Group #25010851 — subgroups

Subgroups — #25010851

Order 11008.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#19133952	C₂	1	2	yes	maximal, complemented
3	#19132667	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132641	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2246	C₂	1	1	yes	maximal, characteristic
6	#14583096	—	2	2	no	—
7	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
8	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₄₃⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₄₃	—	2	1	no	—
11	#14582652	C₂²	1	2	yes	—
12	#14582643	—	2	2	no	—
13	C₂×D₁₆×C₄₃	—	2	1	no	—
14	C₂×D₁₆×C₄₃	—	2	1	no	—
15	C₄₃×#493	C₂²	1	2	yes	—
16	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621883	—	4	2	no	—
18	C₄×Dic₈₆	D₄	1	1	yes	characteristic
19	C₂×C₄₃⋊C₁₆	—	2	1	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₄₃⋊C₈	—	2	1	no	—
23	#13063246	—	4	2	no	—
24	C₈×Dic₄₃	—	2	1	no	—
25	C₂×D₈×C₄₃	—	2	2	no	—
26	C₂×D₈×C₄₃	—	2	2	no	—
27	#127×C₄₃	—	2	2	no	—
28	D₁₆×C₄₃	—	4	2	no	—
29	D₁₆×C₄₃	—	4	2	no	—
30	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₄₃	—	4	2	no	—
34	C₂×D₄×C₄₃	—	4	2	no	—
35	C₂×D₄×C₄₃	—	4	2	no	—
36	C₂×Dic₈₆	D₈	1	1	yes	characteristic
37	D₈×C₄₃	—	4	2	no	—
38	D₈×C₄₃	—	4	2	no	—
39	D₈×C₄₃	—	8	2	no	—
40	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
42	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
43	#10621883	—	8	4	no	—
44	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
45	Dic₄₃⋊C₄	—	2	1	no	—
46	C₄×Dic₄₃	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂×C₄₃⋊C₈	—	4	2	no	—
49	C₁₆×C₄₃	—	2	1	no	—
50	C₁₆×C₄₃	SD₁₆	1	2	yes	complemented
51	C₄₃⋊C₁₆	—	4	1	no	—
52	C₂³×C₄₃	—	8	2	no	—
53	D₄×C₄₃	—	8	2	no	—
54	D₄×C₄₃	—	8	2	no	—
55	D₄×C₄₃	—	8	2	no	—
56	D₄×C₄₃	—	8	2	no	—
57	D₄×C₄₃	—	16	2	no	—
58	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
61	C₂×Dic₄₃	—	2	1	no	—
62	C₂×Dic₄₃	—	2	1	no	—
63	Dic₈₆	—	2	1	no	—
64	Dic₈₆	D₁₆	1	2	yes	complemented
65	C₈×C₄₃	—	2	1	no	—
66	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄₃⋊C₈	—	4	1	no	—
69	C₄₃⋊C₈	—	8	2	no	—
70	#12439	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	—	16	2	no	—
72	C₂²×C₄₃	—	16	2	no	—
73	C₂²×C₄₃	—	16	2	no	—
74	C₂²×C₄₃	#466	1	1	yes	characteristic
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	#460	1	1	yes	characteristic
77	C₄×C₄₃	#472	1	1	yes	characteristic
78	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
79	C₄×C₄₃	#508	1	1	yes	characteristic
80	Dic₄₃	—	2	2	no	—
81	Dic₄₃	—	4	1	no	—
82	#2246	D₄₃	1	1	yes	characteristic
83	Q₈×C₁₆	—	43	1	no	—
84	#1668	—	43	2	no	—
85	#1371	—	43	1	no	—
86	#1361	—	43	2	no	—
87	C₂×C₄₃	—	32	2	no	—
88	C₂×C₄₃	#1668	1	1	yes	characteristic
89	C₂×C₄₃	#2215	1	1	yes	characteristic
90	C₂×C₄₃	#2221	1	1	yes	characteristic
91	C₂×D₁₆	—	2	1	no	—
92	C₂×D₁₆	—	2	1	no	—
93	#493	C₂×D₄₃	1	2	yes	—
94	#460	—	86	2	no	—
95	C₈×Q₈	—	43	1	no	—
96	C₄⋊C₁₆	—	43	1	no	—
97	C₄⋊C₁₆	—	86	1	no	—
98	C₄×C₁₆	—	86	1	no	—
99	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
100	#335	—	43	2	no	—
101	#331	—	86	2	no	—
102	C₄₃	#12439	1	1	yes	characteristic, complemented, Sylow
103	C₂×SD₁₆	—	172	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	#127	—	2	2	no	—
107	C₄×Q₈	—	43	1	no	—
108	D₁₆	—	4	2	no	—
109	D₁₆	—	4	2	no	—
110	C₂×C₁₆	—	86	1	no	—
111	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
113	C₄⋊C₈	—	43	1	no	—
114	C₄⋊C₈	—	86	1	no	—
115	C₄⋊C₈	—	86	2	no	—
116	#102	—	172	2	no	—
117	C₄×C₈	—	86	1	no	—
118	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
119	C₂×Q₈	—	43	1	no	—
120	C₂×D₄	—	4	2	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	4	2	no	—
123	SD₁₆	—	344	4	no	—
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	D₈	—	8	2	no	—
127	C₂×C₈	—	86	1	no	—
128	C₂×C₈	—	172	2	no	—
129	C₂×C₈	#10621900	1	1	yes	characteristic
130	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
131	C₄⋊C₄	—	43	1	no	—
132	C₄⋊C₄	—	86	1	no	—
133	C₄²	—	86	1	no	—
134	C₄²	D₄×D₄₃	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	172	1	no	—
137	C₁₆	#10621883	1	2	yes	complemented
138	C₂³	—	8	2	no	—
139	Q₈	—	43	2	no	—
140	Q₈	—	86	1	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	16	2	no	—
146	C₂×C₄	—	86	1	no	—
147	C₂×C₄	—	86	1	no	—
148	C₂×C₄	#13063296	1	1	yes	characteristic
149	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
150	C₂×C₄	#13063344	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	344	2	no	—
153	C₈	—	172	1	no	—
154	C₈	C₂×#10621883	1	1	yes	characteristic
155	C₈	#13063351	1	1	yes	characteristic
156	C₂²	—	16	2	no	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	#14583132	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	86	2	no	—
162	C₄	—	172	1	no	—
163	C₄	#14583096	1	1	yes	characteristic
164	C₄	#14583166	1	1	yes	characteristic
165	C₄	D₁₆×D₄₃	1	1	yes	characteristic
166	C₄	#14583328	1	1	yes	characteristic
167	C₂	—	32	2	no	—
168	C₂	#19133952	1	1	yes	characteristic
169	C₂	#19136678	1	1	yes	characteristic
170	C₂	#19136712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#19133952

C₂

yes

maximal, complemented

#19132667

C₂

yes

maximal, characteristic, complemented

#19132641

C₂

yes

maximal, complemented

C₄₃×#2246

C₂

yes

maximal, characteristic

#14583096

—

C₈×Dic₈₆

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

—

C₁₆×Dic₄₃

—

#14582652

C₂²

yes

—

#14582643

—

C₂×D₁₆×C₄₃

—

C₂×D₁₆×C₄₃

—

C₄₃×#493

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#10621883

—

C₄×Dic₈₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

characteristic

Dic₄₃⋊C₈

—

#13063246

—

C₈×Dic₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

#127×C₄₃

—

D₁₆×C₄₃

—

D₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×Dic₈₆

D₈

yes

characteristic

D₈×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

#10621883

—

C₄⋊Dic₄₃

D₈

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₁₆×C₄₃

SD₁₆

yes

complemented

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

D₁₆

yes

complemented

C₈×C₄₃

—

C₈×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#12439

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#466

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

#508

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2246

D₄₃

yes

characteristic

Q₈×C₁₆

—

#1668

—

#1371

—

#1361

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×D₁₆

—

C₂×D₁₆

—

#493

C₂×D₄₃

yes

—

#460

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

100

#335

—

101

#331

—

102

C₄₃

#12439

yes

characteristic, complemented, Sylow

103

C₂×SD₁₆

—

172

—

104

C₂×D₈

—

105

C₂×D₈

—

106

#127

—

107

C₄×Q₈

—

108

D₁₆

—

109

D₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

112

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

#102

—

172

—

117

C₄×C₈

—

118

C₄×C₈

C₂²×D₄₃

yes

characteristic

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

SD₁₆

—

344

—

124

D₈

—

125

D₈

—

126

D₈

—

127

C₂×C₈

—

128

C₂×C₈

—

172

—

129

C₂×C₈

#10621900

yes

characteristic

130

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

—

134

C₄²

D₄×D₄₃

yes

characteristic

135

C₁₆

—

136

C₁₆

—

172

—

137

C₁₆

#10621883

yes

complemented

138

C₂³

—

139

Q₈

—

140

Q₈

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#13063296

yes

characteristic

149

C₂×C₄

D₈×D₄₃

yes

characteristic

150

C₂×C₄

#13063344

yes

characteristic

151

C₈

—

152

C₈

—

344

—

153

C₈

—

172

—

154

C₈

C₂×#10621883

yes

characteristic

155

C₈

#13063351

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#14583132

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

172

—

163

C₄

#14583096

yes

characteristic

164

C₄

#14583166

yes

characteristic

165

C₄

D₁₆×D₄₃

yes

characteristic

166

C₄

#14583328

yes

characteristic

167

C₂

—

168

C₂

#19133952

yes

characteristic

169

C₂

#19136678

yes

characteristic

170

C₂

#19136712

yes

characteristic

	subgroup	G/H
center	C₂²	#14583132
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting, prime radical	C₄₃×#2246	C₂
socle	C₂²×C₄₃	#466
hypercenter	#2246	D₄₃
hypocenter	C₄₃	#12439
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#14583132

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting, prime radical

C₄₃×#2246

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. C₁₆×Dic₈₆	C₂	semidirect
2. #19133952	C₂	semidirect
3. #19132667	C₂	semidirect
4. #19132641	C₂	semidirect
50. C₁₆×C₄₃	SD₁₆	semidirect
64. Dic₈₆	D₁₆	semidirect
102. C₄₃	#12439	semidirect
137. C₁₆	#10621883	semidirect

H ≅ G/N

type

1. C₁₆×Dic₈₆

C₂

semidirect

2. #19133952

C₂

semidirect

3. #19132667

C₂

semidirect

4. #19132641

C₂

semidirect

50. C₁₆×C₄₃

SD₁₆

semidirect

64. Dic₈₆

D₁₆

semidirect

102. C₄₃

#12439

semidirect

137. C₁₆

#10621883

semidirect

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

subgroup	index
C₁₆×Dic₈₆	2
#19133952	2
#19132667	2
#19132641	2
C₄₃×#2246	2
#12439	43

subgroup

index

C₁₆×Dic₈₆

C₄₃×#2246

Subgroups — #25010851

Conjugacy classes of subgroups

Complements

Maximal subgroups