yes	—
172	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
173	Q₈	—	344	4	no	—
174	Q₈	—	344	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	172	4	no	—
181	C₂×C₄	—	172	4	no	—
182	C₂×C₄	—	172	4	no	—
183	C₂×C₄	#13063243	1	4	yes	—
184	C₂×C₄	#13063245	1	4	yes	—
185	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
186	C₂×C₄	#13063299	1	2	yes	—
187	C₂×C₄	#13063313	1	1	yes	characteristic
188	C₂×C₄	#13063322	1	1	yes	characteristic
189	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
190	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
191	C₂×C₄	#13063350	1	2	yes	—
192	C₂×C₄	#13063358	1	4	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	344	2	no	—
195	C₂²	#14582801	1	1	yes	characteristic
196	C₂²	C₂×#13063243	1	2	yes	—
197	C₂²	C₂×#13063245	1	1	yes	characteristic
198	C₂²	#14582903	1	1	yes	characteristic
199	C₂²	#14582909	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	344	4	no	—
204	C₄	#14583094	1	2	yes	—
205	C₄	#14583099	1	2	yes	—
206	C₄	#14583268	1	2	yes	—
207	C₄	#14583293	1	2	yes	—
208	C₂	#19133268	1	2	yes	—
209	C₂	#19133313	1	2	yes	—
210	C₂	#19135415	1	1	yes	characteristic
211	C₂	#19135439	1	1	yes	characteristic
212	C₂	#19135522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19142993

C₂

yes

maximal, characteristic

#19134712

C₂

yes

maximal, characteristic

C₂×#14582646

C₂

yes

maximal

C₄₃×#1767

C₂

yes

maximal, characteristic

C₂×#13063325

—

C₂×#13063325

—

C₂×#13063325

C₂²

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#14582769

—

#14582769

C₄

yes

—

#14582654

—

#14582646

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₂×#97×C₄₃

—

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

—

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

—

—

—

—

—

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

D₄

yes

—

#97×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₈

yes

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

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

#101

yes

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#459

yes

—

C₄×C₄₃

#460

yes

—

C₄×C₄₃

#491

yes

—

100

C₄×C₄₃

#497

yes

—

101

Dic₄₃

—

102

#2868

—

103

#1831

—

104

#1767

D₄₃

yes

characteristic

105

C₂×#332

—

106

C₂×C₄₃

#1546

yes

—

107

C₂×C₄₃

#1556

yes

—

108

C₂×C₄₃

#1960

yes

characteristic

109

C₂×C₄₃

#1966

yes

characteristic

110

C₂×C₄₃

#1982

yes

characteristic

111

C₂×C₄⋊Q₈

—

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

—

114

C₂×C₄⋊C₈

—

115

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

116

C₂×#97

—

117

#384

—

118

#384

—

119

C₄³

C₂×D₄₃

yes

characteristic

120

#336

—

121

#336

—

122

#332

—

123

C₄₃

#9705

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

172

—

130

C₄⋊Q₈

—

172

—

131

C₄⋊Q₈

—

172

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

C₄×D₄₃

yes

—

137

C₂×C₄²

D₁₇₂

yes

—

138

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

139

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

140

C₂×C₄²

C₂²×D₄₃

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

D₁₇₂

yes

—

144

#97

—

145

C₂×Q₈

—

172

—

146

C₂×Q₈

—

172

—

147

C₂×Q₈

—

172

—

148

C₂×Q₈

—

172

—

149

C₂²×C₄

—

150

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

—

151

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

—

152

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

153

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

172

—

159

C₂×C₈

—

172

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₄⋊C₄

—

172

—

164

C₄⋊C₄

—

172

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#10621882

yes

—

169

C₄²

#10621884

yes

—

170

C₄²

C₂×D₁₇₂

yes

—

171

C₄²

#10621898

yes

—

172

C₂³

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

173

Q₈

—

344

—

174

Q₈

—

344

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

172

—

181

C₂×C₄

—

172

—

182

C₂×C₄

—

172

—

183

C₂×C₄

#13063243

yes

—

184

C₂×C₄

#13063245

yes

—

185

C₂×C₄

C₄×D₁₇₂

yes

—

186

C₂×C₄

#13063299

yes

—

187

C₂×C₄

#13063313

yes

characteristic

188

C₂×C₄

#13063322

yes

characteristic

189

C₂×C₄

C₂×#10621882

yes

characteristic

190

C₂×C₄

C₂×#10621884

yes

characteristic

191

C₂×C₄

#13063350

yes

—

192

C₂×C₄

#13063358

yes

—

193

C₈

—

194

C₈

—

344

—

195

C₂²

#14582801

yes

characteristic

196

C₂²

C₂×#13063243

yes

—

197

C₂²

C₂×#13063245

yes

characteristic

198

C₂²

#14582903

yes

characteristic

199

C₂²

#14582909

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

344

—

204

C₄

#14583094

yes

—

205

C₄

#14583099

yes

—

206

C₄

#14583268

yes

—

207

C₄

#14583293

yes

—

208

C₂

#19133268

yes

—

209

C₂

#19133313

yes

—

210

C₂

#19135415

yes

characteristic

211

C₂

#19135439

yes

characteristic

212

C₂

#19135522

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1767	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1767	D₄₃
hypocenter	C₄₃	#9705
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1767

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
123. C₄₃	#9705	semidirect

H ≅ G/N

type

123. C₄₃

#9705

semidirect

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

subgroup	index
#19142993	2
#19134712	2
C₂×#14582646	2
C₄₃×#1767	2
#9705	43

subgroup

index

C₄₃×#1767

Subgroups — #24995691

Conjugacy classes of subgroups

Complements

Maximal subgroups