Group #45693355 — subgroups

Subgroups — C₁₃₁×#7471 #45693355

Order 33536.

190 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 118 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×C₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₁₃₁	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
4	C₂×#107×C₁₃₁	—	2	2	no	—
5	C₂×C₃₂×C₁₃₁	—	2	2	no	—
6	C₂×C₃₂×C₁₃₁	C₄	1	2	yes	complemented
7	#366×C₁₃₁	—	2	2	no	—
8	#366×C₁₃₁	C₄	1	4	yes	complemented
9	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₃₁	—	4	2	no	—
11	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
14	C₂×C₁₆×C₁₃₁	Q₈	1	1	yes	characteristic
15	#107×C₁₃₁	—	2	4	no	—
16	#107×C₁₃₁	—	4	2	no	—
17	C₃₂×C₁₃₁	—	2	2	no	—
18	C₂²×C₄×C₁₃₁	—	8	2	no	—
19	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
20	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
21	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
23	C₂×C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
24	C₄⋊C₄×C₁₃₁	—	4	4	no	—
25	C₄⋊C₄×C₁₃₁	—	8	2	no	—
26	C₁₆×C₁₃₁	C₄²	1	1	yes	characteristic
27	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₁₃₁	C₄⋊C₄	1	2	yes	—
29	C₂³×C₁₃₁	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₃₁	—	8	4	no	—
31	C₂×C₄×C₁₃₁	—	16	2	no	—
32	C₂×C₄×C₁₃₁	—	16	2	no	—
33	C₂×C₄×C₁₃₁	#102	1	2	yes	—
34	C₂×C₄×C₁₃₁	#107	1	1	yes	characteristic
35	C₂×C₄×C₁₃₁	SD₃₂	1	2	yes	—
36	C₂×C₄×C₁₃₁	Q₃₂	1	1	yes	characteristic
37	C₈×C₁₃₁	#95	1	1	yes	characteristic
38	C₈×C₁₃₁	#103	1	2	yes	—
39	C₈×C₁₃₁	#106	1	1	yes	characteristic
40	C₂²×C₁₃₁	#357	1	2	yes	—
41	C₂²×C₁₃₁	#366	1	1	yes	characteristic
42	C₂²×C₁₃₁	D₃₂	1	1	yes	characteristic
43	C₂²×C₁₃₁	SD₆₄	1	2	yes	—
44	C₂²×C₁₃₁	Q₆₄	1	1	yes	characteristic
45	C₄×C₁₃₁	—	16	4	no	—
46	C₄×C₁₃₁	#340	1	1	yes	characteristic
47	C₄×C₁₃₁	#358	1	2	yes	—
48	C₄×C₁₃₁	#367	1	1	yes	characteristic
49	C₂×C₁₃₁	#1380	1	1	yes	characteristic
50	C₂×C₁₃₁	#1415	1	2	yes	—
51	C₂×C₁₃₁	#1416	1	2	yes	—
52	C₂×C₁₃₁	#1423	1	1	yes	characteristic
53	C₂×C₁₃₁	#1424	1	1	yes	characteristic
54	#7471	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₁₃₁	#7471	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₃₂	C₂×C₁₃₁	1	1	yes	characteristic
57	C₂×#366	C₂×C₁₃₁	1	2	yes	—
58	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
59	C₂×#107	—	2	2	no	—
60	C₂×C₃₂	—	2	2	no	—
61	C₂×C₃₂	C₄×C₁₃₁	1	2	yes	complemented
62	#366	—	2	2	no	—
63	#366	C₄×C₁₃₁	1	4	yes	complemented
64	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
67	C₂×C₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
68	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
69	C₂×C₁₆	Q₈×C₁₃₁	1	1	yes	characteristic
70	#107	—	2	4	no	—
71	#107	—	4	2	no	—
72	C₃₂	—	2	2	no	—
73	C₂²×C₄	—	8	2	no	—
74	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
76	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
77	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
78	C₂×C₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
79	C₄⋊C₄	—	4	4	no	—
80	C₄⋊C₄	—	8	2	no	—
81	C₁₆	C₄²×C₁₃₁	1	1	yes	characteristic
82	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
83	C₁₆	C₄⋊C₄×C₁₃₁	1	2	yes	—
84	C₂³	D₁₆×C₁₃₁	1	1	yes	characteristic
85	C₂×C₄	—	8	4	no	—
86	C₂×C₄	—	16	2	no	—
87	C₂×C₄	—	16	2	no	—
88	C₂×C₄	#102×C₁₃₁	1	2	yes	—
89	C₂×C₄	#107×C₁₃₁	1	1	yes	characteristic
90	C₂×C₄	SD₃₂×C₁₃₁	1	2	yes	—
91	C₂×C₄	Q₃₂×C₁₃₁	1	1	yes	characteristic
92	C₈	#95×C₁₃₁	1	1	yes	characteristic
93	C₈	#103×C₁₃₁	1	2	yes	—
94	C₈	#106×C₁₃₁	1	1	yes	characteristic
95	C₂²	#357×C₁₃₁	1	2	yes	—
96	C₂²	#366×C₁₃₁	1	1	yes	characteristic
97	C₂²	D₃₂×C₁₃₁	1	1	yes	characteristic
98	C₂²	SD₆₄×C₁₃₁	1	2	yes	—
99	C₂²	Q₆₄×C₁₃₁	1	1	yes	characteristic
100	C₄	—	16	4	no	—
101	C₄	#340×C₁₃₁	1	1	yes	characteristic
102	C₄	#358×C₁₃₁	1	2	yes	—
103	C₄	#367×C₁₃₁	1	1	yes	characteristic
104	C₂	#1380×C₁₃₁	1	1	yes	characteristic
105	C₂	#1415×C₁₃₁	1	2	yes	—
106	C₂	#1416×C₁₃₁	1	2	yes	—
107	C₂	#1423×C₁₃₁	1	1	yes	characteristic
108	C₂	#1424×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃₂×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#366×C₁₃₁

C₂

yes

maximal

C₂²×C₁₆×C₁₃₁

C₂²

yes

characteristic

C₂×#107×C₁₃₁

—

C₂×C₃₂×C₁₃₁

—

C₂×C₃₂×C₁₃₁

C₄

yes

complemented

#366×C₁₃₁

—

#366×C₁₃₁

C₄

yes

complemented

C₂²×C₈×C₁₃₁

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₁₆×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₃₁

C₂×C₄

yes

—

C₂×C₁₆×C₁₃₁

D₄

yes

—

C₂×C₁₆×C₁₃₁

Q₈

yes

characteristic

#107×C₁₃₁

—

#107×C₁₃₁

—

C₃₂×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₈

yes

characteristic

C₂×C₈×C₁₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃₁

SD₁₆

yes

—

C₂×C₈×C₁₃₁

Q₁₆

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₁₆×C₁₃₁

C₄²

yes

characteristic

C₁₆×C₁₃₁

C₂²⋊C₄

yes

characteristic

C₁₆×C₁₃₁

C₄⋊C₄

yes

—

C₂³×C₁₃₁

D₁₆

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

#102

yes

—

C₂×C₄×C₁₃₁

#107

yes

characteristic

C₂×C₄×C₁₃₁

SD₃₂

yes

—

C₂×C₄×C₁₃₁

Q₃₂

yes

C₈×C₁₃₁

yes

C₈×C₁₃₁

yes

—

C₈×C₁₃₁

#106

yes

characteristic

C₂²×C₁₃₁

#357

yes

—

C₂²×C₁₃₁

#366

yes

characteristic

C₂²×C₁₃₁

D₃₂

yes

characteristic

C₂²×C₁₃₁

SD₆₄

yes

—

C₂²×C₁₃₁

Q₆₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

#340

yes

characteristic

C₄×C₁₃₁

#358

yes

—

C₄×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

—

C₂×C₁₃₁

#1416

yes

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#7471

yes

characteristic, direct product factor, Sylow

C₂²×C₃₂

C₂×C₁₃₁

yes

characteristic

C₂×#366

C₂×C₁₃₁

yes

—

C₂²×C₁₆

C₂²×C₁₃₁

yes

characteristic

C₂×#107

—

C₂×C₃₂

—

C₂×C₃₂

C₄×C₁₃₁

yes

complemented

#366

—

#366

C₄×C₁₃₁

yes

complemented

C₂²×C₈

D₄×C₁₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₁₆

C₂×C₄×C₁₃₁

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₃₁

yes

—

C₂×C₁₆

D₄×C₁₃₁

yes

—

C₂×C₁₆

Q₈×C₁₃₁

yes

characteristic

#107

—

#107

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

D₈×C₁₃₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₁₃₁

yes

—

C₂×C₈

C₄⋊C₄×C₁₃₁

yes

characteristic

C₂×C₈

SD₁₆×C₁₃₁

yes

—

C₂×C₈

Q₁₆×C₁₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₁₆

C₄²×C₁₃₁

yes

characteristic

C₁₆

C₂²⋊C₄×C₁₃₁

yes

characteristic

C₁₆

C₄⋊C₄×C₁₃₁

yes

—

C₂³

D₁₆×C₁₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#102×C₁₃₁

yes

—

C₂×C₄

#107×C₁₃₁

yes

characteristic

C₂×C₄

SD₃₂×C₁₃₁

yes

—

C₂×C₄

Q₃₂×C₁₃₁

yes

characteristic

C₈

#95×C₁₃₁

yes

characteristic

C₈

#103×C₁₃₁

yes

—

C₈

#106×C₁₃₁

yes

characteristic

C₂²

#357×C₁₃₁

yes

—

C₂²

#366×C₁₃₁

yes

characteristic

C₂²

D₃₂×C₁₃₁

yes

characteristic

C₂²

SD₆₄×C₁₃₁

yes

—

C₂²

Q₆₄×C₁₃₁

yes

characteristic

100

C₄

—

101

C₄

#340×C₁₃₁

yes

characteristic

102

C₄

#358×C₁₃₁

yes

—

103

C₄

#367×C₁₃₁

yes

characteristic

104

C₂

#1380×C₁₃₁

yes

characteristic

105

C₂

#1415×C₁₃₁

yes

—

106

C₂

#1416×C₁₃₁

yes

—

107

C₂

#1423×C₁₃₁

yes

characteristic

108

C₂

#1424×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	D₁₆
commutator	C₁₆	C₄²×C₁₃₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₃₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₈×C₁₃₁ ◁ C₂²×C₁₆×C₁₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃₁

D₁₆

commutator

C₁₆

C₄²×C₁₃₁

Frattini, cosocle

C₂²×C₁₆

C₂²×C₁₃₁

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₈×C₁₃₁ ◁ C₂²×C₁₆×C₁₃₁ ◁ G

N	H ≅ G/N	type
6. C₂×C₃₂×C₁₃₁	C₄	semidirect
8. #366×C₁₃₁	C₄	semidirect
54. #7471	C₁₃₁	direct (N × H)
55. C₁₃₁	#7471	direct (N × H)
61. C₂×C₃₂	C₄×C₁₃₁	semidirect
63. #366	C₄×C₁₃₁	semidirect

H ≅ G/N

type

6. C₂×C₃₂×C₁₃₁

C₄

semidirect

8. #366×C₁₃₁

C₄

semidirect

54. #7471

C₁₃₁

direct (N × H)

55. C₁₃₁

#7471

direct (N × H)

61. C₂×C₃₂

C₄×C₁₃₁

semidirect

63. #366

C₄×C₁₃₁

semidirect

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

subgroup	index
C₂²×C₃₂×C₁₃₁	2
C₂×#366×C₁₃₁	2
#7471	131

subgroup

index

C₂²×C₃₂×C₁₃₁

C₂×#366×C₁₃₁

#7471

131

Subgroups — C₁₃₁×#7471 #45693355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#7471 #45693355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7471 #45693355