Group #19132995 — subgroups

Subgroups — #19132995

Order 5504.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14582628	C₂	1	4	yes	maximal, complemented
4	C₄₃×#431	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×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂²⋊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	C₂×C₈×C₄₃	—	2	2	no	—
32	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	(C₂×D₄₃)⋊C₄	—	2	2	no	—
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	C₄⋊Dic₄₃	—	2	1	no	—
39	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂×C₄₃⋊C₈	—	4	2	no	—
42	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	4	1	no	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	D₈	1	2	yes	—
50	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
51	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²⋊D₄₃	—	4	4	no	—
55	C₂×Dic₄₃	—	2	2	no	—
56	D₁₇₂	—	2	4	no	—
57	D₁₇₂	—	4	2	no	—
58	D₁₇₂	—	4	2	no	—
59	C₄×D₄₃	—	4	2	no	—
60	C₄×D₄₃	—	4	2	no	—
61	Dic₈₆	—	4	2	no	—
62	C₈×C₄₃	—	4	2	no	—
63	C₄₃⋊C₈	—	4	2	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	#102	1	2	yes	—
67	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×D₄₃	—	4	2	no	—
69	C₂×D₄₃	—	4	2	no	—
70	C₂×D₄₃	—	4	2	no	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	#102	1	2	yes	—
75	C₄×C₄₃	#131	1	2	yes	—
76	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
77	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
78	Dic₄₃	—	4	2	no	—
79	#1480	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	2	no	—
81	C₂×C₄₃	#408	1	1	yes	characteristic
82	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
83	C₂×C₄₃	#421	1	1	yes	characteristic
84	D₄₃	—	8	2	no	—
85	#532	—	43	1	no	—
86	#431	D₄₃	1	1	yes	characteristic, complemented
87	C₂×C₄⋊C₈	—	43	1	no	—
88	D₄⋊C₈	—	43	4	no	—
89	C₄₃	#1480	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	86	1	no	—
91	C₂²×C₈	—	86	1	no	—
92	C₄⋊Q₈	—	43	1	no	—
93	#127	—	43	1	no	—
94	#124	—	86	1	no	—
95	#121	—	43	2	no	—
96	#121	—	86	1	no	—
97	C₄×D₄	—	43	2	no	—
98	C₄×D₄	—	86	1	no	—
99	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	43	2	no	—
102	C₄⋊C₈	—	86	1	no	—
103	C₂²⋊C₈	—	2	1	no	—
104	C₄×C₈	C₂×D₄₃	1	2	yes	—
105	#41	—	172	2	no	—
106	#41	—	172	2	no	—
107	C₂×Q₈	—	86	1	no	—
108	C₂×D₄	—	43	2	no	—
109	C₂×D₄	—	86	2	no	—
110	C₂×D₄	—	86	1	no	—
111	C₂×D₄	—	86	1	no	—
112	C₂²×C₄	—	86	2	no	—
113	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
114	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	86	2	no	—
118	C₂×C₈	—	172	2	no	—
119	C₄⋊C₄	—	43	2	no	—
120	C₄⋊C₄	—	86	1	no	—
121	C₂²⋊C₄	—	86	2	no	—
122	C₂²⋊C₄	—	86	2	no	—
123	C₄²	C₄×D₄₃	1	1	yes	characteristic
124	C₄²	D₁₇₂	1	1	yes	characteristic
125	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
126	C₄²	C₂²×D₄₃	1	1	yes	characteristic
127	C₂³	—	86	2	no	—
128	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
129	Q₈	—	172	2	no	—
130	D₄	—	86	4	no	—
131	D₄	—	172	2	no	—
132	D₄	—	172	4	no	—
133	D₄	—	172	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	86	2	no	—
137	C₂×C₄	—	172	2	no	—
138	C₂×C₄	—	172	2	no	—
139	C₂×C₄	#10621873	1	2	yes	—
140	C₂×C₄	D₃₄₄	1	2	yes	—
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	C₂×C₄×D₄₃	1	1	yes	characteristic
145	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
147	C₈	—	4	2	no	—
148	C₈	—	172	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	172	2	no	—
152	C₂²	—	172	2	no	—
153	C₂²	—	172	2	no	—
154	C₂²	#13063247	1	2	yes	—
155	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	172	2	no	—
160	C₄	#13063247	1	2	yes	—
161	C₄	#13063333	1	2	yes	—
162	C₄	C₂×D₃₄₄	1	1	yes	characteristic
163	C₄	C₂×#10621873	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	344	2	no	—
166	C₂	#14582867	1	1	yes	characteristic
167	C₂	C₂×#13063247	1	1	yes	characteristic
168	C₂	#14582917	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

#14582628

C₂

yes

maximal, complemented

C₄₃×#431

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×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊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

—

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

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#131

yes

—

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

Dic₄₃

—

#1480

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#431

D₄₃

yes

characteristic, complemented

C₂×C₄⋊C₈

—

D₄⋊C₈

—

C₄₃

#1480

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×C₈

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

C₂×D₄₃

yes

—

105

#41

—

172

—

106

#41

—

172

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₄×D₄₃

yes

characteristic

114

C₂²×C₄

D₁₇₂

yes

characteristic

115

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

C₄×D₄₃

yes

characteristic

124

C₄²

D₁₇₂

yes

characteristic

125

C₄²

C₂²⋊D₄₃

yes

characteristic

126

C₄²

C₂²×D₄₃

yes

characteristic

127

C₂³

—

128

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

129

Q₈

—

172

—

130

D₄

—

131

D₄

—

172

—

132

D₄

—

172

—

133

D₄

—

172

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

172

—

138

C₂×C₄

—

172

—

139

C₂×C₄

#10621873

yes

—

140

C₂×C₄

D₃₄₄

yes

—

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

C₂×C₄×D₄₃

yes

characteristic

145

C₂×C₄

C₂×D₁₇₂

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

147

C₈

—

148

C₈

—

172

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

172

—

152

C₂²

—

172

—

153

C₂²

—

172

—

154

C₂²

#13063247

yes

—

155

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

172

—

160

C₄

#13063247

yes

—

161

C₄

#13063333

yes

—

162

C₄

C₂×D₃₄₄

yes

characteristic

163

C₄

C₂×#10621873

yes

characteristic

164

C₂

—

165

C₂

—

344

—

166

C₂

#14582867

yes

characteristic

167

C₂

C₂×#13063247

yes

characteristic

168

C₂

#14582917

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1480
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#431

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. C₂×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
3. #14582628	C₂	semidirect
4. C₄₃×#431	C₂	semidirect
86. #431	D₄₃	semidirect
89. C₄₃	#1480	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₄₃⋊C₈)

C₂

semidirect

3. #14582628

C₂

semidirect

4. C₄₃×#431

C₂

semidirect

86. #431

D₄₃

semidirect

89. C₄₃

#1480

semidirect

Subgroups — #19132995

Conjugacy classes of subgroups

Complements