Group #45692484 — subgroups

Subgroups — (C₃⋊(D₁₉⋊C₃))⋊(C₇⋊D₇) #45692484

Order 33516.

440 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

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

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉⋊C₃×C₃⋊(C₇⋊D₇)

C₂

yes

maximal, characteristic, complemented

C₇²×C₃⋊(D₁₉⋊C₃)

C₂

yes

maximal, characteristic, complemented

C₃×(C₁₉⋊C₃)⋊(C₇⋊D₇)

C₂

yes

maximal, characteristic, complemented

D₅₇⋊(C₇⋊D₇)

C₃

yes

maximal, characteristic, complemented

C₇⋊D₇×D₁₉⋊C₃

—

maximal

C₃×C₇²×C₁₉⋊C₃

C₂²

yes

characteristic, complemented, Hall

C₁₉×C₃⋊(C₇⋊D₇)

C₂×C₃

yes

characteristic, complemented

C₇²×D₅₇

C₂×C₃

yes

characteristic, complemented

C₃×C₇⋊D₁₃₃

C₂×C₃

yes

characteristic, complemented

C₁₉⋊C₃×C₇⋊D₇

—

(C₁₉⋊C₃)⋊(C₇⋊D₇)

—

(C₁₉⋊C₃)⋊(C₇⋊D₇)

D₃

yes

characteristic, complemented

C₇²×D₁₉⋊C₃

—

(C₃⋊(D₁₉⋊C₃))⋊D₇

—

maximal

D₁₉×C₇⋊D₇

—

Hall

C₇²×C₁₉⋊C₃

—

C₇²×C₁₉⋊C₃

C₂×D₃

yes

characteristic, complemented

C₃×C₇²×C₁₉

C₂²×C₃

yes

characteristic, complemented

D₂₁×C₁₉⋊C₃

—

C₇×C₃⋊(D₁₉⋊C₃)

D₇

yes

complemented

C₃×(C₁₉⋊C₃)⋊D₇

—

C₇⋊D₁₃₃

C₃×D₃

yes

characteristic, complemented

C₁₉×C₇⋊D₇

—

C₇²×D₁₉

—

C₃×D₃×C₇⋊D₇

—

maximal, Hall

D₂₁⋊D₁₉

—

D₇×D₁₉⋊C₃

—

C₃×C₇×C₁₉⋊C₃

C₂×D₇

yes

complemented

C₇²×C₁₉

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₃×C₃⋊(C₇⋊D₇)

—

C₃×D₃×C₇²

—

C₃²×C₇⋊D₇

—

C₁₉×D₂₁

—

C₇×D₅₇

C₃×D₇

yes

complemented

C₃×D₁₃₃

—

(C₁₉⋊C₃)⋊D₇

—

(C₁₉⋊C₃)⋊D₇

—

D₇×C₁₉⋊C₃

—

C₇×D₁₉⋊C₃

—

D₃×D₁₉⋊C₃

—

Hall

C₂×C₃×C₇⋊D₇

—

D₃×C₇⋊D₇

—

D₇×D₁₉

—

C₃²×C₇²

—

Hall

C₃×C₇×C₁₉

C₂×C₃×D₇

yes

complemented

C₇×C₁₉⋊C₃

—

C₇×C₁₉⋊C₃

D₃×D₇

yes

complemented

C₃⋊(D₁₉⋊C₃)

C₇⋊D₇

yes

characteristic, complemented

D₃×C₁₉⋊C₃

—

C₃×D₁₉⋊C₃

—

C₂×C₃×C₇²

—

C₃⋊(C₇⋊D₇)

D₁₉⋊C₃

yes

characteristic, complemented

D₃×C₇²

—

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

D₁₃₃

—

C₇×D₁₉

—

D₇×C₁₉

—

C₃×D₃×D₇

—

133

—

D₃×D₁₉

—

C₂×D₁₉⋊C₃

—

147

—

C₂×C₇⋊D₇

—

Hall

C₃×C₁₉⋊C₃

C₂×C₇⋊D₇

yes

characteristic, complemented, Hall

C₃×C₇²

—

C₃×C₇²

—

C₃×C₇²

C₂×D₁₉⋊C₃

yes

characteristic, complemented

C₇×C₁₉

C₃×D₃×D₇

yes

complemented

C₃×D₂₁

—

133

—

C₃×D₃×C₇

—

C₃²×D₇

—

133

—

D₅₇

C₃×C₇⋊D₇

yes

characteristic, complemented

C₃×D₁₉

—

D₃×C₁₉

—

C₂×C₁₉⋊C₃

—

147

—

D₁₉⋊C₃

—

D₁₉⋊C₃

—

D₁₉⋊C₃

—

C₂×C₇²

—

C₇⋊D₇

—

C₇⋊D₇

—

C₂×C₃×D₇

—

399

—

D₃×D₇

—

133

—

C₂×D₁₉

—

147

Hall

C₃²×C₇

—

C₃×C₁₉

C₂×C₃×C₇⋊D₇

yes

characteristic, complemented

C₁₉⋊C₃

—

C₁₉⋊C₃

D₃×C₇⋊D₇

yes

characteristic, complemented

C₇²

D₃×D₁₉⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

D₂₁

—

C₃×D₇

—

266

—

C₃×D₇

—

133

—

C₃×D₇

—

133

—

C₃×D₇

—

399

—

D₃×C₇

—

C₂×C₁₉

—

147

—

D₁₉

—

100

D₁₉

—

101

C₂×C₃×D₃

—

931

Hall

102

C₂×D₇

—

399

—

103

C₃×C₇

—

104

C₃×C₇

—

105

C₃×C₇

D₇×D₁₉⋊C₃

yes

complemented

106

C₁₉

C₃×D₃×C₇⋊D₇

yes

characteristic, complemented, Sylow

107

C₂×C₃²

—

931

—

108

C₃×D₃

—

109

C₃×D₃

—

931

—

110

C₂×C₇

—

111

D₇

—

112

D₇

—

133

—

113

C₂²×C₃

—

2793

—

114

C₂×D₃

—

931

—

115

C₃²

—

Sylow

116

C₇

(C₃⋊(D₁₉⋊C₃))⋊D₇

yes

complemented

117

C₂×C₃

—

118

C₂×C₃

—

1862

—

119

C₂×C₃

—

931

—

120

C₂×C₃

—

931

—

121

C₂×C₃

—

2793

—

122

D₃

—

123

D₃

—

124

C₂²

—

2793

Sylow

125

C₃

—

126

C₃

—

127

C₃

C₇⋊D₇×D₁₉⋊C₃

yes

characteristic, complemented

128

C₂

—

129

C₂

—

147

—

130

C₂

—

931

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇²×C₁₉	C₂²×C₃
LCS	G ▷ C₃×C₇²×C₁₉
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃×C₇²×C₁₉

C₂²×C₃

LCS

G ▷ C₃×C₇²×C₁₉

UCS

C₁

N	H ≅ G/N	type
1. C₁₉⋊C₃×C₃⋊(C₇⋊D₇)	C₂	semidirect
2. C₇²×C₃⋊(D₁₉⋊C₃)	C₂	semidirect
3. C₃×(C₁₉⋊C₃)⋊(C₇⋊D₇)	C₂	semidirect
4. D₅₇⋊(C₇⋊D₇)	C₃	semidirect
6. C₃×C₇²×C₁₉⋊C₃	C₂²	semidirect
7. C₁₉×C₃⋊(C₇⋊D₇)	C₂×C₃	semidirect
8. C₇²×D₅₇	C₂×C₃	semidirect
9. C₃×C₇⋊D₁₃₃	C₂×C₃	semidirect
12. (C₁₉⋊C₃)⋊(C₇⋊D₇)	D₃	semidirect
17. C₇²×C₁₉⋊C₃	C₂×D₃	semidirect
18. C₃×C₇²×C₁₉	C₂²×C₃	semidirect
20. C₇×C₃⋊(D₁₉⋊C₃)	D₇	semidirect
22. C₇⋊D₁₃₃	C₃×D₃	semidirect
28. C₃×C₇×C₁₉⋊C₃	C₂×D₇	semidirect
29. C₇²×C₁₉	C₂×C₃×D₃	semidirect
34. C₇×D₅₇	C₃×D₇	semidirect
45. C₃×C₇×C₁₉	C₂×C₃×D₇	semidirect
47. C₇×C₁₉⋊C₃	D₃×D₇	semidirect
48. C₃⋊(D₁₉⋊C₃)	C₇⋊D₇	semidirect
52. C₃⋊(C₇⋊D₇)	D₁₉⋊C₃	semidirect
65. C₃×C₁₉⋊C₃	C₂×C₇⋊D₇	semidirect
68. C₃×C₇²	C₂×D₁₉⋊C₃	semidirect
69. C₇×C₁₉	C₃×D₃×D₇	semidirect
73. D₅₇	C₃×C₇⋊D₇	semidirect
87. C₃×C₁₉	C₂×C₃×C₇⋊D₇	semidirect
89. C₁₉⋊C₃	D₃×C₇⋊D₇	semidirect
90. C₇²	D₃×D₁₉⋊C₃	semidirect
105. C₃×C₇	D₇×D₁₉⋊C₃	semidirect
106. C₁₉	C₃×D₃×C₇⋊D₇	semidirect
116. C₇	(C₃⋊(D₁₉⋊C₃))⋊D₇	semidirect
127. C₃	C₇⋊D₇×D₁₉⋊C₃	semidirect

H ≅ G/N

type

1. C₁₉⋊C₃×C₃⋊(C₇⋊D₇)

C₂

semidirect

2. C₇²×C₃⋊(D₁₉⋊C₃)

C₂

semidirect

3. C₃×(C₁₉⋊C₃)⋊(C₇⋊D₇)

C₂

semidirect

4. D₅₇⋊(C₇⋊D₇)

C₃

semidirect

6. C₃×C₇²×C₁₉⋊C₃

C₂²

semidirect

7. C₁₉×C₃⋊(C₇⋊D₇)

C₂×C₃

semidirect

8. C₇²×D₅₇

C₂×C₃

semidirect

9. C₃×C₇⋊D₁₃₃

C₂×C₃

semidirect

12. (C₁₉⋊C₃)⋊(C₇⋊D₇)

D₃

semidirect

17. C₇²×C₁₉⋊C₃

C₂×D₃

semidirect

18. C₃×C₇²×C₁₉

C₂²×C₃

semidirect

20. C₇×C₃⋊(D₁₉⋊C₃)

D₇

semidirect

22. C₇⋊D₁₃₃

C₃×D₃

semidirect

28. C₃×C₇×C₁₉⋊C₃

C₂×D₇

semidirect

29. C₇²×C₁₉

C₂×C₃×D₃

semidirect

34. C₇×D₅₇

C₃×D₇

semidirect

45. C₃×C₇×C₁₉

C₂×C₃×D₇

semidirect

47. C₇×C₁₉⋊C₃

D₃×D₇

semidirect

48. C₃⋊(D₁₉⋊C₃)

C₇⋊D₇

semidirect

52. C₃⋊(C₇⋊D₇)

D₁₉⋊C₃

semidirect

65. C₃×C₁₉⋊C₃

C₂×C₇⋊D₇

semidirect

68. C₃×C₇²

C₂×D₁₉⋊C₃

semidirect

69. C₇×C₁₉

C₃×D₃×D₇

semidirect

73. D₅₇

C₃×C₇⋊D₇

semidirect

87. C₃×C₁₉

C₂×C₃×C₇⋊D₇

semidirect

89. C₁₉⋊C₃

D₃×C₇⋊D₇

semidirect

90. C₇²

D₃×D₁₉⋊C₃

semidirect

105. C₃×C₇

D₇×D₁₉⋊C₃

semidirect

106. C₁₉

C₃×D₃×C₇⋊D₇

semidirect

116. C₇

(C₃⋊(D₁₉⋊C₃))⋊D₇

semidirect

127. C₃

C₇⋊D₇×D₁₉⋊C₃

semidirect

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

subgroup	index
C₁₉⋊C₃×C₃⋊(C₇⋊D₇)	2
C₇²×C₃⋊(D₁₉⋊C₃)	2
C₃×(C₁₉⋊C₃)⋊(C₇⋊D₇)	2
D₅₇⋊(C₇⋊D₇)	3
C₇⋊D₇×D₁₉⋊C₃	3
(C₃⋊(D₁₉⋊C₃))⋊D₇	7
C₃×D₃×C₇⋊D₇	19

subgroup

index

C₁₉⋊C₃×C₃⋊(C₇⋊D₇)

C₇²×C₃⋊(D₁₉⋊C₃)

C₃×(C₁₉⋊C₃)⋊(C₇⋊D₇)

D₅₇⋊(C₇⋊D₇)

C₇⋊D₇×D₁₉⋊C₃

(C₃⋊(D₁₉⋊C₃))⋊D₇

C₃×D₃×C₇⋊D₇

Subgroups — (C₃⋊(D₁₉⋊C₃))⋊(C₇⋊D₇) #45692484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C3⋊(D19⋊C3))⋊(C7⋊D7) #45692484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃⋊(D₁₉⋊C₃))⋊(C₇⋊D₇) #45692484