Group #19135906 — subgroups

Subgroups — #19135906

Order 5504.

360 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063247	C₂	1	2	yes	maximal, complemented
3	#14582801	C₂	1	2	yes	maximal, complemented
4	#14582662	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₇₂	C₂²	1	2	yes	—
7	C₂×D₃₄₄	—	2	4	no	—
8	C₂×D₃₄₄	—	2	2	no	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
11	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
12	#13063247	—	2	4	no	—
13	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₃	—	2	2	no	—
17	C₂³×D₄₃	—	2	2	no	—
18	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₃	—	2	2	no	—
22	C₂×D₁₇₂	—	2	2	no	—
23	C₂×D₁₇₂	—	2	2	no	—
24	C₂×D₁₇₂	—	2	2	no	—
25	C₂×D₁₇₂	—	2	2	no	—
26	C₂×D₁₇₂	—	2	2	no	—
27	C₂×D₁₇₂	D₄	1	4	yes	—
28	C₂×C₈×C₄₃	—	2	2	no	—
29	C₂×C₈×C₄₃	—	2	1	no	—
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₂×C₈×C₄₃	D₄	1	2	yes	—
32	C₄²×C₄₃	—	2	1	no	—
33	C₄²×C₄₃	—	2	1	no	—
34	(C₂×D₄₃)⋊C₄	—	4	4	no	—
35	(C₂×D₄₃)⋊C₄	—	4	4	no	—
36	C₄⋊Dic₄₃	—	2	2	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	D₃₄₄	—	2	8	no	—
39	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₄₃	—	2	4	no	—
41	C₂²×D₄₃	—	4	2	no	—
42	C₂²×D₄₃	—	4	2	no	—
43	C₂²×D₄₃	—	4	2	no	—
44	C₂²×D₄₃	—	4	2	no	—
45	C₂²×D₄₃	—	4	2	no	—
46	C₂²×D₄₃	—	4	2	no	—
47	C₂×C₄×C₄₃	—	2	1	no	—
48	C₂×C₄×C₄₃	—	2	1	no	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	D₈	1	2	yes	complemented
52	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	2	yes	—
56	C₂×Dic₄₃	—	4	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	C₂×Dic₄₃	—	4	2	no	—
59	D₁₇₂	—	2	4	no	—
60	D₁₇₂	—	2	4	no	—
61	D₁₇₂	—	4	2	no	—
62	D₁₇₂	—	4	2	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	—	4	1	no	—
65	C₂²×C₄₃	#121	1	1	yes	characteristic
66	C₂²×C₄₃	#121	1	2	yes	—
67	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
68	C₂²×C₄₃	#135	1	1	yes	characteristic
69	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×D₄₃	—	4	4	no	—
71	C₂×D₄₃	—	4	4	no	—
72	C₂×D₄₃	—	4	4	no	—
73	C₂×D₄₃	—	8	2	no	—
74	C₂×D₄₃	—	8	2	no	—
75	C₂×D₄₃	—	8	2	no	—
76	C₂×D₄₃	—	8	2	no	—
77	C₄×C₄₃	—	2	2	no	—
78	C₄×C₄₃	—	4	1	no	—
79	C₄×C₄₃	#120	1	1	yes	characteristic
80	C₄×C₄₃	#123	1	2	yes	—
81	C₄×C₄₃	#124	1	1	yes	characteristic
82	Dic₄₃	—	8	2	no	—
83	#2054	—	43	1	no	maximal, Sylow
84	C₂×C₄₃	#394	1	1	yes	characteristic
85	C₂×C₄₃	#459	1	2	yes	—
86	C₂×C₄₃	#463	1	2	yes	—
87	C₂×C₄₃	#466	1	1	yes	characteristic
88	C₂×C₄₃	#469	1	1	yes	characteristic
89	D₄₃	—	8	4	no	—
90	C₂²×D₈	—	43	1	no	—
91	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
92	C₂×#102	—	43	2	no	—
93	#390	—	43	2	no	—
94	#340	—	43	1	no	—
95	C₄₃	#2054	1	1	yes	characteristic, complemented, Sylow
96	C₂²×D₄	—	43	2	no	—
97	C₂×D₈	—	86	4	no	—
98	C₂×D₈	—	86	2	no	—
99	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
100	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	43	2	no	—
102	C₂×C₂²⋊C₄	—	86	2	no	—
103	C₂×C₂²⋊C₄	—	86	2	no	—
104	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	#102	—	86	4	no	—
107	C₂⁴	—	86	2	no	—
108	C₂×D₄	—	43	4	no	—
109	C₂×D₄	—	86	2	no	—
110	C₂×D₄	—	86	2	no	—
111	C₂×D₄	—	86	2	no	—
112	C₂×D₄	—	86	2	no	—
113	C₂×D₄	—	86	2	no	—
114	C₂²×C₄	—	86	2	no	—
115	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
116	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
118	D₈	—	86	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
123	C₄⋊C₄	—	86	2	no	—
124	C₄⋊C₄	—	86	2	no	—
125	C₂²⋊C₄	—	172	4	no	—
126	C₂²⋊C₄	—	172	4	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₂³	—	86	4	no	—
130	C₂³	—	172	2	no	—
131	C₂³	—	172	2	no	—
132	C₂³	—	172	2	no	—
133	C₂³	—	172	2	no	—
134	C₂³	—	172	2	no	—
135	C₂³	—	172	2	no	—
136	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
137	D₄	—	86	4	no	—
138	D₄	—	86	4	no	—
139	D₄	—	172	2	no	—
140	D₄	—	172	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	172	2	no	—
146	C₂×C₄	—	172	2	no	—
147	C₂×C₄	—	172	2	no	—
148	C₂×C₄	D₃₄₄	1	2	yes	complemented
149	C₂×C₄	#10621898	1	1	yes	characteristic
150	C₂×C₄	D₄×D₄₃	1	2	yes	—
151	C₂×C₄	#10621902	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	4	1	no	—
155	C₂²	—	344	2	no	—
156	C₂²	—	344	2	no	—
157	C₂²	—	344	2	no	—
158	C₂²	—	344	2	no	—
159	C₂²	—	172	4	no	—
160	C₂²	—	172	4	no	—
161	C₂²	—	172	4	no	—
162	C₂²	#13063299	1	1	yes	characteristic
163	C₂²	#13063300	1	2	yes	—
164	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
165	C₂²	#13063347	1	1	yes	characteristic
166	C₂²	#13063353	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	344	2	no	—
170	C₄	#13063293	1	1	yes	characteristic
171	C₄	#13063310	1	2	yes	—
172	C₄	#13063314	1	1	yes	characteristic
173	C₂	—	344	4	no	—
174	C₂	#14582816	1	1	yes	characteristic
175	C₂	#14583095	1	2	yes	—
176	C₂	#14583115	1	2	yes	—
177	C₂	#14583135	1	1	yes	characteristic
178	C₂	#14583152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₃₄₄

C₂

yes

maximal, characteristic

C₂×#13063247

C₂

yes

maximal, complemented

#14582801

C₂

yes

maximal, complemented

#14582662

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

—

C₂×(C₂×D₄₃)⋊C₄

—

#13063247

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

D₃₄₄

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

D₈

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂×D₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

characteristic

C₄×C₄₃

#123

yes

—

C₄×C₄₃

#124

yes

characteristic

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#463

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂²×D₈

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic, complemented

C₂×#102

—

#390

—

#340

—

C₄₃

#2054

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

100

C₂²×C₈

C₂×D₄₃

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₄₃

yes

characteristic

105

C₄⋊C₈

—

106

#102

—

107

C₂⁴

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

D₁₇₂

yes

characteristic

116

C₂²×C₄

D₁₇₂

yes

characteristic

117

C₂²×C₄

C₂²×D₄₃

yes

characteristic

118

D₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₄₃

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

172

—

126

C₂²⋊C₄

—

172

—

127

C₄²

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

172

—

131

C₂³

—

172

—

132

C₂³

—

172

—

133

C₂³

—

172

—

134

C₂³

—

172

—

135

C₂³

—

172

—

136

C₂³

C₂×D₁₇₂

yes

characteristic

137

D₄

—

138

D₄

—

139

D₄

—

172

—

140

D₄

—

172

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

172

—

146

C₂×C₄

—

172

—

147

C₂×C₄

—

172

—

148

C₂×C₄

D₃₄₄

yes

complemented

149

C₂×C₄

#10621898

yes

characteristic

150

C₂×C₄

D₄×D₄₃

yes

—

151

C₂×C₄

#10621902

yes

—

152

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

153

C₈

—

154

C₈

—

155

C₂²

—

344

—

156

C₂²

—

344

—

157

C₂²

—

344

—

158

C₂²

—

344

—

159

C₂²

—

172

—

160

C₂²

—

172

—

161

C₂²

—

172

—

162

C₂²

#13063299

yes

characteristic

163

C₂²

#13063300

yes

—

164

C₂²

C₂×D₃₄₄

yes

characteristic

165

C₂²

#13063347

yes

characteristic

166

C₂²

#13063353

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

344

—

170

C₄

#13063293

yes

characteristic

171

C₄

#13063310

yes

—

172

C₄

#13063314

yes

characteristic

173

C₂

—

344

—

174

C₂

#14582816

yes

characteristic

175

C₂

#14583095

yes

—

176

C₂

#14583115

yes

—

177

C₂

#14583135

yes

characteristic

178

C₂

#14583152

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#2054
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#2054

LCS

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. C₂×#13063247	C₂	semidirect
3. #14582801	C₂	semidirect
4. #14582662	C₂	semidirect
5. C₂×C₄⋊C₈×C₄₃	C₂	semidirect
51. C₂×C₄×C₄₃	D₈	semidirect
91. C₂×C₄⋊C₈	D₄₃	semidirect
95. C₄₃	#2054	semidirect
148. C₂×C₄	D₃₄₄	semidirect

H ≅ G/N

type

2. C₂×#13063247

C₂

semidirect

3. #14582801

C₂

semidirect

4. #14582662

C₂

semidirect

5. C₂×C₄⋊C₈×C₄₃

C₂

semidirect

51. C₂×C₄×C₄₃

D₈

semidirect

91. C₂×C₄⋊C₈

D₄₃

semidirect

95. C₄₃

#2054

semidirect

148. C₂×C₄

D₃₄₄

semidirect

Subgroups — #19135906

Conjugacy classes of subgroups

Complements