Group #19137211 — subgroups

Subgroups — #19137211

Order 5504.

600 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 269 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×#13063284	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582797	C₂	1	4	yes	maximal, complemented
4	#14582795	C₂	1	4	yes	maximal, complemented
5	#14582757	C₂	1	4	yes	maximal
6	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₄₃	C₂²	1	2	yes	—
8	#13063284	—	2	4	no	—
9	C₂×C₄⋊Dic₄₃	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
13	C₂×Dic₄₃⋊C₄	C₂²	1	8	yes	—
14	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
16	C₄₃⋊#95	C₂²	1	8	yes	—
17	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
18	#117×C₄₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
22	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₄₃	D₄	1	4	yes	—
25	C₂²×C₄×C₄₃	Q₈	1	4	yes	—
26	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	4	4	no	—
30	C₂²×Dic₄₃	C₂×C₄	1	8	yes	—
31	C₂²×Dic₄₃	C₂³	1	4	yes	—
32	C₂²×Dic₄₃	C₂³	1	4	yes	—
33	C₄⋊C₄×C₄₃	—	2	4	no	—
34	C₂²⋊C₄×C₄₃	—	2	4	no	—
35	C₄²×C₄₃	—	2	4	no	—
36	C₂²⋊Dic₄₃	—	2	4	no	—
37	C₄⋊Dic₄₃	—	2	4	no	—
38	C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
39	C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
40	Dic₄₃⋊C₄	—	4	8	no	—
41	C₄×Dic₄₃	—	2	4	no	—
42	C₂³×C₄₃	—	2	2	no	—
43	C₂³×C₄₃	—	2	2	no	—
44	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
45	C₂³×C₄₃	C₂×D₄	1	2	yes	—
46	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
47	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₄₃	—	2	4	no	—
49	C₂×C₄×C₄₃	—	2	4	no	—
50	C₂×C₄×C₄₃	—	2	4	no	—
51	C₂×C₄×C₄₃	C₄⋊C₄	1	16	yes	complemented
52	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
58	C₂×Dic₄₃	—	2	8	no	—
59	C₂×Dic₄₃	—	2	4	no	—
60	C₂×Dic₄₃	—	2	4	no	—
61	C₂×Dic₄₃	—	2	4	no	—
62	C₂×Dic₄₃	—	4	8	no	—
63	C₂×Dic₄₃	C₂²×C₄	1	8	yes	—
64	C₂²×C₄₃	—	2	2	no	—
65	C₂²×C₄₃	—	2	2	no	—
66	C₂²×C₄₃	—	2	4	no	—
67	C₂²×C₄₃	—	2	4	no	—
68	C₂²×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
69	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
73	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
74	C₄×C₄₃	—	4	4	no	—
75	C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
77	Dic₄₃	—	2	8	no	—
78	Dic₄₃	—	4	4	no	—
79	#2316	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	4	no	—
81	C₂×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄₃	#518	1	1	yes	characteristic
83	C₂×C₄₃	#519	1	1	yes	characteristic
84	C₂×C₄₃	#537	1	2	yes	—
85	C₂×C₄₃	#544	1	2	yes	—
86	C₂×#117	—	43	1	no	—
87	C₂×#117	D₄₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	—	43	1	no	—
89	#389	—	43	4	no	—
90	#389	—	43	4	no	—
91	#380	—	43	4	no	—
92	C₄₃	#2316	1	1	yes	characteristic, complemented, Sylow
93	C₂³×C₄	—	43	2	no	—
94	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
95	#117	—	2	4	no	—
96	#117	—	86	4	no	—
97	C₂×C₄⋊C₄	—	43	2	no	—
98	C₂×C₄⋊C₄	—	43	2	no	—
99	C₂×C₄⋊C₄	—	43	8	no	—
100	C₂×C₄⋊C₄	—	43	2	no	—
101	C₂×C₄⋊C₄	—	43	8	no	—
102	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
103	C₂×C₂²⋊C₄	—	43	2	no	—
104	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
105	C₂×C₄²	—	43	2	no	—
106	C₂×C₄²	C₂×D₄₃	1	2	yes	—
107	#95	—	43	8	no	—
108	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
109	C₂²×C₄	—	43	4	no	—
110	C₂²×C₄	—	43	8	no	—
111	C₂²×C₄	—	43	4	no	—
112	C₂²×C₄	—	172	4	no	—
113	C₂²×C₄	Dic₈₆	1	4	yes	—
114	C₂²×C₄	C₄×D₄₃	1	4	yes	—
115	C₂²×C₄	C₂²⋊D₄₃	1	4	yes	—
116	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
117	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	43	8	no	—
121	C₄⋊C₄	—	43	8	no	—
122	C₄⋊C₄	—	86	4	no	—
123	C₄⋊C₄	—	172	8	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	86	4	no	—
126	C₄²	—	2	4	no	—
127	C₄²	—	86	4	no	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	C₂×Dic₈₆	1	2	yes	—
131	C₂³	C₂×C₄×D₄₃	1	2	yes	—
132	C₂³	C₂×C₂²⋊D₄₃	1	2	yes	—
133	C₂³	C₂³×D₄₃	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	43	8	no	—
138	C₂×C₄	—	86	8	no	—
139	C₂×C₄	—	86	4	no	—
140	C₂×C₄	—	86	4	no	—
141	C₂×C₄	—	86	4	no	—
142	C₂×C₄	—	172	8	no	—
143	C₂×C₄	Dic₄₃⋊C₄	1	16	yes	complemented
144	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
145	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
146	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
147	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	C₂×Dic₄₃⋊C₄	1	4	yes	—
155	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
156	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
157	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
158	C₂²	#13063371	1	2	yes	—
159	C₂²	#13063372	1	2	yes	—
160	C₄	—	4	4	no	—
161	C₄	—	86	8	no	—
162	C₄	—	172	4	no	—
163	C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
164	C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
165	C₂	—	2	4	no	—
166	C₂	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
167	C₂	#14583373	1	1	yes	characteristic
168	C₂	#14583376	1	1	yes	characteristic
169	C₂	#14583477	1	2	yes	—
170	C₂	#14583527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063284

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

#14582797

C₂

yes

maximal, complemented

#14582795

C₂

yes

maximal, complemented

#14582757

C₂

yes

maximal

C₂×#117×C₄₃

C₂

yes

maximal, characteristic

C₂³×Dic₄₃

C₂²

yes

—

#13063284

—

C₂×C₄⋊Dic₄₃

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

#117×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂×C₄

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂×Q₈

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

Dic₄₃

—

Dic₄₃

—

#2316

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂²×C₄⋊C₄

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#544

yes

—

C₂×#117

—

C₂×#117

D₄₃

yes

characteristic

C₂²×C₄⋊C₄

—

#389

—

#389

—

#380

—

C₄₃

#2316

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₃

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₄₃

yes

—

107

#95

—

108

C₂⁴

C₂²×D₄₃

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

172

—

113

C₂²×C₄

Dic₈₆

yes

—

114

C₂²×C₄

C₄×D₄₃

yes

—

115

C₂²×C₄

C₂²⋊D₄₃

yes

—

116

C₂²×C₄

C₂²×D₄₃

yes

—

117

C₂²×C₄

C₂²×D₄₃

yes

characteristic

118

C₂²×C₄

C₂²×D₄₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

172

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂×Dic₈₆

yes

—

131

C₂³

C₂×C₄×D₄₃

yes

—

132

C₂³

C₂×C₂²⋊D₄₃

yes

—

133

C₂³

C₂³×D₄₃

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

172

—

143

C₂×C₄

Dic₄₃⋊C₄

yes

complemented

144

C₂×C₄

C₂×Dic₈₆

yes

—

145

C₂×C₄

C₂×Dic₈₆

yes

—

146

C₂×C₄

C₂×C₄×D₄₃

yes

—

147

C₂×C₄

C₂×C₄×D₄₃

yes

—

148

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

149

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

C₂×Dic₄₃⋊C₄

yes

—

155

C₂²

C₂²×C₄×D₄₃

yes

characteristic

156

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

157

C₂²

C₂²×Dic₈₆

yes

characteristic

158

C₂²

#13063371

yes

—

159

C₂²

#13063372

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

172

—

163

C₄

C₂×Dic₄₃⋊C₄

yes

—

164

C₄

C₂×Dic₄₃⋊C₄

yes

—

165

C₂

—

166

C₂

C₂²×Dic₄₃⋊C₄

yes

characteristic

167

C₂

#14583373

yes

characteristic

168

C₂

#14583376

yes

characteristic

169

C₂

#14583477

yes

—

170

C₂

#14583527

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#2316
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₂×#117×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#117

N	H ≅ G/N	type
3. #14582797	C₂	semidirect
4. #14582795	C₂	semidirect
9. C₂×C₄⋊Dic₄₃	C₄	semidirect
51. C₂×C₄×C₄₃	C₄⋊C₄	semidirect
92. C₄₃	#2316	semidirect
143. C₂×C₄	Dic₄₃⋊C₄	semidirect

H ≅ G/N

type

3. #14582797

C₂

semidirect

4. #14582795

C₂

semidirect

9. C₂×C₄⋊Dic₄₃

C₄

semidirect

51. C₂×C₄×C₄₃

C₄⋊C₄

semidirect

92. C₄₃

#2316

semidirect

143. C₂×C₄

Dic₄₃⋊C₄

semidirect

Subgroups — #19137211

Conjugacy classes of subgroups

Complements