Group #24996061 — subgroups

Subgroups — #24996061

Order 11008.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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	#19134767	C₂	1	1	yes	maximal, characteristic
2	#19133267	C₂	1	4	yes	maximal
3	#19132878	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₃×#494	C₂	1	1	yes	maximal, characteristic
5	#14582952	C₂²	1	2	yes	—
6	C₂×#13063261	—	2	2	no	—
7	#14582923	C₂²	1	2	yes	—
8	#14582645	—	2	8	no	—
9	#14582620	—	2	1	no	—
10	C₈×C₄₃⋊C₈	—	2	1	no	—
11	C₂²×Q₁₆×C₄₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
13	C₄₃×#494	—	2	1	no	—
14	C₄₃×#494	—	2	1	no	—
15	C₄₃×#494	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621877	—	2	2	no	—
18	#13063261	—	4	4	no	—
19	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄×C₄₃⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₄₃	—	2	2	no	—
24	C₂×Q₁₆×C₄₃	—	2	4	no	—
25	C₂×Q₁₆×C₄₃	—	2	4	no	—
26	C₂×Q₁₆×C₄₃	—	4	4	no	—
27	C₂×Q₁₆×C₄₃	—	4	4	no	—
28	C₂²×C₈×C₄₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₃	—	2	4	no	—
30	C₄⋊Q₈×C₄₃	—	2	2	no	—
31	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
33	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₄₃	—	2	4	no	—
38	C₂×Q₈×C₄₃	—	4	2	no	—
39	C₂×Q₈×C₄₃	—	4	4	no	—
40	C₂×Q₈×C₄₃	—	4	4	no	—
41	C₂²×C₄×C₄₃	—	4	2	no	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	#41	1	2	yes	—
44	Q₁₆×C₄₃	—	4	8	no	—
45	Q₁₆×C₄₃	—	4	8	no	—
46	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₃	D₈	1	4	yes	—
49	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₄₃	—	2	8	no	—
51	C₄⋊C₄×C₄₃	—	4	4	no	—
52	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₄₃	C₂×D₄	1	2	yes	—
54	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
55	#10621877	—	4	4	no	—
56	C₂×C₄₃⋊C₈	—	2	4	no	—
57	C₂×C₄₃⋊C₈	—	4	2	no	—
58	C₂³×C₄₃	#124	1	1	yes	characteristic
59	Q₈×C₄₃	—	4	8	no	—
60	Q₈×C₄₃	—	8	2	no	—
61	Q₈×C₄₃	—	8	2	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	4	4	no	—
64	C₂×C₄×C₄₃	—	8	4	no	—
65	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	#121	1	2	yes	—
68	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
71	C₈×C₄₃	#102	1	4	yes	—
72	C₈×C₄₃	#102	1	4	yes	—
73	C₄₃⋊C₈	—	4	4	no	—
74	C₄₃⋊C₈	—	8	2	no	—
75	#9772	—	43	1	no	maximal, Sylow
76	C₂²×C₄₃	—	2	2	no	—
77	C₂²×C₄₃	#390	1	1	yes	characteristic
78	C₂²×C₄₃	#486	1	2	yes	—
79	C₄×C₄₃	—	2	2	no	—
80	C₄×C₄₃	—	8	4	no	—
81	C₄×C₄₃	C₂×#102	1	2	yes	—
82	C₄×C₄₃	#473	1	4	yes	—
83	C₄×C₄₃	#492	1	1	yes	characteristic
84	C₄×C₄₃	#493	1	1	yes	characteristic
85	C₂×#494	D₄₃	1	1	yes	characteristic
86	#1848	—	43	1	no	—
87	#1546	—	43	4	no	—
88	#1450	—	43	1	no	—
89	C₂×C₄₃	—	2	2	no	—
90	C₂×C₄₃	#1957	1	1	yes	characteristic
91	C₂×C₄₃	#1979	1	2	yes	—
92	C₂²×Q₁₆	—	2	2	no	—
93	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
94	#494	—	2	1	no	—
95	#494	—	2	1	no	—
96	#494	Dic₄₃	1	4	yes	—
97	#431	—	43	2	no	—
98	C₂×#108	—	86	2	no	—
99	#423	—	43	2	no	—
100	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
101	#332	—	86	8	no	—
102	#322	—	86	1	no	—
103	C₈²	—	86	1	no	—
104	C₄₃	#9772	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	86	2	no	—
111	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
117	#108	—	172	4	no	—
118	C₄⋊C₈	—	43	4	no	—
119	C₄⋊C₈	—	86	4	no	—
120	C₂²⋊C₈	—	86	2	no	—
121	C₄×C₈	—	86	2	no	—
122	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
123	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#10621900	1	2	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	172	4	no	—
135	C₂×C₈	—	86	4	no	—
136	C₂×C₈	—	172	2	no	—
137	C₂×C₈	#10621882	1	4	yes	—
138	C₂×C₈	#10621883	1	4	yes	—
139	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
140	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	D₄×D₄₃	1	2	yes	—
144	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
145	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
146	C₂³	#13063312	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
155	C₂×C₄	#13063296	1	2	yes	—
156	C₂×C₄	#13063321	1	1	yes	characteristic
157	C₂×C₄	C₂×#10621882	1	2	yes	—
158	C₂×C₄	C₂×#10621883	1	2	yes	—
159	C₈	—	344	2	no	—
160	C₈	—	172	4	no	—
161	C₈	#13063246	1	4	yes	—
162	C₈	#13063246	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#14582802	1	1	yes	characteristic
165	C₂²	#14583245	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#13063246	1	2	yes	—
169	C₄	#14583171	1	4	yes	—
170	C₄	#14583271	1	1	yes	characteristic
171	C₄	#14583275	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#19135402	1	1	yes	characteristic
174	C₂	#19135511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134767

C₂

yes

maximal, characteristic

#19133267

C₂

yes

maximal

#19132878

C₂

yes

maximal, characteristic

C₂×C₄₃×#494

C₂

yes

maximal, characteristic

#14582952

C₂²

yes

—

C₂×#13063261

—

#14582923

C₂²

yes

—

#14582645

—

#14582620

—

C₈×C₄₃⋊C₈

—

C₂²×Q₁₆×C₄₃

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#494

—

C₄₃×#494

—

C₄₃×#494

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

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

#121

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#102

yes

—

C₈×C₄₃

#102

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9772

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

#486

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#102

yes

—

C₄×C₄₃

#473

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#1957

yes

characteristic

C₂×C₄₃

#1979

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

#494

—

#494

—

#494

Dic₄₃

yes

—

#431

—

C₂×#108

—

#423

—

100

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

101

#332

—

102

#322

—

103

C₈²

—

104

C₄₃

#9772

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

C₂²⋊D₄₃

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₄₃

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₄₃

yes

characteristic

117

#108

—

172

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×Dic₄₃

yes

—

123

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

124

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

#10621900

yes

—

131

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

172

—

135

C₂×C₈

—

136

C₂×C₈

—

172

—

137

C₂×C₈

#10621882

yes

—

138

C₂×C₈

#10621883

yes

—

139

C₂×C₈

C₂²⋊Dic₄₃

yes

—

140

C₂×C₈

C₂²⋊Dic₄₃

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

D₄×D₄₃

yes

—

144

C₄²

C₂²×Dic₄₃

yes

characteristic

145

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

146

C₂³

#13063312

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

154

C₂×C₄

D₄×Dic₄₃

yes

—

155

C₂×C₄

#13063296

yes

—

156

C₂×C₄

#13063321

yes

characteristic

157

C₂×C₄

C₂×#10621882

yes

—

158

C₂×C₄

C₂×#10621883

yes

—

159

C₈

—

344

—

160

C₈

—

172

—

161

C₈

#13063246

yes

—

162

C₈

#13063246

yes

—

163

C₂²

—

164

C₂²

#14582802

yes

characteristic

165

C₂²

#14583245

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#13063246

yes

—

169

C₄

#14583171

yes

—

170

C₄

#14583271

yes

characteristic

171

C₄

#14583275

yes

characteristic

172

C₂

—

173

C₂

#19135402

yes

characteristic

174

C₂

#19135511

yes

—

	subgroup	G/H
center	C₂²	#14582802
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#494	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	#124
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#494	D₄₃
hypocenter	C₄₃	#9772
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#14582802

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₂×C₄₃×#494

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
104. C₄₃	#9772	semidirect

H ≅ G/N

type

104. C₄₃

#9772

semidirect

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

subgroup	index
#19134767	2
#19133267	2
#19132878	2
C₂×C₄₃×#494	2
#9772	43

subgroup

index

C₂×C₄₃×#494

Subgroups — #24996061

Conjugacy classes of subgroups

Complements

Maximal subgroups