Group #14673908 — subgroups

Subgroups — #14673908

Order 2816.

234 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13068791	C₂	1	1	yes	maximal, characteristic
2	#13066732	C₂	1	2	yes	maximal
3	#13066729	C₂	1	2	yes	maximal
4	#13066696	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1946	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
7	C₁₁×#427	C₂²	1	1	yes	characteristic
8	C₁₁×#424	—	2	1	no	—
9	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
10	C₂×#66883	—	2	1	no	—
11	C₂×#66883	C₂²	1	2	yes	—
12	C₂×#66876	C₂²	1	2	yes	—
13	#10622537	—	2	1	no	—
14	C₁₁×#339	—	2	1	no	—
15	C₂×C₁₁×M₁₆	—	2	1	no	—
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₁₁×#117	—	2	1	no	—
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	C₂×#4318	—	2	2	no	—
21	C₁₁×#108	—	2	2	no	—
22	C₁₁×#107	—	2	1	no	—
23	C₁₁×#106	Q₈	1	2	yes	—
24	C₁₁×C₄⋊C₈	—	4	2	no	—
25	C₁₁×#97	—	2	2	no	—
26	#66883	—	2	4	no	—
27	#66883	—	4	2	no	—
28	#66876	—	2	2	no	—
29	#66876	—	2	2	no	—
30	C₂×C₁₁⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
32	#13540	—	11	1	no	maximal, Sylow
33	C₂²×C₄×C₁₁	—	2	1	no	—
34	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
35	C₁₁×M₁₆	—	4	1	no	—
36	C₁₁×M₁₆	—	4	1	no	—
37	C₂×C₈×C₁₁	—	2	1	no	—
38	C₂×C₈×C₁₁	—	2	1	no	—
39	C₂×C₈×C₁₁	—	4	1	no	—
40	C₂×C₈×C₁₁	D₈	1	2	yes	—
41	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
43	C₂×C₈×C₁₁	#41	1	2	yes	—
44	C₂×C₈×C₁₁	#41	1	2	yes	—
45	C₁₁×C₄⋊C₄	—	2	2	no	—
46	C₁₁×C₂²⋊C₄	—	4	1	no	—
47	C₄²×C₁₁	—	2	1	no	—
48	C₄²×C₁₁	—	4	1	no	—
49	C₄²×C₁₁	SD₁₆	1	2	yes	—
50	#4318	—	4	2	no	—
51	#4318	—	4	2	no	—
52	C₂×C₁₁⋊C₈	—	4	2	no	—
53	C₁₁⋊C₁₆	—	4	2	no	—
54	#1946	D₁₁	1	1	yes	characteristic
55	#1849	—	11	1	no	—
56	#1389	—	11	2	no	—
57	#1388	—	11	2	no	—
58	#1375	—	11	1	no	—
59	C₂³×C₁₁	#137	1	1	yes	characteristic
60	C₂×C₄×C₁₁	—	2	2	no	—
61	C₂×C₄×C₁₁	—	4	2	no	—
62	C₂×C₄×C₁₁	—	8	1	no	—
63	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
65	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
69	C₈×C₁₁	—	2	2	no	—
70	C₈×C₁₁	—	8	1	no	—
71	C₈×C₁₁	#125	1	2	yes	—
72	C₈×C₁₁	#126	1	2	yes	—
73	C₁₁⋊C₈	—	8	2	no	—
74	C₂×M₃₂	—	11	2	no	—
75	C₂×#108	—	11	2	no	—
76	C₂×#108	—	22	1	no	—
77	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
78	#427	C₂×D₁₁	1	1	yes	characteristic
79	#424	—	2	1	no	—
80	#423	—	22	1	no	—
81	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
82	#339	—	2	1	no	—
83	C₂²×C₁₁	—	4	1	no	—
84	C₂²×C₁₁	#474	1	1	yes	characteristic
85	C₂²×C₁₁	#483	1	1	yes	characteristic
86	C₂²×C₁₁	#491	1	1	yes	characteristic
87	C₄×C₁₁	—	4	2	no	—
88	C₄×C₁₁	—	8	2	no	—
89	C₄×C₁₁	#400	1	1	yes	characteristic
90	C₄×C₁₁	#475	1	1	yes	characteristic
91	C₄×C₁₁	#480	1	1	yes	characteristic
92	C₄×C₁₁	#486	1	1	yes	characteristic
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	22	2	no	—
95	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
97	#117	—	2	1	no	—
98	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
99	M₃₂	—	22	2	no	—
100	M₃₂	—	22	2	no	—
101	C₂×C₁₆	—	22	2	no	—
102	#108	—	2	2	no	—
103	#108	—	22	4	no	—
104	#108	—	44	2	no	—
105	#107	—	2	1	no	—
106	#106	Dic₂₂	1	2	yes	—
107	C₄⋊C₈	—	4	2	no	—
108	C₄⋊C₈	—	44	2	no	—
109	#97	—	2	2	no	—
110	C₂×C₁₁	—	2	1	no	—
111	C₂×C₁₁	—	2	1	no	—
112	C₂×C₁₁	—	2	1	no	—
113	C₂×C₁₁	#2099	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
116	M₁₆	—	4	1	no	—
117	M₁₆	—	4	1	no	—
118	M₁₆	—	44	2	no	—
119	M₁₆	—	44	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	44	2	no	—
124	C₂×C₈	#4323	1	2	yes	—
125	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
126	C₂×C₈	#4339	1	1	yes	characteristic
127	C₂×C₈	#4341	1	2	yes	—
128	C₂×C₈	#4343	1	2	yes	—
129	C₄⋊C₄	—	2	2	no	—
130	C₂²⋊C₄	—	4	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	4	1	no	—
133	C₄²	#4325	1	2	yes	—
134	C₁₆	—	44	2	no	—
135	C₁₁	#13540	1	1	yes	characteristic, complemented, Sylow
136	C₂³	#67004	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	#66977	1	1	yes	characteristic
141	C₂×C₄	C₂×#4323	1	1	yes	characteristic
142	C₂×C₄	#66988	1	1	yes	characteristic
143	C₂×C₄	C₂×#4325	1	1	yes	characteristic
144	C₂×C₄	#67000	1	1	yes	characteristic
145	C₂×C₄	#67004	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	8	1	no	—
148	C₈	—	88	2	no	—
149	C₈	#66943	1	2	yes	—
150	C₈	#66951	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#10622620	1	1	yes	characteristic
153	C₂²	#10622656	1	1	yes	characteristic
154	C₂²	#10622682	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#10622599	1	1	yes	characteristic
158	C₄	#10622633	1	1	yes	characteristic
159	C₄	#10622646	1	1	yes	characteristic
160	C₄	#10622701	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#13070174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13068791

C₂

yes

maximal, characteristic

#13066732

C₂

yes

maximal

#13066729

C₂

yes

maximal

#13066696

C₂

yes

maximal, characteristic

C₁₁×#1946

C₂

yes

maximal, characteristic

C₂×C₁₁×#108

C₂²

yes

characteristic

C₁₁×#427

C₂²

yes

characteristic

C₁₁×#424

—

C₂×C₁₁×#97

C₂²

yes

characteristic

C₂×#66883

—

C₂×#66883

C₂²

yes

—

C₂×#66876

C₂²

yes

—

#10622537

—

C₁₁×#339

—

C₂×C₁₁×M₁₆

—

C₂²×C₈×C₁₁

D₄

yes

characteristic

C₂²×C₈×C₁₁

C₂³

yes

characteristic

C₁₁×#117

—

C₂×C₄²×C₁₁

D₄

yes

characteristic

C₂×#4318

—

C₁₁×#108

—

C₁₁×#107

—

C₁₁×#106

Q₈

yes

—

C₁₁×C₄⋊C₈

—

C₁₁×#97

—

—

—

—

—

C₂×C₁₁⋊C₁₆

—

C₄⋊(C₁₁⋊C₈)

—

#13540

—

maximal, Sylow

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂×D₄

yes

characteristic

C₁₁×M₁₆

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

D₈

yes

—

C₂×C₈×C₁₁

C₂×Q₈

yes

C₂×C₈×C₁₁

yes

C₂×C₈×C₁₁

yes

—

C₂×C₈×C₁₁

#41

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₂²⋊C₄

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

SD₁₆

yes

—

#4318

—

#4318

—

C₂×C₁₁⋊C₈

—

C₁₁⋊C₁₆

—

#1946

D₁₁

yes

—

—

—

—

C₂³×C₁₁

#137

yes

characteristic

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×D₈

yes

characteristic

C₂×C₄×C₁₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₁

#137

yes

characteristic

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

#125

yes

—

C₈×C₁₁

#126

yes

—

C₁₁⋊C₈

—

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

C₂×#108

C₂×D₁₁

yes

characteristic

#427

C₂×D₁₁

yes

characteristic

#424

—

#423

—

C₂×#97

C₂×D₁₁

yes

characteristic

#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

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₁

yes

characteristic

C₂²×C₈

C₂²×D₁₁

yes

characteristic

#117

—

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

M₃₂

—

100

M₃₂

—

101

C₂×C₁₆

—

102

#108

—

103

#108

—

104

#108

—

105

#107

—

106

#106

Dic₂₂

yes

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#97

—

110

C₂×C₁₁

—

111

C₂×C₁₁

—

112

C₂×C₁₁

—

113

C₂×C₁₁

#2099

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#4323

yes

—

125

C₂×C₈

C₂×Dic₂₂

yes

characteristic

126

C₂×C₈

#4339

yes

characteristic

127

C₂×C₈

#4341

yes

—

128

C₂×C₈

#4343

yes

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

#4325

yes

—

134

C₁₆

—

135

C₁₁

#13540

yes

characteristic, complemented, Sylow

136

C₂³

#67004

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#66977

yes

characteristic

141

C₂×C₄

C₂×#4323

yes

characteristic

142

C₂×C₄

#66988

yes

characteristic

143

C₂×C₄

C₂×#4325

yes

characteristic

144

C₂×C₄

#67000

yes

characteristic

145

C₂×C₄

#67004

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

#66943

yes

—

150

C₈

#66951

yes

—

151

C₂²

—

152

C₂²

#10622620

yes

characteristic

153

C₂²

#10622656

yes

characteristic

154

C₂²

#10622682

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

#10622599

yes

characteristic

158

C₄

#10622633

yes

characteristic

159

C₄

#10622646

yes

characteristic

160

C₄

#10622701

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#13070174

yes

characteristic

	subgroup	G/H
center	C₄	#10622701
commutator, cosocle	C₂²×C₈×C₁₁	C₂³
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1946	C₂
socle	C₂×C₁₁	#2099
prime radical	C₂³×C₁₁	#137
hypercenter	#1946	D₁₁
hypocenter	C₁₁	#13540
LCS	G ▷ C₂²×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#10622701

commutator, cosocle

C₂²×C₈×C₁₁

C₂³

Frattini

C₂²×C₈

C₂²×D₁₁

Fitting

C₁₁×#1946

C₂

C₂×C₁₁

C₂³×C₁₁

D₁₁

C₁₁

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
135. C₁₁	#13540	semidirect

H ≅ G/N

type

135. C₁₁

#13540

semidirect

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

subgroup	index
#13068791	2
#13066732	2
#13066729	2
#13066696	2
C₁₁×#1946	2
#13540	11

subgroup

index

C₁₁×#1946

Subgroups — #14673908

Conjugacy classes of subgroups

Complements

Maximal subgroups