Group #19135356 — subgroups

Subgroups — #19135356

Order 5504.

256 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×#13063256	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#14582661	C₂	1	2	yes	maximal
5	C₄₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₄₃	—	2	1	no	—
10	#13063256	—	2	4	no	—
11	C₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
12	C₄₃⋊#95	—	2	1	no	—
13	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	—	2	2	no	—
18	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₃	—	2	1	no	—
22	C₂²×Dic₄₃	—	2	1	no	—
23	C₄⋊C₄×C₄₃	—	2	2	no	—
24	C₄⋊C₄×C₄₃	—	2	2	no	—
25	C₄⋊C₄×C₄₃	—	4	2	no	—
26	C₄⋊C₄×C₄₃	—	4	2	no	—
27	C₄²×C₄₃	C₂×C₄	1	2	yes	—
28	C₄²×C₄₃	Q₈	1	2	yes	—
29	C₄⋊Dic₄₃	—	2	1	no	—
30	C₄⋊Dic₄₃	—	2	1	no	—
31	C₄⋊Dic₄₃	D₄	1	2	yes	—
32	C₄⋊Dic₄₃	Q₈	1	2	yes	—
33	C₄×Dic₄₃	—	4	2	no	—
34	C₂×C₄₃⋊C₈	—	2	2	no	—
35	C₂×C₄₃⋊C₈	—	2	2	no	—
36	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	—	2	1	no	—
39	C₂×C₄×C₄₃	—	2	1	no	—
40	C₂×C₄×C₄₃	—	4	2	no	—
41	C₂×C₄×C₄₃	—	4	2	no	—
42	C₂×C₄×C₄₃	—	4	2	no	—
43	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	4	1	no	—
56	C₂×Dic₄₃	—	4	1	no	—
57	C₂×Dic₄₃	—	4	1	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	C₄₃⋊C₈	—	2	4	no	—
60	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄₃	#121	1	1	yes	characteristic
62	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₄₃	#137	1	1	yes	characteristic
67	C₄×C₄₃	—	2	2	no	—
68	C₄×C₄₃	—	8	2	no	—
69	C₄×C₄₃	#106	1	4	yes	complemented
70	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
71	C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
72	C₄×C₄₃	#125	1	1	yes	characteristic
73	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
74	Dic₄₃	—	4	2	no	—
75	Dic₄₃	—	4	2	no	—
76	#1948	—	43	1	no	maximal, Sylow
77	C₂×C₄₃	#389	1	1	yes	characteristic
78	C₂×C₄₃	C₂×#106	1	1	yes	characteristic
79	C₂×C₄₃	#428	1	1	yes	characteristic
80	C₂×C₄₃	#460	1	1	yes	characteristic
81	C₂×C₄₃	#461	1	1	yes	characteristic
82	C₂×C₄₃	#475	1	1	yes	characteristic
83	C₂×C₄₃	#476	1	1	yes	characteristic
84	C₂×#106	—	43	2	no	—
85	C₂×C₄⋊C₈	—	43	1	no	—
86	#384	D₄₃	1	1	yes	characteristic
87	C₄×C₄⋊C₄	—	43	1	no	—
88	#340	—	43	2	no	—
89	C₄₃	#1948	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	43	2	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	43	1	no	—
94	C₂×C₄⋊C₄	—	43	1	no	—
95	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
96	C₂×C₄²	—	86	1	no	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	#106	—	86	4	no	—
99	C₄⋊C₈	—	43	4	no	—
100	#95	—	86	1	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	86	1	no	—
103	C₂²×C₄	—	86	1	no	—
104	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
107	C₂×C₈	—	43	4	no	—
108	C₂×C₈	—	86	2	no	—
109	C₂×C₈	—	86	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	43	2	no	—
115	C₄⋊C₄	—	43	2	no	—
116	C₄⋊C₄	—	86	1	no	—
117	C₄⋊C₄	—	86	1	no	—
118	C₄²	—	172	2	no	—
119	C₄²	Dic₈₆	1	2	yes	—
120	C₄²	C₄×D₄₃	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	86	2	no	—
128	C₂×C₄	—	86	2	no	—
129	C₂×C₄	—	172	1	no	—
130	C₂×C₄	—	172	1	no	—
131	C₂×C₄	—	172	1	no	—
132	C₂×C₄	—	172	1	no	—
133	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
134	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
135	C₂×C₄	#10621883	1	2	yes	—
136	C₂×C₄	#10621884	1	2	yes	—
137	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
139	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
140	C₂×C₄	#10621900	1	1	yes	characteristic
141	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
142	C₂×C₄	#10621902	1	1	yes	characteristic
143	C₈	—	86	4	no	—
144	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
145	C₂²	#13063296	1	1	yes	characteristic
146	C₂²	#13063301	1	1	yes	characteristic
147	C₂²	C₂×#10621884	1	1	yes	characteristic
148	C₂²	C₂×#10621883	1	1	yes	characteristic
149	C₂²	#13063351	1	1	yes	characteristic
150	C₂²	#13063356	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	172	2	no	—
154	C₄	—	172	2	no	—
155	C₄	#13063256	1	4	yes	complemented
156	C₄	#13063287	1	1	yes	characteristic
157	C₄	#13063290	1	1	yes	characteristic
158	C₄	#13063316	1	1	yes	characteristic
159	C₄	#13063324	1	1	yes	characteristic
160	C₂	#14582795	1	1	yes	characteristic
161	C₂	C₂×#13063256	1	1	yes	characteristic
162	C₂	#14582939	1	1	yes	characteristic
163	C₂	#14583096	1	1	yes	characteristic
164	C₂	#14583101	1	1	yes	characteristic
165	C₂	#14583181	1	1	yes	characteristic
166	C₂	#14583186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063256

C₂

yes

maximal

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal

C₄₃×#384

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

—

#13063256

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

complemented

C₄₃⋊#95

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

Q₈

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

—

C₄⋊Dic₄₃

Q₈

yes

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#106

yes

complemented

C₄×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

C₄×Q₈

yes

characteristic

C₄×C₄₃

#125

yes

characteristic

C₄×C₄₃

C₄⋊Q₈

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊C₈

—

#384

D₄₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₄₃

#1948

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#106

—

C₄⋊C₈

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

106

C₂²×C₄

C₂²×D₄₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

172

—

119

C₄²

Dic₈₆

yes

—

120

C₄²

C₄×D₄₃

yes

—

121

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

172

—

130

C₂×C₄

—

172

—

131

C₂×C₄

—

172

—

132

C₂×C₄

—

172

—

133

C₂×C₄

Dic₄₃⋊C₄

yes

—

134

C₂×C₄

Dic₄₃⋊C₄

yes

—

135

C₂×C₄

#10621883

yes

—

136

C₂×C₄

#10621884

yes

—

137

C₂×C₄

C₂×Dic₈₆

yes

characteristic

138

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

139

C₂×C₄

D₄×D₄₃

yes

characteristic

140

C₂×C₄

#10621900

yes

characteristic

141

C₂×C₄

Q₈×D₄₃

yes

characteristic

142

C₂×C₄

#10621902

yes

characteristic

143

C₈

—

144

C₂²

C₂×Dic₄₃⋊C₄

yes

characteristic

145

C₂²

#13063296

yes

characteristic

146

C₂²

#13063301

yes

characteristic

147

C₂²

C₂×#10621884

yes

characteristic

148

C₂²

C₂×#10621883

yes

characteristic

149

C₂²

#13063351

yes

characteristic

150

C₂²

#13063356

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

172

—

154

C₄

—

172

—

155

C₄

#13063256

yes

complemented

156

C₄

#13063287

yes

characteristic

157

C₄

#13063290

yes

characteristic

158

C₄

#13063316

yes

characteristic

159

C₄

#13063324

yes

characteristic

160

C₂

#14582795

yes

characteristic

161

C₂

C₂×#13063256

yes

characteristic

162

C₂

#14582939

yes

characteristic

163

C₂

#14583096

yes

characteristic

164

C₂

#14583101

yes

characteristic

165

C₂

#14583181

yes

characteristic

166

C₂

#14583186

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#384	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#384	D₄₃
hypocenter	C₄₃	#1948
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#384

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. C₄⋊(C₄₃⋊C₈)	C₄	semidirect
69. C₄×C₄₃	#106	semidirect
89. C₄₃	#1948	semidirect
155. C₄	#13063256	semidirect

H ≅ G/N

type

11. C₄⋊(C₄₃⋊C₈)

C₄

semidirect

69. C₄×C₄₃

#106

semidirect

89. C₄₃

#1948

semidirect

155. C₄

#13063256

semidirect

Subgroups — #19135356

Conjugacy classes of subgroups

Complements