Group #19140967 — subgroups

Subgroups — #19140967

Order 5504.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13063302	C₂	1	4	yes	maximal, complemented
2	#14582822	C₂	1	4	yes	maximal, complemented
3	#14582819	C₂	1	2	yes	maximal, complemented
4	#14582797	C₂	1	2	yes	maximal, complemented
5	#14582779	C₂	1	2	yes	maximal
6	C₄₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₈₆	C₂²	1	2	yes	—
8	C₂³×Dic₄₃	C₂²	1	2	yes	—
9	#13063302	—	2	16	no	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
13	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
16	C₄₃⋊#95	C₂²	1	4	yes	—
17	C₄₃⋊#95	C₂²	1	2	yes	—
18	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄₃	C₂³	1	2	yes	—
23	C₂×D₄×C₄₃	—	4	1	no	—
24	C₂×D₄×C₄₃	—	4	1	no	—
25	C₂×D₄×C₄₃	Q₈	1	4	yes	complemented
26	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	2	8	no	—
30	C₂²×Dic₄₃	—	4	2	no	—
31	C₂²×Dic₄₃	C₂³	1	4	yes	—
32	C₂²×Dic₄₃	C₂³	1	4	yes	—
33	C₂×Dic₈₆	—	2	8	no	—
34	C₂×Dic₈₆	—	4	2	no	—
35	C₄⋊C₄×C₄₃	—	4	1	no	—
36	C₂²⋊C₄×C₄₃	—	2	8	no	—
37	C₄²×C₄₃	—	4	1	no	—
38	C₂²⋊Dic₄₃	—	2	8	no	—
39	C₂²⋊Dic₄₃	—	4	2	no	—
40	C₄⋊Dic₄₃	—	4	2	no	—
41	C₄⋊Dic₄₃	D₄	1	8	yes	complemented
42	Dic₄₃⋊C₄	—	2	8	no	—
43	Dic₄₃⋊C₄	—	2	8	no	—
44	C₂³×C₄₃	—	2	4	no	—
45	C₂³×C₄₃	—	2	2	no	—
46	C₂³×C₄₃	—	2	2	no	—
47	C₂³×C₄₃	—	2	4	no	—
48	C₂³×C₄₃	C₂×Q₈	1	4	yes	—
49	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
50	D₄×C₄₃	—	4	4	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	4	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂×Dic₄₃	—	2	4	no	—
60	C₂×Dic₄₃	—	2	4	no	—
61	C₂×Dic₄₃	—	2	4	no	—
62	C₂×Dic₄₃	—	2	4	no	—
63	C₂×Dic₄₃	—	2	4	no	—
64	C₂×Dic₄₃	—	4	8	no	—
65	C₂×Dic₄₃	C₂×D₄	1	8	yes	—
66	Dic₈₆	—	4	8	no	—
67	C₂²×C₄₃	—	2	4	no	—
68	C₂²×C₄₃	—	2	8	no	—
69	C₂²×C₄₃	—	4	4	no	—
70	C₂²×C₄₃	—	4	2	no	—
71	C₂²×C₄₃	—	4	2	no	—
72	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
73	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
75	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
77	C₄×C₄₃	—	2	4	no	—
78	C₄×C₄₃	—	2	2	no	—
79	C₄×C₄₃	—	4	2	no	—
80	Dic₄₃	—	2	8	no	—
81	Dic₄₃	—	4	4	no	—
82	#2683	—	43	1	no	maximal, Sylow
83	C₂×C₄₃	—	4	4	no	—
84	C₂×C₄₃	#537	1	2	yes	—
85	C₂×C₄₃	#543	1	1	yes	characteristic
86	C₂×C₄₃	D₄²	1	1	yes	characteristic
87	C₂×C₄₃	D₄×Q₈	1	2	yes	—
88	C₂×C₄₃	#563	1	1	yes	characteristic
89	C₂×C₂²⋊Q₈	—	43	4	no	—
90	#396	—	43	4	no	—
91	#395	—	43	2	no	—
92	#390	D₄₃	1	1	yes	characteristic
93	#389	—	43	2	no	—
94	#386	—	43	2	no	—
95	C₄₃	#2683	1	1	yes	characteristic, complemented, Sylow
96	C₂²×Q₈	—	43	2	no	—
97	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
98	C₂³×C₄	—	43	2	no	—
99	C₂²⋊Q₈	—	86	16	no	—
100	C₂×C₄⋊C₄	—	43	2	no	—
101	C₂×C₄⋊C₄	—	43	4	no	—
102	C₂×C₄⋊C₄	—	43	4	no	—
103	C₂×C₄⋊C₄	—	43	2	no	—
104	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	—	43	4	no	—
106	C₂×C₂²⋊C₄	—	43	2	no	—
107	C₂×C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
108	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
109	#95	—	43	4	no	—
110	#95	—	43	2	no	—
111	C₂⁴	C₂²×D₄₃	1	2	yes	—
112	C₂×Q₈	—	86	8	no	—
113	C₂×Q₈	—	172	2	no	—
114	C₂×D₄	—	4	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	Dic₈₆	1	4	yes	complemented
117	C₂²×C₄	—	43	4	no	—
118	C₂²×C₄	—	43	4	no	—
119	C₂²×C₄	—	86	8	no	—
120	C₂²×C₄	—	172	2	no	—
121	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
124	C₄⋊C₄	—	4	1	no	—
125	C₄⋊C₄	—	43	8	no	—
126	C₄⋊C₄	—	86	8	no	—
127	C₄⋊C₄	—	86	8	no	—
128	C₄⋊C₄	—	172	2	no	—
129	C₂²⋊C₄	—	2	8	no	—
130	C₂²⋊C₄	—	86	8	no	—
131	C₂²⋊C₄	—	172	2	no	—
132	C₄²	—	4	1	no	—
133	C₂³	—	2	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	4	no	—
137	C₂³	C₂×Dic₈₆	1	4	yes	—
138	C₂³	C₂³×D₄₃	1	1	yes	characteristic
139	Q₈	—	172	8	no	—
140	D₄	—	4	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	43	8	no	—
148	C₂×C₄	—	86	4	no	—
149	C₂×C₄	—	86	4	no	—
150	C₂×C₄	—	86	4	no	—
151	C₂×C₄	—	86	4	no	—
152	C₂×C₄	—	86	4	no	—
153	C₂×C₄	—	172	8	no	—
154	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
155	C₂×C₄	D₄×D₄₃	1	4	yes	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	4	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	4	2	no	—
161	C₂²	C₂×D₄×D₄₃	1	2	yes	—
162	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
163	C₂²	#13063370	1	1	yes	characteristic
164	C₂²	#13063370	1	1	yes	characteristic
165	C₂²	#13063372	1	2	yes	—
166	C₄	—	2	4	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	2	no	—
169	C₄	—	86	8	no	—
170	C₄	—	172	4	no	—
171	C₂	—	4	4	no	—
172	C₂	#14583476	1	2	yes	—
173	C₂	#14583525	1	1	yes	characteristic
174	C₂	#14583534	1	1	yes	characteristic
175	C₂	D₄×Dic₈₆	1	2	yes	—
176	C₂	#14583649	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063302

C₂

yes

maximal, complemented

#14582822

C₂

yes

maximal, complemented

#14582819

C₂

yes

maximal, complemented

#14582797

C₂

yes

maximal, complemented

#14582779

C₂

yes

maximal

C₄₃×#390

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

C₂³×Dic₄₃

C₂²

yes

—

#13063302

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×D₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂³

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

Q₈

yes

complemented

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

complemented

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×Q₈

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

—

C₂²×C₄₃

C₂²×Q₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

#2683

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#537

yes

—

C₂×C₄₃

#543

yes

characteristic

C₂×C₄₃

D₄²

yes

characteristic

C₂×C₄₃

D₄×Q₈

yes

—

C₂×C₄₃

#563

yes

characteristic

C₂×C₂²⋊Q₈

—

#396

—

#395

—

#390

D₄₃

yes

characteristic

#389

—

#386

—

C₄₃

#2683

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

C₂×D₄₃

yes

characteristic

C₂³×C₄

—

C₂²⋊Q₈

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

108

C₂×C₄²

C₂×D₄₃

yes

characteristic

109

#95

—

110

#95

—

111

C₂⁴

C₂²×D₄₃

yes

—

112

C₂×Q₈

—

113

C₂×Q₈

—

172

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

Dic₈₆

yes

complemented

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

172

—

121

C₂²×C₄

C₂²×D₄₃

yes

—

122

C₂²×C₄

C₂²×D₄₃

yes

characteristic

123

C₂²×C₄

C₂²×D₄₃

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

172

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

172

—

132

C₄²

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×Dic₈₆

yes

—

138

C₂³

C₂³×D₄₃

yes

characteristic

139

Q₈

—

172

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

172

—

154

C₂×C₄

C₂×Dic₈₆

yes

—

155

C₂×C₄

D₄×D₄₃

yes

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

C₂×D₄×D₄₃

yes

—

162

C₂²

C₂²×Dic₈₆

yes

characteristic

163

C₂²

#13063370

yes

characteristic

164

C₂²

#13063370

yes

characteristic

165

C₂²

#13063372

yes

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

172

—

171

C₂

—

172

C₂

#14583476

yes

—

173

C₂

#14583525

yes

characteristic

174

C₂

#14583534

yes

characteristic

175

C₂

D₄×Dic₈₆

yes

—

176

C₂

#14583649

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#390	C₂
hypercenter	#390	D₄₃
hypocenter	C₄₃	#2683
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #390

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₄₃×#390

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #390

N	H ≅ G/N	type
1. C₂×#13063302	C₂	semidirect
2. #14582822	C₂	semidirect
3. #14582819	C₂	semidirect
4. #14582797	C₂	semidirect
25. C₂×D₄×C₄₃	Q₈	semidirect
41. C₄⋊Dic₄₃	D₄	semidirect
95. C₄₃	#2683	semidirect
116. C₂×D₄	Dic₈₆	semidirect

H ≅ G/N

type

1. C₂×#13063302

C₂

semidirect

2. #14582822

C₂

semidirect

3. #14582819

C₂

semidirect

4. #14582797

C₂

semidirect

25. C₂×D₄×C₄₃

Q₈

semidirect

41. C₄⋊Dic₄₃

D₄

semidirect

95. C₄₃

#2683

semidirect

116. C₂×D₄

Dic₈₆

semidirect

Subgroups — #19140967

Conjugacy classes of subgroups

Complements