Group #25104983 — subgroups

Subgroups — #25104983

Order 11008.

1230 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 355 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	C₂×#14582978	C₂	1	6	yes	maximal, complemented
2	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
3	#19135389	C₂	1	6	yes	maximal, complemented
4	#19134747	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₂²	1	4	yes	—
7	C₂×#13063322	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₇₂	C₂²	1	6	yes	—
9	#14582978	—	2	24	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	6	yes	—
11	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	12	yes	—
12	#14582801	C₄	1	6	yes	—
13	#14582769	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
16	C₄³×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₇₂	C₂×C₄	1	6	yes	—
18	C₂²×C₄×D₄₃	C₂³	1	4	yes	—
19	C₂×#10621872	—	2	24	no	—
20	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
21	#13063322	—	2	4	no	—
22	C₄×D₁₇₂	—	2	12	no	—
23	C₄×D₁₇₂	—	2	12	no	—
24	C₂×C₄⋊Dic₄₃	C₂×C₄	1	6	yes	—
25	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	12	yes	—
26	C₄⋊(C₄₃⋊C₈)	—	2	12	no	—
27	(C₂×D₄₃)⋊C₈	—	2	24	no	—
28	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
29	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
33	C₄×C₈×C₄₃	—	2	6	no	—
34	C₄×C₈×C₄₃	D₄	1	12	yes	complemented
35	C₂³×D₄₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
37	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
39	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
40	C₂×D₁₇₂	—	2	12	no	—
41	C₂×D₁₇₂	—	2	12	no	—
42	C₂×D₁₇₂	—	2	12	no	—
43	C₂×C₄×D₄₃	—	2	12	no	—
44	C₂×C₄×D₄₃	—	2	12	no	—
45	C₂×C₈×C₄₃	—	2	6	no	—
46	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
47	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
48	C₄²×C₄₃	—	2	6	no	—
49	C₄²×C₄₃	M₁₆	1	4	yes	complemented
50	C₄²×C₄₃	C₂×D₄	1	6	yes	—
51	C₄²×C₄₃	#41	1	6	yes	—
52	(C₂×D₄₃)⋊C₄	—	2	24	no	—
53	C₄⋊Dic₄₃	—	2	12	no	—
54	C₂×C₄₃⋊C₈	—	2	12	no	—
55	#10621872	—	4	16	no	—
56	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
57	C₂²×D₄₃	—	2	12	no	—
58	C₂²×D₄₃	—	2	4	no	—
59	C₂²×D₄₃	—	2	12	no	—
60	C₂×C₄×C₄₃	—	2	6	no	—
61	C₂×C₄×C₄₃	C₄×D₄	1	6	yes	—
62	C₂×C₄×C₄₃	C₄×D₄	1	6	yes	—
63	C₂×C₄×C₄₃	C₂×M₁₆	1	6	yes	—
64	C₂×C₄×C₄₃	#131	1	6	yes	—
65	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
66	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
67	C₂×Dic₄₃	—	2	12	no	—
68	D₁₇₂	—	4	24	no	—
69	C₄×D₄₃	—	4	4	no	—
70	C₄×D₄₃	—	4	12	no	—
71	C₈×C₄₃	—	2	6	no	—
72	C₈×C₄₃	#127	1	4	yes	complemented
73	C₄₃⋊C₈	—	4	4	no	—
74	#21681	—	43	1	no	maximal, Sylow
75	C₂²×C₄₃	C₂×C₄×D₄	1	3	yes	—
76	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
77	C₂²×C₄₃	C₂×#131	1	3	yes	—
78	C₂×D₄₃	—	4	4	no	—
79	C₂×D₄₃	—	4	12	no	—
80	C₂×D₄₃	—	4	12	no	—
81	C₄×C₄₃	—	2	6	no	—
82	C₄×C₄₃	#436	1	12	yes	complemented
83	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
84	C₄×C₄₃	#532	1	3	yes	—
85	Dic₄₃	—	4	4	no	—
86	C₂×#436	—	43	6	no	—
87	C₄×#127	—	43	1	no	—
88	#1954	—	43	6	no	—
89	#1842	—	43	1	no	—
90	C₄²×C₈	D₄₃	1	1	yes	characteristic, complemented
91	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
92	C₂×C₄₃	C₂×#436	1	3	yes	—
93	C₂×C₄₃	#2931	1	3	yes	—
94	D₄₃	—	8	4	no	—
95	C₂²×M₁₆	—	43	4	no	—
96	C₂×#127	—	43	1	no	—
97	C₂×C₄×D₄	—	43	6	no	—
98	#436	—	86	24	no	—
99	C₂×C₄⋊C₈	—	43	6	no	—
100	C₂×C₂²⋊C₈	—	43	12	no	—
101	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
102	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
103	#390	—	43	6	no	—
104	#384	—	43	1	no	—
105	C₄³	C₂×D₄₃	1	1	yes	characteristic
106	C₄₃	#21681	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₄	—	43	6	no	—
108	C₂³×C₄	—	43	4	no	—
109	C₂×M₁₆	—	86	24	no	—
110	C₂²×C₈	—	43	4	no	—
111	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
112	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
113	#127	—	86	4	no	—
114	C₄×D₄	—	86	12	no	—
115	C₄×D₄	—	86	12	no	—
116	C₂×C₄⋊C₄	—	43	6	no	—
117	C₂×C₂²⋊C₄	—	43	12	no	—
118	C₂×C₄²	C₄×D₄₃	1	3	yes	—
119	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
121	C₄⋊C₈	—	86	12	no	—
122	C₂²⋊C₈	—	86	24	no	—
123	C₄×C₈	—	2	6	no	—
124	C₄×C₈	D₁₇₂	1	12	yes	complemented
125	C₂⁴	—	43	4	no	—
126	C₂×D₄	—	86	12	no	—
127	C₂×D₄	—	86	12	no	—
128	C₂×D₄	—	86	12	no	—
129	C₂²×C₄	—	43	4	no	—
130	C₂²×C₄	—	86	12	no	—
131	C₂²×C₄	—	86	12	no	—
132	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
133	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
134	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
135	M₁₆	—	172	16	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	86	12	no	—
138	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
139	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
140	C₄⋊C₄	—	86	12	no	—
141	C₂²⋊C₄	—	86	24	no	—
142	C₄²	—	2	6	no	—
143	C₄²	#10621872	1	4	yes	complemented
144	C₄²	C₂×D₁₇₂	1	6	yes	—
145	C₄²	#10621898	1	6	yes	—
146	C₂³	—	86	12	no	—
147	C₂³	—	86	4	no	—
148	C₂³	—	86	12	no	—
149	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
150	D₄	—	172	24	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	86	12	no	—
153	C₂×C₄	—	172	4	no	—
154	C₂×C₄	—	172	12	no	—
155	C₂×C₄	C₄×D₁₇₂	1	6	yes	—
156	C₂×C₄	C₄×D₁₇₂	1	6	yes	—
157	C₂×C₄	C₂×#10621872	1	6	yes	—
158	C₂×C₄	#13063331	1	6	yes	—
159	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
160	C₂×C₄	C₂×#10621898	1	3	yes	—
161	C₈	—	2	6	no	—
162	C₈	—	172	4	no	—
163	C₈	#13063322	1	4	yes	complemented
164	C₂²	—	172	12	no	—
165	C₂²	—	172	4	no	—
166	C₂²	—	172	12	no	—
167	C₂²	C₂×C₄×D₁₇₂	1	3	yes	—
168	C₂²	C₂²×#10621872	1	1	yes	characteristic
169	C₂²	C₂×#13063331	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	172	4	no	—
172	C₄	#14582978	1	12	yes	complemented
173	C₄	C₂×#13063322	1	1	yes	characteristic
174	C₄	#14583449	1	3	yes	—
175	C₂	—	344	4	no	—
176	C₂	C₄×#13063322	1	1	yes	characteristic
177	C₂	C₂×#14582978	1	3	yes	—
178	C₂	#19143308	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582978

C₂

yes

maximal, complemented

C₄×#13063322

C₂

yes

maximal, characteristic

#19135389

C₂

yes

maximal, complemented

#19134747

C₂

yes

maximal, characteristic, complemented

C₄²×C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×#10621872

C₂²

yes

—

C₂×#13063322

C₄

yes

characteristic

C₂×C₄×D₁₇₂

C₂²

yes

—

#14582978

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₈

C₂²

yes

—

#14582801

C₄

yes

—

#14582769

C₄

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×D₁₇₂

C₂×C₄

yes

—

C₂²×C₄×D₄₃

C₂³

yes

—

C₂×#10621872

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

#13063322

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

(C₂×D₄₃)⋊C₈

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

complemented

C₂³×D₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

C₂²×C₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

M₁₆

yes

complemented

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

#10621872

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×C₄×C₄₃

#131

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

—

C₂×C₄×C₄₃

C₂×#41

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#127

yes

complemented

C₄₃⋊C₈

—

#21681

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×D₄

yes

—

C₂²×C₄₃

C₂²×M₁₆

yes

characteristic

C₂²×C₄₃

C₂×#131

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

—

—

—

—

C₄²×C₈

D₄₃

yes

characteristic, complemented

C₂×C₄₃

C₄×#127

yes

characteristic

C₂×C₄₃

C₂×#436

yes

—

C₂×C₄₃

#2931

yes

—

D₄₃

—

C₂²×M₁₆

—

C₂×#127

—

C₂×C₄×D₄

—

#436

—

C₂×C₄⋊C₈

—

100

C₂×C₂²⋊C₈

—

101

C₂×C₄×C₈

C₂×D₄₃

yes

—

102

C₂×C₄×C₈

C₂×D₄₃

yes

—

103

#390

—

104

#384

—

105

C₄³

C₂×D₄₃

yes

characteristic

106

C₄₃

#21681

yes

characteristic, complemented, Sylow

107

C₂²×D₄

—

108

C₂³×C₄

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂²×D₄₃

yes

—

112

C₂²×C₈

C₂²×D₄₃

yes

characteristic

113

#127

—

114

C₄×D₄

—

115

C₄×D₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₄²

C₄×D₄₃

yes

—

119

C₂×C₄²

C₄×D₄₃

yes

characteristic

120

C₂×C₄²

C₂²×D₄₃

yes

—

121

C₄⋊C₈

—

122

C₂²⋊C₈

—

123

C₄×C₈

—

124

C₄×C₈

D₁₇₂

yes

complemented

125

C₂⁴

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×C₄×D₄₃

yes

—

133

C₂²×C₄

C₂×C₄×D₄₃

yes

—

134

C₂²×C₄

C₂³×D₄₃

yes

characteristic

135

M₁₆

—

172

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

C₂×D₁₇₂

yes

—

139

C₂×C₈

C₂×D₁₇₂

yes

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₄²

#10621872

yes

complemented

144

C₄²

C₂×D₁₇₂

yes

—

145

C₄²

#10621898

yes

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₂²×C₄×D₄₃

yes

characteristic

150

D₄

—

172

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

172

—

154

C₂×C₄

—

172

—

155

C₂×C₄

C₄×D₁₇₂

yes

—

156

C₂×C₄

C₄×D₁₇₂

yes

—

157

C₂×C₄

C₂×#10621872

yes

—

158

C₂×C₄

#13063331

yes

—

159

C₂×C₄

C₂²×D₁₇₂

yes

—

160

C₂×C₄

C₂×#10621898

yes

—

161

C₈

—

162

C₈

—

172

—

163

C₈

#13063322

yes

complemented

164

C₂²

—

172

—

165

C₂²

—

172

—

166

C₂²

—

172

—

167

C₂²

C₂×C₄×D₁₇₂

yes

—

168

C₂²

C₂²×#10621872

yes

characteristic

169

C₂²

C₂×#13063331

yes

—

170

C₄

—

171

C₄

—

172

—

172

C₄

#14582978

yes

complemented

173

C₄

C₂×#13063322

yes

characteristic

174

C₄

#14583449

yes

—

175

C₂

—

344

—

176

C₂

C₄×#13063322

yes

characteristic

177

C₂

C₂×#14582978

yes

—

178

C₂

#19143308

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₄²×C₈×C₄₃	C₂
prime radical	C₂×#13063322	C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#21681
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂³×C₄

Fitting

C₄²×C₈×C₄₃

C₂

prime radical

C₂×#13063322

C₄

cosocle

C₂²×C₄×C₄₃

C₂⁴

hypercenter

C₄²×C₈

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₄²×C₈

N	H ≅ G/N	type
1. C₂×#14582978	C₂	semidirect
3. #19135389	C₂	semidirect
4. #19134747	C₂	semidirect
5. C₄²×C₈×C₄₃	C₂	semidirect
34. C₄×C₈×C₄₃	D₄	semidirect
49. C₄²×C₄₃	M₁₆	semidirect
72. C₈×C₄₃	#127	semidirect
82. C₄×C₄₃	#436	semidirect
90. C₄²×C₈	D₄₃	semidirect
106. C₄₃	#21681	semidirect
124. C₄×C₈	D₁₇₂	semidirect
143. C₄²	#10621872	semidirect
163. C₈	#13063322	semidirect
172. C₄	#14582978	semidirect

H ≅ G/N

type

1. C₂×#14582978

C₂

semidirect

3. #19135389

C₂

semidirect

4. #19134747

C₂

semidirect

5. C₄²×C₈×C₄₃

C₂

semidirect

34. C₄×C₈×C₄₃

D₄

semidirect

49. C₄²×C₄₃

M₁₆

semidirect

72. C₈×C₄₃

#127

semidirect

82. C₄×C₄₃

#436

semidirect

90. C₄²×C₈

D₄₃

semidirect

106. C₄₃

#21681

semidirect

124. C₄×C₈

D₁₇₂

semidirect

143. C₄²

#10621872

semidirect

163. C₈

#13063322

semidirect

172. C₄

#14582978

semidirect

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

subgroup	index
C₂×#14582978	2
C₄×#13063322	2
#19135389	2
#19134747	2
C₄²×C₈×C₄₃	2
#21681	43

subgroup

index

C₄²×C₈×C₄₃

Subgroups — #25104983

Conjugacy classes of subgroups

Complements

Maximal subgroups