Group #19133939 — subgroups

Subgroups — #19133939

Order 5504.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#14583273	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583176	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583095	C₂	1	1	yes	maximal, characteristic
4	#14582964	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582649	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621884	—	2	1	no	—
9	C₂×#10621884	—	2	1	no	—
10	C₂×D₃₄₄	—	2	1	no	—
11	C₂×C₈×D₄₃	—	2	1	no	—
12	#13063322	C₂²	1	1	yes	characteristic
13	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
14	#13063258	—	2	1	no	—
15	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	#13063247	—	2	1	no	—
17	#13063245	—	2	1	no	—
18	(C₂×D₄₃)⋊C₈	—	2	1	no	—
19	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×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×Q₈×C₄₃	—	2	1	no	—
24	C₂×D₁₇₂	—	2	1	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₄₃	—	2	1	no	—
29	C₂×C₈×C₄₃	—	2	1	no	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	—	2	1	no	—
32	C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	#10621884	—	4	2	no	—
35	#10621884	—	4	2	no	—
36	(C₂×D₄₃)⋊C₄	—	2	1	no	—
37	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	—	2	1	no	—
39	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
41	D₃₄₄	—	4	2	no	—
42	C₈×D₄₃	—	4	2	no	—
43	C₂²×D₄₃	—	2	1	no	—
44	C₂²×D₄₃	—	4	1	no	—
45	Q₈×C₄₃	—	4	1	no	—
46	Q₈×C₄₃	—	4	1	no	—
47	C₂×C₄×C₄₃	—	4	1	no	—
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
51	C₂×Dic₄₃	—	2	1	no	—
52	D₁₇₂	—	2	1	no	—
53	D₁₇₂	—	4	1	no	—
54	D₁₇₂	—	4	1	no	—
55	D₁₇₂	—	4	1	no	—
56	D₁₇₂	—	4	1	no	—
57	D₁₇₂	—	8	1	no	—
58	D₁₇₂	SD₁₆	1	2	yes	complemented
59	C₄×D₄₃	—	4	1	no	—
60	C₄×D₄₃	—	4	1	no	—
61	C₈×C₄₃	—	4	1	no	—
62	C₈×C₄₃	—	4	1	no	—
63	C₄₃⋊C₈	—	2	1	no	—
64	C₄₃⋊C₈	—	4	1	no	—
65	C₄₃⋊C₈	D₈	1	2	yes	complemented
66	C₂²×C₄₃	#121	1	1	yes	characteristic
67	C₂×D₄₃	—	2	2	no	—
68	C₂×D₄₃	—	4	1	no	—
69	C₂×D₄₃	—	4	1	no	—
70	C₂×D₄₃	—	8	1	no	—
71	C₂×D₄₃	—	8	1	no	—
72	C₂×D₄₃	—	8	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	—	8	1	no	—
75	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
76	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₄₃	#135	1	1	yes	characteristic
78	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₄₃	—	4	1	no	—
80	#1665	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	#459	1	1	yes	characteristic
82	C₂×C₄₃	#465	1	1	yes	characteristic
83	C₂×C₄₃	#471	1	1	yes	characteristic
84	D₄₃	—	4	2	no	—
85	D₄₃	—	16	1	no	—
86	#492	—	43	1	no	—
87	#474	—	43	1	no	—
88	#459	—	43	1	no	—
89	C₈×D₄	—	43	1	no	—
90	#334	—	43	1	no	—
91	#332	D₄₃	1	1	yes	characteristic, complemented
92	#331	—	43	1	no	—
93	C₄₃	#1665	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	86	1	no	—
95	C₂×SD₁₆	—	86	1	no	—
96	C₂×D₈	—	86	1	no	—
97	C₂²×C₈	—	86	1	no	—
98	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
99	#127	—	43	1	no	—
100	C₄×D₄	—	43	1	no	—
101	#107	—	86	1	no	—
102	C₄⋊C₈	—	43	1	no	—
103	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
105	#102	—	86	1	no	—
106	#102	—	86	1	no	—
107	C₂²⋊C₈	—	86	1	no	—
108	C₄×C₈	—	43	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	43	1	no	—
111	C₂×D₄	—	86	1	no	—
112	C₂×D₄	—	86	1	no	—
113	C₂×D₄	—	86	1	no	—
114	C₂²×C₄	—	86	1	no	—
115	SD₁₆	—	172	2	no	—
116	SD₁₆	—	172	2	no	—
117	D₈	—	172	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	43	1	no	—
121	C₂×C₈	—	43	1	no	—
122	C₂×C₈	—	86	1	no	—
123	C₂×C₈	—	172	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	43	1	no	—
127	C₂²⋊C₄	—	86	1	no	—
128	C₄²	C₂²×D₄₃	1	1	yes	characteristic
129	C₂³	—	86	1	no	—
130	C₂³	—	172	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	43	2	no	—
134	D₄	—	86	1	no	—
135	D₄	—	344	1	no	—
136	D₄	—	172	1	no	—
137	D₄	—	172	1	no	—
138	D₄	—	172	1	no	—
139	D₄	—	172	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	86	1	no	—
142	C₂×C₄	—	172	1	no	—
143	C₂×C₄	—	172	1	no	—
144	C₂×C₄	#10621898	1	1	yes	characteristic
145	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
146	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	43	2	no	—
150	C₈	—	86	1	no	—
151	C₈	—	172	1	no	—
152	C₂²	—	86	2	no	—
153	C₂²	—	344	1	no	—
154	C₂²	—	344	1	no	—
155	C₂²	—	344	1	no	—
156	C₂²	—	172	1	no	—
157	C₂²	—	172	1	no	—
158	C₂²	#13063298	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	172	1	no	—
162	C₄	D₈×D₄₃	1	1	yes	characteristic
163	C₄	SD₁₆×D₄₃	1	1	yes	characteristic
164	C₄	#13063348	1	1	yes	characteristic
165	C₄	#13063352	1	1	yes	characteristic
166	C₂	—	172	2	no	—
167	C₂	—	688	1	no	—
168	C₂	#14583093	1	1	yes	characteristic
169	C₂	#14583131	1	1	yes	characteristic
170	C₂	#14583165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583273

C₂

yes

maximal, characteristic, complemented

#14583176

C₂

yes

maximal, characteristic, complemented

#14583095

C₂

yes

maximal, characteristic

#14582964

C₂

yes

maximal, characteristic, complemented

#14582649

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

C₄₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×#10621884

—

C₂×#10621884

—

C₂×D₃₄₄

—

C₂×C₈×D₄₃

—

#13063322

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

characteristic

#13063258

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063247

—

#13063245

—

(C₂×D₄₃)⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621884

—

#10621884

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

D₃₄₄

—

C₈×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

SD₁₆

yes

complemented

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

D₈

yes

complemented

C₂²×C₄₃

#121

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

D₄₃

—

#492

—

#474

—

#459

—

C₈×D₄

—

#334

—

#332

D₄₃

yes

characteristic, complemented

#331

—

C₄₃

#1665

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

#127

—

100

C₄×D₄

—

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₄₃

yes

characteristic

104

C₄⋊C₈

C₂×D₄₃

yes

characteristic

105

#102

—

106

#102

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

SD₁₆

—

172

—

116

SD₁₆

—

172

—

117

D₈

—

172

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

172

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₄₃

yes

characteristic

129

C₂³

—

130

C₂³

—

172

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

D₄

—

344

—

136

D₄

—

172

—

137

D₄

—

172

—

138

D₄

—

172

—

139

D₄

—

172

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

172

—

143

C₂×C₄

—

172

—

144

C₂×C₄

#10621898

yes

characteristic

145

C₂×C₄

D₄×D₄₃

yes

characteristic

146

C₂×C₄

D₄×D₄₃

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

172

—

152

C₂²

—

153

C₂²

—

344

—

154

C₂²

—

344

—

155

C₂²

—

344

—

156

C₂²

—

172

—

157

C₂²

—

172

—

158

C₂²

#13063298

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

172

—

162

C₄

D₈×D₄₃

yes

characteristic

163

C₄

SD₁₆×D₄₃

yes

characteristic

164

C₄

#13063348

yes

characteristic

165

C₄

#13063352

yes

characteristic

166

C₂

—

172

—

167

C₂

—

688

—

168

C₂

#14583093

yes

characteristic

169

C₂

#14583131

yes

characteristic

170

C₂

#14583165

yes

characteristic

	subgroup	G/H
center	C₂²	#13063298
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#332	C₂
socle	C₂²×C₄₃	#121
hypercenter	#332	D₄₃
hypocenter	C₄₃	#1665
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#13063298

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#332

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #14583273	C₂	semidirect
2. #14583176	C₂	semidirect
4. #14582964	C₂	semidirect
5. #14582649	C₂	semidirect
6. #14582642	C₂	semidirect
7. C₄₃×#332	C₂	semidirect
58. D₁₇₂	SD₁₆	semidirect
65. C₄₃⋊C₈	D₈	semidirect
91. #332	D₄₃	semidirect
93. C₄₃	#1665	semidirect

H ≅ G/N

type

1. #14583273

C₂

semidirect

2. #14583176

C₂

semidirect

4. #14582964

C₂

semidirect

5. #14582649

C₂

semidirect

6. #14582642

C₂

semidirect

7. C₄₃×#332

C₂

semidirect

58. D₁₇₂

SD₁₆

semidirect

65. C₄₃⋊C₈

D₈

semidirect

91. #332

D₄₃

semidirect

93. C₄₃

#1665

semidirect

Subgroups — #19133939

Conjugacy classes of subgroups

Complements