Group #45692700 — subgroups

Subgroups — C₃×D₃×C₇⋊D₁₃₃ #45692700

Order 33516.

440 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₃²×C₇⋊D₁₃₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₇²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊(C₇⋊D₁₃₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₇⋊D₁₃₃	—	3	1	no	maximal
5	D₃×C₇⋊D₁₃₃	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃²×C₇²×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₃⋊(C₇⋊D₁₃₃)	C₂×C₃	1	1	yes	characteristic, complemented
8	D₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₇⋊D₁₃₃	—	2	1	no	—
10	C₃×C₇⋊D₁₃₃	—	3	1	no	—
11	C₃×C₇⋊D₁₃₃	D₃	1	1	yes	characteristic, direct product factor
12	C₃×C₇⋊D₁₃₃	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇²×C₁₉	—	3	1	no	—
14	C₃×D₃×D₁₃₃	—	7	8	no	maximal
15	C₂×C₇⋊D₁₃₃	—	3	1	no	Hall
16	C₃×C₇²×C₁₉	—	2	1	no	—
17	C₃×C₇²×C₁₉	C₂×D₃	1	1	yes	characteristic, complemented
18	C₃×C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented
19	C₃×D₃₉₉	—	7	8	no	—
20	C₃×D₃×C₇×C₁₉	D₇	1	8	yes	complemented
21	C₃²×D₁₃₃	—	7	8	no	—
22	C₂×C₇²×C₁₉	—	3	1	no	—
23	C₇⋊D₁₃₃	—	3	1	no	—
24	C₇⋊D₁₃₃	C₃×D₃	1	1	yes	characteristic, direct product factor
25	C₃×D₃×C₇⋊D₇	—	19	1	no	maximal, Hall
26	C₂×C₃×D₁₃₃	—	21	8	no	—
27	D₃×D₁₃₃	—	7	8	no	—
28	C₃²×C₇×C₁₉	C₂×D₇	1	8	yes	complemented
29	C₇²×C₁₉	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
30	C₃×C₃⋊(C₇⋊D₇)	—	19	1	no	—
31	C₃×D₃×C₇²	D₁₉	1	1	yes	characteristic, complemented
32	C₃²×C₇⋊D₇	—	19	1	no	—
33	C₂×C₃×C₇×C₁₉	—	3	8	no	—
34	D₃₉₉	—	7	8	no	—
35	D₃×C₇×C₁₉	C₃×D₇	1	8	yes	complemented
36	C₃×D₁₃₃	—	7	8	no	—
37	C₃×D₁₃₃	—	7	8	no	—
38	C₃×D₁₃₃	—	14	8	no	—
39	C₃×D₁₃₃	—	21	8	no	—
40	C₃×D₃×D₁₉	—	49	1	no	Hall
41	C₂×C₃×C₇⋊D₇	—	57	1	no	—
42	D₃×C₇⋊D₇	—	19	1	no	—
43	C₂×D₁₃₃	—	21	8	no	—
44	C₃²×C₇²	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇×C₁₉	—	2	8	no	—
46	C₃×C₇×C₁₉	D₃×D₇	1	8	yes	complemented
47	C₃×C₇×C₁₉	C₂×C₃×D₇	1	8	yes	complemented
48	C₃×D₅₇	—	49	1	no	—
49	C₃×D₃×C₁₉	C₇⋊D₇	1	1	yes	characteristic, complemented
50	C₃²×D₁₉	—	49	1	no	—
51	C₂×C₃×C₇²	—	3	1	no	—
52	C₃⋊(C₇⋊D₇)	—	19	1	no	—
53	D₃×C₇²	C₃×D₁₉	1	1	yes	characteristic, complemented
54	C₃×C₇⋊D₇	—	19	1	no	—
55	C₃×C₇⋊D₇	—	19	1	no	—
56	C₃×C₇⋊D₇	—	38	1	no	—
57	C₃×C₇⋊D₇	—	57	1	no	—
58	C₂×C₇×C₁₉	—	3	8	no	—
59	D₁₃₃	—	7	8	no	—
60	D₁₃₃	—	21	8	no	—
61	C₃×D₃×D₇	—	133	8	no	—
62	C₂×C₃×D₁₉	—	147	1	no	—
63	D₃×D₁₉	—	49	1	no	—
64	C₂×C₇⋊D₇	—	57	1	no	Hall
65	C₃²×C₁₉	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇²	—	2	1	no	—
67	C₃×C₇²	D₃×D₁₉	1	1	yes	characteristic, complemented
68	C₃×C₇²	C₂×C₃×D₁₉	1	1	yes	characteristic, complemented
69	C₇×C₁₉	C₃×D₃×D₇	1	8	yes	complemented
70	C₃×D₂₁	—	133	8	no	—
71	C₃×D₃×C₇	D₁₃₃	1	8	yes	complemented
72	C₃²×D₇	—	133	8	no	—
73	C₂×C₃×C₁₉	—	3	1	no	—
74	D₅₇	—	49	1	no	—
75	C₃×D₁₉	—	49	1	no	—
76	C₃×D₁₉	—	49	1	no	—
77	C₃×D₁₉	—	98	1	no	—
78	C₃×D₁₉	—	147	1	no	—
79	D₃×C₁₉	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
80	C₂×C₇²	—	3	1	no	—
81	C₇⋊D₇	—	19	1	no	—
82	C₇⋊D₇	—	57	1	no	—
83	C₂×C₃×D₇	—	399	8	no	—
84	D₃×D₇	—	133	8	no	—
85	C₂×D₁₉	—	147	1	no	Hall
86	C₃²×C₇	C₂×D₁₃₃	1	8	yes	complemented
87	C₃×C₁₉	—	2	1	no	—
88	C₃×C₁₉	D₃×C₇⋊D₇	1	1	yes	characteristic, complemented
89	C₃×C₁₉	C₂×C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
90	C₇²	C₃×D₃×D₁₉	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₇	—	3	8	no	—
92	D₂₁	—	133	8	no	—
93	C₃×D₇	—	266	8	no	—
94	C₃×D₇	—	133	8	no	—
95	C₃×D₇	—	133	8	no	—
96	C₃×D₇	—	399	8	no	—
97	D₃×C₇	C₃×D₁₃₃	1	8	yes	complemented
98	C₂×C₁₉	—	3	1	no	—
99	D₁₉	—	49	1	no	—
100	D₁₉	—	147	1	no	—
101	C₂×C₃×D₃	—	931	1	no	Hall
102	C₂×D₇	—	399	8	no	—
103	C₃×C₇	—	2	8	no	—
104	C₃×C₇	D₃×D₁₃₃	1	8	yes	complemented
105	C₃×C₇	C₂×C₃×D₁₃₃	1	8	yes	complemented
106	C₁₉	C₃×D₃×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₃²	—	931	1	no	—
108	C₃×D₃	—	931	1	no	—
109	C₃×D₃	C₇⋊D₁₃₃	1	1	yes	characteristic, direct product factor
110	C₂×C₇	—	3	8	no	—
111	D₇	—	133	8	no	—
112	D₇	—	399	8	no	—
113	C₂²×C₃	—	2793	1	no	—
114	C₂×D₃	—	931	1	no	—
115	C₃²	C₂×C₇⋊D₁₃₃	1	1	yes	characteristic, complemented, Sylow
116	C₇	C₃×D₃×D₁₃₃	1	8	yes	complemented
117	C₂×C₃	—	3	1	no	—
118	C₂×C₃	—	1862	1	no	—
119	C₂×C₃	—	931	1	no	—
120	C₂×C₃	—	931	1	no	—
121	C₂×C₃	—	2793	1	no	—
122	D₃	—	931	1	no	—
123	D₃	C₃×C₇⋊D₁₃₃	1	1	yes	characteristic, direct product factor
124	C₂²	—	2793	1	no	Sylow
125	C₃	—	2	1	no	—
126	C₃	D₃×C₇⋊D₁₃₃	1	1	yes	characteristic, direct product factor
127	C₃	C₂×C₃×C₇⋊D₁₃₃	1	1	yes	characteristic, complemented
128	C₂	—	3	1	no	—
129	C₂	—	931	1	no	—
130	C₂	—	2793	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₇⋊D₁₃₃

C₂

yes

maximal, characteristic, complemented

C₃×D₃×C₇²×C₁₉

C₂

yes

maximal, characteristic, complemented

C₃×C₃⋊(C₇⋊D₁₃₃)

C₂

yes

maximal, characteristic, complemented

C₂×C₃×C₇⋊D₁₃₃

—

maximal

D₃×C₇⋊D₁₃₃

C₃

yes

maximal, characteristic, direct product factor

C₃²×C₇²×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₃⋊(C₇⋊D₁₃₃)

C₂×C₃

yes

characteristic, complemented

D₃×C₇²×C₁₉

C₂×C₃

yes

characteristic, complemented

C₃×C₇⋊D₁₃₃

—

C₃×C₇⋊D₁₃₃

—

C₃×C₇⋊D₁₃₃

D₃

yes

characteristic, direct product factor

C₃×C₇⋊D₁₃₃

C₂×C₃

yes

characteristic, complemented

C₂×C₃×C₇²×C₁₉

—

C₃×D₃×D₁₃₃

—

maximal

C₂×C₇⋊D₁₃₃

—

Hall

C₃×C₇²×C₁₉

—

C₃×C₇²×C₁₉

C₂×D₃

yes

characteristic, complemented

C₃×C₇²×C₁₉

C₂²×C₃

yes

characteristic, complemented

C₃×D₃₉₉

—

C₃×D₃×C₇×C₁₉

D₇

yes

complemented

C₃²×D₁₃₃

—

C₂×C₇²×C₁₉

—

C₇⋊D₁₃₃

—

C₇⋊D₁₃₃

C₃×D₃

yes

characteristic, direct product factor

C₃×D₃×C₇⋊D₇

—

maximal, Hall

C₂×C₃×D₁₃₃

—

D₃×D₁₃₃

—

C₃²×C₇×C₁₉

C₂×D₇

yes

complemented

C₇²×C₁₉

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₃×C₃⋊(C₇⋊D₇)

—

C₃×D₃×C₇²

D₁₉

yes

characteristic, complemented

C₃²×C₇⋊D₇

—

C₂×C₃×C₇×C₁₉

—

D₃₉₉

—

D₃×C₇×C₁₉

C₃×D₇

yes

complemented

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₃×D₁₉

—

Hall

C₂×C₃×C₇⋊D₇

—

D₃×C₇⋊D₇

—

C₂×D₁₃₃

—

C₃²×C₇²

C₂×D₁₉

yes

characteristic, complemented, Hall

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

D₃×D₇

yes

complemented

C₃×C₇×C₁₉

C₂×C₃×D₇

yes

complemented

C₃×D₅₇

—

C₃×D₃×C₁₉

C₇⋊D₇

yes

characteristic, complemented

C₃²×D₁₉

—

C₂×C₃×C₇²

—

C₃⋊(C₇⋊D₇)

—

D₃×C₇²

C₃×D₁₉

yes

characteristic, complemented

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

C₂×C₇×C₁₉

—

D₁₃₃

—

D₁₃₃

—

C₃×D₃×D₇

—

133

—

C₂×C₃×D₁₉

—

147

—

D₃×D₁₉

—

C₂×C₇⋊D₇

—

Hall

C₃²×C₁₉

C₂×C₇⋊D₇

yes

characteristic, complemented, Hall

C₃×C₇²

—

C₃×C₇²

D₃×D₁₉

yes

characteristic, complemented

C₃×C₇²

C₂×C₃×D₁₉

yes

characteristic, complemented

C₇×C₁₉

C₃×D₃×D₇

yes

complemented

C₃×D₂₁

—

133

—

C₃×D₃×C₇

D₁₃₃

yes

complemented

C₃²×D₇

—

133

—

C₂×C₃×C₁₉

—

D₅₇

—

C₃×D₁₉

—

C₃×D₁₉

—

C₃×D₁₉

—

C₃×D₁₉

—

147

—

D₃×C₁₉

C₃×C₇⋊D₇

yes

characteristic, complemented

C₂×C₇²

—

C₇⋊D₇

—

C₇⋊D₇

—

C₂×C₃×D₇

—

399

—

D₃×D₇

—

133

—

C₂×D₁₉

—

147

Hall

C₃²×C₇

C₂×D₁₃₃

yes

complemented

C₃×C₁₉

—

C₃×C₁₉

D₃×C₇⋊D₇

yes

characteristic, complemented

C₃×C₁₉

C₂×C₃×C₇⋊D₇

yes

characteristic, complemented

C₇²

C₃×D₃×D₁₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

D₂₁

—

133

—

C₃×D₇

—

266

—

C₃×D₇

—

133

—

C₃×D₇

—

133

—

C₃×D₇

—

399

—

D₃×C₇

C₃×D₁₃₃

yes

complemented

C₂×C₁₉

—

D₁₉

—

100

D₁₉

—

147

—

101

C₂×C₃×D₃

—

931

Hall

102

C₂×D₇

—

399

—

103

C₃×C₇

—

104

C₃×C₇

D₃×D₁₃₃

yes

complemented

105

C₃×C₇

C₂×C₃×D₁₃₃

yes

complemented

106

C₁₉

C₃×D₃×C₇⋊D₇

yes

characteristic, complemented, Sylow

107

C₂×C₃²

—

931

—

108

C₃×D₃

—

931

—

109

C₃×D₃

C₇⋊D₁₃₃

yes

characteristic, direct product factor

110

C₂×C₇

—

111

D₇

—

133

—

112

D₇

—

399

—

113

C₂²×C₃

—

2793

—

114

C₂×D₃

—

931

—

115

C₃²

C₂×C₇⋊D₁₃₃

yes

characteristic, complemented, Sylow

116

C₇

C₃×D₃×D₁₃₃

yes

complemented

117

C₂×C₃

—

118

C₂×C₃

—

1862

—

119

C₂×C₃

—

931

—

120

C₂×C₃

—

931

—

121

C₂×C₃

—

2793

—

122

D₃

—

931

—

123

D₃

C₃×C₇⋊D₁₃₃

yes

characteristic, direct product factor

124

C₂²

—

2793

Sylow

125

C₃

—

126

C₃

D₃×C₇⋊D₁₃₃

yes

characteristic, direct product factor

127

C₃

C₂×C₃×C₇⋊D₁₃₃

yes

characteristic, complemented

128

C₂

—

129

C₂

—

931

—

130

C₂

—

2793

—

	subgroup	G/H
center, hypercenter	C₃	D₃×C₇⋊D₁₃₃
commutator, cosocle, hypocenter	C₃×C₇²×C₁₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₉	C₂²
LCS	G ▷ C₃×C₇²×C₁₉
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

D₃×C₇⋊D₁₃₃

commutator, cosocle, hypocenter

C₃×C₇²×C₁₉

C₂²×C₃

Frattini

C₁

Fitting, socle

C₃²×C₇²×C₁₉

C₂²

LCS

G ▷ C₃×C₇²×C₁₉

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. C₃²×C₇⋊D₁₃₃	C₂	semidirect
2. C₃×D₃×C₇²×C₁₉	C₂	semidirect
3. C₃×C₃⋊(C₇⋊D₁₃₃)	C₂	semidirect
5. D₃×C₇⋊D₁₃₃	C₃	direct (N × H)
6. C₃²×C₇²×C₁₉	C₂²	semidirect
7. C₃⋊(C₇⋊D₁₃₃)	C₂×C₃	semidirect
8. D₃×C₇²×C₁₉	C₂×C₃	semidirect
11. C₃×C₇⋊D₁₃₃	D₃	direct (N × H)
12. C₃×C₇⋊D₁₃₃	C₂×C₃	semidirect
17. C₃×C₇²×C₁₉	C₂×D₃	semidirect
18. C₃×C₇²×C₁₉	C₂²×C₃	semidirect
20. C₃×D₃×C₇×C₁₉	D₇	semidirect
24. C₇⋊D₁₃₃	C₃×D₃	direct (N × H)
28. C₃²×C₇×C₁₉	C₂×D₇	semidirect
29. C₇²×C₁₉	C₂×C₃×D₃	semidirect
31. C₃×D₃×C₇²	D₁₉	semidirect
35. D₃×C₇×C₁₉	C₃×D₇	semidirect
44. C₃²×C₇²	C₂×D₁₉	semidirect
46. C₃×C₇×C₁₉	D₃×D₇	semidirect
47. C₃×C₇×C₁₉	C₂×C₃×D₇	semidirect
49. C₃×D₃×C₁₉	C₇⋊D₇	semidirect
53. D₃×C₇²	C₃×D₁₉	semidirect
65. C₃²×C₁₉	C₂×C₇⋊D₇	semidirect
67. C₃×C₇²	D₃×D₁₉	semidirect
68. C₃×C₇²	C₂×C₃×D₁₉	semidirect
69. C₇×C₁₉	C₃×D₃×D₇	semidirect
71. C₃×D₃×C₇	D₁₃₃	semidirect
79. D₃×C₁₉	C₃×C₇⋊D₇	semidirect
86. C₃²×C₇	C₂×D₁₃₃	semidirect
88. C₃×C₁₉	D₃×C₇⋊D₇	semidirect
89. C₃×C₁₉	C₂×C₃×C₇⋊D₇	semidirect
90. C₇²	C₃×D₃×D₁₉	semidirect
97. D₃×C₇	C₃×D₁₃₃	semidirect
104. C₃×C₇	D₃×D₁₃₃	semidirect
105. C₃×C₇	C₂×C₃×D₁₃₃	semidirect
106. C₁₉	C₃×D₃×C₇⋊D₇	semidirect
109. C₃×D₃	C₇⋊D₁₃₃	direct (N × H)
115. C₃²	C₂×C₇⋊D₁₃₃	semidirect
116. C₇	C₃×D₃×D₁₃₃	semidirect
123. D₃	C₃×C₇⋊D₁₃₃	direct (N × H)
126. C₃	D₃×C₇⋊D₁₃₃	direct (N × H)
127. C₃	C₂×C₃×C₇⋊D₁₃₃	semidirect

H ≅ G/N

type

1. C₃²×C₇⋊D₁₃₃

C₂

semidirect

2. C₃×D₃×C₇²×C₁₉

C₂

semidirect

3. C₃×C₃⋊(C₇⋊D₁₃₃)

C₂

semidirect

5. D₃×C₇⋊D₁₃₃

C₃

direct (N × H)

6. C₃²×C₇²×C₁₉

C₂²

semidirect

7. C₃⋊(C₇⋊D₁₃₃)

C₂×C₃

semidirect

8. D₃×C₇²×C₁₉

C₂×C₃

semidirect

11. C₃×C₇⋊D₁₃₃

D₃

direct (N × H)

12. C₃×C₇⋊D₁₃₃

C₂×C₃

semidirect

17. C₃×C₇²×C₁₉

C₂×D₃

semidirect

18. C₃×C₇²×C₁₉

C₂²×C₃

semidirect

20. C₃×D₃×C₇×C₁₉

D₇

semidirect

24. C₇⋊D₁₃₃

C₃×D₃

direct (N × H)

28. C₃²×C₇×C₁₉

C₂×D₇

semidirect

29. C₇²×C₁₉

C₂×C₃×D₃

semidirect

31. C₃×D₃×C₇²

D₁₉

semidirect

35. D₃×C₇×C₁₉

C₃×D₇

semidirect

44. C₃²×C₇²

C₂×D₁₉

semidirect

46. C₃×C₇×C₁₉

D₃×D₇

semidirect

47. C₃×C₇×C₁₉

C₂×C₃×D₇

semidirect

49. C₃×D₃×C₁₉

C₇⋊D₇

semidirect

53. D₃×C₇²

C₃×D₁₉

semidirect

65. C₃²×C₁₉

C₂×C₇⋊D₇

semidirect

67. C₃×C₇²

D₃×D₁₉

semidirect

68. C₃×C₇²

C₂×C₃×D₁₉

semidirect

69. C₇×C₁₉

C₃×D₃×D₇

semidirect

71. C₃×D₃×C₇

D₁₃₃

semidirect

79. D₃×C₁₉

C₃×C₇⋊D₇

semidirect

86. C₃²×C₇

C₂×D₁₃₃

semidirect

88. C₃×C₁₉

D₃×C₇⋊D₇

semidirect

89. C₃×C₁₉

C₂×C₃×C₇⋊D₇

semidirect

90. C₇²

C₃×D₃×D₁₉

semidirect

97. D₃×C₇

C₃×D₁₃₃

semidirect

104. C₃×C₇

D₃×D₁₃₃

semidirect

105. C₃×C₇

C₂×C₃×D₁₃₃

semidirect

106. C₁₉

C₃×D₃×C₇⋊D₇

semidirect

109. C₃×D₃

C₇⋊D₁₃₃

direct (N × H)

115. C₃²

C₂×C₇⋊D₁₃₃

semidirect

116. C₇

C₃×D₃×D₁₃₃

semidirect

123. D₃

C₃×C₇⋊D₁₃₃

direct (N × H)

126. C₃

D₃×C₇⋊D₁₃₃

direct (N × H)

127. C₃

C₂×C₃×C₇⋊D₁₃₃

semidirect

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

subgroup	index
C₃²×C₇⋊D₁₃₃	2
C₃×D₃×C₇²×C₁₉	2
C₃×C₃⋊(C₇⋊D₁₃₃)	2
C₂×C₃×C₇⋊D₁₃₃	3
D₃×C₇⋊D₁₃₃	3
C₃×D₃×D₁₃₃	7
C₃×D₃×C₇⋊D₇	19

subgroup

index

C₃²×C₇⋊D₁₃₃

C₃×D₃×C₇²×C₁₉

C₃×C₃⋊(C₇⋊D₁₃₃)

C₂×C₃×C₇⋊D₁₃₃

D₃×C₇⋊D₁₃₃

C₃×D₃×D₁₃₃

C₃×D₃×C₇⋊D₇

Subgroups — C₃×D₃×C₇⋊D₁₃₃ #45692700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×D3×C7⋊D133 #45692700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×C₇⋊D₁₃₃ #45692700