Group #13068656 — subgroups

Subgroups — #13068656

Order 1408.

272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×#4318	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#407	C₂	1	1	yes	maximal, characteristic
3	#10622820	C₂	1	1	yes	maximal, characteristic
4	C₂×#66886	C₂	1	2	yes	maximal, complemented
5	#10622196	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×M₁₆	C₂²	1	2	yes	—
8	C₂×#4318	—	2	4	no	—
9	C₂×#4318	C₄	1	4	yes	—
10	C₂×#4318	C₂²	1	2	yes	—
11	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₁₁×C₂²⋊C₈	—	2	2	no	—
14	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
15	#66886	—	2	4	no	—
16	C₂⁴×C₁₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₁	—	2	2	no	—
18	C₂²×C₄×C₁₁	—	2	2	no	—
19	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₁	D₄	1	2	yes	—
21	C₂²×C₄×C₁₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
23	C₁₁×M₁₆	—	2	2	no	—
24	C₁₁×M₁₆	—	2	2	no	—
25	C₂×C₈×C₁₁	—	2	2	no	—
26	#4318	—	2	8	no	—
27	#4318	—	2	2	no	—
28	#4318	—	2	2	no	—
29	C₂×C₁₁⋊C₈	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₁₁⋊C₈	D₄	1	4	yes	—
34	#1811	—	11	1	no	maximal, Sylow
35	C₂³×C₁₁	—	2	2	no	—
36	C₂³×C₁₁	—	4	2	no	—
37	C₂³×C₁₁	C₄⋊C₄	1	2	yes	—
38	C₂³×C₁₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₁	—	2	2	no	—
40	C₂×C₄×C₁₁	—	2	2	no	—
41	C₂×C₄×C₁₁	—	2	2	no	—
42	C₂×C₄×C₁₁	—	2	4	no	—
43	C₂×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁	#41	1	2	yes	—
48	C₈×C₁₁	—	4	2	no	—
49	C₁₁⋊C₈	—	2	2	no	—
50	C₁₁⋊C₈	—	4	2	no	—
51	C₁₁⋊C₈	C₂²⋊C₄	1	4	yes	—
52	C₂²×M₁₆	—	11	1	no	—
53	C₂×#108	—	11	2	no	—
54	#407	—	11	1	no	—
55	#407	D₁₁	1	1	yes	characteristic
56	#341	—	11	2	no	—
57	C₂²×C₁₁	—	2	2	no	—
58	C₂²×C₁₁	—	2	2	no	—
59	C₂²×C₁₁	—	2	4	no	—
60	C₂²×C₁₁	—	4	4	no	—
61	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₁	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₁₁	—	2	2	no	—
64	C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₁₁	#117	1	1	yes	characteristic
66	C₄×C₁₁	#121	1	2	yes	—
67	C₂³×C₄	C₂×D₁₁	1	1	yes	characteristic
68	C₂×M₁₆	—	11	4	no	—
69	C₂×M₁₆	—	11	2	no	—
70	C₂×M₁₆	—	22	4	no	—
71	C₂×M₁₆	C₂×D₁₁	1	2	yes	—
72	C₂²×C₈	—	11	1	no	—
73	C₂²×C₈	—	11	1	no	—
74	#108	—	22	4	no	—
75	C₂²⋊C₈	—	2	2	no	—
76	C₂²⋊C₈	—	22	2	no	—
77	C₂×C₁₁	—	2	2	no	—
78	C₂×C₁₁	—	4	2	no	—
79	C₂×C₁₁	#380	1	1	yes	characteristic
80	C₂×C₁₁	#430	1	2	yes	—
81	C₂⁴	Dic₂₂	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	Dic₂₂	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₁₁	1	2	yes	—
86	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	22	8	no	—
91	M₁₆	—	22	2	no	—
92	M₁₆	—	22	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	11	2	no	—
95	C₂×C₈	—	11	4	no	—
96	C₂×C₈	—	11	2	no	—
97	C₂×C₈	—	22	2	no	—
98	C₂×C₈	—	22	2	no	—
99	C₁₁	#1811	1	1	yes	characteristic, complemented, Sylow
100	C₂³	—	2	2	no	—
101	C₂³	—	4	2	no	—
102	C₂³	Dic₁₁⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₂₂	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	Dic₁₁⋊C₄	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
110	C₂×C₄	D₄×D₁₁	1	2	yes	—
111	C₂×C₄	#4341	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	11	4	no	—
115	C₈	—	22	2	no	—
116	C₈	—	44	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	4	no	—
121	C₂²	#66931	1	2	yes	—
122	C₂²	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	#66930	1	1	yes	characteristic
125	C₄	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
126	C₄	#66992	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	#10622555	1	1	yes	characteristic
130	C₂	#10622737	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#4318

C₂

yes

maximal, characteristic

C₁₁×#407

C₂

yes

maximal, characteristic

#10622820

C₂

yes

maximal, characteristic

C₂×#66886

C₂

yes

maximal, complemented

#10622196

C₂

yes

maximal, complemented

C₂³×C₄×C₁₁

C₂²

yes

characteristic

C₂×C₁₁×M₁₆

C₂²

yes

—

C₂×#4318

—

C₂×#4318

C₄

yes

—

C₂×#4318

C₂²

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

characteristic

C₁₁×C₂²⋊C₈

—

C₂²⋊(C₁₁⋊C₈)

—

#66886

—

C₂⁴×C₁₁

Q₈

yes

characteristic

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

Q₈

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₁₁×M₁₆

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

—

#4318

—

#4318

—

#4318

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₁⋊C₈

D₄

yes

—

#1811

—

maximal, Sylow

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

C₄⋊C₄

yes

—

C₂³×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

#41

yes

—

C₈×C₁₁

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₁₁⋊C₈

C₂²⋊C₄

yes

—

C₂²×M₁₆

—

C₂×#108

—

#407

—

#407

D₁₁

yes

characteristic

#341

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

—

C₄×C₁₁

C₂×C₂²⋊C₄

yes

C₄×C₁₁

yes

C₄×C₁₁

yes

—

C₂³×C₄

C₂×D₁₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂×C₁₁

—

C₂×C₁₁

—

C₂×C₁₁

#380

yes

characteristic

C₂×C₁₁

#430

yes

—

C₂⁴

Dic₂₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₂₂

yes

characteristic

C₂²×C₄

C₄×D₁₁

yes

—

C₂²×C₄

C₂²⋊D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₁₁

#1811

yes

characteristic, complemented, Sylow

100

C₂³

—

101

C₂³

—

102

C₂³

Dic₁₁⋊C₄

yes

—

103

C₂³

C₂×Dic₂₂

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

Dic₁₁⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₁₁

yes

characteristic

110

C₂×C₄

D₄×D₁₁

yes

—

111

C₂×C₄

#4341

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#66931

yes

—

122

C₂²

C₂×Dic₁₁⋊C₄

yes

characteristic

123

C₄

—

124

C₄

#66930

yes

characteristic

125

C₄

C₂²⋊C₄×D₁₁

yes

characteristic

126

C₄

#66992

yes

—

127

C₂

—

128

C₂

—

129

C₂

#10622555

yes

characteristic

130

C₂

#10622737

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×#407	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	Q₈
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	#407	D₁₁
hypocenter	C₁₁	#1811
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₁

commutator

C₂×C₄×C₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₁₁×#407

C₂

socle

C₂²×C₁₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₁

Q₈

cosocle

C₂²×C₄×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
4. C₂×#66886	C₂	semidirect
5. #10622196	C₂	semidirect
99. C₁₁	#1811	semidirect

H ≅ G/N

type

4. C₂×#66886

C₂

semidirect

5. #10622196

C₂

semidirect

99. C₁₁

#1811

semidirect

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

subgroup	index
C₂²×#4318	2
C₁₁×#407	2
#10622820	2
C₂×#66886	2
#10622196	2
#1811	11

subgroup

index

C₂²×#4318

C₁₁×#407

Subgroups — #13068656

Conjugacy classes of subgroups

Complements

Maximal subgroups