Group #14673922 — subgroups

Subgroups — #14673922

Order 2816.

268 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13069364	C₂	1	1	yes	maximal, characteristic
2	#13066723	C₂	1	2	yes	maximal
3	#13066714	C₂	1	2	yes	maximal
4	#13066698	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#106	—	2	1	no	—
7	C₂×C₁₁×#106	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₁₁×#384	—	2	1	no	—
10	C₂×#66883	C₂²	1	1	yes	characteristic
11	#10622719	C₂²	1	1	yes	characteristic
12	#10622715	—	2	1	no	—
13	C₂×#66876	C₂²	1	2	yes	—
14	#10622161	—	2	1	no	—
15	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₁×C₄⋊C₄	—	4	2	no	—
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	#66979	—	2	1	no	—
21	C₂×#4318	—	2	1	no	—
22	C₁₁×#106	—	2	4	no	—
23	C₁₁×#106	—	4	2	no	—
24	C₄×C₈×C₁₁	—	2	2	no	—
25	#66883	—	2	2	no	—
26	#66882	Q₈	1	2	yes	—
27	#66881	—	2	1	no	—
28	Dic₁₁⋊C₈	—	4	2	no	—
29	#66876	—	2	2	no	—
30	#66876	—	2	2	no	—
31	C₂×C₁₁⋊C₁₆	—	2	2	no	—
32	#13543	—	11	1	no	maximal, Sylow
33	C₂²×C₄×C₁₁	—	2	1	no	—
34	C₂²×C₄×C₁₁	—	4	2	no	—
35	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₁	—	2	1	no	—
37	C₂×C₈×C₁₁	—	2	1	no	—
38	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
39	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
41	C₂×C₈×C₁₁	#41	1	2	yes	—
42	C₂×C₈×C₁₁	#41	1	2	yes	—
43	C₁₁×C₄⋊C₄	—	4	2	no	—
44	C₁₁×C₄⋊C₄	—	4	2	no	—
45	C₁₁×C₄⋊C₄	—	8	4	no	—
46	C₄²×C₁₁	—	2	1	no	—
47	C₄²×C₁₁	Q₁₆	1	2	yes	—
48	C₂²⋊Dic₁₁	—	4	1	no	—
49	C₄⋊Dic₁₁	—	2	2	no	—
50	C₄×Dic₁₁	—	4	1	no	—
51	#4318	—	4	1	no	—
52	#4318	—	4	1	no	—
53	C₂×C₁₁⋊C₈	—	4	1	no	—
54	C₁₁⋊C₁₆	—	4	2	no	—
55	#1945	—	11	1	no	—
56	#1844	D₁₁	1	1	yes	characteristic
57	#1386	—	11	2	no	—
58	#1383	—	11	2	no	—
59	#1376	—	11	1	no	—
60	C₂³×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₁₁	—	2	4	no	—
62	C₂×C₄×C₁₁	—	2	2	no	—
63	C₂×C₄×C₁₁	—	8	4	no	—
64	C₂×C₄×C₁₁	—	8	2	no	—
65	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
67	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂×Dic₁₁	—	4	2	no	—
72	C₈×C₁₁	—	2	2	no	—
73	C₈×C₁₁	#125	1	2	yes	—
74	C₈×C₁₁	#126	1	2	yes	—
75	C₁₁⋊C₈	—	8	1	no	—
76	C₂×M₃₂	—	11	2	no	—
77	C₂×#108	—	11	1	no	—
78	#427	—	11	1	no	—
79	C₂×#106	—	2	1	no	—
80	C₂×#106	C₂×D₁₁	1	2	yes	—
81	#424	—	22	1	no	—
82	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
83	#384	—	2	1	no	—
84	#339	—	22	1	no	—
85	C₂²×C₁₁	—	2	2	no	—
86	C₂²×C₁₁	#476	1	1	yes	characteristic
87	C₂²×C₁₁	#484	1	1	yes	characteristic
88	C₂²×C₁₁	#490	1	1	yes	characteristic
89	C₄×C₁₁	—	2	4	no	—
90	C₄×C₁₁	—	16	2	no	—
91	C₄×C₁₁	#400	1	1	yes	characteristic
92	C₄×C₁₁	#477	1	1	yes	characteristic
93	C₄×C₁₁	#481	1	1	yes	characteristic
94	C₄×C₁₁	#487	1	1	yes	characteristic
95	Dic₁₁	—	8	2	no	—
96	C₂×M₁₆	—	22	1	no	—
97	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
99	#117	—	22	1	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	4	2	no	—
102	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
103	M₃₂	—	22	2	no	—
104	M₃₂	—	22	2	no	—
105	C₂×C₁₆	—	22	2	no	—
106	#108	—	22	2	no	—
107	#107	—	11	2	no	—
108	#106	—	2	4	no	—
109	#106	—	4	2	no	—
110	#106	—	22	1	no	—
111	C₄⋊C₈	—	44	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₂×C₁₁	—	2	1	no	—
114	C₂×C₁₁	—	2	1	no	—
115	C₂×C₁₁	#2100	1	1	yes	characteristic
116	C₂×C₁₁	#2230	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
120	M₁₆	—	44	1	no	—
121	M₁₆	—	44	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	44	1	no	—
125	C₂×C₈	#4325	1	2	yes	—
126	C₂×C₈	#4339	1	2	yes	—
127	C₂×C₈	#4341	1	2	yes	—
128	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
129	C₂×C₈	#4343	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	8	4	no	—
133	C₄⋊C₄	—	22	2	no	—
134	C₂²⋊C₄	—	44	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	44	1	no	—
137	C₄²	#4326	1	2	yes	—
138	C₁₆	—	44	2	no	—
139	C₁₁	#13543	1	1	yes	characteristic, complemented, Sylow
140	C₂³	#66985	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	44	2	no	—
146	C₂×C₄	#66982	1	1	yes	characteristic
147	C₂×C₄	#66985	1	1	yes	characteristic
148	C₂×C₄	#66989	1	1	yes	characteristic
149	C₂×C₄	C₂×#4325	1	1	yes	characteristic
150	C₂×C₄	C₂×#4326	1	1	yes	characteristic
151	C₂×C₄	#66999	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	88	1	no	—
154	C₈	#66942	1	2	yes	—
155	C₈	#66951	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#10622583	1	1	yes	characteristic
158	C₂²	#10622684	1	1	yes	characteristic
159	C₂²	#10622697	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	16	2	no	—
162	C₄	—	88	2	no	—
163	C₄	#10622581	1	1	yes	characteristic
164	C₄	#10622595	1	1	yes	characteristic
165	C₄	#10622694	1	1	yes	characteristic
166	C₄	#10622704	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#13070178	1	1	yes	characteristic
170	C₂	#13070784	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13069364

C₂

yes

maximal, characteristic

#13066723

C₂

yes

maximal

#13066714

C₂

yes

maximal

#13066698

C₂

yes

maximal, characteristic

C₁₁×#1844

C₂

yes

maximal, characteristic

C₂×C₁₁×#106

—

C₂×C₁₁×#106

C₂²

yes

—

C₂×C₄×C₈×C₁₁

C₂²

yes

characteristic

C₁₁×#384

—

C₂×#66883

C₂²

yes

characteristic

#10622719

C₂²

yes

characteristic

#10622715

—

C₂×#66876

C₂²

yes

—

#10622161

—

C₂²×C₈×C₁₁

D₄

yes

characteristic

C₂²×C₈×C₁₁

C₂³

yes

characteristic

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₄²×C₁₁

D₄

yes

characteristic

#66979

—

C₂×#4318

—

C₁₁×#106

—

C₁₁×#106

—

C₄×C₈×C₁₁

—

#66883

—

#66882

Q₈

yes

—

#66881

—

Dic₁₁⋊C₈

—

#66876

—

#66876

—

C₂×C₁₁⋊C₁₆

—

#13543

—

maximal, Sylow

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

SD₁₆

yes

—

C₂×C₈×C₁₁

C₂×Q₈

yes

C₂×C₈×C₁₁

yes

C₂×C₈×C₁₁

yes

—

C₂×C₈×C₁₁

#41

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

—

C₄²×C₁₁

Q₁₆

yes

—

C₂²⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄×Dic₁₁

—

#4318

—

#4318

—

C₂×C₁₁⋊C₈

—

C₁₁⋊C₁₆

—

#1945

—

#1844

D₁₁

yes

—

—

—

C₂³×C₁₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂²⋊Q₈

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₁

—

C₈×C₁₁

—

C₈×C₁₁

#125

yes

—

C₈×C₁₁

#126

yes

—

C₁₁⋊C₈

—

C₂×M₃₂

—

—

—

—

C₂×D₁₁

yes

—

#424

—

C₂×C₄×C₈

C₂×D₁₁

yes

characteristic

#384

—

#339

—

C₂²×C₁₁

—

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

yes

C₄×C₁₁

yes

C₄×C₁₁

yes

C₄×C₁₁

yes

Dic₁₁

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₁

yes

characteristic

C₂²×C₈

C₂²×D₁₁

yes

characteristic

#117

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

103

M₃₂

—

104

M₃₂

—

105

C₂×C₁₆

—

106

#108

—

107

#107

—

108

#106

—

109

#106

—

110

#106

—

111

C₄⋊C₈

—

112

C₄×C₈

—

113

C₂×C₁₁

—

114

C₂×C₁₁

—

115

C₂×C₁₁

#2100

yes

characteristic

116

C₂×C₁₁

#2230

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

120

M₁₆

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

#4325

yes

—

126

C₂×C₈

#4339

yes

—

127

C₂×C₈

#4341

yes

—

128

C₂×C₈

Q₈×D₁₁

yes

characteristic

129

C₂×C₈

#4343

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

#4326

yes

—

138

C₁₆

—

139

C₁₁

#13543

yes

characteristic, complemented, Sylow

140

C₂³

#66985

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#66982

yes

characteristic

147

C₂×C₄

#66985

yes

characteristic

148

C₂×C₄

#66989

yes

characteristic

149

C₂×C₄

C₂×#4325

yes

characteristic

150

C₂×C₄

C₂×#4326

yes

characteristic

151

C₂×C₄

#66999

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

#66942

yes

—

155

C₈

#66951

yes

—

156

C₂²

—

157

C₂²

#10622583

yes

characteristic

158

C₂²

#10622684

yes

characteristic

159

C₂²

#10622697

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#10622581

yes

characteristic

164

C₄

#10622595

yes

characteristic

165

C₄

#10622694

yes

characteristic

166

C₄

#10622704

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#13070178

yes

characteristic

170

C₂

#13070784

yes

—

	subgroup	G/H
center	C₂²	#10622697
commutator, cosocle	C₂²×C₈×C₁₁	C₂³
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1844	C₂
socle	C₂²×C₁₁	#476
prime radical	C₂³×C₁₁	AGL(1,ℤ/8)
hypercenter	#1844	D₁₁
hypocenter	C₁₁	#13543
LCS	G ▷ C₂²×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#10622697

commutator, cosocle

C₂²×C₈×C₁₁

C₂³

Frattini

C₂²×C₈

C₂²×D₁₁

Fitting

C₁₁×#1844

C₂

C₂²×C₁₁

C₂³×C₁₁

D₁₁

C₁₁

G ▷ C₂²×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
139. C₁₁	#13543	semidirect

H ≅ G/N

type

139. C₁₁

#13543

semidirect

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

subgroup	index
#13069364	2
#13066723	2
#13066714	2
#13066698	2
C₁₁×#1844	2
#13543	11

subgroup

index

C₁₁×#1844

Subgroups — #14673922

Conjugacy classes of subgroups

Complements

Maximal subgroups