Group #19133045 — subgroups

Subgroups — #19133045

Order 5504.

276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582626	C₂	1	4	yes	maximal, complemented
4	C₄₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10621898	—	2	1	no	—
6	C₂²×C₄₃⋊C₈	—	2	1	no	—
7	#13063325	C₄	1	1	yes	characteristic
8	#13063322	C₄	1	1	yes	characteristic
9	#13063313	—	2	1	no	—
10	#13063299	—	2	1	no	—
11	#13063299	C₄	1	2	yes	—
12	C₄×D₁₇₂	—	2	1	no	—
13	C₄×D₁₇₂	C₂²	1	2	yes	—
14	C₄×C₄₃⋊C₈	—	2	1	no	—
15	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×M₁₆×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₄₃	—	2	1	no	—
19	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₃	—	2	2	no	—
24	#10621898	—	4	2	no	—
25	#10621898	—	4	2	no	—
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	—	2	1	no	—
28	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₄₃	—	2	2	no	—
30	C₂×Dic₈₆	—	2	1	no	—
31	M₁₆×C₄₃	—	4	2	no	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₄²×C₄₃	D₄	1	1	yes	characteristic
36	C₄²×C₄₃	C₂³	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	C₂×C₄	1	2	yes	—
41	C₂×C₄₃⋊C₈	—	2	4	no	—
42	C₂×C₄₃⋊C₈	—	2	2	no	—
43	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×D₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
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	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²⋊D₄₃	—	4	4	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	D₁₇₂	—	2	4	no	—
58	D₁₇₂	—	4	2	no	—
59	D₁₇₂	—	4	2	no	—
60	C₄×D₄₃	—	4	2	no	—
61	C₄×D₄₃	—	4	2	no	—
62	Dic₈₆	—	4	2	no	—
63	C₈×C₄₃	—	4	2	no	—
64	C₄₃⋊C₈	—	2	4	no	—
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	#102	1	2	yes	—
68	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₄₃	—	4	2	no	—
70	C₂×D₄₃	—	4	2	no	—
71	C₂×D₄₃	—	4	2	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	#102	1	2	yes	—
75	C₄×C₄₃	#104	1	2	yes	—
76	C₄×C₄₃	#131	1	2	yes	—
77	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
78	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	Dic₄₃	—	4	2	no	—
80	#1492	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	—	2	2	no	—
82	C₂×C₄₃	#408	1	1	yes	characteristic
83	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
84	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
85	D₄₃	—	8	2	no	—
86	#532	—	43	1	no	—
87	#423	D₄₃	1	1	yes	characteristic, complemented
88	C₂×C₄×C₈	—	43	1	no	—
89	D₄⋊C₈	—	43	4	no	—
90	C₄₃	#1492	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	86	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂²×C₈	—	86	1	no	—
94	C₄⋊Q₈	—	43	1	no	—
95	#127	—	43	1	no	—
96	#124	—	86	1	no	—
97	#121	—	43	2	no	—
98	#121	—	86	1	no	—
99	C₄×D₄	—	43	2	no	—
100	C₄×D₄	—	86	1	no	—
101	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
104	C₄×C₈	—	43	2	no	—
105	C₄×C₈	—	86	1	no	—
106	#41	—	172	2	no	—
107	#41	—	172	2	no	—
108	C₂×Q₈	—	86	1	no	—
109	C₂×D₄	—	43	2	no	—
110	C₂×D₄	—	86	2	no	—
111	C₂×D₄	—	86	1	no	—
112	C₂×D₄	—	86	1	no	—
113	C₂²×C₄	—	86	2	no	—
114	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
115	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	86	4	no	—
120	C₂×C₈	—	86	2	no	—
121	C₄⋊C₄	—	43	2	no	—
122	C₄⋊C₄	—	86	1	no	—
123	C₂²⋊C₄	—	86	2	no	—
124	C₂²⋊C₄	—	86	2	no	—
125	C₄²	C₄×D₄₃	1	1	yes	characteristic
126	C₄²	D₁₇₂	1	1	yes	characteristic
127	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
128	C₄²	C₂²×D₄₃	1	1	yes	characteristic
129	C₂³	—	86	2	no	—
130	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
131	Q₈	—	172	2	no	—
132	D₄	—	86	4	no	—
133	D₄	—	172	4	no	—
134	D₄	—	172	2	no	—
135	D₄	—	172	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	86	2	no	—
139	C₂×C₄	—	172	2	no	—
140	C₂×C₄	—	172	2	no	—
141	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
142	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
143	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#10621882	1	2	yes	—
145	C₂×C₄	#10621884	1	2	yes	—
146	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
147	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
149	C₈	—	4	2	no	—
150	C₈	—	86	4	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	172	2	no	—
154	C₂²	—	172	2	no	—
155	C₂²	—	172	2	no	—
156	C₂²	#13063245	1	2	yes	—
157	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	172	2	no	—
161	C₄	#13063245	1	2	yes	—
162	C₄	#13063251	1	2	yes	—
163	C₄	#13063333	1	2	yes	—
164	C₄	C₂×#10621882	1	1	yes	characteristic
165	C₄	C₂×#10621884	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	—	344	2	no	—
168	C₂	#14582867	1	1	yes	characteristic
169	C₂	C₂×#13063245	1	1	yes	characteristic
170	C₂	C₂×#13063251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#14582626

C₂

yes

maximal, complemented

C₄₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

—

C₂²×C₄₃⋊C₈

—

#13063325

C₄

yes

characteristic

#13063322

C₄

yes

—

—

C₄

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₂²

yes

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

—

C₂×M₁₆×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

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

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#102

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#104

yes

—

C₄×C₄₃

#131

yes

—

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

Dic₄₃

—

#1492

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#423

D₄₃

yes

characteristic, complemented

C₂×C₄×C₈

—

D₄⋊C₈

—

C₄₃

#1492

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

100

C₄×D₄

—

101

C₂×C₄²

C₂×D₄₃

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₄₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

#41

—

172

—

107

#41

—

172

—

108

C₂×Q₈

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄×D₄₃

yes

characteristic

115

C₂²×C₄

D₁₇₂

yes

characteristic

116

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

C₄×D₄₃

yes

characteristic

126

C₄²

D₁₇₂

yes

characteristic

127

C₄²

C₂²⋊D₄₃

yes

characteristic

128

C₄²

C₂²×D₄₃

yes

characteristic

129

C₂³

—

130

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

131

Q₈

—

172

—

132

D₄

—

133

D₄

—

172

—

134

D₄

—

172

—

135

D₄

—

172

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

172

—

140

C₂×C₄

—

172

—

141

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

142

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

143

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

144

C₂×C₄

#10621882

yes

—

145

C₂×C₄

#10621884

yes

—

146

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

147

C₂×C₄

C₂×D₁₇₂

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

172

—

154

C₂²

—

172

—

155

C₂²

—

172

—

156

C₂²

#13063245

yes

—

157

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

172

—

161

C₄

#13063245

yes

—

162

C₄

#13063251

yes

—

163

C₄

#13063333

yes

—

164

C₄

C₂×#10621882

yes

characteristic

165

C₄

C₂×#10621884

yes

characteristic

166

C₂

—

167

C₂

—

344

—

168

C₂

#14582867

yes

characteristic

169

C₂

C₂×#13063245

yes

characteristic

170

C₂

C₂×#13063251

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#423	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#423	D₄₃
hypocenter	C₄₃	#1492
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#423

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×C₄×C₄₃⋊C₈	C₂	semidirect
3. #14582626	C₂	semidirect
4. C₄₃×#423	C₂	semidirect
87. #423	D₄₃	semidirect
90. C₄₃	#1492	semidirect

H ≅ G/N

type

2. C₂×C₄×C₄₃⋊C₈

C₂

semidirect

3. #14582626

C₂

semidirect

4. C₄₃×#423

C₂

semidirect

87. #423

D₄₃

semidirect

90. C₄₃

#1492

semidirect

Subgroups — #19133045

Conjugacy classes of subgroups

Complements