Group #19133156 — subgroups

Subgroups — #19133156

Order 5504.

264 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14583449	C₂	1	1	yes	maximal, characteristic
2	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582634	C₂	1	4	yes	maximal, complemented
4	C₄₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10621898	—	2	1	no	—
6	#13063325	C₄	1	1	yes	characteristic, complemented
7	#13063322	C₄	1	1	yes	characteristic, complemented
8	#13063313	—	2	1	no	—
9	#13063299	—	2	1	no	—
10	#13063299	C₂²	1	2	yes	—
11	C₄×D₁₇₂	—	2	1	no	—
12	C₄×D₁₇₂	C₄	1	2	yes	complemented
13	C₂×C₄⋊Dic₄₃	—	2	1	no	—
14	C₄₃⋊#95	—	2	1	no	—
15	C₄₃⋊#95	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₄×C₈×C₄₃	—	2	1	no	—
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₃	—	2	2	no	—
24	C₂²×Dic₄₃	—	2	2	no	—
25	#10621898	—	4	2	no	—
26	#10621898	—	4	2	no	—
27	C₂×D₁₇₂	—	2	1	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₄₃	—	2	2	no	—
31	C₂×Dic₈₆	—	2	1	no	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₃	D₄	1	1	yes	characteristic
36	C₄²×C₄₃	D₄	1	1	yes	characteristic
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	C₄⋊Dic₄₃	—	2	1	no	—
40	C₄⋊Dic₄₃	—	4	2	no	—
41	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
42	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	4	1	no	—
45	C₂×C₄×C₄₃	—	4	1	no	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	D₈	1	2	yes	—
51	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
52	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²⋊D₄₃	—	4	2	no	—
55	C₂²⋊D₄₃	—	4	2	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	C₂×Dic₄₃	—	4	2	no	—
59	C₂×Dic₄₃	—	4	2	no	—
60	D₁₇₂	—	4	2	no	—
61	D₁₇₂	—	4	2	no	—
62	D₁₇₂	—	4	2	no	—
63	C₄×D₄₃	—	4	4	no	—
64	Dic₈₆	—	4	2	no	—
65	C₈×C₄₃	—	4	2	no	—
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	—	2	1	no	—
68	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
70	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×D₄₃	—	4	2	no	—
72	C₂×D₄₃	—	4	2	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	2	1	no	—
77	C₄×C₄₃	—	2	1	no	—
78	C₄×C₄₃	#102	1	2	yes	—
79	C₄×C₄₃	#102	1	2	yes	—
80	Dic₄₃	—	4	2	no	—
81	Dic₄₃	—	8	2	no	—
82	#1517	—	43	1	no	maximal, Sylow
83	C₂×C₄₃	—	2	1	no	—
84	C₂×C₄₃	—	2	1	no	—
85	C₂×C₄₃	#410	1	1	yes	characteristic
86	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
87	C₂×C₄₃	#421	1	1	yes	characteristic
88	D₄₃	—	8	2	no	—
89	#532	—	43	1	no	—
90	#431	D₄₃	1	1	yes	characteristic, complemented
91	#382	—	43	1	no	—
92	#327	—	43	4	no	—
93	C₄₃	#1517	1	1	yes	characteristic, complemented, Sylow
94	C₂×#41	—	86	1	no	—
95	C₄⋊Q₈	—	43	1	no	—
96	#127	—	43	1	no	—
97	#124	—	86	1	no	—
98	#121	—	43	2	no	—
99	#121	—	86	1	no	—
100	C₄×D₄	—	43	2	no	—
101	C₄×D₄	—	86	1	no	—
102	C₂×C₄⋊C₄	—	86	1	no	—
103	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	1	no	—
105	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
106	C₄×C₈	—	2	1	no	—
107	#95	—	43	2	no	—
108	#95	—	86	1	no	—
109	#41	—	172	2	no	—
110	#41	—	172	2	no	—
111	C₂×Q₈	—	86	1	no	—
112	C₂×D₄	—	43	2	no	—
113	C₂×D₄	—	86	2	no	—
114	C₂×D₄	—	86	1	no	—
115	C₂×D₄	—	86	1	no	—
116	C₂²×C₄	—	86	2	no	—
117	C₂²×C₄	—	86	2	no	—
118	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
119	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₄⋊C₄	—	43	2	no	—
123	C₄⋊C₄	—	86	1	no	—
124	C₄⋊C₄	—	172	2	no	—
125	C₂²⋊C₄	—	86	2	no	—
126	C₂²⋊C₄	—	86	2	no	—
127	C₄²	C₄×D₄₃	1	1	yes	characteristic
128	C₄²	C₄×D₄₃	1	1	yes	characteristic
129	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
130	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
131	C₂³	—	86	2	no	—
132	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
133	Q₈	—	172	2	no	—
134	D₄	—	172	2	no	—
135	D₄	—	172	2	no	—
136	D₄	—	172	2	no	—
137	D₄	—	172	2	no	—
138	D₄	—	172	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	86	2	no	—
142	C₂×C₄	—	172	2	no	—
143	C₂×C₄	—	172	2	no	—
144	C₂×C₄	—	172	2	no	—
145	C₂×C₄	—	172	4	no	—
146	C₂×C₄	#10621873	1	2	yes	—
147	C₂×C₄	D₃₄₄	1	2	yes	—
148	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
150	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
151	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
154	C₈	—	4	2	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	172	2	no	—
158	C₂²	—	172	2	no	—
159	C₂²	—	172	2	no	—
160	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
161	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
162	C₂²	C₂×#10621873	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	344	2	no	—
168	C₄	—	172	2	no	—
169	C₄	#13063247	1	2	yes	—
170	C₄	#13063247	1	2	yes	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	344	2	no	—
174	C₂	#14582872	1	1	yes	characteristic
175	C₂	C₂×#13063247	1	1	yes	characteristic
176	C₂	#14582915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14582765

C₂

yes

maximal, characteristic, complemented

#14582634

C₂

yes

maximal, complemented

C₄₃×#431

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

—

#13063325

C₄

yes

characteristic, complemented

#13063322

C₄

yes

characteristic, complemented

#13063313

—

#13063299

—

#13063299

C₂²

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₄₃⋊#95

—

C₄₃⋊#95

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#102

yes

—

Dic₄₃

—

Dic₄₃

—

#1517

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#431

D₄₃

yes

characteristic, complemented

#382

—

#327

—

C₄₃

#1517

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

—

—

—

—

—

100

C₄×D₄

—

101

C₄×D₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄²

C₂×D₄₃

yes

characteristic

104

C₄⋊C₈

—

105

C₂²⋊C₈

C₂×D₄₃

yes

—

106

C₄×C₈

—

107

#95

—

108

#95

—

109

#41

—

172

—

110

#41

—

172

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

D₁₇₂

yes

characteristic

119

C₂²×C₄

D₁₇₂

yes

characteristic

120

C₂²×C₄

C₂²×D₄₃

yes

characteristic

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

172

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

C₄×D₄₃

yes

characteristic

128

C₄²

C₄×D₄₃

yes

characteristic

129

C₄²

C₂²⋊D₄₃

yes

characteristic

130

C₄²

C₂²⋊D₄₃

yes

characteristic

131

C₂³

—

132

C₂³

C₂×D₁₇₂

yes

characteristic

133

Q₈

—

172

—

134

D₄

—

172

—

135

D₄

—

172

—

136

D₄

—

172

—

137

D₄

—

172

—

138

D₄

—

172

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

172

—

143

C₂×C₄

—

172

—

144

C₂×C₄

—

172

—

145

C₂×C₄

—

172

—

146

C₂×C₄

#10621873

yes

—

147

C₂×C₄

D₃₄₄

yes

—

148

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

149

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

150

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

151

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

152

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

153

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

154

C₈

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

172

—

158

C₂²

—

172

—

159

C₂²

—

172

—

160

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

161

C₂²

C₂×D₃₄₄

yes

characteristic

162

C₂²

C₂×#10621873

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

344

—

168

C₄

—

172

—

169

C₄

#13063247

yes

—

170

C₄

#13063247

yes

—

171

C₂

—

172

C₂

—

173

C₂

—

344

—

174

C₂

#14582872

yes

characteristic

175

C₂

C₂×#13063247

yes

characteristic

176

C₂

#14582915

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1517
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#431

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. #14582765	C₂	semidirect
3. #14582634	C₂	semidirect
4. C₄₃×#431	C₂	semidirect
6. #13063325	C₄	semidirect
7. #13063322	C₄	semidirect
12. C₄×D₁₇₂	C₄	semidirect
90. #431	D₄₃	semidirect
93. C₄₃	#1517	semidirect

H ≅ G/N

type

2. #14582765

C₂

semidirect

3. #14582634

C₂

semidirect

4. C₄₃×#431

C₂

semidirect

6. #13063325

C₄

semidirect

7. #13063322

C₄

semidirect

12. C₄×D₁₇₂

C₄

semidirect

90. #431

D₄₃

semidirect

93. C₄₃

#1517

semidirect

Subgroups — #19133156

Conjugacy classes of subgroups

Complements