Group #14674255 — subgroups

Subgroups — #14674255

Order 2816.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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×#10622137	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	Dic₁₁⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₁×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₁⋊C₁₆	C₄	1	1	yes	characteristic
8	C₁₁×M₆₄	—	2	2	no	—
9	C₂×C₁₁×C₃₂	C₂²	1	2	yes	—
10	C₁₆×Dic₁₁	C₄	1	2	yes	—
11	C₁₆×Dic₁₁	C₂²	1	2	yes	—
12	#10622137	—	2	2	no	—
13	C₂×C₁₁⋊C₃₂	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
16	C₂²×C₁₁⋊C₈	C₈	1	1	yes	characteristic
17	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
19	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
20	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
21	C₈×Dic₁₁	C₈	1	2	yes	—
22	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₁⋊C₁₆	—	2	2	no	—
24	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₁₁×C₃₂	—	2	2	no	—
26	C₁₁⋊C₃₂	—	2	2	no	—
27	#13627	—	11	1	no	maximal, Sylow
28	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
29	C₂²×Dic₁₁	C₁₆	1	2	yes	—
30	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
33	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₄×Dic₁₁	C₁₆	1	1	yes	characteristic
35	C₄×Dic₁₁	C₁₆	1	1	yes	characteristic
36	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
37	C₂×C₁₁⋊C₈	—	2	2	no	—
38	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
39	C₁₁×C₁₆	C₄⋊C₄	1	2	yes	—
40	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
41	C₁₁×C₁₆	C₂×Q₈	1	1	yes	characteristic
42	C₁₁⋊C₁₆	—	2	2	no	—
43	C₂×M₆₄	—	11	1	no	—
44	C₂×M₆₄	D₁₁	1	1	yes	characteristic
45	C₂×C₄×C₁₆	—	11	1	no	—
46	C₄⋊C₃₂	—	11	4	no	—
47	C₂³×C₁₁	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₁₁	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
51	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
52	C₂×Dic₁₁	—	2	4	no	—
53	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	—
54	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	—
55	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
56	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₈×C₁₁	C₂×M₁₆	1	1	yes	characteristic
58	C₁₁⋊C₈	—	2	2	no	—
59	C₂²×C₁₆	—	11	1	no	—
60	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
61	C₂×C₄×C₈	—	11	1	no	—
62	M₆₄	—	2	2	no	—
63	M₆₄	—	22	2	no	—
64	C₂×C₃₂	—	11	2	no	—
65	C₂×C₃₂	C₂×D₁₁	1	2	yes	—
66	C₄×C₁₆	—	11	2	no	—
67	C₄×C₁₆	—	11	2	no	—
68	C₂²×C₁₁	—	2	2	no	—
69	C₂²×C₁₁	C₄⋊C₁₆	1	2	yes	—
70	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
71	C₄×C₁₁	C₄⋊C₁₆	1	2	yes	—
72	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₁₁	C₂×M₃₂	1	1	yes	characteristic
74	Dic₁₁	—	2	2	no	—
75	Dic₁₁	M₆₄	1	4	yes	complemented
76	C₂²×C₈	—	11	1	no	—
77	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
78	C₂×C₄²	—	11	1	no	—
79	C₂×C₁₆	—	11	2	no	—
80	C₂×C₁₆	—	22	2	no	—
81	C₂×C₁₆	Dic₂₂	1	2	yes	—
82	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
84	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
85	C₄×C₈	—	11	2	no	—
86	C₄×C₈	—	11	2	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	22	2	no	—
89	C₂×C₁₁	—	2	2	no	—
90	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
91	C₂×C₁₁	C₂×M₆₄	1	2	yes	—
92	C₂²×C₄	—	11	2	no	—
93	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
94	C₂×C₈	—	11	2	no	—
95	C₂×C₈	—	22	2	no	—
96	C₂×C₈	C₈×D₁₁	1	1	yes	characteristic
97	C₂×C₈	#4313	1	2	yes	—
98	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
99	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
100	C₄²	—	11	1	no	—
101	C₄²	—	11	1	no	—
102	C₄²	—	11	2	no	—
103	C₁₆	—	22	2	no	—
104	C₁₆	Dic₁₁⋊C₄	1	2	yes	—
105	C₁₆	C₂×Dic₂₂	1	1	yes	characteristic
106	C₁₆	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
107	C₁₁	#13627	1	1	yes	characteristic, complemented, Sylow
108	C₂³	C₁₆×D₁₁	1	1	yes	characteristic
109	C₂×C₄	—	11	2	no	—
110	C₂×C₄	—	11	2	no	—
111	C₂×C₄	—	22	4	no	—
112	C₂×C₄	C₁₆×D₁₁	1	1	yes	characteristic
113	C₂×C₄	#66862	1	2	yes	—
114	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
116	C₈	—	22	2	no	—
117	C₈	Dic₁₁⋊C₈	1	2	yes	—
118	C₈	C₂×#4313	1	1	yes	characteristic
119	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	Dic₁₁⋊C₁₆	1	2	yes	—
122	C₂²	C₂×C₁₆×D₁₁	1	1	yes	characteristic
123	C₄	—	11	4	no	—
124	C₄	—	22	2	no	—
125	C₄	Dic₁₁⋊C₁₆	1	2	yes	—
126	C₄	C₂×#66862	1	1	yes	characteristic
127	C₄	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×Dic₁₁⋊C₁₆	1	1	yes	characteristic
130	C₂	D₁₁×M₆₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10622137

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₁₁

C₂

yes

maximal, characteristic

Dic₁₁⋊C₃₂

C₂

yes

maximal, complemented

C₂×C₁₁×M₆₄

C₂

yes

maximal, characteristic

C₂²×C₁₁×C₁₆

C₂²

yes

characteristic

C₂×C₈×Dic₁₁

C₄

yes

characteristic

C₂²×C₁₁⋊C₁₆

C₄

yes

characteristic

C₁₁×M₆₄

—

C₂×C₁₁×C₃₂

C₂²

yes

—

C₁₆×Dic₁₁

C₄

yes

—

C₁₆×Dic₁₁

C₂²

yes

—

#10622137

—

C₂×C₁₁⋊C₃₂

C₂²

yes

—

C₂²×C₈×C₁₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₁

C₈

yes

characteristic

C₂²×C₁₁⋊C₈

C₈

yes

characteristic

C₂×C₁₁×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₁×C₁₆

D₄

yes

—

C₂×C₁₁×C₁₆

Q₈

yes

—

C₂×C₁₁×C₁₆

C₂³

yes

characteristic

C₈×Dic₁₁

C₈

yes

—

C₈×Dic₁₁

C₂×C₄

yes

—

C₂×C₁₁⋊C₁₆

—

C₂×C₁₁⋊C₁₆

C₂×C₄

yes

—

C₁₁×C₃₂

—

C₁₁⋊C₃₂

—

#13627

—

maximal, Sylow

C₂²×C₄×C₁₁

C₂×C₈

yes

characteristic

C₂²×Dic₁₁

C₁₆

yes

—

C₂×C₈×C₁₁

C₄⋊C₄

yes

—

C₂×C₈×C₁₁

C₂×C₈

yes

characteristic

C₂×C₈×C₁₁

M₁₆

yes

—

C₂×C₈×C₁₁

C₂²×C₄

yes

characteristic

C₄×Dic₁₁

C₁₆

yes

characteristic

C₄×Dic₁₁

C₁₆

yes

characteristic

C₄×Dic₁₁

C₂×C₈

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

C₂×C₈

yes

—

C₁₁×C₁₆

C₄⋊C₄

yes

—

C₁₁×C₁₆

C₂×D₄

yes

characteristic

C₁₁×C₁₆

C₂×Q₈

yes

characteristic

C₁₁⋊C₁₆

—

C₂×M₆₄

—

C₂×M₆₄

D₁₁

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂³×C₁₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₁

C₄⋊C₈

yes

—

C₂×C₄×C₁₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₁

M₃₂

yes

—

C₂×C₄×C₁₁

C₂²×C₈

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂×C₁₆

yes

—

C₂×Dic₁₁

C₂×C₁₆

yes

—

C₈×C₁₁

C₄⋊C₈

yes

—

C₈×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₁

C₂×M₁₆

yes

characteristic

C₁₁⋊C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₁

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₁₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₁₁

—

C₂²×C₁₁

C₄⋊C₁₆

yes

—

C₂²×C₁₁

C₂²×C₁₆

yes

characteristic

C₄×C₁₁

C₄⋊C₁₆

yes

—

C₄×C₁₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₁

C₂×M₃₂

yes

characteristic

Dic₁₁

—

Dic₁₁

M₆₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₁₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Dic₂₂

yes

—

C₂×C₁₆

C₄×D₁₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₁

yes

—

C₂×C₁₆

C₂²×D₁₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂×C₁₁

—

C₂×C₁₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₁₁

C₂×M₆₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₈×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₁₁

yes

characteristic

C₂×C₈

#4313

yes

—

C₂×C₈

Dic₁₁⋊C₄

yes

—

C₂×C₈

C₂×C₄×D₁₁

yes

characteristic

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

Dic₁₁⋊C₄

yes

—

105

C₁₆

C₂×Dic₂₂

yes

characteristic

106

C₁₆

C₂×C₂²⋊D₁₁

yes

characteristic

107

C₁₁

#13627

yes

characteristic, complemented, Sylow

108

C₂³

C₁₆×D₁₁

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₁₆×D₁₁

yes

characteristic

113

C₂×C₄

#66862

yes

—

114

C₂×C₄

Dic₁₁⋊C₈

yes

—

115

C₂×C₄

C₂×C₈×D₁₁

yes

characteristic

116

C₈

—

117

C₈

Dic₁₁⋊C₈

yes

—

118

C₈

C₂×#4313

yes

characteristic

119

C₈

C₂×Dic₁₁⋊C₄

yes

characteristic

120

C₂²

—

121

C₂²

Dic₁₁⋊C₁₆

yes

—

122

C₂²

C₂×C₁₆×D₁₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

Dic₁₁⋊C₁₆

yes

—

126

C₄

C₂×#66862

yes

characteristic

127

C₄

C₂×Dic₁₁⋊C₈

yes

characteristic

128

C₂

—

129

C₂

C₂×Dic₁₁⋊C₁₆

yes

characteristic

130

C₂

D₁₁×M₆₄

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₁
commutator, socle	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₂×C₁₁×M₆₄	C₂
prime radical	C₂³×C₁₁	C₂×C₁₆
cosocle	C₂×C₁₁×C₁₆	C₂³
hypercenter	C₂×M₆₄	D₁₁
hypocenter	C₁₁	#13627
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₁₁

commutator, socle

C₂²×C₁₁

C₂²×C₁₆

Fitting

C₂×C₁₁×M₆₄

C₂

prime radical

C₂³×C₁₁

C₂×C₁₆

cosocle

C₂×C₁₁×C₁₆

C₂³

hypercenter

C₂×M₆₄

D₁₁

C₁₁

G ▷ C₂²×C₁₁ ▷ C₁₁

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. Dic₁₁⋊C₃₂	C₂	semidirect
75. Dic₁₁	M₆₄	semidirect
107. C₁₁	#13627	semidirect

H ≅ G/N

type

3. Dic₁₁⋊C₃₂

C₂

semidirect

75. Dic₁₁

M₆₄

semidirect

107. C₁₁

#13627

semidirect

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

subgroup	index
C₂×#10622137	2
C₂×C₁₆×Dic₁₁	2
Dic₁₁⋊C₃₂	2
C₂×C₁₁×M₆₄	2
#13627	11

subgroup

index

C₂×#10622137

C₂×C₁₆×Dic₁₁

Dic₁₁⋊C₃₂

C₂×C₁₁×M₆₄

#13627

Subgroups — #14674255

Conjugacy classes of subgroups

Complements

Maximal subgroups