Group #24996364 — subgroups

Subgroups — #24996364

Order 11008.

558 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 211 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	#19135556	C₂	1	4	yes	maximal
3	#19134755	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₂	—	2	2	no	—
6	C₂×#13063325	C₂²	1	2	yes	—
7	#14583279	—	2	2	no	—
8	#14583279	C₄	1	4	yes	complemented
9	C₂×#13063259	—	2	2	no	—
10	#14582769	C₂²	1	2	yes	—
11	#14582662	—	2	8	no	—
12	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₈²×C₄₃	—	2	2	no	—
15	C₂²×Dic₈₆	—	2	2	no	—
16	C₂×Dic₁₇₂	—	2	8	no	—
17	C₂×Dic₁₇₂	—	4	4	no	—
18	C₂×Dic₁₇₂	—	4	4	no	—
19	#13063325	—	2	4	no	—
20	#13063325	—	2	2	no	—
21	#13063325	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₄₃	—	2	4	no	—
23	C₂×C₄⋊Dic₄₃	—	2	4	no	—
24	C₂×C₄⋊Dic₄₃	—	2	2	no	—
25	#13063259	—	4	4	no	—
26	C₂²×C₈×C₄₃	D₄	1	4	yes	—
27	C₂²×C₈×C₄₃	D₄	1	2	yes	—
28	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₄₃	—	2	2	no	—
30	C₄×C₈×C₄₃	—	2	2	no	—
31	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₃	D₄	1	2	yes	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
35	C₂²×Dic₄₃	—	4	2	no	—
36	C₂²×Dic₄₃	—	4	2	no	—
37	C₂×Dic₈₆	—	2	4	no	—
38	C₂×Dic₈₆	—	4	4	no	—
39	C₂×Dic₈₆	—	4	4	no	—
40	C₂×Dic₈₆	—	4	2	no	—
41	C₂×C₈×C₄₃	—	2	4	no	—
42	C₂×C₈×C₄₃	—	2	4	no	—
43	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₄₃	D₈	1	4	yes	—
45	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
46	C₂×C₈×C₄₃	Q₁₆	1	8	yes	complemented
47	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₃	#41	1	2	yes	—
50	C₄⋊Dic₄₃	—	2	8	no	—
51	C₄⋊Dic₄₃	—	4	4	no	—
52	C₄⋊Dic₄₃	—	4	4	no	—
53	C₄⋊Dic₄₃	—	4	8	no	—
54	Dic₁₇₂	—	4	16	no	—
55	C₂³×C₄₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₄₃	#121	1	2	yes	—
59	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
63	C₂×C₄×C₄₃	#135	1	4	yes	—
64	C₂×Dic₄₃	—	4	4	no	—
65	C₂×Dic₄₃	—	8	2	no	—
66	C₂×Dic₄₃	—	8	2	no	—
67	C₂×Dic₄₃	—	8	2	no	—
68	C₂×Dic₄₃	—	8	4	no	—
69	Dic₈₆	—	4	8	no	—
70	Dic₈₆	—	8	4	no	—
71	C₈×C₄₃	—	2	8	no	—
72	C₈×C₄₃	#102	1	8	yes	—
73	#9825	—	43	1	no	maximal, Sylow
74	C₂²×C₄₃	#390	1	1	yes	characteristic
75	C₂²×C₄₃	#492	1	1	yes	characteristic
76	C₂²×C₄₃	#493	1	1	yes	characteristic
77	C₂²×C₄₃	#494	1	2	yes	—
78	C₂²×C₄₃	#495	1	2	yes	—
79	C₄×C₄₃	C₂×#102	1	2	yes	—
80	C₄×C₄₃	C₄×Q₁₆	1	4	yes	—
81	C₄×C₄₃	#467	1	4	yes	—
82	C₄×C₄₃	#486	1	2	yes	—
83	Dic₄₃	—	8	4	no	—
84	Dic₄₃	—	16	2	no	—
85	C₂×#494	—	43	1	no	—
86	#1990	—	43	4	no	—
87	#1845	—	43	1	no	—
88	C₂×C₈²	D₄₃	1	1	yes	characteristic
89	C₂×C₄₃	#1707	1	2	yes	—
90	C₂×C₄₃	#1709	1	1	yes	characteristic
91	C₂×C₄₃	#1713	1	1	yes	characteristic
92	C₂×C₄₃	#1957	1	1	yes	characteristic
93	C₂×C₄₃	#1969	1	2	yes	—
94	C₂²×Q₁₆	—	86	2	no	—
95	C₂×C₄⋊Q₈	—	43	2	no	—
96	#494	—	43	4	no	—
97	#494	—	86	2	no	—
98	C₂×#107	—	86	2	no	—
99	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
101	#384	—	43	2	no	—
102	#340	—	86	8	no	—
103	C₈²	—	2	2	no	—
104	C₄₃	#9825	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	86	2	no	—
106	C₂×Q₁₆	—	86	8	no	—
107	C₂×Q₁₆	—	172	4	no	—
108	C₂×Q₁₆	—	172	4	no	—
109	C₂²×C₈	D₁₇₂	1	4	yes	—
110	C₂²×C₈	D₁₇₂	1	2	yes	—
111	C₄⋊Q₈	—	43	4	no	—
112	C₄⋊Q₈	—	86	4	no	—
113	C₄⋊Q₈	—	86	2	no	—
114	C₂×C₄⋊C₄	—	86	4	no	—
115	C₂×C₄⋊C₄	—	86	4	no	—
116	C₂×C₄⋊C₄	—	86	2	no	—
117	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
118	#107	—	172	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₄×D₄₃	1	2	yes	—
122	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
123	C₂×Q₈	—	86	4	no	—
124	C₂×Q₈	—	172	4	no	—
125	C₂×Q₈	—	172	4	no	—
126	C₂×Q₈	—	172	2	no	—
127	C₂²×C₄	—	172	2	no	—
128	C₂²×C₄	—	172	2	no	—
129	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
131	Q₁₆	—	172	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	#10621873	1	4	yes	—
135	C₂×C₈	D₃₄₄	1	4	yes	—
136	C₂×C₈	Dic₁₇₂	1	8	yes	complemented
137	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	86	8	no	—
139	C₄⋊C₄	—	172	4	no	—
140	C₄⋊C₄	—	172	8	no	—
141	C₄⋊C₄	—	172	4	no	—
142	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
143	C₄²	#10621898	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₂³	#13063322	1	1	yes	characteristic
146	Q₈	—	344	4	no	—
147	Q₈	—	172	8	no	—
148	C₂×C₄	—	344	2	no	—
149	C₂×C₄	—	344	2	no	—
150	C₂×C₄	—	344	2	no	—
151	C₂×C₄	—	344	4	no	—
152	C₂×C₄	—	172	4	no	—
153	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
155	C₂×C₄	#13063299	1	2	yes	—
156	C₂×C₄	#13063313	1	1	yes	characteristic
157	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
158	C₂×C₄	C₂×#10621873	1	2	yes	—
159	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
160	C₂×C₄	#13063347	1	4	yes	—
161	C₈	—	2	8	no	—
162	C₈	#13063247	1	8	yes	—
163	C₂²	#14582801	1	1	yes	characteristic
164	C₂²	#14583272	1	1	yes	characteristic
165	C₂²	#14583276	1	1	yes	characteristic
166	C₂²	#14583279	1	2	yes	—
167	C₂²	#14583283	1	2	yes	—
168	C₄	—	344	4	no	—
169	C₄	—	688	2	no	—
170	C₄	C₂×#13063247	1	2	yes	—
171	C₄	C₄×Dic₁₇₂	1	4	yes	—
172	C₄	#14583140	1	4	yes	—
173	C₄	#14583246	1	2	yes	—
174	C₂	#19134192	1	2	yes	—
175	C₂	#19134201	1	1	yes	characteristic
176	C₂	#19134213	1	1	yes	characteristic
177	C₂	#19135401	1	1	yes	characteristic
178	C₂	#19135456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583279

C₂

yes

maximal, characteristic

#19135556

C₂

yes

maximal

#19134755

C₂

yes

maximal, characteristic

C₂×C₈²×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

—

C₂×#13063325

C₂²

yes

—

#14583279

—

#14583279

C₄

yes

complemented

C₂×#13063259

—

#14582769

C₂²

yes

—

#14582662

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

—

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

#13063325

—

#13063325

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

complemented

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

#102

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#495

yes

—

C₄×C₄₃

C₂×#102

yes

—

C₄×C₄₃

C₄×Q₁₆

yes

—

C₄×C₄₃

#467

yes

—

C₄×C₄₃

#486

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×#494

—

#1990

—

#1845

—

C₂×C₈²

D₄₃

yes

characteristic

C₂×C₄₃

#1707

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×#107

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

100

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

101

#384

—

102

#340

—

103

C₈²

—

104

C₄₃

#9825

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

172

—

108

C₂×Q₁₆

—

172

—

109

C₂²×C₈

D₁₇₂

yes

—

110

C₂²×C₈

D₁₇₂

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

C₂²×D₄₃

yes

characteristic

118

#107

—

172

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₄×D₄₃

yes

—

122

C₄×C₈

C₂²⋊D₄₃

yes

—

123

C₂×Q₈

—

124

C₂×Q₈

—

172

—

125

C₂×Q₈

—

172

—

126

C₂×Q₈

—

172

—

127

C₂²×C₄

—

172

—

128

C₂²×C₄

—

172

—

129

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

130

C₂²×C₄

C₂×D₁₇₂

yes

—

131

Q₁₆

—

172

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

#10621873

yes

—

135

C₂×C₈

D₃₄₄

yes

—

136

C₂×C₈

Dic₁₇₂

yes

complemented

137

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

172

—

140

C₄⋊C₄

—

172

—

141

C₄⋊C₄

—

172

—

142

C₄²

C₂×C₄×D₄₃

yes

characteristic

143

C₄²

#10621898

yes

—

144

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₂³

#13063322

yes

characteristic

146

Q₈

—

344

—

147

Q₈

—

172

—

148

C₂×C₄

—

344

—

149

C₂×C₄

—

344

—

150

C₂×C₄

—

344

—

151

C₂×C₄

—

344

—

152

C₂×C₄

—

172

—

153

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

154

C₂×C₄

C₄×D₁₇₂

yes

—

155

C₂×C₄

#13063299

yes

—

156

C₂×C₄

#13063313

yes

characteristic

157

C₂×C₄

C₂×D₃₄₄

yes

—

158

C₂×C₄

C₂×#10621873

yes

—

159

C₂×C₄

C₂×Dic₁₇₂

yes

—

160

C₂×C₄

#13063347

yes

—

161

C₈

—

162

C₈

#13063247

yes

—

163

C₂²

#14582801

yes

characteristic

164

C₂²

#14583272

yes

characteristic

165

C₂²

#14583276

yes

characteristic

166

C₂²

#14583279

yes

—

167

C₂²

#14583283

yes

—

168

C₄

—

344

—

169

C₄

—

688

—

170

C₄

C₂×#13063247

yes

—

171

C₄

C₄×Dic₁₇₂

yes

—

172

C₄

#14583140

yes

—

173

C₄

#14583246

yes

—

174

C₂

#19134192

yes

—

175

C₂

#19134201

yes

characteristic

176

C₂

#19134213

yes

characteristic

177

C₂

#19135401

yes

characteristic

178

C₂

#19135456

yes

—

	subgroup	G/H
center	C₂³	#13063322
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#9825
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13063322

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₂×C₈²×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

#127

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₂×C₈²

D₄₃

hypocenter

C₄₃

#9825

LCS

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #14583279	C₄	semidirect
46. C₂×C₈×C₄₃	Q₁₆	semidirect
104. C₄₃	#9825	semidirect
136. C₂×C₈	Dic₁₇₂	semidirect

H ≅ G/N

type

8. #14583279

C₄

semidirect

46. C₂×C₈×C₄₃

Q₁₆

semidirect

104. C₄₃

#9825

semidirect

136. C₂×C₈

Dic₁₇₂

semidirect

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

subgroup	index
C₂×#14583279	2
#19135556	2
#19134755	2
C₂×C₈²×C₄₃	2
#9825	43

subgroup

index

C₂×C₈²×C₄₃

Subgroups — #24996364

Conjugacy classes of subgroups

Complements

Maximal subgroups