Group #19133764 — subgroups

Subgroups — #19133764

Order 5504.

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

normal subgroups only

This group has 194 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	#14583623	C₂	1	1	yes	maximal, characteristic
2	#14583191	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583186	C₂	1	1	yes	maximal, characteristic
4	#14582645	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582631	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582628	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	#13063320	—	2	2	no	—
9	#13063314	—	2	1	no	—
10	#13063305	—	2	1	no	—
11	Q₈×Dic₄₃	—	2	1	no	—
12	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
13	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
14	#13063283	—	2	1	no	—
15	#13063258	—	2	1	no	—
16	#13063256	—	2	1	no	—
17	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	#13063248	—	2	1	no	—
20	#13063245	—	2	1	no	—
21	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	#103×C₄₃	—	2	2	no	—
24	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	C₂×Q₈×C₄₃	—	2	1	no	—
27	C₂×D₁₇₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₄₃	—	2	1	no	—
29	C₂×Dic₈₆	D₄	1	1	yes	characteristic
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	—	2	1	no	—
32	C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₄⋊C₄×C₄₃	—	2	1	no	—
34	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	(C₂×D₄₃)⋊C₄	—	2	2	no	—
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	(C₂×D₄₃)⋊C₄	—	2	1	no	—
39	C₄⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
41	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
42	Dic₄₃⋊C₄	—	2	1	no	—
43	C₄×Dic₄₃	—	2	2	no	—
44	C₄×Dic₄₃	—	2	1	no	—
45	C₂×C₄₃⋊C₈	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂²×D₄₃	—	2	1	no	—
48	Q₈×C₄₃	—	4	2	no	—
49	Q₈×C₄₃	—	4	1	no	—
50	Q₈×C₄₃	—	4	1	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	4	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	—	2	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	—	2	1	no	—
64	Dic₈₆	—	2	1	no	—
65	C₈×C₄₃	—	4	1	no	—
66	C₄₃⋊C₈	—	4	1	no	—
67	C₄₃⋊C₈	—	4	1	no	—
68	C₂²×C₄₃	#120	1	1	yes	characteristic
69	C₂×D₄₃	—	4	1	no	—
70	C₂×D₄₃	—	4	1	no	—
71	C₂×D₄₃	—	4	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	4	2	no	—
74	C₄×C₄₃	—	8	1	no	—
75	C₄×C₄₃	#135	1	1	yes	characteristic
76	C₄×C₄₃	#135	1	1	yes	characteristic
77	C₄×C₄₃	#137	1	1	yes	characteristic
78	C₄×C₄₃	#137	1	1	yes	characteristic
79	Dic₄₃	—	4	1	no	—
80	Dic₄₃	—	4	1	no	—
81	Dic₄₃	—	4	1	no	—
82	#1638	—	43	1	no	maximal, Sylow
83	C₂×C₄₃	#452	1	1	yes	characteristic
84	C₂×C₄₃	#452	1	1	yes	characteristic
85	C₂×C₄₃	#454	1	1	yes	characteristic
86	D₄₃	—	8	1	no	—
87	#556	—	43	1	no	—
88	#487	D₄₃	1	1	yes	characteristic, complemented
89	#477	—	43	1	no	—
90	#476	—	43	1	no	—
91	#332	—	43	1	no	—
92	Q₈⋊C₈	—	43	1	no	—
93	D₄⋊C₈	—	43	1	no	—
94	C₄₃	#1638	1	1	yes	characteristic, complemented, Sylow
95	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
96	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
97	#126	—	86	2	no	—
98	#124	—	86	1	no	—
99	C₂²⋊Q₈	—	86	1	no	—
100	C₄×Q₈	—	43	1	no	—
101	C₄×Q₈	—	86	1	no	—
102	C₄×D₄	—	43	1	no	—
103	#117	—	86	1	no	—
104	#107	—	86	1	no	—
105	#106	—	86	1	no	—
106	C₄⋊C₈	—	43	1	no	—
107	C₄⋊C₈	—	43	1	no	—
108	#103	—	2	2	no	—
109	#103	—	86	1	no	—
110	#102	—	86	1	no	—
111	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	—	43	1	no	—
115	C₂×D₄	—	43	1	no	—
116	C₂²×C₄	—	86	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	86	1	no	—
119	C₂×C₈	—	86	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	43	1	no	—
124	C₄⋊C₄	—	43	1	no	—
125	C₄⋊C₄	—	86	2	no	—
126	C₄⋊C₄	—	86	1	no	—
127	C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
128	C₂²⋊C₄	—	86	2	no	—
129	C₂²⋊C₄	—	86	2	no	—
130	C₂²⋊C₄	—	86	1	no	—
131	C₄²	—	86	2	no	—
132	C₄²	—	86	1	no	—
133	C₄²	C₂²×D₄₃	1	1	yes	characteristic
134	C₂³	—	86	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	86	1	no	—
139	Q₈	—	86	1	no	—
140	D₄	—	86	1	no	—
141	D₄	—	86	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	86	1	no	—
145	C₂×C₄	—	86	1	no	—
146	C₂×C₄	—	86	1	no	—
147	C₂×C₄	—	172	1	no	—
148	C₂×C₄	—	172	1	no	—
149	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
150	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
152	C₈	—	4	1	no	—
153	C₈	—	172	1	no	—
154	C₈	—	172	1	no	—
155	C₂²	—	172	1	no	—
156	C₂²	—	172	1	no	—
157	C₂²	—	172	1	no	—
158	C₂²	#13063293	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	172	1	no	—
163	C₄	—	172	1	no	—
164	C₄	—	172	1	no	—
165	C₄	#13063346	1	1	yes	characteristic
166	C₄	#13063348	1	1	yes	characteristic
167	C₄	#13063354	1	1	yes	characteristic
168	C₄	#13063357	1	1	yes	characteristic
169	C₂	—	344	1	no	—
170	C₂	#14583058	1	1	yes	characteristic
171	C₂	#14583060	1	1	yes	characteristic
172	C₂	#14583071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583623

C₂

yes

maximal, characteristic

#14583191

C₂

yes

maximal, characteristic, complemented

#14583186

C₂

yes

maximal, characteristic

#14582645

C₂

yes

maximal, characteristic, complemented

#14582631

C₂

yes

maximal, characteristic, complemented

#14582628

C₂

yes

maximal, characteristic

C₄₃×#487

C₂

yes

maximal, characteristic, complemented

#13063320

—

#13063314

—

#13063305

—

Q₈×Dic₄₃

—

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

—

—

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063248

—

#13063245

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

#103×C₄₃

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

D₄

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂²×D₄₃

—

Q₈×C₄₃

—

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

—

C₈×C₄₃

—

C₄₃⋊C₈

—

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

—

#556

—

#487

D₄₃

yes

characteristic, complemented

#477

—

#476

—

#332

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₄₃

#1638

yes

characteristic, complemented, Sylow

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

#126

—

#124

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

102

C₄×D₄

—

103

#117

—

104

#107

—

105

#106

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

#103

—

109

#103

—

110

#102

—

111

C₄×C₈

C₂×D₄₃

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂²×C₄

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₂²⋊D₄₃

yes

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×D₄₃

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

172

—

148

C₂×C₄

—

172

—

149

C₂×C₄

D₄×D₄₃

yes

characteristic

150

C₂×C₄

D₄×D₄₃

yes

characteristic

151

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

152

C₈

—

153

C₈

—

172

—

154

C₈

—

172

—

155

C₂²

—

172

—

156

C₂²

—

172

—

157

C₂²

—

172

—

158

C₂²

#13063293

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

172

—

163

C₄

—

172

—

164

C₄

—

172

—

165

C₄

#13063346

yes

characteristic

166

C₄

#13063348

yes

characteristic

167

C₄

#13063354

yes

characteristic

168

C₄

#13063357

yes

characteristic

169

C₂

—

344

—

170

C₂

#14583058

yes

characteristic

171

C₂

#14583060

yes

characteristic

172

C₂

#14583071

yes

characteristic

	subgroup	G/H
center	C₂²	#13063293
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#487	C₂
socle	C₂²×C₄₃	#120
hypercenter	#487	D₄₃
hypocenter	C₄₃	#1638
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#13063293

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#487

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. #14583191	C₂	semidirect
4. #14582645	C₂	semidirect
5. #14582631	C₂	semidirect
7. C₄₃×#487	C₂	semidirect
88. #487	D₄₃	semidirect
94. C₄₃	#1638	semidirect

H ≅ G/N

type

2. #14583191

C₂

semidirect

4. #14582645

C₂

semidirect

5. #14582631

C₂

semidirect

7. C₄₃×#487

C₂

semidirect

88. #487

D₄₃

semidirect

94. C₄₃

#1638

semidirect

Subgroups — #19133764

Conjugacy classes of subgroups

Complements