Group #19134622 — subgroups

Subgroups — #19134622

Order 5504.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂³×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063258	C₂	1	2	yes	maximal, complemented
3	#14582759	C₂	1	1	yes	maximal, characteristic
4	#14582661	C₂	1	2	yes	maximal, complemented
5	C₄₃×#380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	—	2	4	no	—
7	C₂²×C₄₃⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
12	#13063258	—	2	4	no	—
13	C₄₃⋊#95	—	2	1	no	—
14	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
17	#95×C₄₃	—	2	1	no	—
18	C₂⁴×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	—	2	2	no	—
20	C₂²×C₄×C₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₃	—	2	2	no	—
27	C₄⋊C₄×C₄₃	—	2	2	no	—
28	C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂²⋊C₄×C₄₃	—	4	2	no	—
30	C₂²⋊Dic₄₃	—	4	2	no	—
31	C₄⋊Dic₄₃	—	2	2	no	—
32	C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄₃⋊C₈	—	2	2	no	—
34	C₂×C₄₃⋊C₈	—	2	8	no	—
35	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄₃⋊C₈	D₄	1	4	yes	—
38	C₂³×C₄₃	—	2	2	no	—
39	C₂³×C₄₃	—	2	2	no	—
40	C₂³×C₄₃	C₄⋊C₄	1	2	yes	—
41	C₂³×C₄₃	D₈	1	2	yes	—
42	C₂³×C₄₃	Q₁₆	1	2	yes	—
43	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	4	no	—
48	C₂×C₄×C₄₃	—	4	2	no	—
49	C₂×C₄×C₄₃	—	4	2	no	—
50	C₂×C₄×C₄₃	—	4	2	no	—
51	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	2	yes	—
56	C₂×Dic₄₃	—	4	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	C₂×Dic₄₃	—	4	2	no	—
59	C₄₃⋊C₈	—	2	2	no	—
60	C₄₃⋊C₈	C₂²⋊C₄	1	4	yes	complemented
61	C₂²×C₄₃	—	2	2	no	—
62	C₂²×C₄₃	—	2	2	no	—
63	C₂²×C₄₃	—	2	4	no	—
64	C₂²×C₄₃	—	2	4	no	—
65	C₂²×C₄₃	#107	1	4	yes	complemented
66	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₃	#121	1	2	yes	—
68	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
69	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₄₃	#135	1	2	yes	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	—	8	2	no	—
73	C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₄₃	#117	1	1	yes	characteristic
75	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
76	Dic₄₃	—	8	2	no	—
77	#1808	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	—	2	4	no	—
79	C₂×C₄₃	#380	1	1	yes	characteristic
80	C₂×C₄₃	C₂×#107	1	1	yes	characteristic
81	C₂×C₄₃	#427	1	1	yes	characteristic
82	C₂×C₄₃	#466	1	2	yes	—
83	C₂×C₄₃	#467	1	2	yes	—
84	C₂³×C₈	—	43	1	no	—
85	C₂×#107	—	43	2	no	—
86	#380	—	43	1	no	—
87	#380	D₄₃	1	1	yes	characteristic
88	#340	—	43	2	no	—
89	C₄₃	#1808	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
91	C₂²×C₈	—	43	1	no	—
92	C₂²×C₈	—	43	1	no	—
93	C₂²×C₈	—	43	4	no	—
94	C₂²×C₈	—	86	4	no	—
95	C₂×C₄⋊C₄	—	43	2	no	—
96	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
97	C₂×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₂²⋊C₄	—	86	1	no	—
99	#107	—	86	4	no	—
100	#95	—	2	1	no	—
101	#95	—	86	1	no	—
102	C₂⁴	C₂²⋊D₄₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	86	2	no	—
107	C₂²×C₄	Dic₈₆	1	2	yes	—
108	C₂²×C₄	C₄×D₄₃	1	2	yes	—
109	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
111	C₂×C₈	—	43	2	no	—
112	C₂×C₈	—	43	4	no	—
113	C₂×C₈	—	43	2	no	—
114	C₂×C₈	—	86	8	no	—
115	C₂×C₈	—	86	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	86	2	no	—
119	C₄⋊C₄	—	86	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	172	2	no	—
122	C₂³	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	Dic₄₃⋊C₄	1	2	yes	—
125	C₂³	#10621882	1	2	yes	—
126	C₂³	#10621885	1	2	yes	—
127	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	172	2	no	—
136	C₂×C₄	—	172	2	no	—
137	C₂×C₄	—	172	2	no	—
138	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
139	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
141	C₂×C₄	D₄×D₄₃	1	2	yes	—
142	C₂×C₄	#10621900	1	2	yes	—
143	C₈	—	43	4	no	—
144	C₈	—	86	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	#13063258	1	4	yes	complemented
150	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
151	C₂²	#13063297	1	2	yes	—
152	C₂²	C₂×#10621882	1	1	yes	characteristic
153	C₂²	C₂×#10621885	1	1	yes	characteristic
154	C₂²	#13063345	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	8	2	no	—
157	C₄	—	344	2	no	—
158	C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
159	C₄	#13063281	1	1	yes	characteristic
160	C₄	#13063302	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	#14582757	1	1	yes	characteristic
163	C₂	C₂×#13063258	1	1	yes	characteristic
164	C₂	#14582935	1	1	yes	characteristic
165	C₂	#14583133	1	2	yes	—
166	C₂	#14583138	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄₃⋊C₈

C₂

yes

maximal, characteristic

C₂×#13063258

C₂

yes

maximal, complemented

#14582759

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, complemented

C₄₃×#380

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

C₄

yes

complemented

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

—

#13063258

—

C₄₃⋊#95

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₃

—

#95×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

characteristic

C₂²×C₄×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₄⋊C₄

yes

—

C₂³×C₄₃

D₈

yes

—

C₂³×C₄₃

Q₁₆

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

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₂²⋊C₄

yes

complemented

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#107

yes

complemented

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₂²×C₄₃

#135

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

#117

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

#1808

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#467

yes

—

C₂³×C₈

—

C₂×#107

—

#380

—

#380

D₄₃

yes

characteristic

#340

—

C₄₃

#1808

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#107

—

100

#95

—

101

#95

—

102

C₂⁴

C₂²⋊D₄₃

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

Dic₈₆

yes

—

108

C₂²×C₄

C₄×D₄₃

yes

—

109

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

110

C₂²×C₄

C₂²×D₄₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

172

—

122

C₂³

—

123

C₂³

—

124

C₂³

Dic₄₃⋊C₄

yes

—

125

C₂³

#10621882

yes

—

126

C₂³

#10621885

yes

—

127

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

172

—

136

C₂×C₄

—

172

—

137

C₂×C₄

—

172

—

138

C₂×C₄

Dic₄₃⋊C₄

yes

—

139

C₂×C₄

C₂×Dic₈₆

yes

characteristic

140

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

141

C₂×C₄

D₄×D₄₃

yes

—

142

C₂×C₄

#10621900

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#13063258

yes

complemented

150

C₂²

C₂×Dic₄₃⋊C₄

yes

characteristic

151

C₂²

#13063297

yes

—

152

C₂²

C₂×#10621882

yes

characteristic

153

C₂²

C₂×#10621885

yes

characteristic

154

C₂²

#13063345

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

344

—

158

C₄

C₂²⋊C₄×D₄₃

yes

characteristic

159

C₄

#13063281

yes

characteristic

160

C₄

#13063302

yes

—

161

C₂

—

162

C₂

#14582757

yes

characteristic

163

C₂

C₂×#13063258

yes

characteristic

164

C₂

#14582935

yes

characteristic

165

C₂

#14583133

yes

—

166

C₂

#14583138

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#380	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#380	D₄₃
hypocenter	C₄₃	#1808
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#380

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
2. C₂×#13063258	C₂	semidirect
4. #14582661	C₂	semidirect
7. C₂²×C₄₃⋊C₈	C₄	semidirect
60. C₄₃⋊C₈	C₂²⋊C₄	semidirect
65. C₂²×C₄₃	#107	semidirect
89. C₄₃	#1808	semidirect
149. C₂²	#13063258	semidirect

H ≅ G/N

type

2. C₂×#13063258

C₂

semidirect

4. #14582661

C₂

semidirect

7. C₂²×C₄₃⋊C₈

C₄

semidirect

60. C₄₃⋊C₈

C₂²⋊C₄

semidirect

65. C₂²×C₄₃

#107

semidirect

89. C₄₃

#1808

semidirect

149. C₂²

#13063258

semidirect

Subgroups — #19134622

Conjugacy classes of subgroups

Complements