Group #25014648 — subgroups

Subgroups — #25014648

Order 11008.

286 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14582729	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19135344	C₂	1	2	yes	maximal
4	#19132690	C₂	1	2	yes	maximal
5	C₄₃×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13063261	C₂²	1	2	yes	—
8	#14582926	—	2	2	no	—
9	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
10	#14582729	—	2	2	no	—
11	C₁₆×Dic₄₃	—	2	2	no	—
12	#14582658	—	2	2	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#427	C₂²	1	2	yes	—
15	C₄₃×#367	C₄	1	2	yes	complemented
16	C₄₃×#366	C₄	1	2	yes	complemented
17	C₂×#10621877	—	2	2	no	—
18	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	C₂×C₄₃⋊C₁₆	—	2	1	no	—
22	C₂×C₄₃⋊C₁₆	Q₈	1	2	yes	—
23	#13063261	—	2	2	no	—
24	#13063261	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	4	4	no	—
26	C₈×Dic₄₃	—	2	1	no	—
27	C₈×Dic₄₃	—	2	1	no	—
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	#117×C₄₃	—	2	2	no	—
30	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₄₃	Q₈	1	2	yes	—
34	#107×C₄₃	C₂×C₄	1	4	yes	—
35	#106×C₄₃	—	2	2	no	—
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
43	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₂²⋊C₄×C₄₃	—	4	2	no	—
46	C₄²×C₄₃	—	4	2	no	—
47	C₄×Dic₄₃	—	2	1	no	—
48	C₄×Dic₄₃	Q₁₆	1	2	yes	—
49	#10621877	—	4	2	no	—
50	#10621877	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	2	1	no	—
52	C₂×C₄₃⋊C₈	—	2	1	no	—
53	C₂×C₄₃⋊C₈	—	4	2	no	—
54	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
55	C₁₆×C₄₃	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₄₃	C₄⋊C₄	1	2	yes	—
57	C₄₃⋊C₁₆	—	2	2	no	—
58	C₂³×C₄₃	#135	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	4	4	no	—
60	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
66	C₂×Dic₄₃	—	2	4	no	—
67	C₂×Dic₄₃	—	2	2	no	—
68	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
70	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₈×C₄₃	#125	1	1	yes	characteristic
72	C₄₃⋊C₈	—	2	2	no	—
73	C₄₃⋊C₈	—	8	2	no	—
74	#13169	—	43	1	no	maximal, Sylow
75	C₂²×C₄₃	—	2	2	no	—
76	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
77	C₂²×C₄₃	#467	1	1	yes	characteristic
78	C₂²×C₄₃	#499	1	1	yes	characteristic
79	C₄×C₄₃	—	8	4	no	—
80	C₄×C₄₃	#389	1	1	yes	characteristic
81	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
82	C₄×C₄₃	#466	1	1	yes	characteristic
83	C₄×C₄₃	#500	1	1	yes	characteristic
84	Dic₄₃	—	2	4	no	—
85	C₂×#368	—	43	1	no	—
86	#2158	D₄₃	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	43	1	no	—
88	#1945	—	43	2	no	—
89	#1382	—	43	2	no	—
90	C₂×C₄₃	—	2	1	no	—
91	C₂×C₄₃	—	2	1	no	—
92	C₂×C₄₃	#1942	1	1	yes	characteristic
93	C₂×C₄₃	#2178	1	2	yes	—
94	C₂²×C₁₆	—	43	1	no	—
95	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
96	C₂×#108	—	43	2	no	—
97	#427	C₂×D₄₃	1	2	yes	—
98	#424	—	86	2	no	—
99	C₂×C₄×C₈	—	43	1	no	—
100	#368	—	86	2	no	—
101	#367	Dic₄₃	1	2	yes	complemented
102	#366	Dic₄₃	1	2	yes	complemented
103	C₄×C₁₆	—	86	2	no	—
104	#339	—	86	2	no	—
105	C₄₃	#13169	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	—	86	2	no	—
107	C₂²×C₈	—	43	1	no	—
108	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
109	#117	—	2	2	no	—
110	C₂×C₄²	—	43	1	no	—
111	C₂×C₁₆	—	43	2	no	—
112	C₂×C₁₆	—	86	1	no	—
113	C₂×C₁₆	—	86	1	no	—
114	C₂×C₁₆	Dic₈₆	1	2	yes	—
115	C₂×C₁₆	D₁₇₂	1	2	yes	—
116	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
117	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
118	#108	—	86	2	no	—
119	#108	—	86	2	no	—
120	#107	C₂×Dic₄₃	1	4	yes	—
121	#106	—	2	2	no	—
122	C₄⋊C₈	—	172	4	no	—
123	C₄×C₈	—	86	1	no	—
124	C₄×C₈	—	86	1	no	—
125	C₂²×C₄	—	86	1	no	—
126	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
127	M₁₆	—	172	2	no	—
128	M₁₆	—	172	2	no	—
129	C₂×C₈	—	43	2	no	—
130	C₂×C₈	—	86	1	no	—
131	C₂×C₈	—	86	1	no	—
132	C₂×C₈	—	172	2	no	—
133	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
134	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
135	C₂×C₈	#10621900	1	1	yes	characteristic
136	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
137	C₂×C₈	#10621902	1	1	yes	characteristic
138	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	4	2	no	—
142	C₄²	—	43	2	no	—
143	C₄²	—	86	1	no	—
144	C₁₆	—	86	2	no	—
145	C₁₆	C₄⋊Dic₄₃	1	2	yes	—
146	C₁₆	C₄⋊Dic₄₃	1	2	yes	—
147	C₂³	#13063344	1	1	yes	characteristic
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	86	4	no	—
150	C₂×C₄	—	86	2	no	—
151	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
152	C₂×C₄	#13063300	1	1	yes	characteristic
153	C₂×C₄	#13063324	1	1	yes	characteristic
154	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
156	C₂×C₄	#13063348	1	1	yes	characteristic
157	C₈	—	86	2	no	—
158	C₈	—	344	2	no	—
159	C₈	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
160	C₈	Q₈×Dic₄₃	1	1	yes	characteristic
161	C₈	#13063305	1	1	yes	characteristic
162	C₈	#13063316	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	D₈×Dic₄₃	1	1	yes	characteristic
165	C₂²	#14583141	1	1	yes	characteristic
166	C₂²	#14583300	1	1	yes	characteristic
167	C₄	—	8	4	no	—
168	C₄	—	86	4	no	—
169	C₄	#14582798	1	1	yes	characteristic
170	C₄	Q₁₆×Dic₄₃	1	1	yes	characteristic
171	C₄	#14583136	1	1	yes	characteristic
172	C₄	#14583303	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#19135329	1	1	yes	characteristic
176	C₂	#19136476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582729

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₄₃

C₂

yes

maximal, characteristic

#19135344

C₂

yes

maximal

#19132690

C₂

yes

maximal

C₄₃×#2158

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₂×#13063261

C₂²

yes

—

#14582926

—

C₂×C₈×Dic₄₃

C₂²

yes

characteristic

#14582729

—

C₁₆×Dic₄₃

—

#14582658

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₄₃×#427

C₂²

yes

—

C₄₃×#367

C₄

yes

complemented

C₄₃×#366

C₄

yes

complemented

C₂×#10621877

—

C₂²×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

Q₈

yes

—

#13063261

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#117×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

Q₈

yes

—

#107×C₄₃

C₂×C₄

yes

—

#106×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

C₂×C₈×C₄₃

yes

C₂×C₈×C₄₃

yes

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

Q₁₆

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₈

yes

—

C₁₆×C₄₃

C₄⋊C₄

yes

—

C₁₆×C₄₃

C₄⋊C₄

yes

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#135

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₃

C₄×Q₈

yes

characteristic

C₈×C₄₃

C₂²⋊Q₈

yes

characteristic

C₈×C₄₃

#125

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#13169

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₄×D₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

#389

yes

characteristic

C₄×C₄₃

C₄×Q₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

C₂×#368

—

#2158

D₄₃

yes

characteristic

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1942

yes

characteristic

C₂×C₄₃

#2178

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

C₂×#108

—

#427

C₂×D₄₃

yes

—

#424

—

C₂×C₄×C₈

—

100

#368

—

101

#367

Dic₄₃

yes

complemented

102

#366

Dic₄₃

yes

complemented

103

C₄×C₁₆

—

104

#339

—

105

C₄₃

#13169

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₄₃

yes

characteristic

109

#117

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

Dic₈₆

yes

—

115

C₂×C₁₆

D₁₇₂

yes

—

116

C₂×C₁₆

C₂×Dic₄₃

yes

characteristic

117

C₂×C₁₆

C₂×Dic₄₃

yes

characteristic

118

#108

—

119

#108

—

120

#107

C₂×Dic₄₃

yes

—

121

#106

—

122

C₄⋊C₈

—

172

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

—

126

C₂²×C₄

D₄×D₄₃

yes

characteristic

127

M₁₆

—

172

—

128

M₁₆

—

172

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

172

—

133

C₂×C₈

C₂×Dic₈₆

yes

characteristic

134

C₂×C₈

C₂×D₁₇₂

yes

characteristic

135

C₂×C₈

#10621900

yes

characteristic

136

C₂×C₈

Q₈×D₄₃

yes

characteristic

137

C₂×C₈

#10621902

yes

characteristic

138

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₁₆

—

145

C₁₆

C₄⋊Dic₄₃

yes

—

146

C₁₆

C₄⋊Dic₄₃

yes

—

147

C₂³

#13063344

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

D₄×Dic₄₃

yes

characteristic

152

C₂×C₄

#13063300

yes

characteristic

153

C₂×C₄

#13063324

yes

characteristic

154

C₂×C₄

D₈×D₄₃

yes

characteristic

155

C₂×C₄

Q₁₆×D₄₃

yes

characteristic

156

C₂×C₄

#13063348

yes

characteristic

157

C₈

—

158

C₈

—

344

—

159

C₈

C₂×C₄⋊Dic₄₃

yes

characteristic

160

C₈

Q₈×Dic₄₃

yes

characteristic

161

C₈

#13063305

yes

characteristic

162

C₈

#13063316

yes

characteristic

163

C₂²

—

164

C₂²

D₈×Dic₄₃

yes

characteristic

165

C₂²

#14583141

yes

characteristic

166

C₂²

#14583300

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

#14582798

yes

characteristic

170

C₄

Q₁₆×Dic₄₃

yes

characteristic

171

C₄

#14583136

yes

characteristic

172

C₄

#14583303

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#19135329

yes

characteristic

176

C₂

#19136476

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2158	C₂
socle	C₂²×C₄₃	C₄×D₈
prime radical	C₂³×C₄₃	#135
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2158	D₄₃
hypocenter	C₄₃	#13169
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₄₃

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#2158

C₂

socle

C₂²×C₄₃

C₄×D₈

prime radical

C₂³×C₄₃

#135

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
15. C₄₃×#367	C₄	semidirect
16. C₄₃×#366	C₄	semidirect
101. #367	Dic₄₃	semidirect
102. #366	Dic₄₃	semidirect
105. C₄₃	#13169	semidirect

H ≅ G/N

type

15. C₄₃×#367

C₄

semidirect

16. C₄₃×#366

C₄

semidirect

101. #367

Dic₄₃

semidirect

102. #366

Dic₄₃

semidirect

105. C₄₃

#13169

semidirect

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

subgroup	index
C₂×#14582729	2
C₂×C₁₆×Dic₄₃	2
#19135344	2
#19132690	2
C₄₃×#2158	2
#13169	43

subgroup

index

C₂×C₁₆×Dic₄₃

C₄₃×#2158

Subgroups — #25014648

Conjugacy classes of subgroups

Complements

Maximal subgroups