Group #13068786 — subgroups

Subgroups — #13068786

Order 1408.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₁₁×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#66886	C₂	1	2	yes	maximal
3	#10622731	C₂	1	2	yes	maximal
4	C₂×#66883	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
9	C₂×#4318	C₂²	1	2	yes	—
10	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₁₁×#108	—	2	2	no	—
12	#66886	—	2	4	no	—
13	#66883	—	2	2	no	—
14	Dic₁₁⋊C₈	—	2	4	no	—
15	C₈×Dic₁₁	C₄	1	2	yes	—
16	C₈×Dic₁₁	C₄	1	2	yes	—
17	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₁	Q₈	1	2	yes	—
19	C₁₁×M₁₆	—	2	2	no	—
20	C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₈×C₁₁	—	2	2	no	—
22	C₂×C₈×C₁₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₁	D₄	1	2	yes	—
24	C₂×C₈×C₁₁	Q₈	1	2	yes	—
25	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₁	D₄	1	2	yes	—
28	#4318	—	2	2	no	—
29	#4318	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₁₁⋊C₈	D₄	1	2	yes	—
33	C₂×C₁₁⋊C₈	Q₈	1	2	yes	—
34	#1848	—	11	1	no	maximal, Sylow
35	C₂³×C₁₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₁₁	—	2	4	no	—
43	C₂×Dic₁₁	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₁₁	C₄⋊C₄	1	2	yes	—
45	C₈×C₁₁	—	4	2	no	—
46	C₈×C₁₁	C₄⋊C₄	1	4	yes	—
47	C₁₁⋊C₈	—	4	2	no	—
48	C₁₁⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂×#108	—	11	2	no	—
50	C₂×#108	—	11	1	no	—
51	C₂×#108	D₁₁	1	1	yes	characteristic
52	#423	—	11	2	no	—
53	C₂×C₄×C₈	—	11	1	no	—
54	C₂²×C₁₁	—	2	2	no	—
55	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₁₁	#127	1	1	yes	characteristic
61	C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₁	—	2	2	no	—
63	Dic₁₁	#108	1	4	yes	complemented
64	C₂×M₁₆	—	11	2	no	—
65	C₂×M₁₆	C₂×D₁₁	1	2	yes	—
66	C₂²×C₈	—	11	1	no	—
67	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
68	C₂×C₄²	—	11	1	no	—
69	#108	—	2	2	no	—
70	#108	—	22	4	no	—
71	#108	—	22	2	no	—
72	C₄⋊C₈	—	22	4	no	—
73	C₄×C₈	—	11	2	no	—
74	C₄×C₈	—	11	2	no	—
75	C₂×C₁₁	—	2	1	no	—
76	C₂×C₁₁	—	2	1	no	—
77	C₂×C₁₁	#384	1	1	yes	characteristic
78	C₂×C₁₁	C₂×#108	1	2	yes	—
79	C₂²×C₄	—	11	2	no	—
80	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
81	M₁₆	—	2	2	no	—
82	M₁₆	—	2	2	no	—
83	M₁₆	—	22	2	no	—
84	M₁₆	—	22	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	11	2	no	—
87	C₂×C₈	—	11	2	no	—
88	C₂×C₈	—	11	2	no	—
89	C₂×C₈	—	22	2	no	—
90	C₂×C₈	Dic₂₂	1	2	yes	—
91	C₂×C₈	C₄×D₁₁	1	2	yes	—
92	C₂×C₈	C₂²⋊D₁₁	1	2	yes	—
93	C₄²	—	11	1	no	—
94	C₄²	—	11	1	no	—
95	C₄²	—	11	2	no	—
96	C₁₁	#1848	1	1	yes	characteristic, complemented, Sylow
97	C₂³	Q₈×D₁₁	1	1	yes	characteristic
98	C₂×C₄	—	11	2	no	—
99	C₂×C₄	—	11	2	no	—
100	C₂×C₄	—	22	4	no	—
101	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
103	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	11	4	no	—
109	C₈	—	44	2	no	—
110	C₈	Dic₁₁⋊C₄	1	4	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	#66941	1	1	yes	characteristic
113	C₂²	C₄⋊C₄×D₁₁	1	1	yes	characteristic
114	C₂²	#66997	1	1	yes	characteristic
115	C₄	—	11	4	no	—
116	C₄	—	22	2	no	—
117	C₄	#66941	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
119	C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
120	C₄	#66993	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	D₁₁×#108	1	2	yes	—
124	C₂	#10622589	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×#108

C₂

yes

maximal, characteristic

C₂×#66886

C₂

yes

maximal

#10622731

C₂

yes

maximal

C₂×#66883

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₁

C₂

yes

maximal, characteristic

C₂×C₁₁×M₁₆

C₂²

yes

—

C₂²×C₈×C₁₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₁

C₂²

yes

characteristic

C₂×#4318

C₂²

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₁₁×#108

—

#66886

—

#66883

—

Dic₁₁⋊C₈

—

C₈×Dic₁₁

C₄

yes

—

C₈×Dic₁₁

C₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

Q₈

yes

—

C₁₁×M₁₆

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

C₂×C₄

yes

—

C₂×C₈×C₁₁

D₄

yes

—

C₂×C₈×C₁₁

Q₈

yes

—

C₄×Dic₁₁

C₂×C₄

yes

characteristic

C₄×Dic₁₁

C₂×C₄

yes

characteristic

C₄×Dic₁₁

D₄

yes

—

#4318

—

#4318

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₁⋊C₈

D₄

yes

—

C₂×C₁₁⋊C₈

Q₈

yes

—

#1848

—

maximal, Sylow

C₂³×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

C₄⋊C₄

yes

—

C₂×Dic₁₁

C₄⋊C₄

yes

—

C₈×C₁₁

—

C₈×C₁₁

C₄⋊C₄

yes

—

C₁₁⋊C₈

—

C₁₁⋊C₈

C₄⋊C₄

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

D₁₁

yes

characteristic

#423

—

C₂×C₄×C₈

—

C₂²×C₁₁

—

C₂²×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁

#127

yes

characteristic

C₄×C₁₁

C₄⋊Q₈

yes

characteristic

Dic₁₁

—

Dic₁₁

#108

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₁

yes

characteristic

C₂×C₄²

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₁

—

C₂×C₁₁

—

C₂×C₁₁

#384

yes

characteristic

C₂×C₁₁

C₂×#108

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₂

yes

—

C₂×C₈

C₄×D₁₁

yes

—

C₂×C₈

C₂²⋊D₁₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₁₁

#1848

yes

characteristic, complemented, Sylow

C₂³

Q₈×D₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×Dic₂₂

yes

characteristic

102

C₂×C₄

C₂×C₄×D₁₁

yes

characteristic

103

C₂×C₄

D₄×D₁₁

yes

characteristic

104

C₂×C₄

D₄×D₁₁

yes

characteristic

105

C₂×C₄

Q₈×D₁₁

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

Dic₁₁⋊C₄

yes

—

111

C₂²

—

112

C₂²

#66941

yes

characteristic

113

C₂²

C₄⋊C₄×D₁₁

yes

characteristic

114

C₂²

#66997

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

#66941

yes

characteristic

118

C₄

C₄⋊C₄×D₁₁

yes

characteristic

119

C₄

C₂×Dic₁₁⋊C₄

yes

characteristic

120

C₄

#66993

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

D₁₁×#108

yes

—

124

C₂

#10622589

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₂×C₁₁×#108	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₂×#108	D₁₁
hypocenter	C₁₁	#1848
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₁

commutator

C₂×C₄×C₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₂×C₁₁×#108

C₂

socle

C₂²×C₁₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₁

C₂×Q₈

cosocle

C₂²×C₄×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
63. Dic₁₁	#108	semidirect
96. C₁₁	#1848	semidirect

H ≅ G/N

type

63. Dic₁₁

#108

semidirect

96. C₁₁

#1848

semidirect

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

subgroup	index
C₂×C₁₁×#108	2
C₂×#66886	2
#10622731	2
C₂×#66883	2
C₂×C₈×Dic₁₁	2
#1848	11

subgroup

index

C₂×C₁₁×#108

C₂×C₈×Dic₁₁

Subgroups — #13068786

Conjugacy classes of subgroups

Complements

Maximal subgroups