Group #19133718 — subgroups

Subgroups — #19133718

Order 5504.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#14583634	C₂	1	1	yes	maximal, characteristic
2	#14583272	C₂	1	1	yes	maximal, characteristic
3	#14583195	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583194	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582642	C₂	1	1	yes	maximal, characteristic
6	#14582631	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621873	—	2	2	no	—
9	#13063325	C₂²	1	1	yes	characteristic
10	#13063322	C₂²	1	1	yes	characteristic
11	#13063314	—	2	1	no	—
12	#13063313	—	2	1	no	—
13	#13063310	—	2	2	no	—
14	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
15	C₄×D₁₇₂	—	2	1	no	—
16	#13063287	—	2	1	no	—
17	#13063259	—	2	1	no	—
18	#13063258	—	2	1	no	—
19	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	#13063250	—	2	1	no	—
21	#13063249	—	2	1	no	—
22	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	—	2	1	no	—
28	C₂×D₁₇₂	D₄	1	2	yes	—
29	C₂×C₄×D₄₃	—	2	2	no	—
30	C₂×Dic₈₆	—	2	1	no	—
31	C₂×Dic₈₆	D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₃	—	2	1	no	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₄⋊C₄×C₄₃	—	2	1	no	—
35	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₄²×C₄₃	—	2	1	no	—
37	C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	(C₂×D₄₃)⋊C₄	—	2	2	no	—
41	C₄⋊Dic₄₃	—	2	1	no	—
42	C₄⋊Dic₄₃	—	2	1	no	—
43	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
44	Dic₄₃⋊C₄	—	2	1	no	—
45	C₄×Dic₄₃	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	#10621873	—	4	4	no	—
48	C₂²×D₄₃	—	2	2	no	—
49	Q₈×C₄₃	—	2	1	no	—
50	Q₈×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₄₃	—	2	1	no	—
57	C₂×Dic₄₃	—	2	1	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	D₁₇₂	—	4	2	no	—
60	D₁₇₂	—	4	1	no	—
61	D₁₇₂	—	4	1	no	—
62	D₁₇₂	—	4	1	no	—
63	D₁₇₂	—	4	1	no	—
64	C₄×D₄₃	—	8	2	no	—
65	Dic₈₆	—	2	1	no	—
66	Dic₈₆	—	2	1	no	—
67	Dic₈₆	—	4	2	no	—
68	C₈×C₄₃	—	2	2	no	—
69	C₈×C₄₃	—	4	1	no	—
70	C₄₃⋊C₈	—	4	1	no	—
71	C₂²×C₄₃	#120	1	1	yes	characteristic
72	C₂×D₄₃	—	4	2	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₂×D₄₃	—	4	2	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	4	1	no	—
77	C₄×C₄₃	—	4	1	no	—
78	C₄×C₄₃	#135	1	1	yes	characteristic
79	C₄×C₄₃	#135	1	1	yes	characteristic
80	C₄×C₄₃	#137	1	1	yes	characteristic
81	C₄×C₄₃	#137	1	1	yes	characteristic
82	Dic₄₃	—	4	1	no	—
83	Dic₄₃	—	4	1	no	—
84	Dic₄₃	—	8	1	no	—
85	#1630	—	43	1	no	maximal, Sylow
86	C₂×C₄₃	#452	1	1	yes	characteristic
87	C₂×C₄₃	#452	1	1	yes	characteristic
88	C₂×C₄₃	#453	1	1	yes	characteristic
89	D₄₃	—	8	2	no	—
90	#559	—	43	1	no	—
91	#492	—	43	1	no	—
92	#477	—	43	1	no	—
93	#477	—	43	1	no	—
94	#331	—	43	1	no	—
95	Q₈⋊C₈	—	43	1	no	—
96	Q₈⋊C₈	D₄₃	1	1	yes	characteristic, complemented
97	C₄₃	#1630	1	1	yes	characteristic, complemented, Sylow
98	C₂×SD₁₆	—	86	2	no	—
99	C₄⋊Q₈	—	43	1	no	—
100	#127	—	43	1	no	—
101	#124	—	86	1	no	—
102	#124	—	86	1	no	—
103	#123	—	86	2	no	—
104	C₄×Q₈	—	43	1	no	—
105	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
106	C₄×D₄	—	86	1	no	—
107	C₄×D₄	—	86	1	no	—
108	#107	—	86	1	no	—
109	#107	—	86	1	no	—
110	C₄⋊C₈	—	43	1	no	—
111	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
112	#103	—	86	1	no	—
113	#103	—	86	1	no	—
114	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
115	C₂×Q₈	—	43	1	no	—
116	C₂×Q₈	—	86	1	no	—
117	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
118	C₂×D₄	—	43	2	no	—
119	C₂×D₄	—	86	1	no	—
120	C₂×D₄	—	86	1	no	—
121	C₂²×C₄	—	86	2	no	—
122	SD₁₆	—	172	4	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	86	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	43	1	no	—
128	C₄⋊C₄	—	86	1	no	—
129	C₄⋊C₄	—	86	1	no	—
130	C₄⋊C₄	—	86	1	no	—
131	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
132	C₂²⋊C₄	—	86	2	no	—
133	C₂²⋊C₄	—	86	2	no	—
134	C₂²⋊C₄	—	86	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	86	1	no	—
137	C₄²	C₂²×D₄₃	1	1	yes	characteristic
138	C₂³	—	86	2	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	2	1	no	—
141	Q₈	—	86	1	no	—
142	Q₈	—	86	1	no	—
143	Q₈	—	172	2	no	—
144	D₄	—	172	2	no	—
145	D₄	—	172	1	no	—
146	D₄	—	172	1	no	—
147	D₄	—	172	1	no	—
148	D₄	—	172	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	86	1	no	—
152	C₂×C₄	—	86	1	no	—
153	C₂×C₄	—	344	2	no	—
154	C₂×C₄	—	172	1	no	—
155	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
156	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
157	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	4	1	no	—
160	C₈	—	172	1	no	—
161	C₂²	—	172	2	no	—
162	C₂²	—	172	2	no	—
163	C₂²	—	172	2	no	—
164	C₂²	#13063295	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	344	1	no	—
169	C₄	—	172	1	no	—
170	C₄	—	172	1	no	—
171	C₄	#13063346	1	1	yes	characteristic
172	C₄	#13063347	1	1	yes	characteristic
173	C₄	#13063357	1	1	yes	characteristic
174	C₄	#13063358	1	1	yes	characteristic
175	C₂	—	344	2	no	—
176	C₂	#14583063	1	1	yes	characteristic
177	C₂	#14583064	1	1	yes	characteristic
178	C₂	#14583068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583634

C₂

yes

maximal, characteristic

#14583272

C₂

yes

maximal, characteristic

#14583195

C₂

yes

maximal, characteristic, complemented

#14583194

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic

#14582631

C₂

yes

maximal, characteristic, complemented

C₄₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#10621873

—

#13063325

C₂²

yes

characteristic

#13063322

C₂²

yes

—

—

—

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×D₁₇₂

—

#13063287

—

#13063259

—

#13063258

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063250

—

#13063249

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

#10621873

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₄₃

yes

characteristic, complemented

C₄₃

#1630

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₄⋊Q₈

—

100

#127

—

101

#124

—

102

#124

—

103

#123

—

104

C₄×Q₈

—

105

C₄×Q₈

C₂×D₄₃

yes

characteristic

106

C₄×D₄

—

107

C₄×D₄

—

108

#107

—

109

#107

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₄₃

yes

characteristic

112

#103

—

113

#103

—

114

C₄×C₈

C₂×D₄₃

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

D₁₇₂

yes

characteristic

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

SD₁₆

—

172

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

D₁₇₂

yes

characteristic

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²×D₄₃

yes

characteristic

138

C₂³

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

172

—

144

D₄

—

172

—

145

D₄

—

172

—

146

D₄

—

172

—

147

D₄

—

172

—

148

D₄

—

172

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

344

—

154

C₂×C₄

—

172

—

155

C₂×C₄

C₂×D₁₇₂

yes

characteristic

156

C₂×C₄

D₄×D₄₃

yes

characteristic

157

C₂×C₄

D₄×D₄₃

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

172

—

161

C₂²

—

172

—

162

C₂²

—

172

—

163

C₂²

—

172

—

164

C₂²

#13063295

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

344

—

169

C₄

—

172

—

170

C₄

—

172

—

171

C₄

#13063346

yes

characteristic

172

C₄

#13063347

yes

characteristic

173

C₄

#13063357

yes

characteristic

174

C₄

#13063358

yes

characteristic

175

C₂

—

344

—

176

C₂

#14583063

yes

characteristic

177

C₂

#14583064

yes

characteristic

178

C₂

#14583068

yes

characteristic

	subgroup	G/H
center	C₂²	#13063295
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×Q₈⋊C₈	C₂
socle	C₂²×C₄₃	#120
hypercenter	Q₈⋊C₈	D₄₃
hypocenter	C₄₃	#1630
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13063295

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×Q₈⋊C₈

C₂

socle

C₂²×C₄₃

#120

hypercenter

Q₈⋊C₈

D₄₃

hypocenter

C₄₃

#1630

LCS

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
3. #14583195	C₂	semidirect
4. #14583194	C₂	semidirect
6. #14582631	C₂	semidirect
7. C₄₃×Q₈⋊C₈	C₂	semidirect
96. Q₈⋊C₈	D₄₃	semidirect
97. C₄₃	#1630	semidirect

H ≅ G/N

type

3. #14583195

C₂

semidirect

4. #14583194

C₂

semidirect

6. #14582631

C₂

semidirect

7. C₄₃×Q₈⋊C₈

C₂

semidirect

96. Q₈⋊C₈

D₄₃

semidirect

97. C₄₃

#1630

semidirect

Subgroups — #19133718

Conjugacy classes of subgroups

Complements