Group #19134654 — subgroups

Subgroups — #19134654

Order 5504.

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

normal subgroups only

This group has 264 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×#13063261	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063257	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582900	C₂	1	2	yes	maximal
4	#14582667	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#10621877	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
8	#13063261	—	2	2	no	—
9	#13063257	—	2	1	no	—
10	#13063257	—	2	1	no	—
11	#13063249	—	2	2	no	—
12	#13063246	—	2	2	no	—
13	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
14	#131×C₄₃	—	2	2	no	—
15	#131×C₄₃	C₄	1	4	yes	complemented
16	C₂×M₁₆×C₄₃	—	2	1	no	—
17	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₃	—	2	1	no	—
19	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	#41×C₄₃	—	2	1	no	—
21	#41×C₄₃	—	2	1	no	—
22	#41×C₄₃	C₂×C₄	1	2	yes	—
23	#41×C₄₃	Q₈	1	2	yes	—
24	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	—	2	1	no	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	2	2	no	—
30	M₁₆×C₄₃	—	2	1	no	—
31	M₁₆×C₄₃	—	2	4	no	—
32	M₁₆×C₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₂×C₈×C₄₃	—	2	4	no	—
35	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	D₄	1	2	yes	—
39	C₄⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	—	2	2	no	—
41	#10621877	—	2	2	no	—
42	#10621877	—	2	2	no	—
43	C₂×C₄₃⋊C₈	—	2	2	no	—
44	C₂³×C₄₃	—	2	1	no	—
45	C₂³×C₄₃	#41	1	1	yes	characteristic
46	Q₈×C₄₃	—	2	1	no	—
47	Q₈×C₄₃	C₄⋊C₄	1	2	yes	—
48	D₄×C₄₃	—	2	2	no	—
49	D₄×C₄₃	—	2	2	no	—
50	D₄×C₄₃	—	2	1	no	—
51	D₄×C₄₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
61	C₂×Dic₄₃	—	4	2	no	—
62	C₂×Dic₄₃	—	8	2	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
65	C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
66	C₄₃⋊C₈	—	4	2	no	—
67	C₂²×C₄₃	—	2	2	no	—
68	C₂²×C₄₃	—	2	2	no	—
69	C₂²×C₄₃	—	4	2	no	—
70	C₂²×C₄₃	#117	1	1	yes	characteristic
71	C₂²×C₄₃	#121	1	1	yes	characteristic
72	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₄×C₄₃	—	2	2	no	—
74	C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₄₃	#121	1	1	yes	characteristic
77	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
78	Dic₄₃	—	8	2	no	—
79	#1815	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	1	no	—
81	C₂×C₄₃	—	2	1	no	—
82	C₂×C₄₃	—	4	2	no	—
83	C₂×C₄₃	#380	1	1	yes	characteristic
84	C₂×C₄₃	#471	1	2	yes	—
85	C₂×#131	D₄₃	1	1	yes	characteristic
86	C₂×#108	—	43	1	no	—
87	C₂×#106	—	43	1	no	—
88	#417	—	43	2	no	—
89	#343	—	43	2	no	—
90	C₄₃	#1815	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
92	#131	—	2	2	no	—
93	#131	Dic₄₃	1	4	yes	complemented
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	43	2	no	—
96	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	43	2	no	—
100	#108	—	86	2	no	—
101	#106	—	86	1	no	—
102	#106	—	86	1	no	—
103	#103	—	86	2	no	—
104	#102	—	86	2	no	—
105	#41	—	2	1	no	—
106	#41	—	2	1	no	—
107	#41	Dic₈₆	1	2	yes	—
108	#41	C₂×Dic₄₃	1	2	yes	—
109	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	D₁₇₂	1	1	yes	characteristic
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	86	2	no	—
114	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	4	no	—
117	M₁₆	—	86	2	no	—
118	M₁₆	—	86	2	no	—
119	M₁₆	C₂×Dic₄₃	1	2	yes	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	86	2	no	—
123	C₂×C₈	C₂×Dic₄₃	1	2	yes	—
124	C₂×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
127	C₄⋊C₄	—	86	2	no	—
128	C₄⋊C₄	—	86	2	no	—
129	C₂³	—	2	1	no	—
130	C₂³	#10621898	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	C₄⋊Dic₄₃	1	2	yes	—
133	D₄	—	2	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	1	no	—
136	D₄	C₄⋊Dic₄₃	1	2	yes	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	344	2	no	—
141	C₂×C₄	—	172	2	no	—
142	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
143	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
144	C₂×C₄	#10621898	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	172	2	no	—
150	C₈	C₂²⋊Dic₄₃	1	2	yes	—
151	C₈	C₂²⋊Dic₄₃	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#13063284	1	1	yes	characteristic
156	C₂²	#13063299	1	1	yes	characteristic
157	C₂²	#13063304	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	344	2	no	—
160	C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
161	C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
162	C₄	#13063299	1	1	yes	characteristic
163	C₄	#13063304	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	4	2	no	—
167	C₂	#14582759	1	1	yes	characteristic
168	C₂	#14583162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063261

C₂

yes

maximal, characteristic, complemented

C₂×#13063257

C₂

yes

maximal, characteristic, complemented

#14582900

C₂

yes

maximal

#14582667

C₂

yes

maximal, complemented

C₂×#131×C₄₃

C₂

yes

maximal, characteristic

C₂×#10621877

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

—

—

—

—

—

C₂×#41×C₄₃

C₂²

yes

characteristic

#131×C₄₃

—

#131×C₄₃

C₄

yes

complemented

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

#41×C₄₃

—

#41×C₄₃

—

#41×C₄₃

C₂×C₄

yes

—

#41×C₄₃

Q₈

yes

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

#41

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

C₄⋊C₄

yes

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊C₄

yes

—

C₈×C₄₃

C₂²⋊C₄

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₃

#121

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

Dic₄₃

—

#1815

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#380

yes

characteristic

C₂×C₄₃

#471

yes

—

C₂×#131

D₄₃

yes

—

—

—

—

C₄₃

#1815

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₃

yes

characteristic

#131

—

#131

Dic₄₃

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

100

#108

—

101

#106

—

102

#106

—

103

#103

—

104

#102

—

105

#41

—

106

#41

—

107

#41

Dic₈₆

yes

—

108

#41

C₂×Dic₄₃

yes

—

109

C₂×Q₈

D₁₇₂

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

D₁₇₂

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²×D₄₃

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

C₂×Dic₄₃

yes

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂×Dic₄₃

yes

—

124

C₂×C₈

C₂²⋊D₄₃

yes

characteristic

125

C₂×C₈

C₂²⋊D₄₃

yes

characteristic

126

C₂×C₈

C₂²⋊D₄₃

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂³

—

130

C₂³

#10621898

yes

characteristic

131

Q₈

—

132

Q₈

C₄⋊Dic₄₃

yes

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

C₄⋊Dic₄₃

yes

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

344

—

141

C₂×C₄

—

172

—

142

C₂×C₄

C₂×Dic₈₆

yes

characteristic

143

C₂×C₄

C₂×D₁₇₂

yes

characteristic

144

C₂×C₄

#10621898

yes

characteristic

145

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

148

C₈

—

149

C₈

—

172

—

150

C₈

C₂²⋊Dic₄₃

yes

—

151

C₈

C₂²⋊Dic₄₃

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#13063284

yes

characteristic

156

C₂²

#13063299

yes

characteristic

157

C₂²

#13063304

yes

characteristic

158

C₄

—

159

C₄

—

344

—

160

C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

161

C₄

C₂×C₄⋊Dic₄₃

yes

characteristic

162

C₄

#13063299

yes

characteristic

163

C₄

#13063304

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

167

C₂

#14582759

yes

characteristic

168

C₂

#14583162

yes

—

	subgroup	G/H
center	C₂²	#13063299
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#131×C₄₃	C₂
socle	C₂²×C₄₃	#121
hypercenter	C₂×#131	D₄₃
hypocenter	C₄₃	#1815
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

subgroup

G/H

center

C₂²

#13063299

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#131×C₄₃

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#13063261	C₂	semidirect
2. C₂×#13063257	C₂	semidirect
4. #14582667	C₂	semidirect
15. #131×C₄₃	C₄	semidirect
90. C₄₃	#1815	semidirect
93. #131	Dic₄₃	semidirect

H ≅ G/N

type

1. C₂×#13063261

C₂

semidirect

2. C₂×#13063257

C₂

semidirect

4. #14582667

C₂

semidirect

15. #131×C₄₃

C₄

semidirect

90. C₄₃

#1815

semidirect

93. #131

Dic₄₃

semidirect

Subgroups — #19134654

Conjugacy classes of subgroups

Complements