Group #19134690 — subgroups

Subgroups — #19134690

Order 5504.

300 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₂²⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14582759	C₂	1	1	yes	maximal, characteristic
3	#14582661	C₂	1	2	yes	maximal, complemented
4	#14582661	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
9	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
10	C₄₃⋊#95	—	2	1	no	—
11	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
12	#117×C₄₃	—	2	2	no	—
13	#117×C₄₃	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	—	2	1	no	—
18	C₂⁴×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	—	2	4	no	—
20	C₂²×C₄×C₄₃	—	2	1	no	—
21	C₂²×C₄×C₄₃	—	2	1	no	—
22	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₃	D₄	1	2	yes	—
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₃	—	2	2	no	—
27	C₄⋊C₄×C₄₃	—	2	1	no	—
28	C₄⋊C₄×C₄₃	—	2	1	no	—
29	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
30	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂²⋊C₄×C₄₃	—	2	4	no	—
32	C₄²×C₄₃	—	2	4	no	—
33	C₂²⋊Dic₄₃	—	4	2	no	—
34	C₄⋊Dic₄₃	—	2	4	no	—
35	C₂×C₄₃⋊C₈	—	2	2	no	—
36	C₂×C₄₃⋊C₈	—	2	4	no	—
37	C₂³×C₄₃	—	2	2	no	—
38	C₂³×C₄₃	—	2	2	no	—
39	C₂³×C₄₃	—	2	1	no	—
40	C₂³×C₄₃	—	2	1	no	—
41	C₂³×C₄₃	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	4	no	—
45	C₂×C₄×C₄₃	—	2	4	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	—	4	2	no	—
49	C₂×C₄×C₄₃	—	4	1	no	—
50	C₂×C₄×C₄₃	—	4	1	no	—
51	C₂×C₄×C₄₃	C₂²⋊C₄	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	#41	1	4	yes	—
55	C₂×Dic₄₃	—	4	2	no	—
56	C₂×Dic₄₃	—	4	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	C₄₃⋊C₈	—	4	2	no	—
59	C₂²×C₄₃	—	2	2	no	—
60	C₂²×C₄₃	—	2	2	no	—
61	C₂²×C₄₃	—	2	2	no	—
62	C₂²×C₄₃	—	4	2	no	—
63	C₂²×C₄₃	—	4	2	no	—
64	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₃	#123	1	2	yes	—
66	C₂²×C₄₃	#135	1	2	yes	—
67	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₄₃	#137	1	1	yes	characteristic
69	C₄×C₄₃	—	2	2	no	—
70	C₄×C₄₃	—	4	2	no	—
71	C₄×C₄₃	—	4	2	no	—
72	C₄×C₄₃	#117	1	2	yes	—
73	C₄×C₄₃	#123	1	2	yes	—
74	Dic₄₃	—	8	2	no	—
75	#1825	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	—	4	2	no	—
77	C₂×C₄₃	#381	1	1	yes	characteristic
78	C₂×C₄₃	#416	1	1	yes	characteristic
79	C₂×C₄₃	#417	1	1	yes	characteristic
80	C₂×C₄₃	#482	1	2	yes	—
81	C₂×C₄₃	#485	1	2	yes	—
82	C₂×#117	D₄₃	1	1	yes	characteristic
83	C₂×C₂²⋊C₈	—	43	1	no	—
84	#380	—	43	1	no	—
85	#340	—	43	2	no	—
86	#340	—	43	2	no	—
87	C₄₃	#1825	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
89	C₂²×C₈	—	43	2	no	—
90	#117	—	2	2	no	—
91	#117	Dic₄₃	1	4	yes	complemented
92	C₂×C₄⋊C₄	—	43	2	no	—
93	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	2	1	no	—
96	C₂×C₂²⋊C₄	—	86	1	no	—
97	C₂×C₄²	—	2	1	no	—
98	C₂²⋊C₈	—	86	2	no	—
99	#95	—	86	1	no	—
100	C₂⁴	C₂²⋊D₄₃	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	86	2	no	—
105	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
106	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
108	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
109	C₂×C₈	—	86	4	no	—
110	C₂×C₈	—	86	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	86	4	no	—
114	C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
115	C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
116	C₂²⋊C₄	—	2	4	no	—
117	C₂²⋊C₄	—	172	2	no	—
118	C₄²	—	2	4	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	2	1	no	—
122	C₂³	—	2	1	no	—
123	C₂³	C₂²⋊Dic₄₃	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	172	2	no	—
134	C₂×C₄	—	172	2	no	—
135	C₂×C₄	—	172	2	no	—
136	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
137	C₂×C₄	#10621898	1	4	yes	—
138	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
140	C₈	—	172	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
147	C₂²	#13063308	1	2	yes	—
148	C₂²	#13063345	1	2	yes	—
149	C₂²	#13063350	1	1	yes	characteristic
150	C₂²	#13063355	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	344	2	no	—
155	C₄	#13063284	1	2	yes	—
156	C₄	#13063308	1	2	yes	—
157	C₂	—	4	2	no	—
158	C₂	#14582762	1	1	yes	characteristic
159	C₂	#14582895	1	1	yes	characteristic
160	C₂	#14582900	1	1	yes	characteristic
161	C₂	#14583223	1	2	yes	—
162	C₂	#14583240	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

#14582759

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, complemented

#14582661

C₂

yes

maximal, complemented

C₂×#117×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

—

C₂²⋊(C₄₃⋊C₈)

—

C₄₃⋊#95

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

#117×C₄₃

—

#117×C₄₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

—

C₂⁴×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²⋊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

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#123

yes

—

C₂²×C₄₃

#135

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#117

yes

—

C₄×C₄₃

#123

yes

—

Dic₄₃

—

#1825

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#485

yes

—

C₂×#117

D₄₃

yes

characteristic

C₂×C₂²⋊C₈

—

#380

—

#340

—

#340

—

C₄₃

#1825

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₃

yes

characteristic

C₂²×C₈

—

#117

—

#117

Dic₄₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

#95

—

100

C₂⁴

C₂²⋊D₄₃

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂×Dic₄₃

yes

—

106

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

107

C₂²×C₄

C₂²⋊D₄₃

yes

—

108

C₂²×C₄

C₂²×D₄₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×Dic₄₃

yes

—

115

C₄⋊C₄

C₂×Dic₄₃

yes

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

172

—

118

C₄²

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

C₂²⋊Dic₄₃

yes

—

124

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

—

172

—

136

C₂×C₄

C₂²⋊Dic₄₃

yes

—

137

C₂×C₄

#10621898

yes

—

138

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

140

C₈

—

172

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

147

C₂²

#13063308

yes

—

148

C₂²

#13063345

yes

—

149

C₂²

#13063350

yes

characteristic

150

C₂²

#13063355

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

344

—

155

C₄

#13063284

yes

—

156

C₄

#13063308

yes

—

157

C₂

—

158

C₂

#14582762

yes

characteristic

159

C₂

#14582895

yes

characteristic

160

C₂

#14582900

yes

characteristic

161

C₂

#14583223

yes

—

162

C₂

#14583240

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#1825
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#117×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #14582661	C₂	semidirect
4. #14582661	C₂	semidirect
13. #117×C₄₃	C₄	semidirect
87. C₄₃	#1825	semidirect
91. #117	Dic₄₃	semidirect

H ≅ G/N

type

3. #14582661

C₂

semidirect

4. #14582661

C₂

semidirect

13. #117×C₄₃

C₄

semidirect

87. C₄₃

#1825

semidirect

91. #117

Dic₄₃

semidirect

Subgroups — #19134690

Conjugacy classes of subgroups

Complements