Group #19133867 — subgroups

Subgroups — #19133867

Order 5504.

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

normal subgroups only

This group has 200 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	#14583622	C₂	1	1	yes	maximal, characteristic
2	#14583297	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583180	C₂	1	1	yes	maximal, characteristic
4	#14583109	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582646	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582628	C₂	1	1	yes	maximal, characteristic
7	C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621885	—	2	1	no	—
9	#13063325	C₂²	1	1	yes	characteristic
10	#13063325	C₂²	1	1	yes	characteristic
11	#13063320	—	2	2	no	—
12	#13063313	—	2	1	no	—
13	#13063301	—	2	1	no	—
14	C₄×Dic₈₆	—	2	1	no	—
15	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
16	#13063283	—	2	1	no	—
17	#13063259	—	2	1	no	—
18	#13063257	—	2	2	no	—
19	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	#13063248	—	2	1	no	—
21	#13063247	—	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	D₄	1	1	yes	characteristic
27	C₂×C₄×D₄₃	—	2	1	no	—
28	C₂×Dic₈₆	—	2	1	no	—
29	C₂×Dic₈₆	—	2	1	no	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₂×C₈×C₄₃	—	2	1	no	—
32	C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	—	2	1	no	—
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	#10621885	—	4	2	no	—
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	(C₂×D₄₃)⋊C₄	—	2	1	no	—
40	C₄⋊Dic₄₃	—	2	1	no	—
41	C₄⋊Dic₄₃	—	2	1	no	—
42	C₄⋊Dic₄₃	—	2	1	no	—
43	C₄⋊Dic₄₃	D₄	1	2	yes	—
44	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
45	Dic₄₃⋊C₄	—	2	2	no	—
46	C₄×Dic₄₃	—	2	2	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂²×D₄₃	—	2	1	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	2	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	D₁₇₂	—	2	1	no	—
60	D₁₇₂	—	2	1	no	—
61	C₄×D₄₃	—	4	1	no	—
62	C₄×D₄₃	—	4	1	no	—
63	Dic₈₆	—	4	1	no	—
64	Dic₈₆	—	4	1	no	—
65	Dic₈₆	—	4	1	no	—
66	Dic₈₆	—	4	1	no	—
67	C₈×C₄₃	—	2	2	no	—
68	C₈×C₄₃	—	4	1	no	—
69	C₄₃⋊C₈	—	4	1	no	—
70	C₂²×C₄₃	#120	1	1	yes	characteristic
71	C₂×D₄₃	—	4	1	no	—
72	C₂×D₄₃	—	4	1	no	—
73	C₂×D₄₃	—	4	1	no	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	4	1	no	—
76	C₄×C₄₃	—	4	1	no	—
77	C₄×C₄₃	#135	1	1	yes	characteristic
78	C₄×C₄₃	#135	1	1	yes	characteristic
79	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₄×C₄₃	#137	1	1	yes	characteristic
81	Dic₄₃	—	4	1	no	—
82	Dic₄₃	—	4	2	no	—
83	Dic₄₃	—	8	1	no	—
84	#1652	—	43	1	no	maximal, Sylow
85	C₂×C₄₃	#449	1	1	yes	characteristic
86	C₂×C₄₃	#452	1	1	yes	characteristic
87	C₂×C₄₃	#456	1	1	yes	characteristic
88	D₄₃	—	8	1	no	—
89	#556	—	43	1	no	—
90	#498	—	43	1	no	—
91	#474	—	43	1	no	—
92	#462	—	43	1	no	—
93	#332	—	43	1	no	—
94	Q₈⋊C₈	D₄₃	1	1	yes	characteristic, complemented
95	D₄⋊C₈	—	43	1	no	—
96	C₄₃	#1652	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₁₆	—	86	1	no	—
98	C₄⋊Q₈	—	43	1	no	—
99	C₄⋊Q₈	—	43	1	no	—
100	#126	—	86	2	no	—
101	#124	—	86	1	no	—
102	C₂²⋊Q₈	—	86	1	no	—
103	C₄×Q₈	—	86	1	no	—
104	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
105	C₄×D₄	—	43	1	no	—
106	#117	—	86	1	no	—
107	#107	—	86	1	no	—
108	#106	—	86	2	no	—
109	C₄⋊C₈	—	43	1	no	—
110	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
111	#103	—	86	1	no	—
112	#102	—	86	1	no	—
113	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
114	C₂×Q₈	—	86	1	no	—
115	C₂×Q₈	—	86	1	no	—
116	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
117	C₂×D₄	—	43	1	no	—
118	C₂²×C₄	—	86	1	no	—
119	Q₁₆	—	172	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	86	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	43	2	no	—
125	C₄⋊C₄	—	43	1	no	—
126	C₄⋊C₄	—	86	2	no	—
127	C₄⋊C₄	—	86	1	no	—
128	C₄⋊C₄	—	86	1	no	—
129	C₄⋊C₄	—	86	1	no	—
130	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
131	C₂²⋊C₄	—	86	2	no	—
132	C₂²⋊C₄	—	86	2	no	—
133	C₂²⋊C₄	—	86	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	86	2	no	—
136	C₄²	C₂²×D₄₃	1	1	yes	characteristic
137	C₂³	—	86	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	172	1	no	—
141	Q₈	—	172	1	no	—
142	Q₈	—	172	1	no	—
143	Q₈	—	172	1	no	—
144	D₄	—	86	1	no	—
145	D₄	—	86	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	86	1	no	—
149	C₂×C₄	—	86	2	no	—
150	C₂×C₄	—	172	1	no	—
151	C₂×C₄	—	172	1	no	—
152	C₂×C₄	—	172	1	no	—
153	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
154	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
155	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	4	1	no	—
158	C₈	—	172	1	no	—
159	C₂²	—	172	1	no	—
160	C₂²	—	172	1	no	—
161	C₂²	—	172	1	no	—
162	C₂²	#13063295	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	344	1	no	—
167	C₄	—	172	1	no	—
168	C₄	—	172	2	no	—
169	C₄	#13063346	1	1	yes	characteristic
170	C₄	#13063347	1	1	yes	characteristic
171	C₄	#13063353	1	1	yes	characteristic
172	C₄	#13063357	1	1	yes	characteristic
173	C₂	—	344	1	no	—
174	C₂	#14583046	1	1	yes	characteristic
175	C₂	#14583064	1	1	yes	characteristic
176	C₂	#14583083	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583622

C₂

yes

maximal, characteristic

#14583297

C₂

yes

maximal, characteristic, complemented

#14583180

C₂

yes

maximal, characteristic

#14583109

C₂

yes

maximal, characteristic, complemented

#14582646

C₂

yes

maximal, characteristic, complemented

#14582628

C₂

yes

maximal, characteristic

C₄₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#10621885

—

#13063325

C₂²

yes

characteristic

#13063325

C₂²

yes

—

—

—

C₄×Dic₈₆

—

C₄×D₁₇₂

C₂²

yes

—

—

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063248

—

#13063247

—

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

D₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621885

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

—

C₄⋊Dic₄₃

D₄

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

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

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

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

D₄₃

yes

characteristic, complemented

D₄⋊C₈

—

C₄₃

#1652

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

#126

—

101

#124

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

C₂×D₄₃

yes

characteristic

105

C₄×D₄

—

106

#117

—

107

#107

—

108

#106

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₄₃

yes

characteristic

111

#103

—

112

#102

—

113

C₄×C₈

C₂×D₄₃

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

D₁₇₂

yes

characteristic

117

C₂×D₄

—

118

C₂²×C₄

—

119

Q₁₆

—

172

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

D₁₇₂

yes

characteristic

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₄₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

172

—

141

Q₈

—

172

—

142

Q₈

—

172

—

143

Q₈

—

172

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

172

—

151

C₂×C₄

—

172

—

152

C₂×C₄

—

172

—

153

C₂×C₄

C₂×D₁₇₂

yes

characteristic

154

C₂×C₄

D₄×D₄₃

yes

characteristic

155

C₂×C₄

D₄×D₄₃

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

172

—

159

C₂²

—

172

—

160

C₂²

—

172

—

161

C₂²

—

172

—

162

C₂²

#13063295

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

344

—

167

C₄

—

172

—

168

C₄

—

172

—

169

C₄

#13063346

yes

characteristic

170

C₄

#13063347

yes

characteristic

171

C₄

#13063353

yes

characteristic

172

C₄

#13063357

yes

characteristic

173

C₂

—

344

—

174

C₂

#14583046

yes

characteristic

175

C₂

#14583064

yes

characteristic

176

C₂

#14583083

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	#1652
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

#1652

LCS

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
2. #14583297	C₂	semidirect
4. #14583109	C₂	semidirect
5. #14582646	C₂	semidirect
7. C₄₃×Q₈⋊C₈	C₂	semidirect
94. Q₈⋊C₈	D₄₃	semidirect
96. C₄₃	#1652	semidirect

H ≅ G/N

type

2. #14583297

C₂

semidirect

4. #14583109

C₂

semidirect

5. #14582646

C₂

semidirect

7. C₄₃×Q₈⋊C₈

C₂

semidirect

94. Q₈⋊C₈

D₄₃

semidirect

96. C₄₃

#1652

semidirect

Subgroups — #19133867

Conjugacy classes of subgroups

Complements