Group #25015191 — subgroups

Subgroups — #25015191

Order 11008.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#19136351	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
3	#19132703	C₂	1	4	yes	maximal
4	C₂×C₄₃×#499	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₂²	1	2	yes	—
6	C₂×#13063261	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
9	#14582664	—	2	1	no	—
10	C₂×#125×C₄₃	—	2	1	no	—
11	C₄₃×#499	—	2	2	no	—
12	C₄₃×#499	C₄	1	4	yes	—
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₂×#106×C₄₃	—	2	1	no	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#10621877	—	2	2	no	—
17	#13063265	—	2	4	no	—
18	C₂×C₄₃⋊C₁₆	—	2	2	no	—
19	#13063261	—	2	2	no	—
20	#13063261	Q₈	1	4	yes	—
21	#13063236	—	4	1	no	—
22	C₄×C₄₃⋊C₈	—	4	1	no	—
23	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
25	#125×C₄₃	—	2	4	no	—
26	#125×C₄₃	—	4	1	no	—
27	#125×C₄₃	—	4	1	no	—
28	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
30	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
31	#107×C₄₃	—	2	2	no	—
32	#107×C₄₃	C₂×C₄	1	4	yes	—
33	#106×C₄₃	—	2	4	no	—
34	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₃	D₄	1	2	yes	—
36	C₂²×C₄×C₄₃	—	4	2	no	—
37	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	—	2	2	no	—
40	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₄₃	#41	1	2	yes	—
45	C₄⋊C₄×C₄₃	—	2	4	no	—
46	C₄⋊C₄×C₄₃	—	4	4	no	—
47	C₄⋊C₄×C₄₃	—	4	4	no	—
48	C₄⋊C₄×C₄₃	—	4	2	no	—
49	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₄₃	D₈	1	1	yes	characteristic
51	C₄²×C₄₃	D₈	1	1	yes	characteristic
52	#10621877	—	2	4	no	—
53	#10621877	—	4	2	no	—
54	C₂×C₄₃⋊C₈	—	4	2	no	—
55	C₄₃⋊C₁₆	—	4	2	no	—
56	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₃	—	4	2	no	—
58	C₂×C₄×C₄₃	—	4	4	no	—
59	C₂×C₄×C₄₃	—	8	4	no	—
60	C₂×C₄×C₄₃	#102	1	2	yes	—
61	C₂×C₄×C₄₃	#102	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#137	1	2	yes	—
66	C₈×C₄₃	—	2	2	no	—
67	C₈×C₄₃	C₄×Q₈	1	2	yes	—
68	C₈×C₄₃	#124	1	1	yes	characteristic
69	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₄₃⋊C₈	—	4	4	no	—
71	#13280	—	43	1	no	maximal, Sylow
72	C₂²×C₄₃	—	2	2	no	—
73	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
74	C₂²×C₄₃	#475	1	2	yes	—
75	C₄×C₄₃	—	2	2	no	—
76	C₄×C₄₃	—	2	2	no	—
77	C₄×C₄₃	—	8	4	no	—
78	C₄×C₄₃	#391	1	1	yes	characteristic
79	C₄×C₄₃	#419	1	1	yes	characteristic
80	C₄×C₄₃	#474	1	2	yes	—
81	C₂×#499	D₄₃	1	1	yes	characteristic
82	#2150	—	43	1	no	—
83	C₄×#108	—	43	1	no	—
84	#1387	—	43	4	no	—
85	C₂×C₄₃	—	2	2	no	—
86	C₂×C₄₃	#1988	1	1	yes	characteristic
87	C₂×C₄₃	#2148	1	2	yes	—
88	C₂×#125	—	2	1	no	—
89	C₂×M₃₂	—	43	2	no	—
90	#499	—	2	2	no	—
91	#499	Dic₄₃	1	4	yes	—
92	C₂×#108	—	43	2	no	—
93	C₂×#107	C₂×D₄₃	1	2	yes	—
94	C₂×#106	—	2	1	no	—
95	C₄×M₁₆	—	86	1	no	—
96	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	86	2	no	—
98	#341	—	86	1	no	—
99	C₄₃	#13280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	86	2	no	—
101	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	1	no	—
105	#125	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
109	M₃₂	—	86	4	no	—
110	C₂×C₁₆	—	86	2	no	—
111	#108	—	43	4	no	—
112	#108	—	86	2	no	—
113	#107	—	2	2	no	—
114	#107	C₂×Dic₄₃	1	4	yes	—
115	#106	—	2	4	no	—
116	#97	—	172	1	no	—
117	C₄×C₈	—	172	1	no	—
118	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
119	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#10621883	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
123	M₁₆	—	86	4	no	—
124	M₁₆	—	172	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	172	2	no	—
127	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
128	C₂×C₈	Q₈×D₄₃	1	2	yes	—
129	C₂×C₈	#10621902	1	2	yes	—
130	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	#10621882	1	1	yes	characteristic
137	C₄²	#10621882	1	1	yes	characteristic
138	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
139	C₁₆	—	172	2	no	—
140	C₂³	C₂×#10621883	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#13063246	1	2	yes	—
145	C₂×C₄	#13063246	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
147	C₂×C₄	#13063301	1	2	yes	—
148	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
149	C₂×C₄	#13063356	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	172	4	no	—
152	C₈	Q₈×Dic₄₃	1	2	yes	—
153	C₈	#13063314	1	1	yes	characteristic
154	C₈	#13063323	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#13063246	1	1	yes	characteristic
157	C₂²	#14583181	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#14582806	1	1	yes	characteristic
162	C₄	#14582908	1	1	yes	characteristic
163	C₄	#14583176	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#19135548	1	1	yes	characteristic
166	C₂	#19136344	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136351

C₂

yes

maximal, characteristic

C₄×#13063261

C₂

yes

maximal, characteristic

#19132703

C₂

yes

maximal

C₂×C₄₃×#499

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

C₂²

yes

—

C₄×#10621877

—

C₄⋊(C₄₃⋊C₁₆)

—

#14582664

—

C₂×#125×C₄₃

—

C₄₃×#499

—

C₄₃×#499

C₄

yes

—

C₂×#107×C₄₃

C₂²

yes

—

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063261

Q₈

yes

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#125×C₄₃

—

#125×C₄₃

—

#125×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#107×C₄₃

C₂×C₄

yes

—

#106×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#137

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₄×Q₈

yes

—

C₈×C₄₃

#124

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

characteristic

C₄₃⋊C₈

—

#13280

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#102

yes

characteristic

C₂²×C₄₃

#475

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×#499

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#1988

yes

characteristic

C₂×C₄₃

#2148

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

Dic₄₃

yes

—

C₂×#108

—

C₂×#107

C₂×D₄₃

yes

—

C₂×#106

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄⋊C₁₆

—

#341

—

C₄₃

#13280

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

102

C₂²×C₈

C₂²×D₄₃

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

112

#108

—

113

#107

—

114

#107

C₂×Dic₄₃

yes

—

115

#106

—

116

#97

—

172

—

117

C₄×C₈

—

172

—

118

C₄×C₈

C₂×Dic₄₃

yes

—

119

C₄×C₈

C₂²⋊D₄₃

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

#10621883

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

172

—

125

C₂×C₈

—

126

C₂×C₈

—

172

—

127

C₂×C₈

C₂²⋊Dic₄₃

yes

—

128

C₂×C₈

Q₈×D₄₃

yes

—

129

C₂×C₈

#10621902

yes

—

130

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

#10621882

yes

characteristic

137

C₄²

#10621882

yes

characteristic

138

C₄²

C₂²⋊Dic₄₃

yes

—

139

C₁₆

—

172

—

140

C₂³

C₂×#10621883

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#13063246

yes

—

145

C₂×C₄

#13063246

yes

—

146

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

147

C₂×C₄

#13063301

yes

—

148

C₂×C₄

C₂×#10621882

yes

characteristic

149

C₂×C₄

#13063356

yes

—

150

C₈

—

151

C₈

—

172

—

152

C₈

Q₈×Dic₄₃

yes

—

153

C₈

#13063314

yes

characteristic

154

C₈

#13063323

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#13063246

yes

characteristic

157

C₂²

#14583181

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#14582806

yes

characteristic

162

C₄

#14582908

yes

characteristic

163

C₄

#14583176

yes

—

164

C₂

—

165

C₂

#19135548

yes

characteristic

166

C₂

#19136344

yes

—

	subgroup	G/H
center	C₂²	C₂×#13063246
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#499	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#499	D₄₃
hypocenter	C₄₃	#13280
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

C₂×#13063246

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#499

C₂

socle

C₂²×C₄₃

C₂×#102

prime radical

C₂³×C₄₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
99. C₄₃	#13280	semidirect

H ≅ G/N

type

99. C₄₃

#13280

semidirect

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

subgroup	index
#19136351	2
C₄×#13063261	2
#19132703	2
C₂×C₄₃×#499	2
#13280	43

subgroup

index

C₂×C₄₃×#499

Subgroups — #25015191

Conjugacy classes of subgroups

Complements

Maximal subgroups