Group #24994117 — subgroups

Subgroups — #24994117

Order 11008.

384 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 95 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	#19144977	C₂	1	1	yes	maximal, characteristic
2	#19143701	C₂	1	1	yes	maximal, characteristic
3	#19133311	C₂	1	2	yes	maximal
4	#19132468	C₂	1	2	yes	maximal
5	C₄₃×#1537	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	1	yes	characteristic
7	#14583269	—	2	2	no	—
8	#14583251	C₄	1	4	yes	—
9	#14583030	—	2	4	no	—
10	#14582960	—	2	2	no	—
11	#14582956	C₂²	1	2	yes	—
12	#14582907	—	2	2	no	—
13	C₄×#10621877	C₂²	1	1	yes	characteristic
14	#14582644	—	2	4	no	—
15	C₄₃×#533	C₂²	1	1	yes	characteristic
16	C₄₃×#432	C₂²	1	2	yes	—
17	C₄₃×#330	—	2	2	no	—
18	C₂²×Dic₈₆	—	2	1	no	—
19	C₂×#10621877	—	2	1	no	—
20	#13063325	—	2	2	no	—
21	#13063325	—	2	1	no	—
22	#13063325	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₄₃	—	2	2	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
25	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
26	#13063248	—	2	8	no	—
27	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
28	#13063236	—	2	2	no	—
29	#13063236	—	2	1	no	—
30	#13063236	—	2	2	no	—
31	C₄×C₄₃⋊C₈	—	2	2	no	—
32	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
34	#125×C₄₃	D₄	1	2	yes	—
35	C₂²⋊Q₈×C₄₃	—	2	2	no	—
36	C₄×Q₈×C₄₃	—	2	2	no	—
37	#117×C₄₃	D₄	1	2	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₃	—	2	2	no	—
40	C₂²⋊C₈×C₄₃	—	2	2	no	—
41	#97×C₄₃	—	2	2	no	—
42	C₂×Q₈×C₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	#41	1	2	yes	—
45	C₂²×Dic₄₃	—	4	1	no	—
46	C₂×Dic₈₆	—	2	2	no	—
47	C₂×Dic₈₆	—	4	4	no	—
48	C₂×Dic₈₆	—	4	1	no	—
49	C₂×C₈×C₄₃	—	4	2	no	—
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₄⋊C₄×C₄₃	—	2	4	no	—
52	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
53	C₂²⋊C₄×C₄₃	—	2	2	no	—
54	C₄²×C₄₃	—	2	2	no	—
55	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₃	C₂×D₄	1	2	yes	—
57	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Dic₄₃	—	2	4	no	—
59	C₄⋊Dic₄₃	—	4	2	no	—
60	#10621877	—	4	2	no	—
61	C₂×C₄₃⋊C₈	—	2	2	no	—
62	C₂×C₄₃⋊C₈	—	2	4	no	—
63	C₂³×C₄₃	#123	1	1	yes	characteristic
64	Q₈×C₄₃	—	8	2	no	—
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	4	no	—
67	C₂×C₄×C₄₃	—	4	2	no	—
68	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#121	1	2	yes	—
72	C₂×Dic₄₃	—	4	2	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	Dic₈₆	—	4	4	no	—
75	Dic₈₆	—	8	2	no	—
76	C₈×C₄₃	—	8	2	no	—
77	C₄₃⋊C₈	—	4	2	no	—
78	C₄₃⋊C₈	—	4	4	no	—
79	#9422	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	—	2	1	no	—
81	C₂²×C₄₃	—	2	2	no	—
82	C₂²×C₄₃	#386	1	1	yes	characteristic
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	4	4	no	—
87	C₄×C₄₃	#454	1	2	yes	—
88	Dic₄₃	—	8	2	no	—
89	#3139	—	43	1	no	—
90	#2987	—	43	1	no	—
91	#1556	—	43	2	no	—
92	#1537	D₄₃	1	1	yes	characteristic
93	#1291	—	43	2	no	—
94	C₂×C₄₃	—	4	1	no	—
95	C₂×C₄₃	#1888	1	1	yes	characteristic
96	C₂×C₄₃	#1902	1	2	yes	—
97	#533	C₂×D₄₃	1	1	yes	characteristic
98	C₂×C₄⋊Q₈	—	43	1	no	—
99	#491	—	86	2	no	—
100	#487	—	43	4	no	—
101	#446	—	86	4	no	—
102	#433	—	86	2	no	—
103	#432	—	43	2	no	—
104	#432	C₂×D₄₃	1	2	yes	—
105	#419	—	86	2	no	—
106	C₄×M₁₆	—	43	1	no	—
107	#332	—	86	4	no	—
108	#330	—	2	2	no	—
109	C₄₃	#9422	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	86	1	no	—
111	C₂×M₁₆	—	86	1	no	—
112	C₄⋊Q₈	—	43	2	no	—
113	C₄⋊Q₈	—	86	2	no	—
114	C₄⋊Q₈	—	86	1	no	—
115	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
116	#125	C₂²⋊D₄₃	1	2	yes	—
117	C₂²⋊Q₈	—	2	2	no	—
118	C₄×Q₈	—	2	2	no	—
119	#117	D₁₇₂	1	2	yes	—
120	C₂×C₄⋊C₄	—	86	2	no	—
121	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	43	4	no	—
124	C₄⋊C₈	—	86	4	no	—
125	#103	—	86	8	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	86	2	no	—
128	#97	—	2	2	no	—
129	#97	—	86	2	no	—
130	#97	—	86	2	no	—
131	#97	—	86	1	no	—
132	C₄×C₈	—	43	2	no	—
133	C₄×C₈	—	86	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	86	2	no	—
136	C₂×Q₈	—	172	4	no	—
137	C₂×Q₈	—	172	1	no	—
138	C₂²×C₄	—	172	1	no	—
139	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
140	C₂²×C₄	#10621900	1	2	yes	—
141	M₁₆	—	172	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	86	2	no	—
144	C₂×C₈	—	86	4	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	86	4	no	—
148	C₄⋊C₄	—	172	2	no	—
149	C₄⋊C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
153	C₄²	D₄×D₄₃	1	2	yes	—
154	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
155	C₂³	#13063309	1	1	yes	characteristic
156	Q₈	—	8	2	no	—
157	Q₈	—	344	2	no	—
158	Q₈	—	172	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	344	2	no	—
163	C₂×C₄	—	172	2	no	—
164	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
166	C₂×C₄	#13063295	1	1	yes	characteristic
167	C₂×C₄	#13063297	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	172	2	no	—
170	C₈	—	172	4	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#14582776	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	344	2	no	—
179	C₄	#14583074	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#19134996	1	1	yes	characteristic
182	C₂	#19135066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19144977

C₂

yes

maximal, characteristic

#19143701

C₂

yes

maximal, characteristic

#19133311

C₂

yes

maximal

#19132468

C₂

yes

maximal

C₄₃×#1537

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

characteristic

#14583269

—

#14583251

C₄

yes

—

#14583030

—

#14582960

—

#14582956

C₂²

yes

—

#14582907

—

C₄×#10621877

C₂²

yes

characteristic

#14582644

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#432

C₂²

yes

—

C₄₃×#330

—

C₂²×Dic₈₆

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063248

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

—

#125×C₄₃

D₄

yes

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#123

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9422

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#386

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#454

yes

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic

#1291

—

C₂×C₄₃

—

C₂×C₄₃

#1888

yes

characteristic

C₂×C₄₃

#1902

yes

—

#533

C₂×D₄₃

yes

characteristic

C₂×C₄⋊Q₈

—

#491

—

100

#487

—

101

#446

—

102

#433

—

103

#432

—

104

#432

C₂×D₄₃

yes

—

105

#419

—

106

C₄×M₁₆

—

107

#332

—

108

#330

—

109

C₄₃

#9422

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

C₄×D₄₃

yes

—

116

#125

C₂²⋊D₄₃

yes

—

117

C₂²⋊Q₈

—

118

C₄×Q₈

—

119

#117

D₁₇₂

yes

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₄₃

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

#103

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

128

#97

—

129

#97

—

130

#97

—

131

#97

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

172

—

137

C₂×Q₈

—

172

—

138

C₂²×C₄

—

172

—

139

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

140

C₂²×C₄

#10621900

yes

—

141

M₁₆

—

172

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

172

—

149

C₄⋊C₄

(C₂×D₄₃)⋊C₄

yes

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

C₂×C₄×D₄₃

yes

characteristic

153

C₄²

D₄×D₄₃

yes

—

154

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

155

C₂³

#13063309

yes

characteristic

156

Q₈

—

157

Q₈

—

344

—

158

Q₈

—

172

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

344

—

163

C₂×C₄

—

172

—

164

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

165

C₂×C₄

(C₂²⋊D₄₃)⋊C₄

yes

—

166

C₂×C₄

#13063295

yes

characteristic

167

C₂×C₄

#13063297

yes

—

168

C₈

—

169

C₈

—

172

—

170

C₈

—

172

—

171

C₂²

—

172

C₂²

—

173

C₂²

#14582776

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

344

—

179

C₄

#14583074

yes

—

180

C₂

—

181

C₂

#19134996

yes

characteristic

182

C₂

#19135066

yes

—

	subgroup	G/H
center	C₂²	#14582776
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1537	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂³×C₄₃	#123
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1537	D₄₃
hypocenter	C₄₃	#9422
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#14582776

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1537

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

N	H ≅ G/N	type
109. C₄₃	#9422	semidirect

H ≅ G/N

type

109. C₄₃

#9422

semidirect

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

subgroup	index
#19144977	2
#19143701	2
#19133311	2
#19132468	2
C₄₃×#1537	2
#9422	43

subgroup

index

C₄₃×#1537

Subgroups — #24994117

Conjugacy classes of subgroups

Complements

Maximal subgroups