Group #19134031 — subgroups

Subgroups — #19134031

Order 5504.

182 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 51 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	#14583245	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583105	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583091	C₂	1	1	yes	maximal, characteristic
4	#14582964	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582651	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582644	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621873	—	2	1	no	—
9	C₂×#10621882	—	2	1	no	—
10	C₂×C₈×D₄₃	—	2	1	no	—
11	#13063325	C₂²	1	1	yes	characteristic
12	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#13063250	—	2	1	no	—
15	#13063246	—	2	1	no	—
16	#13063246	—	2	1	no	—
17	#13063246	—	2	1	no	—
18	(C₂×D₄₃)⋊C₈	—	2	1	no	—
19	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₄₃	—	2	1	no	—
24	C₂×D₄×C₄₃	—	2	1	no	—
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₁₇₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₄₃	—	2	1	no	—
28	C₂×Dic₈₆	—	2	1	no	—
29	C₂×C₈×C₄₃	—	2	1	no	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	#10621882	—	4	2	no	—
33	(C₂×D₄₃)⋊C₄	—	2	1	no	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄₃⋊C₈	—	2	1	no	—
38	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	#10621873	—	4	2	no	—
41	C₈×D₄₃	—	4	2	no	—
42	C₂³×C₄₃	—	4	1	no	—
43	C₂²×D₄₃	—	2	1	no	—
44	D₄×C₄₃	—	4	1	no	—
45	D₄×C₄₃	—	4	1	no	—
46	D₄×C₄₃	—	4	1	no	—
47	D₄×C₄₃	—	4	1	no	—
48	D₄×C₄₃	—	8	1	no	—
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
52	C₂×Dic₄₃	—	2	1	no	—
53	C₂×Dic₄₃	—	4	1	no	—
54	D₁₇₂	—	2	1	no	—
55	D₁₇₂	D₈	1	2	yes	—
56	C₄×D₄₃	—	4	1	no	—
57	C₄×D₄₃	—	4	1	no	—
58	Dic₈₆	—	4	1	no	—
59	Dic₈₆	—	4	1	no	—
60	C₈×C₄₃	—	4	1	no	—
61	C₈×C₄₃	—	4	1	no	—
62	C₄₃⋊C₈	—	2	1	no	—
63	C₄₃⋊C₈	—	2	1	no	—
64	C₄₃⋊C₈	—	4	1	no	—
65	C₂²×C₄₃	—	8	1	no	—
66	C₂²×C₄₃	—	8	1	no	—
67	C₂²×C₄₃	—	8	1	no	—
68	C₂²×C₄₃	#121	1	1	yes	characteristic
69	C₂×D₄₃	—	2	2	no	—
70	C₂×D₄₃	—	4	1	no	—
71	C₂×D₄₃	—	4	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
74	C₄×C₄₃	#135	1	1	yes	characteristic
75	C₄×C₄₃	#135	1	1	yes	characteristic
76	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₄₃	—	4	1	no	—
78	Dic₄₃	—	8	1	no	—
79	#1681	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	16	1	no	—
81	C₂×C₄₃	#463	1	1	yes	characteristic
82	C₂×C₄₃	#466	1	1	yes	characteristic
83	C₂×C₄₃	#471	1	1	yes	characteristic
84	D₄₃	—	4	2	no	—
85	#486	—	43	1	no	—
86	#461	—	43	1	no	—
87	#459	—	43	1	no	—
88	C₈×D₄	—	43	1	no	—
89	#335	—	43	1	no	—
90	#332	—	43	1	no	—
91	#331	D₄₃	1	1	yes	characteristic, complemented
92	C₄₃	#1681	1	1	yes	characteristic, complemented, Sylow
93	C₂×SD₁₆	—	86	1	no	—
94	C₂×D₈	—	86	1	no	—
95	C₂²×C₈	—	86	1	no	—
96	C₄⋊Q₈	—	43	1	no	—
97	#127	C₂×D₄₃	1	1	yes	characteristic
98	C₄×D₄	—	43	1	no	—
99	C₄⋊C₈	—	43	1	no	—
100	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
101	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
102	#103	—	86	1	no	—
103	#102	—	86	1	no	—
104	#102	—	86	1	no	—
105	#102	—	86	1	no	—
106	C₂²⋊C₈	—	86	1	no	—
107	C₄×C₈	—	43	1	no	—
108	C₂×Q₈	—	86	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	43	1	no	—
113	C₂²×C₄	—	86	1	no	—
114	SD₁₆	—	172	2	no	—
115	D₈	—	172	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	43	1	no	—
119	C₂×C₈	—	43	1	no	—
120	C₂×C₈	—	86	1	no	—
121	C₂×C₈	—	172	2	no	—
122	C₄⋊C₄	—	43	1	no	—
123	C₄⋊C₄	—	86	1	no	—
124	C₄⋊C₄	—	86	1	no	—
125	C₂²⋊C₄	—	86	1	no	—
126	C₄²	C₂²×D₄₃	1	1	yes	characteristic
127	C₂³	—	4	1	no	—
128	C₂³	—	86	1	no	—
129	Q₈	—	172	1	no	—
130	Q₈	—	172	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	8	1	no	—
136	D₄	—	43	2	no	—
137	D₄	—	86	1	no	—
138	C₂×C₄	—	86	1	no	—
139	C₂×C₄	—	172	1	no	—
140	C₂×C₄	—	172	1	no	—
141	C₂×C₄	—	172	1	no	—
142	C₂×C₄	#10621898	1	1	yes	characteristic
143	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
144	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	86	1	no	—
148	C₈	—	86	1	no	—
149	C₈	—	172	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	86	2	no	—
154	C₂²	—	172	1	no	—
155	C₂²	—	172	1	no	—
156	C₂²	#13063298	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	344	1	no	—
159	C₄	—	172	1	no	—
160	C₄	D₈×D₄₃	1	1	yes	characteristic
161	C₄	#13063346	1	1	yes	characteristic
162	C₄	#13063346	1	1	yes	characteristic
163	C₄	#13063349	1	1	yes	characteristic
164	C₂	—	16	1	no	—
165	C₂	—	172	2	no	—
166	C₂	#14583113	1	1	yes	characteristic
167	C₂	#14583134	1	1	yes	characteristic
168	C₂	#14583161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583245

C₂

yes

maximal, characteristic, complemented

#14583105

C₂

yes

maximal, characteristic, complemented

#14583091

C₂

yes

maximal, characteristic

#14582964

C₂

yes

maximal, characteristic, complemented

#14582651

C₂

yes

maximal, characteristic, complemented

#14582644

C₂

yes

maximal, characteristic, complemented

C₄₃×#331

C₂

yes

maximal, characteristic, complemented

C₂×#10621873

—

C₂×#10621882

—

C₂×C₈×D₄₃

—

#13063325

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

—

(C₂×D₄₃)⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

#127×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621882

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

#10621873

—

C₈×D₄₃

—

C₂³×C₄₃

—

C₂²×D₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

D₈

yes

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#1681

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#486

—

#461

—

#459

—

C₈×D₄

—

#335

—

#332

—

#331

D₄₃

yes

characteristic, complemented

C₄₃

#1681

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

C₂×D₄₃

yes

characteristic

C₄×D₄

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₄₃

yes

characteristic

101

C₄⋊C₈

C₂×D₄₃

yes

characteristic

102

#103

—

103

#102

—

104

#102

—

105

#102

—

106

C₂²⋊C₈

—

107

C₄×C₈

—

108

C₂×Q₈

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

SD₁₆

—

172

—

115

D₈

—

172

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

172

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

C₂²×D₄₃

yes

characteristic

127

C₂³

—

128

C₂³

—

129

Q₈

—

172

—

130

Q₈

—

172

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

172

—

140

C₂×C₄

—

172

—

141

C₂×C₄

—

172

—

142

C₂×C₄

#10621898

yes

characteristic

143

C₂×C₄

D₄×D₄₃

yes

characteristic

144

C₂×C₄

D₄×D₄₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

172

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

172

—

155

C₂²

—

172

—

156

C₂²

#13063298

yes

characteristic

157

C₄

—

158

C₄

—

344

—

159

C₄

—

172

—

160

C₄

D₈×D₄₃

yes

characteristic

161

C₄

#13063346

yes

characteristic

162

C₄

#13063346

yes

characteristic

163

C₄

#13063349

yes

characteristic

164

C₂

—

165

C₂

—

172

—

166

C₂

#14583113

yes

characteristic

167

C₂

#14583134

yes

characteristic

168

C₂

#14583161

yes

characteristic

	subgroup	G/H
center	C₂²	#13063298
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#331	C₂
socle	C₂²×C₄₃	#121
hypercenter	#331	D₄₃
hypocenter	C₄₃	#1681
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#13063298

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#331

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #14583245	C₂	semidirect
2. #14583105	C₂	semidirect
4. #14582964	C₂	semidirect
5. #14582651	C₂	semidirect
6. #14582644	C₂	semidirect
7. C₄₃×#331	C₂	semidirect
91. #331	D₄₃	semidirect
92. C₄₃	#1681	semidirect

H ≅ G/N

type

1. #14583245

C₂

semidirect

2. #14583105

C₂

semidirect

4. #14582964

C₂

semidirect

5. #14582651

C₂

semidirect

6. #14582644

C₂

semidirect

7. C₄₃×#331

C₂

semidirect

91. #331

D₄₃

semidirect

92. C₄₃

#1681

semidirect

Subgroups — #19134031

Conjugacy classes of subgroups

Complements