Group #14674259 — subgroups

Subgroups — #14674259

Order 2816.

206 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂²×C₁₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₁⋊C₃₂)	C₂	1	4	yes	maximal, complemented
4	C₁₁×#2108	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₁₁×#405	C₄	1	1	yes	characteristic
8	C₁₁×#346	C₄	1	2	yes	—
9	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
10	#10622137	—	2	2	no	—
11	C₂×C₁₁⋊C₃₂	—	2	2	no	—
12	C₂×C₁₁⋊C₃₂	C₂²	1	2	yes	—
13	C₂×C₁₁⋊C₃₂	C₂²	1	2	yes	—
14	C₂×C₁₁×M₁₆	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₁₁×#117	C₈	1	1	yes	characteristic
17	C₁₁×M₃₂	—	2	2	no	—
18	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
21	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
22	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
23	C₁₁×#97	C₈	1	2	yes	—
24	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
25	C₁₁⋊C₃₂	—	2	2	no	—
26	C₁₁⋊C₃₂	—	2	2	no	—
27	#13628	—	11	1	no	maximal, Sylow
28	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
29	C₁₁×M₁₆	—	2	2	no	—
30	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
34	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
35	C₁₁×C₄⋊C₄	C₁₆	1	2	yes	—
36	C₁₁×C₂²⋊C₄	C₁₆	1	2	yes	—
37	C₄²×C₁₁	C₂×C₈	1	2	yes	—
38	C₁₁×C₁₆	—	2	2	no	—
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	C₂×M₆₄	—	11	1	no	—
43	C₂²×C₃₂	—	11	1	no	—
44	#2108	D₁₁	1	1	yes	characteristic
45	C₄⋊C₃₂	—	11	4	no	—
46	C₂³×C₁₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₁₁	C₂×C₁₆	1	4	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	C₈×C₁₁	—	2	2	no	—
53	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
54	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₁₁	C₂×M₁₆	1	1	yes	characteristic
56	C₂×M₃₂	Dic₁₁	1	1	yes	characteristic
57	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
58	#405	Dic₁₁	1	1	yes	characteristic
59	M₆₄	—	22	2	no	—
60	C₂×C₃₂	—	11	2	no	—
61	C₂×C₃₂	—	11	2	no	—
62	C₂×C₃₂	—	22	2	no	—
63	#346	Dic₁₁	1	2	yes	—
64	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
65	C₂²×C₁₁	—	2	2	no	—
66	C₂²×C₁₁	C₄⋊C₁₆	1	2	yes	—
67	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
68	C₄×C₁₁	—	2	4	no	—
69	C₄×C₁₁	C₄⋊C₁₆	1	2	yes	—
70	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₁₁	C₂×M₃₂	1	1	yes	characteristic
72	C₂×M₁₆	C₁₁⋊C₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
74	#117	C₁₁⋊C₈	1	1	yes	characteristic
75	M₃₂	—	2	2	no	—
76	C₂×C₁₆	Dic₂₂	1	2	yes	—
77	C₂×C₁₆	D₄₄	1	2	yes	—
78	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
79	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
80	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
81	#97	C₁₁⋊C₈	1	2	yes	—
82	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
83	C₃₂	—	22	2	no	—
84	C₃₂	—	22	2	no	—
85	C₂×C₁₁	—	2	2	no	—
86	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
87	C₂×C₁₁	#2258	1	2	yes	—
88	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
91	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
92	C₂×C₈	#4318	1	2	yes	—
93	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
94	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
95	C₄⋊C₄	C₁₁⋊C₁₆	1	2	yes	—
96	C₂²⋊C₄	C₁₁⋊C₁₆	1	2	yes	—
97	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
98	C₁₆	—	2	2	no	—
99	C₁₆	C₄⋊Dic₁₁	1	2	yes	—
100	C₁₆	C₂×Dic₂₂	1	1	yes	characteristic
101	C₁₆	C₂×D₄₄	1	1	yes	characteristic
102	C₁₁	#13628	1	1	yes	characteristic, complemented, Sylow
103	C₂³	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
104	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
105	C₂×C₄	C₂×C₁₁⋊C₁₆	1	4	yes	—
106	C₂×C₄	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
107	C₂×C₄	#66876	1	2	yes	—
108	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
111	C₈	C₂×#4318	1	1	yes	characteristic
112	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊(C₁₁⋊C₁₆)	1	2	yes	—
115	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊(C₁₁⋊C₁₆)	1	2	yes	—
118	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
119	C₄	C₂×#66876	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
122	C₂	#13070910	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10622137

C₂

yes

maximal, characteristic

C₂²×C₁₁⋊C₃₂

C₂

yes

maximal, characteristic

C₄⋊(C₁₁⋊C₃₂)

C₂

yes

maximal, complemented

C₁₁×#2108

C₂

yes

maximal, characteristic

C₂×C₁₁×M₃₂

C₄

yes

characteristic

C₂²×C₁₁×C₁₆

C₂²

yes

characteristic

C₁₁×#405

C₄

yes

characteristic

C₁₁×#346

C₄

yes

—

C₄×C₁₁×C₁₆

C₂²

yes

—

#10622137

—

C₂×C₁₁⋊C₃₂

—

C₂×C₁₁⋊C₃₂

C₂²

yes

—

C₂×C₁₁⋊C₃₂

C₂²

yes

—

C₂×C₁₁×M₁₆

C₈

yes

characteristic

C₂²×C₈×C₁₁

C₂×C₄

yes

characteristic

C₁₁×#117

C₈

yes

characteristic

C₁₁×M₃₂

—

C₂×C₁₁×C₁₆

C₂×C₄

yes

—

C₂×C₁₁×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₁×C₁₆

D₄

yes

—

C₂×C₁₁×C₁₆

Q₈

yes

—

C₂×C₁₁×C₁₆

C₂³

yes

characteristic

C₁₁×#97

C₈

yes

—

C₄×C₈×C₁₁

C₂×C₄

yes

—

C₁₁⋊C₃₂

—

C₁₁⋊C₃₂

—

#13628

—

maximal, Sylow

C₂²×C₄×C₁₁

C₂×C₈

yes

characteristic

C₁₁×M₁₆

—

C₂×C₈×C₁₁

C₄⋊C₄

yes

—

C₂×C₈×C₁₁

C₂×C₈

yes

—

C₂×C₈×C₁₁

C₂×C₈

yes

characteristic

C₂×C₈×C₁₁

M₁₆

yes

—

C₂×C₈×C₁₁

C₂²×C₄

yes

characteristic

C₁₁×C₄⋊C₄

C₁₆

yes

—

C₁₁×C₂²⋊C₄

C₁₆

yes

—

C₄²×C₁₁

C₂×C₈

yes

—

C₁₁×C₁₆

—

C₁₁×C₁₆

C₄⋊C₄

yes

—

C₁₁×C₁₆

C₂×D₄

yes

characteristic

C₁₁×C₁₆

C₂×Q₈

yes

characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

D₁₁

yes

characteristic

C₄⋊C₃₂

—

C₂³×C₁₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₁

C₄⋊C₈

yes

—

C₂×C₄×C₁₁

C₂×C₁₆

yes

—

C₂×C₄×C₁₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₁

M₃₂

yes

—

C₂×C₄×C₁₁

C₂²×C₈

yes

characteristic

C₈×C₁₁

—

C₈×C₁₁

C₄⋊C₈

yes

—

C₈×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₁

C₂×M₁₆

yes

characteristic

C₂×M₃₂

Dic₁₁

yes

characteristic

C₂²×C₁₆

C₂×D₁₁

yes

characteristic

#405

Dic₁₁

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

Dic₁₁

yes

—

C₄×C₁₆

C₂×D₁₁

yes

—

C₂²×C₁₁

—

C₂²×C₁₁

C₄⋊C₁₆

yes

—

C₂²×C₁₁

C₂²×C₁₆

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

C₄⋊C₁₆

yes

—

C₄×C₁₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₁

C₂×M₃₂

yes

characteristic

C₂×M₁₆

C₁₁⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₁₁

yes

characteristic

#117

C₁₁⋊C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

Dic₂₂

yes

—

C₂×C₁₆

D₄₄

yes

—

C₂×C₁₆

C₂×Dic₁₁

yes

—

C₂×C₁₆

C₂×Dic₁₁

yes

characteristic

C₂×C₁₆

C₂²×D₁₁

yes

characteristic

#97

C₁₁⋊C₈

yes

—

C₄×C₈

C₂×Dic₁₁

yes

—

C₃₂

—

C₃₂

—

C₂×C₁₁

—

C₂×C₁₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₁₁

#2258

yes

—

C₂²×C₄

C₂×C₁₁⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

C₂×C₁₁⋊C₈

yes

—

C₂×C₈

C₂×C₁₁⋊C₈

yes

characteristic

C₂×C₈

#4318

yes

—

C₂×C₈

C₄⋊Dic₁₁

yes

—

C₂×C₈

C₂²×Dic₁₁

yes

characteristic

C₄⋊C₄

C₁₁⋊C₁₆

yes

—

C₂²⋊C₄

C₁₁⋊C₁₆

yes

—

C₄²

C₂×C₁₁⋊C₈

yes

—

C₁₆

—

C₁₆

C₄⋊Dic₁₁

yes

—

100

C₁₆

C₂×Dic₂₂

yes

characteristic

101

C₁₆

C₂×D₄₄

yes

characteristic

102

C₁₁

#13628

yes

characteristic, complemented, Sylow

103

C₂³

C₂×C₁₁⋊C₁₆

yes

characteristic

104

C₂×C₄

C₄⋊(C₁₁⋊C₈)

yes

—

105

C₂×C₄

C₂×C₁₁⋊C₁₆

yes

—

106

C₂×C₄

C₂×C₁₁⋊C₁₆

yes

characteristic

107

C₂×C₄

#66876

yes

—

108

C₂×C₄

C₂²×C₁₁⋊C₈

yes

characteristic

109

C₈

—

110

C₈

C₄⋊(C₁₁⋊C₈)

yes

—

111

C₈

C₂×#4318

yes

characteristic

112

C₈

C₂×C₄⋊Dic₁₁

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊(C₁₁⋊C₁₆)

yes

—

115

C₂²

C₂²×C₁₁⋊C₁₆

yes

characteristic

116

C₄

—

117

C₄

C₄⋊(C₁₁⋊C₁₆)

yes

—

118

C₄

C₂×C₄⋊(C₁₁⋊C₈)

yes

characteristic

119

C₄

C₂×#66876

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊(C₁₁⋊C₁₆)

yes

characteristic

122

C₂

#13070910

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₁
commutator, socle	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₁₁×#2108	C₂
prime radical	C₂³×C₁₁	C₂×C₁₆
cosocle	C₂×C₁₁×C₁₆	C₂³
hypercenter	#2108	D₁₁
hypocenter	C₁₁	#13628
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₁₁

commutator, socle

C₂²×C₁₁

C₂²×C₁₆

Fitting

C₁₁×#2108

C₂

prime radical

C₂³×C₁₁

C₂×C₁₆

cosocle

C₂×C₁₁×C₁₆

C₂³

D₁₁

C₁₁

G ▷ C₂²×C₁₁ ▷ C₁₁

C₁ ◁ C₂×C₁₆ ◁ #2108

N	H ≅ G/N	type
3. C₄⋊(C₁₁⋊C₃₂)	C₂	semidirect
102. C₁₁	#13628	semidirect

H ≅ G/N

type

3. C₄⋊(C₁₁⋊C₃₂)

C₂

semidirect

102. C₁₁

#13628

semidirect

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

subgroup	index
C₂×#10622137	2
C₂²×C₁₁⋊C₃₂	2
C₄⋊(C₁₁⋊C₃₂)	2
C₁₁×#2108	2
#13628	11

subgroup

index

C₂×#10622137

C₂²×C₁₁⋊C₃₂

C₄⋊(C₁₁⋊C₃₂)

C₁₁×#2108

#13628

Subgroups — #14674259

Conjugacy classes of subgroups

Complements

Maximal subgroups