Group #14665579 — subgroups

Subgroups — #14665579

Order 2816.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13070525	C₂	1	2	yes	maximal
2	#13066686	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
4	C₈×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₁	C₂²	1	2	yes	—
7	C₄×C₁₁×Q₁₆	C₄	1	1	yes	characteristic
8	C₁₁×#335	C₄	1	1	yes	characteristic
9	C₁₁×Q₈⋊C₈	C₄	1	2	yes	—
10	C₈²×C₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
12	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
13	#10622099	—	2	2	no	—
14	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×Q₁₆	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
18	C₁₁×#107	C₈	1	1	yes	characteristic
19	C₁₁×C₄⋊C₈	—	2	2	no	—
20	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
21	C₁₁×#103	C₈	1	2	yes	—
22	C₄×C₈×C₁₁	—	2	2	no	—
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₁₁⋊C₁₆	—	2	1	no	—
27	C₂×C₁₁⋊C₁₆	—	2	2	no	—
28	C₂×C₁₁⋊C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₁⋊C₁₆	D₄	1	1	yes	characteristic
30	#12054	—	11	1	no	maximal, Sylow
31	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
32	C₁₁×Q₁₆	—	2	2	no	—
33	C₂×C₈×C₁₁	—	2	2	no	—
34	C₂×C₈×C₁₁	—	2	1	no	—
35	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
39	C₁₁×C₄⋊C₄	—	2	2	no	—
40	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
41	C₄²×C₁₁	—	2	2	no	—
42	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₁₁⋊C₁₆	—	2	2	no	—
44	C₁₁⋊C₁₆	—	2	1	no	—
45	C₁₁⋊C₁₆	—	4	2	no	—
46	C₁₁⋊C₁₆	Q₁₆	1	2	yes	complemented
47	#2183	—	11	2	no	—
48	C₈×Q₁₆	D₁₁	1	1	yes	characteristic
49	#1370	—	11	1	no	—
50	Q₈⋊C₁₆	—	11	2	no	—
51	C₈×C₁₆	—	11	1	no	—
52	Q₈×C₁₁	—	2	2	no	—
53	Q₈×C₁₁	—	2	2	no	—
54	C₂×C₄×C₁₁	—	2	2	no	—
55	C₂×C₄×C₁₁	—	2	2	no	—
56	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
59	C₈×C₁₁	—	2	2	no	—
60	C₈×C₁₁	—	2	1	no	—
61	C₈×C₁₁	—	2	1	no	—
62	C₈×C₁₁	—	4	2	no	—
63	C₈×C₁₁	M₃₂	1	2	yes	—
64	C₈×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₁₁	#135	1	1	yes	characteristic
66	C₈×Q₈	C₂×D₁₁	1	2	yes	—
67	C₄×Q₁₆	Dic₁₁	1	1	yes	characteristic
68	C₄⋊C₁₆	—	11	2	no	—
69	C₄⋊C₁₆	—	22	2	no	—
70	#346	—	22	2	no	—
71	C₄×C₁₆	—	11	1	no	—
72	C₄×C₁₆	—	11	1	no	—
73	#335	Dic₁₁	1	1	yes	characteristic
74	Q₈⋊C₈	Dic₁₁	1	2	yes	—
75	C₈²	C₂×D₁₁	1	1	yes	characteristic
76	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
77	C₄×C₁₁	—	2	1	no	—
78	C₄×C₁₁	—	4	2	no	—
79	C₄×C₁₁	—	4	2	no	—
80	C₄×C₁₁	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₁₁	#443	1	1	yes	characteristic
82	C₄×C₁₁	C₂×M₃₂	1	1	yes	characteristic
83	C₄×C₁₁	#504	1	1	yes	characteristic
84	C₂×Q₁₆	C₁₁⋊C₈	1	1	yes	characteristic
85	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
86	C₂×C₁₆	—	11	1	no	—
87	C₂×C₁₆	—	11	1	no	—
88	C₂×C₁₆	—	22	1	no	—
89	C₂×C₁₆	—	22	2	no	—
90	#107	C₁₁⋊C₈	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
93	#103	C₁₁⋊C₈	1	2	yes	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
98	C₂×C₁₁	C₈×Q₁₆	1	1	yes	characteristic
99	C₂×C₁₁	#2169	1	1	yes	characteristic
100	C₂×C₁₁	#2170	1	1	yes	characteristic
101	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
102	Q₁₆	—	2	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
106	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
107	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
108	C₂×C₈	#4341	1	1	yes	characteristic
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
111	C₄²	—	2	2	no	—
112	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
113	C₁₆	—	11	2	no	—
114	C₁₆	—	22	2	no	—
115	C₁₆	—	22	1	no	—
116	C₁₆	—	44	2	no	—
117	C₁₁	#12054	1	1	yes	characteristic, complemented, Sylow
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
123	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
124	C₂×C₄	#67001	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	4	2	no	—
129	C₈	#66876	1	2	yes	—
130	C₈	Q₁₆×D₁₁	1	1	yes	characteristic
131	C₈	#66972	1	1	yes	characteristic
132	C₂²	D₄×C₁₁⋊C₈	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	2	no	—
136	C₄	C₂×#66876	1	1	yes	characteristic
137	C₄	#10622753	1	1	yes	characteristic
138	C₄	Q₁₆×Dic₁₁	1	1	yes	characteristic
139	C₄	#10622809	1	1	yes	characteristic
140	C₂	Q₁₆×C₁₁⋊C₈	1	1	yes	characteristic
141	C₂	#13070448	1	1	yes	characteristic
142	C₂	#13070453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13070525

C₂

yes

maximal

#13066686

C₂

yes

maximal, characteristic

Q₈⋊(C₁₁⋊C₁₆)

C₂

yes

maximal

C₈×C₁₁⋊C₁₆

C₂

yes

maximal, characteristic

C₈×C₁₁×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₁

C₂²

yes

—

C₄×C₁₁×Q₁₆

C₄

yes

characteristic

C₁₁×#335

C₄

yes

characteristic

C₁₁×Q₈⋊C₈

C₄

yes

—

C₈²×C₁₁

C₂²

yes

characteristic

C₄⋊(C₁₁⋊C₁₆)

—

C₄⋊(C₁₁⋊C₁₆)

C₂²

yes

—

#10622099

—

C₄×C₁₁⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₁×Q₁₆

C₈

yes

characteristic

C₄×Q₈×C₁₁

C₂×C₄

yes

—

C₁₁×#107

C₈

yes

characteristic

C₁₁×C₄⋊C₈

—

C₁₁×C₄⋊C₈

C₂×C₄

yes

—

C₁₁×#103

C₈

yes

—

C₄×C₈×C₁₁

—

C₄×C₈×C₁₁

C₂×C₄

yes

characteristic

C₄×C₈×C₁₁

C₂×C₄

yes

characteristic

C₄×C₈×C₁₁

C₂³

yes

characteristic

C₂×C₁₁⋊C₁₆

—

C₂×C₁₁⋊C₁₆

—

C₂×C₁₁⋊C₁₆

D₄

yes

characteristic

C₂×C₁₁⋊C₁₆

D₄

yes

characteristic

#12054

—

maximal, Sylow

C₂×Q₈×C₁₁

C₂×C₈

yes

—

C₁₁×Q₁₆

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

C₂×C₈

yes

characteristic

C₂×C₈×C₁₁

C₂×C₈

yes

characteristic

C₂×C₈×C₁₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₁

#41

yes

characteristic

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₁

—

C₄²×C₁₁

C₂²×C₄

yes

characteristic

C₁₁⋊C₁₆

—

C₁₁⋊C₁₆

—

C₁₁⋊C₁₆

—

C₁₁⋊C₁₆

Q₁₆

yes

complemented

#2183

—

C₈×Q₁₆

D₁₁

yes

characteristic

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

Q₈×C₁₁

—

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₄×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₁

#131

yes

characteristic

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

M₃₂

yes

—

C₈×C₁₁

C₂×Q₁₆

yes

characteristic

C₈×C₁₁

#135

yes

characteristic

C₈×Q₈

C₂×D₁₁

yes

—

C₄×Q₁₆

Dic₁₁

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₁₁

yes

characteristic

Q₈⋊C₈

Dic₁₁

yes

—

C₈²

C₂×D₁₁

yes

characteristic

C₂²×C₁₁

C₈×D₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₄×Q₁₆

yes

characteristic

C₄×C₁₁

#443

yes

characteristic

C₄×C₁₁

C₂×M₃₂

yes

characteristic

C₄×C₁₁

#504

yes

characteristic

C₂×Q₁₆

C₁₁⋊C₈

yes

characteristic

C₄×Q₈

C₂×Dic₁₁

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

C₁₁⋊C₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₁₁

yes

—

#103

C₁₁⋊C₈

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₁₁

yes

characteristic

C₄×C₈

C₂×Dic₁₁

yes

characteristic

C₄×C₈

C₂²×D₁₁

yes

characteristic

C₂×C₁₁

C₈×Q₁₆

yes

characteristic

C₂×C₁₁

#2169

yes

characteristic

100

C₂×C₁₁

#2170

yes

characteristic

101

C₂×Q₈

C₂×C₁₁⋊C₈

yes

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×C₁₁⋊C₈

yes

characteristic

106

C₂×C₈

C₂×C₁₁⋊C₈

yes

characteristic

107

C₂×C₈

D₄×D₁₁

yes

characteristic

108

C₂×C₈

#4341

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×C₁₁⋊C₈

yes

—

111

C₄²

—

112

C₄²

C₂²×Dic₁₁

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₁

#12054

yes

characteristic, complemented, Sylow

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²×C₁₁⋊C₈

yes

characteristic

123

C₂×C₄

D₄×Dic₁₁

yes

characteristic

124

C₂×C₄

#67001

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#66876

yes

—

130

C₈

Q₁₆×D₁₁

yes

characteristic

131

C₈

#66972

yes

characteristic

132

C₂²

D₄×C₁₁⋊C₈

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×#66876

yes

characteristic

137

C₄

#10622753

yes

characteristic

138

C₄

Q₁₆×Dic₁₁

yes

characteristic

139

C₄

#10622809

yes

characteristic

140

C₂

Q₁₆×C₁₁⋊C₈

yes

characteristic

141

C₂

#13070448

yes

characteristic

142

C₂

#13070453

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₈×C₁₁×Q₁₆	C₂
socle, prime radical	C₂²×C₁₁	C₈×D₄
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₈×Q₁₆	D₁₁
hypocenter	C₁₁	#12054
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₁₁

commutator

C₂×C₄×C₁₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₁

Fitting

C₈×C₁₁×Q₁₆

C₂

socle, prime radical

C₂²×C₁₁

C₈×D₄

cosocle

C₄×C₈×C₁₁

C₂³

hypercenter

C₈×Q₁₆

D₁₁

hypocenter

C₁₁

#12054

LCS

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
46. C₁₁⋊C₁₆	Q₁₆	semidirect
117. C₁₁	#12054	semidirect

H ≅ G/N

type

46. C₁₁⋊C₁₆

Q₁₆

semidirect

117. C₁₁

#12054

semidirect

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

subgroup	index
#13070525	2
#13066686	2
Q₈⋊(C₁₁⋊C₁₆)	2
C₈×C₁₁⋊C₁₆	2
C₈×C₁₁×Q₁₆	2
#12054	11

subgroup

index

#13070525

#13066686

Q₈⋊(C₁₁⋊C₁₆)

C₈×C₁₁⋊C₁₆

C₈×C₁₁×Q₁₆

#12054

Subgroups — #14665579

Conjugacy classes of subgroups

Complements

Maximal subgroups