Group #25001584 — subgroups

Subgroups — #25001584

Order 11008.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#19135535	C₂	1	1	yes	maximal, characteristic
2	#19134770	C₂	1	1	yes	maximal, characteristic
3	#19133370	C₂	1	2	yes	maximal
4	#19133025	C₂	1	2	yes	maximal
5	C₄₃×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂×#13063261	—	2	2	no	—
8	#14582923	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	#14582653	—	2	2	no	—
11	#14582652	—	2	4	no	—
12	#14582631	—	2	4	no	—
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₆	Q₈	1	1	yes	characteristic
16	C₂×#10621877	—	2	2	no	—
17	C₂²×C₄₃⋊C₈	—	2	1	no	—
18	C₄×Dic₈₆	—	2	2	no	—
19	C₄×Dic₈₆	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
21	C₂×Dic₄₃⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₄₃	—	2	1	no	—
23	#13063261	—	4	4	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
27	C₂²×C₈×C₄₃	D₄	1	2	yes	—
28	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₃	—	2	2	no	—
30	C₂²⋊C₈×C₄₃	—	2	2	no	—
31	#97×C₄₃	—	2	2	no	—
32	C₄×C₈×C₄₃	D₄	1	4	yes	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
36	C₂²×Dic₄₃	—	2	1	no	—
37	C₂²×Dic₄₃	—	2	1	no	—
38	C₂×Dic₈₆	—	2	2	no	—
39	C₂×Dic₈₆	—	4	2	no	—
40	C₂×C₈×C₄₃	—	2	2	no	—
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	4	no	—
43	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
44	C₄²×C₄₃	C₂×D₄	1	2	yes	—
45	C₄²×C₄₃	#41	1	2	yes	—
46	C₄⋊Dic₄₃	—	2	2	no	—
47	Dic₄₃⋊C₄	—	4	2	no	—
48	C₄×Dic₄₃	—	4	1	no	—
49	C₄×Dic₄₃	—	4	1	no	—
50	#10621877	—	4	4	no	—
51	C₂×C₄₃⋊C₈	—	4	1	no	—
52	C₂×C₄₃⋊C₈	—	4	2	no	—
53	C₂×C₄₃⋊C₈	—	4	2	no	—
54	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	—	4	1	no	—
57	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₄₃	#123	1	2	yes	—
59	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×Dic₄₃	—	2	2	no	—
64	C₂×Dic₄₃	—	4	1	no	—
65	C₂×Dic₄₃	—	4	1	no	—
66	C₂×Dic₄₃	—	4	2	no	—
67	C₂×Dic₄₃	—	4	1	no	—
68	Dic₈₆	—	2	4	no	—
69	Dic₈₆	—	4	2	no	—
70	C₈×C₄₃	—	2	4	no	—
71	C₈×C₄₃	—	4	4	no	—
72	C₄₃⋊C₈	—	8	1	no	—
73	C₄₃⋊C₈	—	8	2	no	—
74	#10662	—	43	1	no	maximal, Sylow
75	C₂²×C₄₃	—	2	1	no	—
76	C₂²×C₄₃	—	2	1	no	—
77	C₂²×C₄₃	#399	1	1	yes	characteristic
78	C₂²×C₄₃	#477	1	2	yes	—
79	C₄×C₄₃	—	2	2	no	—
80	C₄×C₄₃	—	2	2	no	—
81	C₄×C₄₃	#484	1	2	yes	—
82	C₄×C₄₃	#494	1	1	yes	characteristic
83	C₄×C₄₃	#496	1	1	yes	characteristic
84	Dic₄₃	—	4	2	no	—
85	Dic₄₃	—	8	1	no	—
86	#1985	—	43	1	no	—
87	#1849	—	43	1	no	—
88	#1571	—	43	2	no	—
89	#1487	—	43	2	no	—
90	#1275	D₄₃	1	1	yes	characteristic
91	C₂×C₄₃	—	2	2	no	—
92	C₂×C₄₃	#2091	1	1	yes	characteristic
93	C₂×C₄₃	#2095	1	1	yes	characteristic
94	C₂×C₄₃	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	—	43	1	no	—
96	#431	C₂×D₄₃	1	2	yes	—
97	C₂×#108	—	86	2	no	—
98	#423	—	43	2	no	—
99	C₂×C₄⋊C₈	—	43	1	no	—
100	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
101	#336	—	86	2	no	—
102	#335	—	86	4	no	—
103	Q₈⋊C₈	—	86	4	no	—
104	C₄₃	#10662	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	43	1	no	—
106	C₂×M₁₆	—	86	2	no	—
107	C₂²×C₈	—	86	1	no	—
108	C₂²×C₈	D₁₇₂	1	2	yes	—
109	C₄×Q₈	—	86	2	no	—
110	C₄×Q₈	—	86	2	no	—
111	C₂×C₄⋊C₄	—	43	1	no	—
112	C₂×C₄⋊C₄	—	86	1	no	—
113	C₂×C₄²	—	86	1	no	—
114	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
115	#108	—	172	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	86	2	no	—
118	C₄⋊C₈	—	86	2	no	—
119	C₄⋊C₈	—	86	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	D₁₇₂	1	4	yes	—
123	C₂×Q₈	—	86	2	no	—
124	C₂×Q₈	—	172	2	no	—
125	C₂²×C₄	—	86	1	no	—
126	C₂²×C₄	—	86	1	no	—
127	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
128	C₂²×C₄	#10621900	1	1	yes	characteristic
129	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
130	M₁₆	—	172	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	172	1	no	—
135	C₂×C₈	—	172	2	no	—
136	C₂×C₈	—	172	2	no	—
137	C₂×C₈	Dic₁₇₂	1	4	yes	—
138	C₄⋊C₄	—	86	2	no	—
139	C₄⋊C₄	—	172	2	no	—
140	C₄²	—	172	1	no	—
141	C₄²	—	172	1	no	—
142	C₄²	C₂×D₁₇₂	1	2	yes	—
143	C₄²	#10621900	1	2	yes	—
144	C₂³	#13063305	1	1	yes	characteristic
145	Q₈	—	86	4	no	—
146	Q₈	—	172	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	86	2	no	—
150	C₂×C₄	—	172	2	no	—
151	C₂×C₄	—	172	1	no	—
152	C₂×C₄	—	172	1	no	—
153	C₂×C₄	—	172	1	no	—
154	C₂×C₄	#13063305	1	2	yes	—
155	C₂×C₄	#13063309	1	2	yes	—
156	C₂×C₄	#13063309	1	1	yes	characteristic
157	C₂×C₄	#13063322	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
159	C₂×C₄	#13063353	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	344	1	no	—
163	C₈	—	344	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#14582836	1	1	yes	characteristic
167	C₂²	#14583195	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	344	1	no	—
171	C₄	—	172	2	no	—
172	C₄	#14583235	1	2	yes	—
173	C₄	#14583279	1	1	yes	characteristic
174	C₄	#14583288	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#19136112	1	1	yes	characteristic
177	C₂	#19136132	1	1	yes	characteristic
178	C₂	#19136149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135535

C₂

yes

maximal, characteristic

#19134770

C₂

yes

maximal, characteristic

#19133370

C₂

yes

maximal

#19133025

C₂

yes

maximal

C₄₃×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₂²

yes

characteristic

C₂×#13063261

—

#14582923

C₂²

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

C₄₃×#431

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

Q₈

yes

characteristic

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

Q₈

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10662

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#399

yes

characteristic

C₂²×C₄₃

#477

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#484

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×Q₈

—

#431

C₂×D₄₃

yes

—

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

100

C₂×#97

C₂×D₄₃

yes

characteristic

101

#336

—

102

#335

—

103

Q₈⋊C₈

—

104

C₄₃

#10662

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

D₁₇₂

yes

—

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂²×D₄₃

yes

characteristic

115

#108

—

172

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

D₁₇₂

yes

—

123

C₂×Q₈

—

124

C₂×Q₈

—

172

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

128

C₂²×C₄

#10621900

yes

characteristic

129

C₂²×C₄

Q₈×D₄₃

yes

characteristic

130

M₁₆

—

172

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

172

—

135

C₂×C₈

—

172

—

136

C₂×C₈

—

172

—

137

C₂×C₈

Dic₁₇₂

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

172

—

140

C₄²

—

172

—

141

C₄²

—

172

—

142

C₄²

C₂×D₁₇₂

yes

—

143

C₄²

#10621900

yes

—

144

C₂³

#13063305

yes

characteristic

145

Q₈

—

146

Q₈

—

172

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

172

—

151

C₂×C₄

—

172

—

152

C₂×C₄

—

172

—

153

C₂×C₄

—

172

—

154

C₂×C₄

#13063305

yes

—

155

C₂×C₄

#13063309

yes

—

156

C₂×C₄

#13063309

yes

characteristic

157

C₂×C₄

#13063322

yes

characteristic

158

C₂×C₄

C₂×Dic₁₇₂

yes

—

159

C₂×C₄

#13063353

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

344

—

163

C₈

—

344

—

164

C₂²

—

165

C₂²

—

166

C₂²

#14582836

yes

characteristic

167

C₂²

#14583195

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

344

—

171

C₄

—

172

—

172

C₄

#14583235

yes

—

173

C₄

#14583279

yes

characteristic

174

C₄

#14583288

yes

characteristic

175

C₂

—

176

C₂

#19136112

yes

characteristic

177

C₂

#19136132

yes

characteristic

178

C₂

#19136149

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063322
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1275	C₂
socle	C₂²×C₄₃	#399
prime radical	C₂³×C₄₃	C₂²⋊Q₈
hypercenter	#1275	D₄₃
hypocenter	C₄₃	#10662
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#13063322

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1275

C₂

socle

C₂²×C₄₃

#399

prime radical

C₂³×C₄₃

C₂²⋊Q₈

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
104. C₄₃	#10662	semidirect

H ≅ G/N

type

104. C₄₃

#10662

semidirect

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

subgroup	index
#19135535	2
#19134770	2
#19133370	2
#19133025	2
C₄₃×#1275	2
#10662	43

subgroup

index

C₄₃×#1275

Subgroups — #25001584

Conjugacy classes of subgroups

Complements

Maximal subgroups