Group #19135678 — subgroups

Subgroups — #19135678

Order 5504.

406 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 99 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×#10621884	C₂	1	2	yes	maximal
2	C₂×#13063249	C₂	1	2	yes	maximal, complemented
3	#14582801	C₂	1	1	yes	maximal, characteristic
4	#14582654	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
7	C₂×#10621884	—	2	4	no	—
8	C₂×#10621884	—	2	4	no	—
9	C₂×#10621884	—	2	4	no	—
10	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
13	#13063249	—	2	4	no	—
14	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₄₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₂³×D₄₃	—	2	1	no	—
19	C₂×Q₈×C₄₃	—	2	2	no	—
20	C₂×Q₈×C₄₃	—	2	2	no	—
21	C₂×Q₈×C₄₃	—	2	2	no	—
22	C₂×Q₈×C₄₃	—	2	2	no	—
23	C₂×Q₈×C₄₃	—	2	2	no	—
24	C₂×Q₈×C₄₃	D₄	1	4	yes	—
25	C₂²×C₄×C₄₃	—	2	2	no	—
26	C₂²×C₄×C₄₃	D₄	1	2	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₄₃	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	2	no	—
31	C₂×D₁₇₂	—	2	2	no	—
32	C₂×D₁₇₂	D₄	1	2	yes	—
33	C₄⋊C₄×C₄₃	—	2	8	no	—
34	C₄²×C₄₃	—	2	2	no	—
35	#10621884	—	2	16	no	—
36	(C₂×D₄₃)⋊C₄	—	4	4	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	C₂×C₄₃⋊C₈	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂×C₄₃⋊C₈	D₄	1	4	yes	—
41	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₄₃	—	2	2	no	—
43	C₂²×D₄₃	—	4	1	no	—
44	C₂²×D₄₃	—	4	1	no	—
45	C₂²×D₄₃	—	4	2	no	—
46	C₂²×D₄₃	—	4	2	no	—
47	Q₈×C₄₃	—	2	4	no	—
48	Q₈×C₄₃	—	2	4	no	—
49	Q₈×C₄₃	—	4	2	no	—
50	Q₈×C₄₃	—	4	2	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	2	4	no	—
54	C₂×C₄×C₄₃	—	4	2	no	—
55	C₂×C₄×C₄₃	—	4	2	no	—
56	C₂×C₄×C₄₃	SD₁₆	1	4	yes	complemented
57	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
61	C₂×Dic₄₃	—	4	1	no	—
62	C₂×Dic₄₃	—	4	2	no	—
63	D₁₇₂	—	2	4	no	—
64	D₁₇₂	—	4	2	no	—
65	C₄₃⋊C₈	—	2	4	no	—
66	C₂²×C₄₃	#120	1	1	yes	characteristic
67	C₂²×C₄₃	#121	1	1	yes	characteristic
68	C₂²×C₄₃	#127	1	1	yes	characteristic
69	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
70	C₂²×C₄₃	#137	1	2	yes	—
71	C₂×D₄₃	—	4	2	no	—
72	C₂×D₄₃	—	4	4	no	—
73	C₂×D₄₃	—	8	2	no	—
74	C₂×D₄₃	—	8	1	no	—
75	C₂×D₄₃	—	8	1	no	—
76	C₄×C₄₃	—	2	4	no	—
77	C₄×C₄₃	—	4	4	no	—
78	C₄×C₄₃	#120	1	2	yes	—
79	C₄×C₄₃	#121	1	2	yes	—
80	Dic₄₃	—	8	1	no	—
81	#2013	—	43	1	no	maximal, Sylow
82	C₂×C₄₃	#392	1	1	yes	characteristic
83	C₂×C₄₃	#448	1	2	yes	—
84	C₂×C₄₃	#461	1	2	yes	—
85	C₂×C₄₃	#492	1	1	yes	characteristic
86	C₂×C₄₃	#497	1	1	yes	characteristic
87	D₄₃	—	8	2	no	—
88	C₂²×SD₁₆	—	43	2	no	—
89	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
90	C₂×#103	—	43	2	no	—
91	#390	—	43	1	no	—
92	#336	—	43	1	no	—
93	C₄₃	#2013	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
95	C₂²×D₄	—	43	1	no	—
96	C₂×SD₁₆	—	86	4	no	—
97	C₂×SD₁₆	—	86	4	no	—
98	C₂×SD₁₆	—	86	4	no	—
99	C₂²×C₈	—	43	2	no	—
100	C₄⋊Q₈	—	2	4	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	43	1	no	—
103	C₂×C₂²⋊C₄	—	86	2	no	—
104	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
105	#103	—	86	4	no	—
106	C₂⁴	—	86	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	C₂²⋊D₄₃	1	4	yes	—
113	C₂×D₄	—	43	2	no	—
114	C₂×D₄	—	86	2	no	—
115	C₂×D₄	—	86	1	no	—
116	C₂×D₄	—	86	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	86	1	no	—
119	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
121	SD₁₆	—	86	16	no	—
122	C₂×C₈	—	43	4	no	—
123	C₂×C₈	—	86	2	no	—
124	C₂×C₈	—	86	2	no	—
125	C₄⋊C₄	—	2	8	no	—
126	C₄⋊C₄	—	86	2	no	—
127	C₂²⋊C₄	—	172	4	no	—
128	C₄²	—	2	2	no	—
129	C₂³	—	86	2	no	—
130	C₂³	—	172	1	no	—
131	C₂³	—	172	1	no	—
132	C₂³	—	172	2	no	—
133	C₂³	—	172	2	no	—
134	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	D₄	—	86	4	no	—
140	D₄	—	172	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	172	1	no	—
147	C₂×C₄	—	172	2	no	—
148	C₂×C₄	#10621884	1	4	yes	complemented
149	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
150	C₂×C₄	D₄×D₄₃	1	2	yes	—
151	C₂×C₄	#10621900	1	1	yes	characteristic
152	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
153	C₈	—	86	4	no	—
154	C₂²	—	344	1	no	—
155	C₂²	—	344	1	no	—
156	C₂²	—	344	2	no	—
157	C₂²	—	172	2	no	—
158	C₂²	—	172	4	no	—
159	C₂²	#13063294	1	1	yes	characteristic
160	C₂²	#13063296	1	1	yes	characteristic
161	C₂²	#13063321	1	1	yes	characteristic
162	C₂²	C₂×#10621884	1	2	yes	—
163	C₂²	#13063356	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	—	344	1	no	—
167	C₄	#13063293	1	2	yes	—
168	C₄	#13063297	1	2	yes	—
169	C₂	—	344	2	no	—
170	C₂	#14582807	1	1	yes	characteristic
171	C₂	#14583038	1	2	yes	—
172	C₂	#14583101	1	2	yes	—
173	C₂	#14583273	1	1	yes	characteristic
174	C₂	#14583291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621884

C₂

yes

maximal

C₂×#13063249

C₂

yes

maximal, complemented

#14582801

C₂

yes

maximal, characteristic

#14582654

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

—

—

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

#13063249

—

C₂²×Q₈×C₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

#10621884

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

SD₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×SD₁₆

yes

—

C₂²×C₄₃

#137

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

—

C₄×C₄₃

#121

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#461

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

D₄₃

yes

characteristic, complemented

C₂×#103

—

#390

—

#336

—

C₄₃

#2013

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₃

yes

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

—

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₄₃

yes

characteristic

105

#103

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

C₂²⋊D₄₃

yes

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊D₄₃

yes

—

120

C₂²×C₄

C₂²×D₄₃

yes

characteristic

121

SD₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

172

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

172

—

131

C₂³

—

172

—

132

C₂³

—

172

—

133

C₂³

—

172

—

134

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

D₄

—

140

D₄

—

172

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

172

—

147

C₂×C₄

—

172

—

148

C₂×C₄

#10621884

yes

complemented

149

C₂×C₄

D₄×D₄₃

yes

characteristic

150

C₂×C₄

D₄×D₄₃

yes

—

151

C₂×C₄

#10621900

yes

characteristic

152

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

153

C₈

—

154

C₂²

—

344

—

155

C₂²

—

344

—

156

C₂²

—

344

—

157

C₂²

—

172

—

158

C₂²

—

172

—

159

C₂²

#13063294

yes

characteristic

160

C₂²

#13063296

yes

characteristic

161

C₂²

#13063321

yes

characteristic

162

C₂²

C₂×#10621884

yes

—

163

C₂²

#13063356

yes

—

164

C₄

—

165

C₄

—

166

C₄

—

344

—

167

C₄

#13063293

yes

—

168

C₄

#13063297

yes

—

169

C₂

—

344

—

170

C₂

#14582807

yes

characteristic

171

C₂

#14583038

yes

—

172

C₂

#14583101

yes

—

173

C₂

#14583273

yes

characteristic

174

C₂

#14583291

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#2013
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₃

hypocenter

C₄₃

#2013

LCS

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#13063249	C₂	semidirect
4. #14582654	C₂	semidirect
5. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
56. C₂×C₄×C₄₃	SD₁₆	semidirect
89. C₂×C₄⋊Q₈	D₄₃	semidirect
93. C₄₃	#2013	semidirect
148. C₂×C₄	#10621884	semidirect

H ≅ G/N

type

2. C₂×#13063249

C₂

semidirect

4. #14582654

C₂

semidirect

5. C₂×C₄⋊Q₈×C₄₃

C₂

semidirect

56. C₂×C₄×C₄₃

SD₁₆

semidirect

89. C₂×C₄⋊Q₈

D₄₃

semidirect

93. C₄₃

#2013

semidirect

148. C₂×C₄

#10621884

semidirect

Subgroups — #19135678

Conjugacy classes of subgroups

Complements