Group #24983527 — subgroups

Subgroups — #24983527

Order 11008.

246 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582652	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063261	C₄	1	2	yes	—
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
6	C₄×#10621877	—	2	1	no	—
7	#14582664	—	2	1	no	—
8	#14582652	—	2	1	no	—
9	#14582652	C₄	1	2	yes	—
10	C₂×M₃₂×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#346	C₄	1	2	yes	—
13	C₄×C₁₆×C₄₃	—	2	1	no	—
14	C₂×#10621877	—	2	2	no	—
15	C₂²×C₄₃⋊C₈	—	4	1	no	—
16	#13063261	—	2	2	no	—
17	#13063261	C₈	1	4	yes	complemented
18	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
19	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
20	#13063236	—	4	1	no	—
21	C₄×C₄₃⋊C₈	—	4	1	no	—
22	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
25	M₃₂×C₄₃	—	2	4	no	—
26	C₂×C₁₆×C₄₃	—	2	2	no	—
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₄₃	—	2	2	no	—
35	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
41	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₄₃	D₈	1	1	yes	characteristic
44	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
45	#10621877	—	2	4	no	—
46	#10621877	—	4	2	no	—
47	C₂×C₄₃⋊C₈	—	4	2	no	—
48	C₂×C₄₃⋊C₈	—	4	2	no	—
49	C₂×C₄₃⋊C₈	—	8	2	no	—
50	C₁₆×C₄₃	—	2	4	no	—
51	C₂³×C₄₃	#103	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	4	1	no	—
55	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#104	1	2	yes	—
61	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
63	C₈×C₄₃	—	2	1	no	—
64	C₈×C₄₃	—	2	1	no	—
65	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
66	C₈×C₄₃	#97	1	1	yes	characteristic
67	C₈×C₄₃	C₄⋊C₈	1	2	yes	—
68	C₄₃⋊C₈	—	4	4	no	—
69	C₄₃⋊C₈	—	8	2	no	—
70	#7176	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	—	2	1	no	—
72	C₂²×C₄₃	—	2	1	no	—
73	C₂²×C₄₃	Q₈⋊C₈	1	2	yes	—
74	C₂²×C₄₃	#340	1	1	yes	characteristic
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	2	1	no	—
77	C₄×C₄₃	—	2	1	no	—
78	C₄×C₄₃	D₄⋊C₈	1	2	yes	—
79	C₄×C₄₃	#336	1	1	yes	characteristic
80	C₄×C₄₃	#339	1	1	yes	characteristic
81	C₄×C₄₃	#361	1	1	yes	characteristic
82	C₄×C₄₃	#362	1	1	yes	characteristic
83	C₄×M₃₂	D₄₃	1	1	yes	characteristic
84	C₄×#108	—	43	1	no	—
85	C₂×#335	—	43	1	no	—
86	C₂×C₄₃	—	2	2	no	—
87	C₂×C₄₃	#1279	1	1	yes	characteristic
88	C₂×C₄₃	#1387	1	1	yes	characteristic
89	C₂×C₄₃	#1388	1	1	yes	characteristic
90	C₂×M₃₂	—	2	1	no	—
91	C₂×#108	—	43	2	no	—
92	C₂×C₄⋊C₈	—	86	1	no	—
93	C₄×M₁₆	—	86	1	no	—
94	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
95	#346	Dic₄₃	1	2	yes	—
96	C₄×C₁₆	—	2	1	no	—
97	#341	—	86	1	no	—
98	#335	—	43	2	no	—
99	#335	—	86	1	no	—
100	C₄₃	#7176	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	86	2	no	—
102	C₂²×C₈	—	172	1	no	—
103	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
105	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
106	M₃₂	—	2	4	no	—
107	C₂×C₁₆	—	2	2	no	—
108	#108	—	43	4	no	—
109	#108	—	86	2	no	—
110	C₄⋊C₈	—	86	2	no	—
111	C₄⋊C₈	—	172	1	no	—
112	#97	—	172	1	no	—
113	C₄×C₈	—	172	1	no	—
114	C₄×C₈	Dic₈₆	1	1	yes	characteristic
115	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
118	C₂²×C₄	#10621873	1	1	yes	characteristic
119	C₂²×C₄	Dic₁₇₂	1	1	yes	characteristic
120	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
121	M₁₆	—	86	4	no	—
122	M₁₆	—	172	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	344	2	no	—
125	C₂×C₈	—	172	2	no	—
126	C₂×C₈	—	172	2	no	—
127	C₂×C₈	C₈×D₄₃	1	2	yes	—
128	C₂×C₈	#10621872	1	2	yes	—
129	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
130	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
131	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
133	C₄²	#10621873	1	1	yes	characteristic
134	C₄²	D₃₄₄	1	1	yes	characteristic
135	C₄²	C₄⋊Dic₄₃	1	1	yes	characteristic
136	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₂³	#13063250	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
143	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
144	C₂×C₄	#13063247	1	1	yes	characteristic
145	C₂×C₄	#13063247	1	1	yes	characteristic
146	C₂×C₄	#13063250	1	1	yes	characteristic
147	C₂×C₄	#13063253	1	2	yes	—
148	C₂×C₄	#13063257	1	1	yes	characteristic
149	C₂×C₄	#13063259	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	344	2	no	—
153	C₈	—	172	4	no	—
154	C₈	C₈×Dic₄₃	1	1	yes	characteristic
155	C₈	#13063235	1	1	yes	characteristic
156	C₈	Dic₄₃⋊C₈	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#14582631	1	2	yes	—
160	C₂²	#14582662	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#14582628	1	2	yes	—
165	C₄	#14582653	1	1	yes	characteristic
166	C₄	#14582659	1	1	yes	characteristic
167	C₄	#14582713	1	1	yes	characteristic
168	C₄	#14582716	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#19132435	1	1	yes	characteristic
171	C₂	#19132704	1	1	yes	characteristic
172	C₂	#19132707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063261

C₂

yes

maximal, characteristic

C₂×#14582652

C₂

yes

maximal, characteristic

C₄×M₃₂×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063261

C₄

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

—

—

—

—

C₄

yes

—

C₂×M₃₂×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#346

C₄

yes

—

C₄×C₁₆×C₄₃

—

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

#13063261

—

#13063261

C₈

yes

complemented

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

M₃₂×C₄₃

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

SD₁₆

yes

characteristic

C₂²×C₄×C₄₃

Q₁₆

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

characteristic

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₂³×C₄₃

#103

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₄×C₈

yes

characteristic

C₈×C₄₃

#97

yes

characteristic

C₈×C₄₃

C₄⋊C₈

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#7176

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

Q₈⋊C₈

yes

—

C₂²×C₄₃

#340

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

D₄⋊C₈

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×M₃₂

D₄₃

yes

characteristic

C₄×#108

—

C₂×#335

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×M₃₂

—

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#346

Dic₄₃

yes

—

C₄×C₁₆

—

#341

—

#335

—

#335

—

100

C₄₃

#7176

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

—

172

—

103

C₂²×C₈

C₄×D₄₃

yes

characteristic

104

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

105

C₂×C₄²

D₁₇₂

yes

characteristic

106

M₃₂

—

107

C₂×C₁₆

—

108

#108

—

109

#108

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

172

—

112

#97

—

172

—

113

C₄×C₈

—

172

—

114

C₄×C₈

Dic₈₆

yes

characteristic

115

C₄×C₈

C₄×D₄₃

yes

characteristic

116

C₄×C₈

C₂×Dic₄₃

yes

characteristic

117

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

118

C₂²×C₄

#10621873

yes

characteristic

119

C₂²×C₄

Dic₁₇₂

yes

characteristic

120

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

121

M₁₆

—

122

M₁₆

—

172

—

123

C₂×C₈

—

124

C₂×C₈

—

344

—

125

C₂×C₈

—

172

—

126

C₂×C₈

—

172

—

127

C₂×C₈

C₈×D₄₃

yes

—

128

C₂×C₈

#10621872

yes

—

129

C₂×C₈

C₄×Dic₄₃

yes

characteristic

130

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

131

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

132

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

133

C₄²

#10621873

yes

characteristic

134

C₄²

D₃₄₄

yes

characteristic

135

C₄²

C₄⋊Dic₄₃

yes

characteristic

136

C₄²

(C₂×D₄₃)⋊C₄

yes

characteristic

137

C₁₆

—

138

C₂³

#13063250

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₄₃⋊#95

yes

characteristic

143

C₂×C₄

(C₂×D₄₃)⋊C₈

yes

—

144

C₂×C₄

#13063247

yes

characteristic

145

C₂×C₄

#13063247

yes

characteristic

146

C₂×C₄

#13063250

yes

characteristic

147

C₂×C₄

#13063253

yes

—

148

C₂×C₄

#13063257

yes

characteristic

149

C₂×C₄

#13063259

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

344

—

153

C₈

—

172

—

154

C₈

C₈×Dic₄₃

yes

characteristic

155

C₈

#13063235

yes

characteristic

156

C₈

Dic₄₃⋊C₈

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

#14582631

yes

—

160

C₂²

#14582662

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

#14582628

yes

—

165

C₄

#14582653

yes

characteristic

166

C₄

#14582659

yes

characteristic

167

C₄

#14582713

yes

characteristic

168

C₄

#14582716

yes

characteristic

169

C₂

—

170

C₂

#19132435

yes

characteristic

171

C₂

#19132704

yes

characteristic

172

C₂

#19132707

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063247
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	#340
prime radical	C₂³×C₄₃	#103
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#7176
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂×C₄

#13063247

commutator

C₈×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₄×M₃₂×C₄₃

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄×C₈×C₄₃

C₂²

hypercenter

C₄×M₃₂

D₄₃

hypocenter

C₄₃

#7176

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
17. #13063261	C₈	semidirect
100. C₄₃	#7176	semidirect

H ≅ G/N

type

17. #13063261

C₈

semidirect

100. C₄₃

#7176

semidirect

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

subgroup	index
C₄×#13063261	2
C₂×#14582652	2
C₄×M₃₂×C₄₃	2
#7176	43

subgroup

index

C₄×#13063261

C₂×#14582652

C₄×M₃₂×C₄₃

#7176

Subgroups — #24983527

Conjugacy classes of subgroups

Complements

Maximal subgroups