Group #14671821 — subgroups

Subgroups — #14671821

Order 2816.

286 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 115 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	#13070400	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	#13069362	C₂	1	2	yes	maximal
4	#13066713	C₂	1	2	yes	maximal
5	C₂×C₁₁×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#108	C₂²	1	2	yes	—
8	#10622732	—	2	2	no	—
9	#10622719	C₂²	1	2	yes	—
10	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₁₁×#368	—	2	2	no	—
13	#10622193	—	2	2	no	—
14	#10622143	C₄	1	2	yes	complemented
15	#10622142	C₄	1	2	yes	complemented
16	C₁₆×Dic₁₁	—	2	2	no	—
17	C₂×C₁₁×M₁₆	—	2	2	no	—
18	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
19	#66979	—	2	2	no	—
20	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
21	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
25	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
26	C₁₁×#108	—	2	2	no	—
27	C₁₁×#108	—	2	2	no	—
28	#66882	C₂×C₄	1	4	yes	—
29	#66881	—	2	2	no	—
30	Dic₁₁⋊C₈	—	4	4	no	—
31	C₈×Dic₁₁	—	2	1	no	—
32	C₈×Dic₁₁	—	2	1	no	—
33	C₂×C₁₁⋊C₁₆	—	2	1	no	—
34	C₂×C₁₁⋊C₁₆	—	2	1	no	—
35	C₂×C₁₁⋊C₁₆	Q₈	1	2	yes	—
36	#13169	—	11	1	no	maximal, Sylow
37	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₁	—	2	1	no	—
39	C₁₁×M₁₆	—	4	2	no	—
40	C₁₁×M₁₆	—	4	2	no	—
41	C₂×C₈×C₁₁	—	4	2	no	—
42	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
48	C₂²⋊Dic₁₁	—	4	2	no	—
49	C₄⋊Dic₁₁	—	2	4	no	—
50	C₄×Dic₁₁	—	2	1	no	—
51	C₄×Dic₁₁	—	4	2	no	—
52	C₄×Dic₁₁	Q₁₆	1	2	yes	—
53	C₂×C₁₁⋊C₈	—	2	1	no	—
54	C₂×C₁₁⋊C₈	—	2	1	no	—
55	C₂×C₁₁⋊C₈	D₈	1	2	yes	—
56	C₁₁×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₁₁×C₁₆	C₄⋊C₄	1	2	yes	—
58	C₁₁⋊C₁₆	—	2	2	no	—
59	C₂×#368	D₁₁	1	1	yes	characteristic
60	#2158	—	11	1	no	—
61	C₂×C₄×C₁₆	—	11	1	no	—
62	#1945	—	11	2	no	—
63	#1382	—	11	2	no	—
64	C₂³×C₁₁	#135	1	1	yes	characteristic
65	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
71	C₂×Dic₁₁	—	2	4	no	—
72	C₂×Dic₁₁	—	2	2	no	—
73	C₂×Dic₁₁	—	4	4	no	—
74	C₈×C₁₁	—	8	2	no	—
75	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₁₁	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₁₁	#125	1	1	yes	characteristic
79	C₁₁⋊C₈	—	2	2	no	—
80	C₂²×C₁₆	—	11	1	no	—
81	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
82	C₂×#108	C₂×D₁₁	1	2	yes	—
83	#427	—	11	2	no	—
84	#424	—	22	2	no	—
85	C₂×C₄×C₈	—	11	1	no	—
86	#368	—	2	2	no	—
87	#367	—	11	2	no	—
88	#366	—	11	2	no	—
89	C₄×C₁₆	—	22	2	no	—
90	#339	—	22	2	no	—
91	C₂²×C₁₁	—	2	2	no	—
92	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
93	C₂²×C₁₁	#467	1	1	yes	characteristic
94	C₂²×C₁₁	#499	1	1	yes	characteristic
95	C₄×C₁₁	#389	1	1	yes	characteristic
96	C₄×C₁₁	C₄×Q₁₆	1	1	yes	characteristic
97	C₄×C₁₁	#466	1	1	yes	characteristic
98	C₄×C₁₁	#500	1	1	yes	characteristic
99	Dic₁₁	—	2	4	no	—
100	Dic₁₁	—	8	4	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	11	1	no	—
103	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
104	#117	—	22	2	no	—
105	C₂×C₄²	—	11	1	no	—
106	C₂×C₁₆	—	11	2	no	—
107	C₂×C₁₆	—	22	1	no	—
108	C₂×C₁₆	—	22	1	no	—
109	C₂×C₁₆	Dic₂₂	1	2	yes	—
110	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
111	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	2	2	no	—
115	#107	—	11	4	no	—
116	#106	—	22	2	no	—
117	C₄⋊C₈	—	44	4	no	—
118	C₄×C₈	—	22	1	no	—
119	C₄×C₈	—	22	1	no	—
120	C₂×C₁₁	—	2	1	no	—
121	C₂×C₁₁	—	2	1	no	—
122	C₂×C₁₁	#1942	1	1	yes	characteristic
123	C₂×C₁₁	#2178	1	2	yes	—
124	C₂²×C₄	—	22	1	no	—
125	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
126	M₁₆	—	4	2	no	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	11	2	no	—
130	C₂×C₈	—	22	1	no	—
131	C₂×C₈	—	22	1	no	—
132	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
134	C₂×C₈	#4341	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
136	C₂×C₈	#4343	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
138	C₄⋊C₄	—	22	4	no	—
139	C₂²⋊C₄	—	44	2	no	—
140	C₄²	—	11	2	no	—
141	C₄²	—	22	1	no	—
142	C₄²	—	44	2	no	—
143	C₁₆	—	22	2	no	—
144	C₁₆	Dic₁₁⋊C₄	1	2	yes	—
145	C₁₆	Dic₁₁⋊C₄	1	2	yes	—
146	C₁₁	#13169	1	1	yes	characteristic, complemented, Sylow
147	C₂³	#66972	1	1	yes	characteristic
148	C₂×C₄	—	22	4	no	—
149	C₂×C₄	—	22	2	no	—
150	C₂×C₄	—	44	4	no	—
151	C₂×C₄	#66941	1	1	yes	characteristic
152	C₂×C₄	#66946	1	1	yes	characteristic
153	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
154	C₂×C₄	#66965	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
156	C₂×C₄	#66991	1	1	yes	characteristic
157	C₈	—	8	2	no	—
158	C₈	—	22	2	no	—
159	C₈	#66940	1	1	yes	characteristic
160	C₈	#66943	1	1	yes	characteristic
161	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
162	C₈	#66999	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#10622485	1	1	yes	characteristic
165	C₂²	#10622491	1	1	yes	characteristic
166	C₂²	#10622801	1	1	yes	characteristic
167	C₄	—	22	4	no	—
168	C₄	—	88	4	no	—
169	C₄	#10622486	1	1	yes	characteristic
170	C₄	#10622487	1	1	yes	characteristic
171	C₄	#10622586	1	1	yes	characteristic
172	C₄	#10622770	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#13069347	1	1	yes	characteristic
176	C₂	#13070496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13070400

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₁₁

C₂

yes

maximal, characteristic

#13069362

C₂

yes

maximal

#13066713

C₂

yes

maximal

C₂×C₁₁×#368

C₂

yes

maximal, characteristic

C₂²×C₁₁×C₁₆

C₂²

yes

characteristic

C₂×C₁₁×#108

C₂²

yes

—

#10622732

—

#10622719

C₂²

yes

—

C₂×C₈×Dic₁₁

C₂²

yes

characteristic

C₂²×C₁₁⋊C₁₆

C₂²

yes

characteristic

C₁₁×#368

—

#10622193

—

#10622143

C₄

yes

complemented

#10622142

C₄

yes

complemented

C₁₆×Dic₁₁

—

C₂×C₁₁×M₁₆

—

C₂²×C₈×C₁₁

C₂³

yes

characteristic

#66979

—

C₂×C₄×Dic₁₁

D₄

yes

characteristic

C₂²×C₁₁⋊C₈

D₄

yes

characteristic

C₂×C₁₁×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₁×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₁×C₁₆

D₄

yes

—

C₂×C₁₁×C₁₆

Q₈

yes

—

C₁₁×#108

—

C₁₁×#108

—

#66882

C₂×C₄

yes

—

#66881

—

Dic₁₁⋊C₈

—

C₈×Dic₁₁

—

C₈×Dic₁₁

—

C₂×C₁₁⋊C₁₆

—

C₂×C₁₁⋊C₁₆

—

C₂×C₁₁⋊C₁₆

Q₈

yes

—

#13169

—

maximal, Sylow

C₂²×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂²×Dic₁₁

—

C₁₁×M₁₆

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₁

C₂×Q₈

yes

C₂×C₈×C₁₁

yes

C₂×C₈×C₁₁

yes

C₂²⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄×Dic₁₁

—

C₄×Dic₁₁

—

C₄×Dic₁₁

Q₁₆

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

D₈

yes

—

C₁₁×C₁₆

C₄⋊C₄

yes

—

C₁₁×C₁₆

C₄⋊C₄

yes

—

C₁₁⋊C₁₆

—

C₂×#368

D₁₁

yes

characteristic

#2158

—

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₂³×C₁₁

#135

yes

characteristic

C₂×C₄×C₁₁

C₄×D₄

yes

characteristic

C₂×C₄×C₁₁

#121

yes

characteristic

C₂×C₄×C₁₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₁

#135

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₈×C₁₁

—

C₈×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₁

C₄×Q₈

yes

characteristic

C₈×C₁₁

C₂²⋊Q₈

yes

characteristic

C₈×C₁₁

#125

yes

characteristic

C₁₁⋊C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₁

yes

characteristic

C₂×#108

C₂×D₁₁

yes

—

#427

—

#424

—

C₂×C₄×C₈

—

#368

—

#367

—

#366

—

C₄×C₁₆

—

#339

—

C₂²×C₁₁

—

C₂²×C₁₁

C₄×D₈

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₄×C₁₁

yes

C₄×C₁₁

C₄×Q₁₆

yes

C₄×C₁₁

yes

C₄×C₁₁

yes

Dic₁₁

—

100

Dic₁₁

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₁₁

yes

characteristic

104

#117

—

105

C₂×C₄²

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

Dic₂₂

yes

—

110

C₂×C₁₆

C₄×D₁₁

yes

characteristic

111

C₂×C₁₆

C₄×D₁₁

yes

characteristic

112

C₂×C₁₆

C₂²⋊D₁₁

yes

—

113

#108

—

114

#108

—

115

#107

—

116

#106

—

117

C₄⋊C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂×C₁₁

—

121

C₂×C₁₁

—

122

C₂×C₁₁

#1942

yes

characteristic

123

C₂×C₁₁

#2178

yes

—

124

C₂²×C₄

—

125

C₂²×C₄

D₄×D₁₁

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₂×Dic₂₂

yes

characteristic

133

C₂×C₈

C₂×C₄×D₁₁

yes

characteristic

134

C₂×C₈

#4341

yes

characteristic

135

C₂×C₈

Q₈×D₁₁

yes

characteristic

136

C₂×C₈

#4343

yes

characteristic

137

C₂×C₈

C₂×C₂²⋊D₁₁

yes

characteristic

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₁₆

—

144

C₁₆

Dic₁₁⋊C₄

yes

—

145

C₁₆

Dic₁₁⋊C₄

yes

—

146

C₁₁

#13169

yes

characteristic, complemented, Sylow

147

C₂³

#66972

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#66941

yes

characteristic

152

C₂×C₄

#66946

yes

characteristic

153

C₂×C₄

D₈×D₁₁

yes

characteristic

154

C₂×C₄

#66965

yes

characteristic

155

C₂×C₄

Q₁₆×D₁₁

yes

characteristic

156

C₂×C₄

#66991

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

#66940

yes

characteristic

160

C₈

#66943

yes

characteristic

161

C₈

C₂×Dic₁₁⋊C₄

yes

characteristic

162

C₈

#66999

yes

characteristic

163

C₂²

—

164

C₂²

#10622485

yes

characteristic

165

C₂²

#10622491

yes

characteristic

166

C₂²

#10622801

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

#10622486

yes

characteristic

170

C₄

#10622487

yes

characteristic

171

C₄

#10622586

yes

characteristic

172

C₄

#10622770

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#13069347

yes

characteristic

176

C₂

#13070496

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₁
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#368	C₂
socle	C₂²×C₁₁	C₄×D₈
prime radical	C₂³×C₁₁	#135
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#368	D₁₁
hypocenter	C₁₁	#13169
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₁₁

commutator

C₂×C₈×C₁₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₁

Fitting

C₂×C₁₁×#368

C₂

socle

C₂²×C₁₁

C₄×D₈

prime radical

C₂³×C₁₁

#135

cosocle

C₂²×C₈×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
14. #10622143	C₄	semidirect
15. #10622142	C₄	semidirect
146. C₁₁	#13169	semidirect

H ≅ G/N

type

14. #10622143

C₄

semidirect

15. #10622142

C₄

semidirect

146. C₁₁

#13169

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13070400	2
C₂×C₁₆×Dic₁₁	2
#13069362	2
#13066713	2
C₂×C₁₁×#368	2
#13169	11

subgroup

index

C₂×C₁₆×Dic₁₁

C₂×C₁₁×#368

Subgroups — #14671821

Conjugacy classes of subgroups

Complements

Maximal subgroups