Group #19135959 — subgroups

Subgroups — #19135959

Order 5504.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#13063260	C₂	1	1	yes	maximal, characteristic
2	#14582926	C₂	1	1	yes	maximal, characteristic
3	#14582907	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063248	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063243	C₂	1	1	yes	maximal, characteristic
6	#14582668	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#428	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂×#10621877	C₂²	1	1	yes	characteristic
10	C₂×#10621877	C₂²	1	1	yes	characteristic
11	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	#13063260	—	2	2	no	—
13	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
14	#13063248	—	2	1	no	—
15	#13063248	—	2	1	no	—
16	#13063248	—	2	2	no	—
17	#13063243	—	2	2	no	—
18	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	#117×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	#107×C₄₃	—	2	1	no	—
22	#106×C₄₃	—	2	1	no	—
23	C₂²×C₄×C₄₃	—	2	1	no	—
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂×Dic₈₆	—	2	1	no	—
27	C₂×Dic₈₆	—	2	1	no	—
28	C₂×Dic₈₆	—	2	1	no	—
29	C₂×Dic₈₆	—	2	1	no	—
30	C₂×Dic₈₆	—	2	1	no	—
31	C₂×Dic₈₆	Q₈	1	2	yes	—
32	M₁₆×C₄₃	—	2	1	no	—
33	M₁₆×C₄₃	D₄	1	2	yes	—
34	C₂×C₈×C₄₃	—	2	1	no	—
35	C₄⋊C₄×C₄₃	—	2	1	no	—
36	C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₄₃	—	2	1	no	—
40	C₄²×C₄₃	—	2	1	no	—
41	#10621877	—	2	1	no	—
42	#10621877	—	2	1	no	—
43	#10621877	—	2	1	no	—
44	#10621877	—	2	1	no	—
45	C₂×C₄₃⋊C₈	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂×C₄₃⋊C₈	—	2	1	no	—
49	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
50	C₂³×C₄₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	4	1	no	—
54	C₂×C₄×C₄₃	—	8	1	no	—
55	C₂×C₄×C₄₃	C₂×D₄	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×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
61	C₂×Dic₄₃	—	2	2	no	—
62	C₂×Dic₄₃	—	4	1	no	—
63	C₂×Dic₄₃	—	4	1	no	—
64	Dic₈₆	—	2	2	no	—
65	Dic₈₆	—	2	2	no	—
66	Dic₈₆	—	4	1	no	—
67	Dic₈₆	—	4	1	no	—
68	C₈×C₄₃	—	2	2	no	—
69	C₄₃⋊C₈	—	2	2	no	—
70	C₄₃⋊C₈	—	4	1	no	—
71	C₄₃⋊C₈	—	4	1	no	—
72	C₂²×C₄₃	—	2	1	no	—
73	C₂²×C₄₃	—	2	1	no	—
74	C₂²×C₄₃	#121	1	1	yes	characteristic
75	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₄₃	#126	1	1	yes	characteristic
77	C₄×C₄₃	—	4	1	no	—
78	C₄×C₄₃	—	4	1	no	—
79	C₄×C₄₃	—	8	1	no	—
80	C₄×C₄₃	#121	1	1	yes	characteristic
81	C₄×C₄₃	#121	1	1	yes	characteristic
82	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₄₃	—	4	2	no	—
85	#2064	—	43	1	no	maximal, Sylow
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	#396	1	1	yes	characteristic
89	C₂×C₄₃	#471	1	1	yes	characteristic
90	C₂×C₄₃	#473	1	1	yes	characteristic
91	C₂×#108	—	43	1	no	—
92	#428	D₄₃	1	1	yes	characteristic
93	#424	—	43	1	no	—
94	#419	—	43	1	no	—
95	C₂×#103	—	43	1	no	—
96	C₂×#101	—	43	1	no	—
97	#343	—	43	1	no	—
98	C₄₃	#2064	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	43	1	no	—
100	C₂×M₁₆	—	43	1	no	—
101	C₂×M₁₆	—	43	1	no	—
102	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
103	C₂²×C₈	—	43	1	no	—
104	#117	C₂×D₄₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
106	#108	—	86	2	no	—
107	#107	—	2	1	no	—
108	#106	—	2	1	no	—
109	C₄⋊C₈	—	86	2	no	—
110	#103	—	86	2	no	—
111	#103	—	86	1	no	—
112	#103	—	86	1	no	—
113	#101	—	86	2	no	—
114	C₂×Q₈	—	43	2	no	—
115	C₂×Q₈	—	86	1	no	—
116	C₂×Q₈	—	86	1	no	—
117	C₂×Q₈	—	86	1	no	—
118	C₂×Q₈	—	86	1	no	—
119	C₂×Q₈	—	86	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	86	1	no	—
122	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	86	1	no	—
125	M₁₆	—	86	1	no	—
126	M₁₆	—	86	1	no	—
127	M₁₆	—	86	1	no	—
128	M₁₆	D₁₇₂	1	2	yes	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	43	2	no	—
131	C₂×C₈	—	86	1	no	—
132	C₂×C₈	—	86	1	no	—
133	C₂×C₈	—	86	1	no	—
134	C₂×C₈	—	86	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	#10621900	1	1	yes	characteristic
142	Q₈	—	86	2	no	—
143	Q₈	—	86	2	no	—
144	Q₈	—	172	1	no	—
145	Q₈	—	172	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	86	2	no	—
151	C₂×C₄	—	172	1	no	—
152	C₂×C₄	—	172	1	no	—
153	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
154	C₂×C₄	#10621898	1	1	yes	characteristic
155	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
157	C₂×C₄	#10621902	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	86	2	no	—
161	C₈	—	172	1	no	—
162	C₈	—	172	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#13063297	1	1	yes	characteristic
166	C₂²	#13063305	1	1	yes	characteristic
167	C₂²	#13063320	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	1	no	—
171	C₄	—	172	2	no	—
172	C₄	#13063299	1	1	yes	characteristic
173	C₄	#13063300	1	1	yes	characteristic
174	C₄	#13063301	1	1	yes	characteristic
175	C₄	#13063303	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#14582821	1	1	yes	characteristic
179	C₂	#14583160	1	1	yes	characteristic
180	C₂	#14583172	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063260

C₂

yes

maximal, characteristic

#14582926

C₂

yes

maximal, characteristic

#14582907

C₂

yes

maximal, characteristic

C₂×#13063248

C₂

yes

maximal, characteristic

C₂×#13063243

C₂

yes

maximal, characteristic

#14582668

C₂

yes

maximal, characteristic

C₄₃×#428

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂×#10621877

C₂²

yes

characteristic

C₂×#10621877

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

#13063260

—

C₄⋊(C₄₃⋊C₈)

—

—

—

—

—

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

#107×C₄₃

—

#106×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

Q₈

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

—

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

#41

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

#126

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

Dic₄₃

—

#2064

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

D₄₃

yes

—

—

—

—

—

C₄₃

#2064

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₄₃

yes

characteristic

103

C₂²×C₈

—

104

#117

C₂×D₄₃

yes

characteristic

105

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

106

#108

—

107

#107

—

108

#106

—

109

C₄⋊C₈

—

110

#103

—

111

#103

—

112

#103

—

113

#101

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²×D₄₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

128

M₁₆

D₁₇₂

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

#10621900

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

172

—

145

Q₈

—

172

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

172

—

152

C₂×C₄

—

172

—

153

C₂×C₄

C₂×D₁₇₂

yes

characteristic

154

C₂×C₄

#10621898

yes

characteristic

155

C₂×C₄

D₄×D₄₃

yes

characteristic

156

C₂×C₄

Q₈×D₄₃

yes

characteristic

157

C₂×C₄

#10621902

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

172

—

162

C₈

—

172

—

163

C₂²

—

164

C₂²

—

165

C₂²

#13063297

yes

characteristic

166

C₂²

#13063305

yes

characteristic

167

C₂²

#13063320

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

—

172

C₄

#13063299

yes

characteristic

173

C₄

#13063300

yes

characteristic

174

C₄

#13063301

yes

characteristic

175

C₄

#13063303

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#14582821

yes

characteristic

179

C₂

#14583160

yes

characteristic

180

C₂

#14583172

yes

characteristic

	subgroup	G/H
center	C₂²	#13063297
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#428	C₂
socle	C₂²×C₄₃	#121
hypercenter	#428	D₄₃
hypocenter	C₄₃	#2064
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#13063297

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#428

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
98. C₄₃	#2064	semidirect

H ≅ G/N

type

98. C₄₃

#2064

semidirect

Subgroups — #19135959

Conjugacy classes of subgroups

Complements