Group #13068658 — subgroups

Subgroups — #13068658

Order 1408.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10622820	C₂	1	2	yes	maximal
3	C₂×#66883	C₂	1	2	yes	maximal, complemented
4	#10622190	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₁₁×M₁₆	—	2	4	no	—
7	C₂×C₁₁×M₁₆	C₄	1	4	yes	—
8	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×#4318	C₂²	1	4	yes	—
11	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
12	#66883	—	2	4	no	—
13	C₂⁴×C₁₁	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₁	—	2	2	no	—
15	C₂²×C₄×C₁₁	—	2	2	no	—
16	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₁	D₄	1	2	yes	—
18	C₂²×C₄×C₁₁	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
20	C₁₁×M₁₆	—	2	8	no	—
21	C₂×C₈×C₁₁	—	2	2	no	—
22	C₂×C₈×C₁₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₁	D₄	1	4	yes	—
25	#4318	—	2	4	no	—
26	#4318	—	2	4	no	—
27	C₂×C₁₁⋊C₈	—	2	4	no	—
28	#1811	—	11	1	no	maximal, Sylow
29	C₂³×C₁₁	—	2	2	no	—
30	C₂³×C₁₁	—	4	2	no	—
31	C₂³×C₁₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₁	—	2	2	no	—
34	C₂×C₄×C₁₁	—	2	2	no	—
35	C₂×C₄×C₁₁	—	2	2	no	—
36	C₂×C₄×C₁₁	—	2	4	no	—
37	C₂×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₁	#41	1	2	yes	—
42	C₈×C₁₁	—	2	2	no	—
43	C₈×C₁₁	C₂²⋊C₄	1	4	yes	—
44	C₁₁⋊C₈	—	4	4	no	—
45	C₂²×M₁₆	D₁₁	1	1	yes	characteristic
46	C₂×#108	—	11	2	no	—
47	#407	—	11	2	no	—
48	#341	—	11	2	no	—
49	C₂²×C₁₁	—	2	2	no	—
50	C₂²×C₁₁	—	2	2	no	—
51	C₂²×C₁₁	—	2	4	no	—
52	C₂²×C₁₁	—	4	4	no	—
53	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₁	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₁	—	2	2	no	—
56	C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₁	#117	1	1	yes	characteristic
58	C₄×C₁₁	#121	1	2	yes	—
59	C₂³×C₄	C₂×D₁₁	1	1	yes	characteristic
60	C₂×M₁₆	—	2	4	no	—
61	C₂×M₁₆	—	11	4	no	—
62	C₂×M₁₆	Dic₁₁	1	4	yes	—
63	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
65	#108	—	22	4	no	—
66	C₂²⋊C₈	—	22	4	no	—
67	C₂×C₁₁	—	2	2	no	—
68	C₂×C₁₁	—	4	2	no	—
69	C₂×C₁₁	#380	1	1	yes	characteristic
70	C₂×C₁₁	#430	1	2	yes	—
71	C₂⁴	Dic₂₂	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	Dic₂₂	1	1	yes	characteristic
75	C₂²×C₄	D₄₄	1	2	yes	—
76	C₂²×C₄	C₂×Dic₁₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
78	M₁₆	—	2	8	no	—
79	M₁₆	—	22	4	no	—
80	M₁₆	—	22	4	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	22	4	no	—
83	C₂×C₈	C₂×Dic₁₁	1	2	yes	—
84	C₂×C₈	C₂×Dic₁₁	1	2	yes	—
85	C₂×C₈	C₂²⋊D₁₁	1	4	yes	—
86	C₁₁	#1811	1	1	yes	characteristic, complemented, Sylow
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₁₁	1	2	yes	—
90	C₂³	C₂×Dic₂₂	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	C₄⋊Dic₁₁	1	2	yes	—
96	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
97	C₂×C₄	#4339	1	2	yes	—
98	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	44	4	no	—
102	C₈	C₂²⋊Dic₁₁	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	#66977	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	#66979	1	1	yes	characteristic
111	C₄	#66983	1	2	yes	—
112	C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#10622738	1	2	yes	—
116	C₂	#10622825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₁×M₁₆

C₂

yes

maximal, characteristic

#10622820

C₂

yes

maximal

C₂×#66883

C₂

yes

maximal, complemented

#10622190

C₂

yes

maximal, complemented

C₂³×C₄×C₁₁

C₂²

yes

characteristic

C₂×C₁₁×M₁₆

—

C₂×C₁₁×M₁₆

C₄

yes

—

C₂²×C₈×C₁₁

C₂²

yes

characteristic

C₂²×C₈×C₁₁

C₂²

yes

characteristic

C₂×#4318

C₂²

yes

—

C₂²⋊(C₁₁⋊C₈)

—

#66883

—

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

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

C₂×C₄

yes

—

C₂×C₈×C₁₁

C₂×C₄

yes

—

C₂×C₈×C₁₁

D₄

yes

—

#4318

—

#4318

—

C₂×C₁₁⋊C₈

—

#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

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₈×C₁₁

—

C₈×C₁₁

C₂²⋊C₄

yes

—

C₁₁⋊C₈

—

C₂²×M₁₆

D₁₁

yes

—

—

—

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

Dic₁₁

yes

—

C₂²×C₈

C₂×D₁₁

yes

characteristic

C₂²×C₈

C₂×D₁₁

yes

characteristic

#108

—

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

D₄₄

yes

—

C₂²×C₄

C₂×Dic₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₁

yes

—

C₂×C₈

C₂×Dic₁₁

yes

—

C₂×C₈

C₂²⋊D₁₁

yes

—

C₁₁

#1811

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₁₁

yes

—

C₂³

C₂×Dic₂₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₁

yes

—

C₂×C₄

C₂×D₄₄

yes

characteristic

C₂×C₄

#4339

yes

—

C₂×C₄

C₂²×Dic₁₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

100

C₈

—

101

C₈

—

102

C₈

C₂²⋊Dic₁₁

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#66977

yes

—

108

C₂²

C₂×C₄⋊Dic₁₁

yes

characteristic

109

C₄

—

110

C₄

#66979

yes

characteristic

111

C₄

#66983

yes

—

112

C₄

C₂×C₂²⋊Dic₁₁

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#10622738

yes

—

116

C₂

#10622825

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₂²×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	Q₈
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₂²×M₁₆	D₁₁
hypocenter	C₁₁	#1811
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₄

commutator

C₂×C₄×C₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₂²×C₁₁×M₁₆

C₂

socle

C₂²×C₁₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₁

Q₈

cosocle

C₂²×C₄×C₁₁

C₂³

hypercenter

C₂²×M₁₆

D₁₁

hypocenter

C₁₁

#1811

LCS

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
3. C₂×#66883	C₂	semidirect
4. #10622190	C₂	semidirect
86. C₁₁	#1811	semidirect

H ≅ G/N

type

3. C₂×#66883

C₂

semidirect

4. #10622190

C₂

semidirect

86. C₁₁

#1811

semidirect

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

subgroup	index
C₂²×C₁₁×M₁₆	2
#10622820	2
C₂×#66883	2
#10622190	2
#1811	11

subgroup

index

C₂²×C₁₁×M₁₆

Subgroups — #13068658

Conjugacy classes of subgroups

Complements

Maximal subgroups