Group #25110447 — subgroups

Subgroups — #25110447

Order 11008.

826 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 227 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	C₂×#14583279	C₂	1	1	yes	maximal, characteristic
2	#19137314	C₂	1	2	yes	maximal
3	#19135415	C₂	1	2	yes	maximal
4	#19134935	C₂	1	8	yes	maximal
5	#19134726	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₂	C₂²	1	4	yes	—
8	C₂×#13063325	C₂²	1	2	yes	—
9	C₂×C₄×Dic₈₆	C₂²	1	4	yes	—
10	#14583279	C₄	1	8	yes	complemented
11	C₂×#13063250	C₂²	1	8	yes	—
12	#14582805	C₂²	1	4	yes	—
13	#14582784	—	2	8	no	—
14	#14582765	C₂²	1	2	yes	—
15	#14582662	C₂²	1	4	yes	—
16	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#376	C₂²	1	1	yes	characteristic
19	C₄₃×#336	C₂²	1	4	yes	—
20	C₂²×Dic₈₆	C₂³	1	4	yes	—
21	C₂×Dic₁₇₂	—	4	8	no	—
22	C₂×Dic₁₇₂	C₂×C₄	1	16	yes	—
23	#13063325	—	2	4	no	—
24	#13063325	C₂×C₄	1	8	yes	—
25	C₄×Dic₈₆	—	4	8	no	—
26	C₂×C₄⋊Dic₄₃	—	2	4	no	—
27	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
28	C₂×Dic₄₃⋊C₄	—	2	4	no	—
29	C₂×C₄×Dic₄₃	—	2	4	no	—
30	#13063250	—	4	8	no	—
31	C₄₃⋊#95	—	2	8	no	—
32	C₄₃⋊#95	—	2	4	no	—
33	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	#97×C₄₃	—	2	2	no	—
38	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
39	#95×C₄₃	D₄	1	4	yes	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
43	C₂²×Dic₄₃	—	2	4	no	—
44	C₂²×Dic₄₃	—	2	4	no	—
45	C₂×Dic₈₆	—	2	8	no	—
46	C₂×Dic₈₆	—	2	8	no	—
47	C₂×Dic₈₆	—	2	4	no	—
48	C₂×Dic₈₆	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₄₃	—	2	4	no	—
50	C₂×C₈×C₄₃	—	2	2	no	—
51	C₂×C₈×C₄₃	—	2	4	no	—
52	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
53	C₄²×C₄₃	—	2	4	no	—
54	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
55	C₄²×C₄₃	#41	1	2	yes	—
56	C₄⋊Dic₄₃	—	2	16	no	—
57	C₄⋊Dic₄₃	—	2	8	no	—
58	Dic₄₃⋊C₄	—	4	8	no	—
59	C₄×Dic₄₃	—	4	8	no	—
60	Dic₁₇₂	—	4	16	no	—
61	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	2	4	no	—
64	C₂×C₄×C₄₃	—	2	8	no	—
65	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
66	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
69	C₂×Dic₄₃	—	2	8	no	—
70	C₂×Dic₄₃	—	4	4	no	—
71	C₂×Dic₄₃	—	4	4	no	—
72	C₂×Dic₄₃	—	4	4	no	—
73	C₂×Dic₄₃	—	4	8	no	—
74	Dic₈₆	—	2	16	no	—
75	Dic₈₆	—	4	8	no	—
76	C₈×C₄₃	—	2	4	no	—
77	C₈×C₄₃	—	4	2	no	—
78	#22277	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
80	C₂²×C₄₃	#535	1	2	yes	—
81	C₂²×C₄₃	#578	1	4	yes	—
82	C₄×C₄₃	—	2	4	no	—
83	C₄×C₄₃	—	4	4	no	—
84	C₄×C₄₃	#518	1	2	yes	—
85	C₄×C₄₃	#540	1	2	yes	—
86	Dic₄₃	—	4	8	no	—
87	Dic₄₃	—	8	4	no	—
88	C₂×#494	—	43	1	no	—
89	#2327	—	43	2	no	—
90	#1960	—	43	2	no	—
91	#1878	—	43	8	no	—
92	#1835	—	43	1	no	—
93	#1746	D₄₃	1	1	yes	characteristic
94	C₂×C₄₃	#2328	1	1	yes	characteristic
95	C₂×C₄₃	#2950	1	2	yes	—
96	C₂×C₄₃	#3223	1	4	yes	—
97	C₂²×Q₁₆	—	43	4	no	—
98	C₂×C₄⋊Q₈	—	43	2	no	—
99	C₂×C₄×Q₈	—	43	4	no	—
100	#494	—	43	8	no	—
101	C₂×#103	—	43	8	no	—
102	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
104	#391	—	43	4	no	—
105	#387	—	86	8	no	—
106	#382	—	43	2	no	—
107	#376	C₂×D₄₃	1	1	yes	characteristic
108	#340	—	43	4	no	—
109	#336	C₂×D₄₃	1	4	yes	—
110	C₄₃	#22277	1	1	yes	characteristic, complemented, Sylow
111	C₂²×Q₈	—	43	4	no	—
112	C₂×Q₁₆	—	43	16	no	—
113	C₂×Q₁₆	—	172	8	no	—
114	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
115	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
116	C₄⋊Q₈	—	43	8	no	—
117	C₄⋊Q₈	—	86	4	no	—
118	C₄×Q₈	—	172	8	no	—
119	C₂×C₄⋊C₄	—	43	4	no	—
120	C₂×C₄⋊C₄	—	86	4	no	—
121	C₂×C₄⋊C₄	—	86	4	no	—
122	C₂×C₄²	—	86	4	no	—
123	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
124	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
125	#103	—	172	8	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	C₄×D₄₃	1	4	yes	—
128	#95	—	86	4	no	—
129	#95	—	86	8	no	—
130	#95	D₁₇₂	1	4	yes	—
131	C₂×Q₈	—	43	8	no	—
132	C₂×Q₈	—	86	4	no	—
133	C₂×Q₈	—	86	8	no	—
134	C₂×Q₈	—	86	8	no	—
135	C₂²×C₄	—	86	4	no	—
136	C₂²×C₄	—	86	4	no	—
137	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
138	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
139	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
140	Q₁₆	—	172	16	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
145	C₄⋊C₄	—	86	16	no	—
146	C₄⋊C₄	—	86	8	no	—
147	C₄⋊C₄	—	172	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	172	8	no	—
150	C₄²	C₂×C₄×D₄₃	1	2	yes	—
151	C₄²	#10621898	1	2	yes	—
152	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
153	Q₈	—	86	16	no	—
154	Q₈	—	172	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	86	8	no	—
159	C₂×C₄	—	172	8	no	—
160	C₂×C₄	—	172	4	no	—
161	C₂×C₄	—	172	4	no	—
162	C₂×C₄	—	172	4	no	—
163	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
164	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
165	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
166	C₂×C₄	#13063371	1	4	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	4	2	no	—
169	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
170	C₂²	#14583468	1	2	yes	—
171	C₂²	#14583716	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	4	4	no	—
174	C₄	—	344	4	no	—
175	C₄	—	172	8	no	—
176	C₄	#14583373	1	2	yes	—
177	C₄	#14583504	1	2	yes	—
178	C₂	#19137320	1	1	yes	characteristic
179	C₂	#19143464	1	2	yes	—
180	C₂	#19145680	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583279

C₂

yes

maximal, characteristic

#19137314

C₂

yes

maximal

#19135415

C₂

yes

maximal

#19134935

C₂

yes

maximal

#19134726

C₂

yes

maximal, characteristic

C₄₃×#1746

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

—

C₂×#13063325

C₂²

yes

—

C₂×C₄×Dic₈₆

C₂²

yes

—

#14583279

C₄

yes

complemented

C₂×#13063250

C₂²

yes

—

#14582805

C₂²

yes

—

#14582784

—

#14582765

C₂²

yes

—

#14582662

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#376

C₂²

yes

characteristic

C₄₃×#336

C₂²

yes

—

C₂²×Dic₈₆

C₂³

yes

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂×C₄

yes

—

#13063325

—

#13063325

C₂×C₄

yes

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063250

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#97×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

#95×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂²×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×#41

yes

characteristic

C₂×C₄×C₄₃

H(5,𝔽₂)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

#22277

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#535

yes

—

C₂²×C₄₃

#578

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#518

yes

—

C₄×C₄₃

#540

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

—

D₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#3223

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

100

#494

—

101

C₂×#103

—

102

C₂×#97

C₂×D₄₃

yes

characteristic

103

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

104

#391

—

105

#387

—

106

#382

—

107

#376

C₂×D₄₃

yes

characteristic

108

#340

—

109

#336

C₂×D₄₃

yes

—

110

C₄₃

#22277

yes

characteristic, complemented, Sylow

111

C₂²×Q₈

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

172

—

114

C₂²×C₈

C₂²×D₄₃

yes

—

115

C₂²×C₈

C₂²×D₄₃

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄×Q₈

—

172

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂²×D₄₃

yes

—

124

C₂×C₄²

C₂²×D₄₃

yes

characteristic

125

#103

—

172

—

126

#97

—

127

C₄×C₈

C₄×D₄₃

yes

—

128

#95

—

129

#95

—

130

#95

D₁₇₂

yes

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₁₇₂

yes

—

138

C₂²×C₄

C₂×D₁₇₂

yes

—

139

C₂²×C₄

C₂³×D₄₃

yes

characteristic

140

Q₁₆

—

172

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

C₂×C₄×D₄₃

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

172

—

148

C₄²

—

149

C₄²

—

172

—

150

C₄²

C₂×C₄×D₄₃

yes

—

151

C₄²

#10621898

yes

—

152

C₂³

C₂²×D₁₇₂

yes

characteristic

153

Q₈

—

154

Q₈

—

172

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

172

—

160

C₂×C₄

—

172

—

161

C₂×C₄

—

172

—

162

C₂×C₄

—

172

—

163

C₂×C₄

C₄×D₁₇₂

yes

—

164

C₂×C₄

C₂²×C₄×D₄₃

yes

characteristic

165

C₂×C₄

C₂×#10621898

yes

characteristic

166

C₂×C₄

#13063371

yes

—

167

C₈

—

168

C₈

—

169

C₂²

C₂×C₄×D₁₇₂

yes

characteristic

170

C₂²

#14583468

yes

—

171

C₂²

#14583716

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

344

—

175

C₄

—

172

—

176

C₄

#14583373

yes

—

177

C₄

#14583504

yes

—

178

C₂

#19137320

yes

characteristic

179

C₂

#19143464

yes

—

180

C₂

#19145680

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1746	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1746	D₄₃
hypocenter	C₄₃	#22277
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1746

subgroup

G/H

center

C₂³

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1746

C₂

socle, prime radical

C₂³×C₄₃

C₂²×D₄

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ #376 ◁ #1746

N	H ≅ G/N	type
10. #14583279	C₄	semidirect
110. C₄₃	#22277	semidirect

H ≅ G/N

type

10. #14583279

C₄

semidirect

110. C₄₃

#22277

semidirect

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

subgroup	index
C₂×#14583279	2
#19137314	2
#19135415	2
#19134935	2
#19134726	2
C₄₃×#1746	2
#22277	43

subgroup

index

C₄₃×#1746

Subgroups — #25110447

Conjugacy classes of subgroups

Complements

Maximal subgroups