Group #45691645 — subgroups

Subgroups — C₂×(C₇⋊C₉)⋊D₁₃₃ #45691645

Order 33516.

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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×C₇×C₁₉×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₇⋊C₉)⋊D₁₃₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₇⋊D₁₃₃	C₃	1	1	yes	maximal, characteristic
4	C₇×C₁₉×C₇⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇⋊D₁₃₃	C₂×C₃	1	2	yes	—
6	C₂×C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic
7	C₂×C₉×D₁₃₃	—	7	1	no	maximal
8	C₂×(C₇⋊C₉)⋊D₁₉	—	7	1	no	maximal
9	C₂×C₇⋊D₁₃₃	C₉	1	1	yes	characteristic, complemented, Hall
10	C₃×C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic
11	C₉×D₁₃₃	—	7	2	no	—
12	C₂×C₁₉×C₇⋊C₉	D₇	1	1	yes	characteristic, complemented
13	(C₇⋊C₉)⋊D₁₉	—	7	2	no	—
14	C₂×C₇×C₉×C₁₉	—	7	1	no	—
15	C₂×C₇²×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented
16	C₇⋊D₁₃₃	C₂×C₉	1	2	yes	complemented
17	C₂×(C₇⋊C₉)⋊D₇	—	19	1	no	maximal, Hall
18	C₂×C₃×D₁₃₃	—	7	1	no	—
19	C₂×C₃×D₁₃₃	—	7	1	no	—
20	C₂×C₃×D₁₃₃	—	21	2	no	—
21	C₇×C₉×C₁₉	—	7	1	no	—
22	C₁₉×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
23	C₇²×C₁₉	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
24	C₂×C₇×C₇⋊C₉	D₁₉	1	1	yes	characteristic, complemented
25	(C₇⋊C₉)⋊D₇	—	19	2	no	—
26	C₂×C₃×C₇×C₁₉	—	3	2	no	—
27	C₂×C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic
28	C₂×C₃×C₇×C₁₉	C₃×D₇	1	1	yes	characteristic
29	C₃×D₁₃₃	—	7	2	no	—
30	C₃×D₁₃₃	—	7	2	no	—
31	C₃×D₁₃₃	—	21	4	no	—
32	C₂×C₉×D₁₉	—	49	1	no	Hall
33	C₂×C₃×C₇⋊D₇	—	19	1	no	—
34	C₂×D₁₃₃	—	7	1	no	—
35	C₂×D₁₃₃	—	7	1	no	—
36	C₂×D₁₃₃	—	21	2	no	—
37	C₇×C₇⋊C₉	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
38	C₃×C₇×C₁₉	—	3	2	no	—
39	C₃×C₇×C₁₉	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
40	C₃×C₇×C₁₉	C₂×C₃×D₇	1	1	yes	characteristic
41	C₂×C₉×C₁₉	—	7	1	no	—
42	C₉×D₁₉	—	49	2	no	—
43	C₂×C₃×C₇²	C₃×D₁₉	1	1	yes	characteristic
44	C₃×C₇⋊D₇	—	19	2	no	—
45	C₂×C₇×C₁₉	—	3	2	no	—
46	C₂×C₇×C₁₉	D₇⋊C₉	1	1	yes	characteristic, complemented
47	C₂×C₇×C₁₉	C₉×D₇	1	1	yes	characteristic, complemented
48	D₁₃₃	—	7	2	no	—
49	D₁₃₃	—	7	2	no	—
50	D₁₃₃	—	21	4	no	—
51	C₂×C₉×D₇	—	133	1	no	—
52	C₂×D₇⋊C₉	—	133	1	no	—
53	C₂×C₃×D₁₉	—	49	1	no	—
54	C₂×C₇⋊D₇	—	19	1	no	Hall
55	C₉×C₁₉	—	7	1	no	Hall
56	C₃×C₇²	C₂×C₃×D₁₉	1	1	yes	characteristic
57	C₇×C₁₉	—	3	2	no	—
58	C₇×C₁₉	C₂×D₇⋊C₉	1	1	yes	characteristic, complemented
59	C₇×C₁₉	C₂×C₉×D₇	1	1	yes	characteristic, complemented
60	C₂×C₇×C₉	—	7	1	no	—
61	C₉×D₇	—	133	2	no	—
62	C₂×C₇⋊C₉	D₁₃₃	1	1	yes	characteristic, complemented
63	D₇⋊C₉	—	133	2	no	—
64	C₂×C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic
65	C₃×D₁₉	—	49	2	no	—
66	C₂×C₇²	C₉×D₁₉	1	1	yes	characteristic, complemented
67	C₇⋊D₇	—	19	2	no	—
68	C₂×C₃×D₇	—	133	1	no	—
69	C₂×C₃×D₇	—	133	1	no	—
70	C₂×C₃×D₇	—	399	2	no	—
71	C₂×D₁₉	—	49	1	no	Hall
72	C₇×C₉	—	7	1	no	—
73	C₇⋊C₉	C₂×D₁₃₃	1	1	yes	characteristic, complemented
74	C₃×C₁₉	C₂×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic
75	C₇²	C₂×C₉×D₁₉	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃×C₇	—	3	2	no	—
77	C₂×C₃×C₇	ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic
78	C₂×C₃×C₇	C₃×D₁₃₃	1	1	yes	characteristic
79	C₃×D₇	—	133	2	no	—
80	C₃×D₇	—	133	2	no	—
81	C₃×D₇	—	399	4	no	—
82	C₂×C₁₉	(C₇⋊C₉)⋊D₇	1	1	yes	characteristic, complemented
83	D₁₉	—	49	2	no	—
84	C₂²×C₉	—	931	1	no	Hall
85	C₂×D₇	—	133	1	no	—
86	C₂×D₇	—	133	1	no	—
87	C₂×D₇	—	399	2	no	—
88	C₃×C₇	—	3	2	no	—
89	C₃×C₇	C₂×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic
90	C₃×C₇	C₂×C₃×D₁₃₃	1	1	yes	characteristic
91	C₁₉	C₂×(C₇⋊C₉)⋊D₇	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₉	—	7	1	no	—
93	C₂×C₉	—	931	2	no	—
94	C₂×C₇	—	3	2	no	—
95	C₂×C₇	(C₇⋊C₉)⋊D₁₉	1	1	yes	characteristic, complemented
96	C₂×C₇	C₉×D₁₃₃	1	1	yes	characteristic, complemented
97	D₇	—	133	2	no	—
98	D₇	—	133	2	no	—
99	D₇	—	399	4	no	—
100	C₂²×C₃	—	931	1	no	—
101	C₉	—	7	1	no	Sylow
102	C₇	—	3	2	no	—
103	C₇	C₂×(C₇⋊C₉)⋊D₁₉	1	1	yes	characteristic, complemented
104	C₇	C₂×C₉×D₁₃₃	1	1	yes	characteristic, complemented
105	C₂×C₃	—	931	2	no	—
106	C₂×C₃	ΓL(1,𝔽₈)⋊D₁₃₃	1	1	yes	characteristic
107	C₂²	—	931	1	no	Sylow
108	C₃	C₂×ΓL(1,𝔽₈)⋊D₁₃₃	1	1	yes	characteristic
109	C₂	—	931	2	no	—
110	C₂	(C₇⋊C₉)⋊D₁₃₃	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₁₉×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

(C₇⋊C₉)⋊D₁₃₃

C₂

yes

maximal, direct product factor

C₂×C₃×C₇⋊D₁₃₃

C₃

yes

maximal, characteristic

C₇×C₁₉×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₇⋊D₁₃₃

C₂×C₃

yes

—

C₂×C₃×C₇²×C₁₉

C₂×C₃

yes

characteristic

C₂×C₉×D₁₃₃

—

maximal

C₂×(C₇⋊C₉)⋊D₁₉

—

maximal

C₂×C₇⋊D₁₃₃

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₉

C₂²×C₃

yes

characteristic

C₉×D₁₃₃

—

C₂×C₁₉×C₇⋊C₉

D₇

yes

characteristic, complemented

(C₇⋊C₉)⋊D₁₉

—

C₂×C₇×C₉×C₁₉

—

C₂×C₇²×C₁₉

C₂×C₉

yes

characteristic, complemented

C₇⋊D₁₃₃

C₂×C₉

yes

complemented

C₂×(C₇⋊C₉)⋊D₇

—

maximal, Hall

C₂×C₃×D₁₃₃

—

C₂×C₃×D₁₃₃

—

C₂×C₃×D₁₃₃

—

C₇×C₉×C₁₉

—

C₁₉×C₇⋊C₉

C₂×D₇

yes

characteristic, complemented

C₇²×C₁₉

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₇×C₇⋊C₉

D₁₉

yes

characteristic, complemented

(C₇⋊C₉)⋊D₇

—

C₂×C₃×C₇×C₁₉

—

C₂×C₃×C₇×C₁₉

AGL(1,𝔽₇)

yes

characteristic

C₂×C₃×C₇×C₁₉

C₃×D₇

yes

characteristic

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₂×C₉×D₁₉

—

Hall

C₂×C₃×C₇⋊D₇

—

C₂×D₁₃₃

—

C₂×D₁₃₃

—

C₂×D₁₃₃

—

C₇×C₇⋊C₉

C₂×D₁₉

yes

characteristic, complemented, Hall

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

C₂×AGL(1,𝔽₇)

yes

characteristic

C₃×C₇×C₁₉

C₂×C₃×D₇

yes

characteristic

C₂×C₉×C₁₉

—

C₉×D₁₉

—

C₂×C₃×C₇²

C₃×D₁₉

yes

characteristic

C₃×C₇⋊D₇

—

C₂×C₇×C₁₉

—

C₂×C₇×C₁₉

D₇⋊C₉

yes

characteristic, complemented

C₂×C₇×C₁₉

C₉×D₇

yes

characteristic, complemented

D₁₃₃

—

D₁₃₃

—

D₁₃₃

—

C₂×C₉×D₇

—

133

—

C₂×D₇⋊C₉

—

133

—

C₂×C₃×D₁₉

—

C₂×C₇⋊D₇

—

Hall

C₉×C₁₉

—

Hall

C₃×C₇²

C₂×C₃×D₁₉

yes

characteristic

C₇×C₁₉

—

C₇×C₁₉

C₂×D₇⋊C₉

yes

characteristic, complemented

C₇×C₁₉

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₇×C₉

—

C₉×D₇

—

133

—

C₂×C₇⋊C₉

D₁₃₃

yes

characteristic, complemented

D₇⋊C₉

—

133

—

C₂×C₃×C₁₉

ΓL(1,𝔽₈)⋊D₇

yes

characteristic

C₃×D₁₉

—

C₂×C₇²

C₉×D₁₉

yes

characteristic, complemented

C₇⋊D₇

—

C₂×C₃×D₇

—

133

—

C₂×C₃×D₇

—

133

—

C₂×C₃×D₇

—

399

—

C₂×D₁₉

—

Hall

C₇×C₉

—

C₇⋊C₉

C₂×D₁₃₃

yes

characteristic, complemented

C₃×C₁₉

C₂×ΓL(1,𝔽₈)⋊D₇

yes

characteristic

C₇²

C₂×C₉×D₁₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

ΓL(1,𝔽₈)⋊D₁₉

yes

characteristic

C₂×C₃×C₇

C₃×D₁₃₃

yes

characteristic

C₃×D₇

—

133

—

C₃×D₇

—

133

—

C₃×D₇

—

399

—

C₂×C₁₉

(C₇⋊C₉)⋊D₇

yes

characteristic, complemented

D₁₉

—

C₂²×C₉

—

931

Hall

C₂×D₇

—

133

—

C₂×D₇

—

133

—

C₂×D₇

—

399

—

C₃×C₇

—

C₃×C₇

C₂×ΓL(1,𝔽₈)⋊D₁₉

yes

characteristic

C₃×C₇

C₂×C₃×D₁₃₃

yes

characteristic

C₁₉

C₂×(C₇⋊C₉)⋊D₇

yes

characteristic, complemented, Sylow

C₂×C₉

—

C₂×C₉

—

931

—

C₂×C₇

—

C₂×C₇

(C₇⋊C₉)⋊D₁₉

yes

characteristic, complemented

C₂×C₇

C₉×D₁₃₃

yes

characteristic, complemented

D₇

—

133

—

D₇

—

133

—

D₇

—

399

—

100

C₂²×C₃

—

931

—

101

C₉

—

Sylow

102

C₇

—

103

C₇

C₂×(C₇⋊C₉)⋊D₁₉

yes

characteristic, complemented

104

C₇

C₂×C₉×D₁₃₃

yes

characteristic, complemented

105

C₂×C₃

—

931

—

106

C₂×C₃

ΓL(1,𝔽₈)⋊D₁₃₃

yes

characteristic

107

C₂²

—

931

Sylow

108

C₃

C₂×ΓL(1,𝔽₈)⋊D₁₃₃

yes

characteristic

109

C₂

—

931

—

110

C₂

(C₇⋊C₉)⋊D₁₃₃

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)⋊D₁₃₃
commutator, hypocenter	C₇²×C₁₉	C₂²×C₉
Frattini	C₃	C₂×ΓL(1,𝔽₈)⋊D₁₃₃
Fitting, socle	C₂×C₃×C₇²×C₁₉	C₂×C₃
prime radical	C₂×C₃×C₇⋊D₁₃₃	C₃
cosocle	C₃×C₇²×C₁₉	C₂²×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₂×C₃

subgroup

G/H

center, hypercenter

C₂×C₃

ΓL(1,𝔽₈)⋊D₁₃₃

commutator, hypocenter

C₇²×C₁₉

C₂²×C₉

Frattini

C₃

C₂×ΓL(1,𝔽₈)⋊D₁₃₃

Fitting, socle

C₂×C₃×C₇²×C₁₉

C₂×C₃

prime radical

C₂×C₃×C₇⋊D₁₃₃

C₃

cosocle

C₃×C₇²×C₁₉

C₂²×C₃

LCS

G ▷ C₇²×C₁₉

UCS

C₁ ◁ C₂×C₃

N	H ≅ G/N	type
1. C₂×C₇×C₁₉×C₇⋊C₉	C₂	semidirect
2. (C₇⋊C₉)⋊D₁₃₃	C₂	direct (N × H)
4. C₇×C₁₉×C₇⋊C₉	C₂²	semidirect
9. C₂×C₇⋊D₁₃₃	C₉	semidirect
12. C₂×C₁₉×C₇⋊C₉	D₇	semidirect
15. C₂×C₇²×C₁₉	C₂×C₉	semidirect
16. C₇⋊D₁₃₃	C₂×C₉	semidirect
22. C₁₉×C₇⋊C₉	C₂×D₇	semidirect
23. C₇²×C₁₉	C₂²×C₉	semidirect
24. C₂×C₇×C₇⋊C₉	D₁₉	semidirect
37. C₇×C₇⋊C₉	C₂×D₁₉	semidirect
46. C₂×C₇×C₁₉	D₇⋊C₉	semidirect
47. C₂×C₇×C₁₉	C₉×D₇	semidirect
58. C₇×C₁₉	C₂×D₇⋊C₉	semidirect
59. C₇×C₁₉	C₂×C₉×D₇	semidirect
62. C₂×C₇⋊C₉	D₁₃₃	semidirect
66. C₂×C₇²	C₉×D₁₉	semidirect
73. C₇⋊C₉	C₂×D₁₃₃	semidirect
75. C₇²	C₂×C₉×D₁₉	semidirect
82. C₂×C₁₉	(C₇⋊C₉)⋊D₇	semidirect
91. C₁₉	C₂×(C₇⋊C₉)⋊D₇	semidirect
95. C₂×C₇	(C₇⋊C₉)⋊D₁₉	semidirect
96. C₂×C₇	C₉×D₁₃₃	semidirect
103. C₇	C₂×(C₇⋊C₉)⋊D₁₉	semidirect
104. C₇	C₂×C₉×D₁₃₃	semidirect
110. C₂	(C₇⋊C₉)⋊D₁₃₃	direct (N × H)

H ≅ G/N

type

1. C₂×C₇×C₁₉×C₇⋊C₉

C₂

semidirect

2. (C₇⋊C₉)⋊D₁₃₃

C₂

direct (N × H)

4. C₇×C₁₉×C₇⋊C₉

C₂²

semidirect

9. C₂×C₇⋊D₁₃₃

C₉

semidirect

12. C₂×C₁₉×C₇⋊C₉

D₇

semidirect

15. C₂×C₇²×C₁₉

C₂×C₉

semidirect

16. C₇⋊D₁₃₃

C₂×C₉

semidirect

22. C₁₉×C₇⋊C₉

C₂×D₇

semidirect

23. C₇²×C₁₉

C₂²×C₉

semidirect

24. C₂×C₇×C₇⋊C₉

D₁₉

semidirect

37. C₇×C₇⋊C₉

C₂×D₁₉

semidirect

46. C₂×C₇×C₁₉

D₇⋊C₉

semidirect

47. C₂×C₇×C₁₉

C₉×D₇

semidirect

58. C₇×C₁₉

C₂×D₇⋊C₉

semidirect

59. C₇×C₁₉

C₂×C₉×D₇

semidirect

62. C₂×C₇⋊C₉

D₁₃₃

semidirect

66. C₂×C₇²

C₉×D₁₉

semidirect

73. C₇⋊C₉

C₂×D₁₃₃

semidirect

75. C₇²

C₂×C₉×D₁₉

semidirect

82. C₂×C₁₉

(C₇⋊C₉)⋊D₇

semidirect

91. C₁₉

C₂×(C₇⋊C₉)⋊D₇

semidirect

95. C₂×C₇

(C₇⋊C₉)⋊D₁₉

semidirect

96. C₂×C₇

C₉×D₁₃₃

semidirect

103. C₇

C₂×(C₇⋊C₉)⋊D₁₉

semidirect

104. C₇

C₂×C₉×D₁₃₃

semidirect

110. C₂

(C₇⋊C₉)⋊D₁₃₃

direct (N × H)

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

subgroup	index
C₂×C₇×C₁₉×C₇⋊C₉	2
(C₇⋊C₉)⋊D₁₃₃	2
C₂×C₃×C₇⋊D₁₃₃	3
C₂×C₉×D₁₃₃	7
C₂×(C₇⋊C₉)⋊D₁₉	7
C₂×(C₇⋊C₉)⋊D₇	19

subgroup

index

C₂×C₇×C₁₉×C₇⋊C₉

(C₇⋊C₉)⋊D₁₃₃

C₂×C₃×C₇⋊D₁₃₃

C₂×C₉×D₁₃₃

C₂×(C₇⋊C₉)⋊D₁₉

C₂×(C₇⋊C₉)⋊D₇

Subgroups — C₂×(C₇⋊C₉)⋊D₁₃₃ #45691645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×(C7⋊C9)⋊D133 #45691645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₇⋊C₉)⋊D₁₃₃ #45691645