Group #24989591 — subgroups

Subgroups — #24989591

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	#19143530	C₂	1	1	yes	maximal, characteristic
2	#19133356	C₂	1	2	yes	maximal
3	#19133233	C₂	1	1	yes	maximal, characteristic
4	#19132505	C₂	1	2	yes	maximal
5	C₄₃×#1537	C₂	1	1	yes	maximal, characteristic
6	#14583030	—	2	2	no	—
7	Q₈×C₄₃⋊C₈	—	2	2	no	—
8	#14582956	C₂²	1	2	yes	—
9	#14582926	C₄	1	2	yes	—
10	#14582923	C₂²	1	1	yes	characteristic
11	#14582855	C₄	1	2	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	#14582651	—	2	2	no	—
14	#14582649	—	2	2	no	—
15	#14582641	—	2	2	no	—
16	C₄₃×#533	C₂²	1	1	yes	characteristic
17	C₄₃×#432	C₂²	1	2	yes	—
18	C₄₃×#330	—	2	2	no	—
19	C₂×#10621877	C₂×C₄	1	2	yes	—
20	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
26	#13063236	—	2	2	no	—
27	#13063236	—	2	2	no	—
28	C₄×C₄₃⋊C₈	—	2	2	no	—
29	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
30	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
31	C₄⋊Q₈×C₄₃	Q₈	1	2	yes	—
32	#125×C₄₃	D₄	1	2	yes	—
33	C₂²⋊Q₈×C₄₃	—	2	2	no	—
34	C₄×Q₈×C₄₃	—	2	2	no	—
35	#117×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₃	—	2	2	no	—
38	C₂²⋊C₈×C₄₃	—	2	2	no	—
39	#97×C₄₃	—	2	2	no	—
40	C₂×Q₈×C₄₃	—	2	2	no	—
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	#41	1	2	yes	—
43	C₂×C₈×C₄₃	—	4	2	no	—
44	C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
47	C₂²⋊C₄×C₄₃	—	2	2	no	—
48	C₄²×C₄₃	—	2	2	no	—
49	C₄²×C₄₃	C₂×D₄	1	2	yes	—
50	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	#10621877	—	4	2	no	—
53	C₂×C₄₃⋊C₈	—	2	2	no	—
54	C₂×C₄₃⋊C₈	—	2	2	no	—
55	C₂×C₄₃⋊C₈	—	4	1	no	—
56	C₂×C₄₃⋊C₈	—	4	2	no	—
57	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×C₄₃	#117	1	1	yes	characteristic
61	Q₈×C₄₃	—	4	4	no	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	2	4	no	—
65	C₂×C₄×C₄₃	—	4	2	no	—
66	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#121	1	2	yes	—
69	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₄₃	—	8	2	no	—
71	C₄₃⋊C₈	—	2	1	no	—
72	C₄₃⋊C₈	—	2	2	no	—
73	C₄₃⋊C₈	—	4	2	no	—
74	C₄₃⋊C₈	—	8	2	no	—
75	C₄₃⋊C₈	#101	1	2	yes	—
76	#8631	—	43	1	no	maximal, Sylow
77	C₂²×C₄₃	—	2	1	no	—
78	C₂²×C₄₃	—	2	2	no	—
79	C₂²×C₄₃	#380	1	1	yes	characteristic
80	C₄×C₄₃	—	2	1	no	—
81	C₄×C₄₃	—	2	2	no	—
82	C₄×C₄₃	—	2	2	no	—
83	C₄×C₄₃	—	4	4	no	—
84	C₄×C₄₃	C₂×#101	1	1	yes	characteristic
85	C₄×C₄₃	#413	1	1	yes	characteristic
86	#2960	—	43	1	no	—
87	#1568	—	43	2	no	—
88	#1537	—	43	1	no	—
89	#1537	D₄₃	1	1	yes	characteristic
90	#1307	—	43	2	no	—
91	C₂×C₄₃	—	4	1	no	—
92	C₂×C₄₃	#1801	1	1	yes	characteristic
93	C₂×C₄₃	#1814	1	2	yes	—
94	#533	C₂×D₄₃	1	1	yes	characteristic
95	#446	—	86	2	no	—
96	C₈×Q₈	—	86	2	no	—
97	#432	—	43	2	no	—
98	#432	C₂×D₄₃	1	2	yes	—
99	#424	—	43	2	no	—
100	#423	—	43	1	no	—
101	#405	—	43	2	no	—
102	C₂×C₄×C₈	—	43	1	no	—
103	#335	—	86	2	no	—
104	#334	—	86	2	no	—
105	#330	—	2	2	no	—
106	#330	—	86	2	no	—
107	C₄₃	#8631	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	43	2	no	—
109	C₂²×C₈	—	43	1	no	—
110	C₂²×C₈	—	43	1	no	—
111	C₄⋊Q₈	Dic₈₆	1	2	yes	—
112	#125	C₂²⋊D₄₃	1	2	yes	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	C₄×D₄₃	1	2	yes	—
116	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	86	4	no	—
119	C₄⋊C₈	—	86	2	no	—
120	C₄⋊C₈	—	86	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	86	2	no	—
123	#97	—	2	2	no	—
124	#97	—	86	2	no	—
125	#97	—	86	2	no	—
126	C₄×C₈	—	43	2	no	—
127	C₄×C₈	—	43	2	no	—
128	C₄×C₈	—	86	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
131	C₂²×C₄	#10621900	1	2	yes	—
132	M₁₆	—	172	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	43	1	no	—
135	C₂×C₈	—	43	1	no	—
136	C₂×C₈	—	43	2	no	—
137	C₂×C₈	—	86	2	no	—
138	C₂×C₈	—	86	2	no	—
139	C₂×C₈	—	172	1	no	—
140	C₂×C₈	—	172	2	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	Dic₄₃⋊C₄	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
147	C₄²	D₄×D₄₃	1	2	yes	—
148	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
149	C₂³	#13063281	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#13063297	1	2	yes	—
158	C₂×C₄	#13063302	1	2	yes	—
159	C₈	—	8	2	no	—
160	C₈	—	43	2	no	—
161	C₈	—	86	1	no	—
162	C₈	—	86	2	no	—
163	C₈	—	344	2	no	—
164	C₈	—	172	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#14582757	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#101×D₄₃	1	1	yes	characteristic
173	C₄	#14582882	1	1	yes	characteristic
174	C₂	—	4	1	no	—
175	C₂	#19134589	1	1	yes	characteristic
176	C₂	#19134647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19143530

C₂

yes

maximal, characteristic

#19133356

C₂

yes

maximal

#19133233

C₂

yes

maximal, characteristic

#19132505

C₂

yes

maximal

C₄₃×#1537

C₂

yes

maximal, characteristic

#14583030

—

Q₈×C₄₃⋊C₈

—

#14582956

C₂²

yes

—

#14582926

C₄

yes

—

#14582923

C₂²

yes

characteristic

#14582855

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

—

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#432

C₂²

yes

—

C₄₃×#330

—

C₂×#10621877

C₂×C₄

yes

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄⋊Q₈×C₄₃

Q₈

yes

—

#125×C₄₃

D₄

yes

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₄⋊C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₄₃

#117

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#101

yes

—

#8631

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#380

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

—

—

D₄₃

yes

characteristic

#1307

—

C₂×C₄₃

—

C₂×C₄₃

#1801

yes

characteristic

C₂×C₄₃

#1814

yes

—

#533

C₂×D₄₃

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#432

C₂×D₄₃

yes

—

#424

—

100

#423

—

101

#405

—

102

C₂×C₄×C₈

—

103

#335

—

104

#334

—

105

#330

—

106

#330

—

107

C₄₃

#8631

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₄⋊Q₈

Dic₈₆

yes

—

112

#125

C₂²⋊D₄₃

yes

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

C₄×D₄₃

yes

—

116

C₂×C₄²

C₂²×D₄₃

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

123

#97

—

124

#97

—

125

#97

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₂×Q₈

—

130

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

131

C₂²×C₄

#10621900

yes

—

132

M₁₆

—

172

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

172

—

140

C₂×C₈

—

172

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

Dic₄₃⋊C₄

yes

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

C₂×Dic₈₆

yes

characteristic

147

C₄²

D₄×D₄₃

yes

—

148

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

149

C₂³

#13063281

yes

characteristic

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊C₄×D₄₃

yes

characteristic

156

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

157

C₂×C₄

#13063297

yes

—

158

C₂×C₄

#13063302

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

344

—

164

C₈

—

172

—

165

C₂²

—

166

C₂²

—

167

C₂²

#14582757

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#101×D₄₃

yes

characteristic

173

C₄

#14582882

yes

characteristic

174

C₂

—

175

C₂

#19134589

yes

characteristic

176

C₂

#19134647

yes

—

	subgroup	G/H
center	C₂²	#14582757
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1537	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂³×C₄₃	#117
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1537	D₄₃
hypocenter	C₄₃	#8631
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#14582757

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1537

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

N	H ≅ G/N	type
107. C₄₃	#8631	semidirect

H ≅ G/N

type

107. C₄₃

#8631

semidirect

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

subgroup	index
#19143530	2
#19133356	2
#19133233	2
#19132505	2
C₄₃×#1537	2
#8631	43

subgroup

index

C₄₃×#1537

Subgroups — #24989591

Conjugacy classes of subgroups

Complements

Maximal subgroups