Group #25016732 — subgroups

Subgroups — #25016732

Order 11008.

234 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19134770	C₂	1	1	yes	maximal, characteristic
2	#19132711	C₂	1	2	yes	maximal
3	#19132708	C₂	1	2	yes	maximal
4	#19132675	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1946	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	2	yes	—
7	C₂×#13063261	—	2	1	no	—
8	C₂×#13063261	C₂²	1	2	yes	—
9	#14582923	—	2	1	no	—
10	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×#427	C₂²	1	1	yes	characteristic
12	C₄₃×#424	—	2	1	no	—
13	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#339	—	2	1	no	—
15	C₂×#10621877	—	2	2	no	—
16	#13063265	—	2	2	no	—
17	#13063265	—	2	2	no	—
18	C₂×C₄₃⋊C₁₆	—	2	2	no	—
19	#13063261	—	2	4	no	—
20	#13063261	—	4	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₄₃	—	2	1	no	—
23	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
25	#117×C₄₃	—	2	1	no	—
26	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
27	#108×C₄₃	—	2	2	no	—
28	#107×C₄₃	—	2	1	no	—
29	#106×C₄₃	Q₈	1	2	yes	—
30	C₄⋊C₈×C₄₃	—	4	2	no	—
31	#97×C₄₃	—	2	2	no	—
32	C₂²×C₄×C₄₃	—	2	1	no	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	M₁₆×C₄₃	—	4	1	no	—
35	M₁₆×C₄₃	—	4	1	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	—	4	1	no	—
39	C₂×C₈×C₄₃	D₈	1	2	yes	—
40	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₄₃	#41	1	2	yes	—
43	C₂×C₈×C₄₃	#41	1	2	yes	—
44	C₄⋊C₄×C₄₃	—	2	2	no	—
45	C₂²⋊C₄×C₄₃	—	4	1	no	—
46	C₄²×C₄₃	—	2	1	no	—
47	C₄²×C₄₃	—	4	1	no	—
48	C₄²×C₄₃	SD₁₆	1	2	yes	—
49	#10621877	—	4	2	no	—
50	#10621877	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	4	2	no	—
52	C₄₃⋊C₁₆	—	4	2	no	—
53	C₂³×C₄₃	#137	1	1	yes	characteristic
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	4	2	no	—
56	C₂×C₄×C₄₃	—	8	1	no	—
57	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	—	8	1	no	—
65	C₈×C₄₃	#125	1	2	yes	—
66	C₈×C₄₃	#126	1	2	yes	—
67	C₄₃⋊C₈	—	8	2	no	—
68	#13540	—	43	1	no	maximal, Sylow
69	C₂²×C₄₃	—	4	1	no	—
70	C₂²×C₄₃	#474	1	1	yes	characteristic
71	C₂²×C₄₃	#483	1	1	yes	characteristic
72	C₂²×C₄₃	#491	1	1	yes	characteristic
73	C₄×C₄₃	—	4	2	no	—
74	C₄×C₄₃	—	8	2	no	—
75	C₄×C₄₃	#400	1	1	yes	characteristic
76	C₄×C₄₃	#475	1	1	yes	characteristic
77	C₄×C₄₃	#480	1	1	yes	characteristic
78	C₄×C₄₃	#486	1	1	yes	characteristic
79	#1946	D₄₃	1	1	yes	characteristic
80	#1849	—	43	1	no	—
81	#1389	—	43	2	no	—
82	#1388	—	43	2	no	—
83	#1375	—	43	1	no	—
84	C₂×C₄₃	—	2	1	no	—
85	C₂×C₄₃	—	2	1	no	—
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	#2099	1	1	yes	characteristic
88	C₂×M₃₂	—	43	2	no	—
89	C₂×#108	—	43	2	no	—
90	C₂×#108	—	86	1	no	—
91	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
92	#427	C₂×D₄₃	1	1	yes	characteristic
93	#424	—	2	1	no	—
94	#423	—	86	1	no	—
95	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
96	#339	—	2	1	no	—
97	C₄₃	#13540	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	86	2	no	—
100	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
104	M₃₂	—	86	2	no	—
105	M₃₂	—	86	2	no	—
106	C₂×C₁₆	—	86	2	no	—
107	#108	—	2	2	no	—
108	#108	—	86	4	no	—
109	#108	—	172	2	no	—
110	#107	—	2	1	no	—
111	#106	Dic₈₆	1	2	yes	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	172	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	1	no	—
119	M₁₆	—	172	2	no	—
120	M₁₆	—	172	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	172	2	no	—
125	C₂×C₈	#10621882	1	2	yes	—
126	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
127	C₂×C₈	#10621898	1	1	yes	characteristic
128	C₂×C₈	#10621900	1	2	yes	—
129	C₂×C₈	#10621902	1	2	yes	—
130	C₄⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	4	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	#10621884	1	2	yes	—
135	C₁₆	—	172	2	no	—
136	C₂³	#13063355	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	#13063304	1	1	yes	characteristic
141	C₂×C₄	#13063306	1	1	yes	characteristic
142	C₂×C₄	#13063314	1	1	yes	characteristic
143	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
144	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
145	C₂×C₄	#13063355	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	8	1	no	—
148	C₈	—	344	2	no	—
149	C₈	#13063316	1	2	yes	—
150	C₈	#13063320	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#14583179	1	1	yes	characteristic
153	C₂²	#14583227	1	1	yes	characteristic
154	C₂²	#14583269	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#14582838	1	1	yes	characteristic
158	C₄	#14583184	1	1	yes	characteristic
159	C₄	#14583210	1	1	yes	characteristic
160	C₄	#14583247	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#19136153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134770

C₂

yes

maximal, characteristic

#19132711

C₂

yes

maximal

#19132708

C₂

yes

maximal

#19132675

C₂

yes

maximal, characteristic

C₄₃×#1946

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

—

C₂×#13063261

C₂²

yes

—

#14582923

—

C₂×#108×C₄₃

C₂²

yes

characteristic

C₄₃×#427

C₂²

yes

characteristic

C₄₃×#424

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₄₃×#339

—

C₂×#10621877

—

#13063265

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#117×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#108×C₄₃

—

#107×C₄₃

—

#106×C₄₃

Q₈

yes

—

C₄⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

C₂×C₈×C₄₃

yes

C₂×C₈×C₄₃

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#137

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

#137

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#125

yes

—

C₈×C₄₃

#126

yes

—

C₄₃⋊C₈

—

#13540

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

D₄₃

yes

—

—

—

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#2099

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

C₂×#108

C₂×D₄₃

yes

characteristic

#427

C₂×D₄₃

yes

characteristic

#424

—

#423

—

C₂×#97

C₂×D₄₃

yes

characteristic

#339

—

C₄₃

#13540

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

101

C₂²×C₈

C₂²×D₄₃

yes

characteristic

102

#117

—

103

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

108

#108

—

109

#108

—

172

—

110

#107

—

111

#106

Dic₈₆

yes

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

172

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

172

—

120

M₁₆

—

172

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

172

—

125

C₂×C₈

#10621882

yes

—

126

C₂×C₈

C₂×Dic₈₆

yes

characteristic

127

C₂×C₈

#10621898

yes

characteristic

128

C₂×C₈

#10621900

yes

—

129

C₂×C₈

#10621902

yes

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

#10621884

yes

—

135

C₁₆

—

172

—

136

C₂³

#13063355

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#13063304

yes

characteristic

141

C₂×C₄

#13063306

yes

characteristic

142

C₂×C₄

#13063314

yes

characteristic

143

C₂×C₄

C₂×#10621882

yes

characteristic

144

C₂×C₄

C₂×#10621884

yes

characteristic

145

C₂×C₄

#13063355

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

344

—

149

C₈

#13063316

yes

—

150

C₈

#13063320

yes

—

151

C₂²

—

152

C₂²

#14583179

yes

characteristic

153

C₂²

#14583227

yes

characteristic

154

C₂²

#14583269

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

#14582838

yes

characteristic

158

C₄

#14583184

yes

characteristic

159

C₄

#14583210

yes

characteristic

160

C₄

#14583247

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#19136153

yes

characteristic

	subgroup	G/H
center	C₄	#14583247
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1946	C₂
socle	C₂×C₄₃	#2099
prime radical	C₂³×C₄₃	#137
hypercenter	#1946	D₄₃
hypocenter	C₄₃	#13540
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#14583247

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1946

C₂

C₂×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
97. C₄₃	#13540	semidirect

H ≅ G/N

type

97. C₄₃

#13540

semidirect

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

subgroup	index
#19134770	2
#19132711	2
#19132708	2
#19132675	2
C₄₃×#1946	2
#13540	43

subgroup

index

C₄₃×#1946

Subgroups — #25016732

Conjugacy classes of subgroups

Complements

Maximal subgroups