Group #45692413 — subgroups

Subgroups — C₁₉⋊C₃×D₂₁⋊D₇ #45692413

Order 33516.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₇×D₂₁×C₁₉⋊C₃	C₂	1	2	yes	maximal, complemented
2	C₃×C₁₉⋊C₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×D₂₁⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
4	D₇²×C₁₉⋊C₃	—	3	1	no	maximal
5	C₃×C₇²×C₁₉⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×C₁₉×D₂₁	C₂×C₃	1	2	yes	complemented
7	C₃×C₁₉×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₉⋊C₃×C₇⋊D₇	—	2	1	no	—
9	C₁₉⋊C₃×C₇⋊D₇	D₃	1	1	yes	characteristic, complemented
10	C₇×D₇×C₁₉⋊C₃	—	3	2	no	—
11	D₃×D₇×C₁₉⋊C₃	—	7	2	no	maximal
12	D₇²×C₁₉	—	3	1	no	Hall
13	C₇²×C₁₉⋊C₃	—	2	1	no	—
14	C₇²×C₁₉⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
15	C₃×C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented
16	D₂₁×C₁₉⋊C₃	D₇	1	2	yes	complemented
17	D₃×C₇×C₁₉⋊C₃	—	7	2	no	—
18	C₃×D₇×C₁₉⋊C₃	—	7	2	no	—
19	C₃×D₇×C₁₉⋊C₃	—	14	3	no	—
20	C₁₉×C₇⋊D₇	C₃×D₃	1	1	yes	characteristic, complemented
21	C₇×D₇×C₁₉	—	3	2	no	—
22	C₃×D₂₁⋊D₇	—	19	1	no	maximal, Hall
23	D₃×D₇×C₁₉	—	7	2	no	—
24	C₂×D₇×C₁₉⋊C₃	—	21	2	no	—
25	C₃×C₇×C₁₉⋊C₃	—	2	3	no	—
26	C₃×C₇×C₁₉⋊C₃	C₂×D₇	1	2	yes	complemented
27	C₇²×C₁₉	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
28	C₃×C₇×D₂₁	—	19	2	no	—
29	C₃²×C₇⋊D₇	—	19	1	no	—
30	C₁₉×D₂₁	C₃×D₇	1	2	yes	complemented
31	D₃×C₇×C₁₉	—	7	2	no	—
32	C₃×D₇×C₁₉	—	7	2	no	—
33	C₃×D₇×C₁₉	—	14	3	no	—
34	C₂×C₇×C₁₉⋊C₃	—	21	2	no	—
35	D₇×C₁₉⋊C₃	—	3	2	no	—
36	D₇×C₁₉⋊C₃	—	7	2	no	—
37	D₇×C₁₉⋊C₃	—	14	3	no	—
38	D₇×C₁₉⋊C₃	—	14	2	no	—
39	D₇×C₁₉⋊C₃	—	14	6	no	—
40	C₂×D₃×C₁₉⋊C₃	—	49	1	no	Hall
41	D₂₁⋊D₇	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
42	C₃×D₇²	—	57	1	no	—
43	C₂×D₇×C₁₉	—	21	2	no	—
44	C₃²×C₇²	—	19	1	no	Hall
45	C₃×C₇×C₁₉	—	2	3	no	—
46	C₃×C₇×C₁₉	C₂×C₃×D₇	1	2	yes	complemented
47	C₇×C₁₉⋊C₃	—	2	6	no	—
48	C₇×C₁₉⋊C₃	—	2	3	no	—
49	C₇×C₁₉⋊C₃	—	2	2	no	—
50	C₇×C₁₉⋊C₃	D₃×D₇	1	2	yes	complemented
51	C₂×C₃×C₁₉⋊C₃	—	49	1	no	—
52	D₃×C₁₉⋊C₃	—	7	2	no	—
53	C₇×D₂₁	C₂×C₁₉⋊C₃	1	2	yes	complemented
54	C₃×C₇⋊D₇	—	19	1	no	—
55	C₃×C₇⋊D₇	—	38	1	no	—
56	C₃×C₇⋊D₇	C₂×C₁₉⋊C₃	1	1	yes	characteristic, complemented
57	C₃×C₇×D₇	—	57	2	no	—
58	C₂×C₇×C₁₉	—	21	2	no	—
59	D₇×C₁₉	—	3	2	no	—
60	D₇×C₁₉	—	7	2	no	—
61	D₇×C₁₉	—	14	3	no	—
62	C₃×D₃×D₇	—	133	2	no	—
63	C₂×D₃×C₁₉	—	49	1	no	—
64	C₂²×C₁₉⋊C₃	—	147	1	no	—
65	D₇²	—	3	1	no	Hall
66	C₃×C₁₉⋊C₃	D₇²	1	1	yes	characteristic, complemented, Hall
67	C₃×C₇²	—	19	1	no	—
68	C₃×C₇²	—	38	1	no	—
69	C₃×C₇²	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, complemented
70	C₇×C₁₉	—	2	3	no	—
71	C₇×C₁₉	C₃×D₃×D₇	1	2	yes	complemented
72	C₃×D₂₁	—	19	2	no	—
73	C₃×D₃×C₇	—	133	2	no	—
74	C₃²×D₇	—	266	3	no	—
75	C₃²×D₇	—	133	2	no	—
76	C₂×C₃×C₁₉	—	49	1	no	—
77	D₃×C₁₉	—	7	2	no	—
78	C₂×C₁₉⋊C₃	—	21	2	no	—
79	C₂×C₁₉⋊C₃	—	49	1	no	—
80	C₂×C₁₉⋊C₃	—	98	1	no	—
81	C₇⋊D₇	D₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
82	C₇×D₇	—	3	2	no	—
83	C₂×C₃×D₇	—	399	2	no	—
84	D₃×D₇	—	7	2	no	—
85	C₂²×C₁₉	—	147	1	no	Hall
86	C₃²×C₇	—	19	2	no	—
87	C₃²×C₇	—	38	3	no	—
88	C₃×C₁₉	C₃×D₇²	1	1	yes	characteristic, complemented
89	C₁₉⋊C₃	—	2	1	no	—
90	C₁₉⋊C₃	D₂₁⋊D₇	1	1	yes	characteristic, direct product factor
91	C₇²	C₂×D₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₃×C₇	—	399	2	no	—
93	D₂₁	D₇×C₁₉⋊C₃	1	2	yes	complemented
94	C₃×D₇	—	7	2	no	—
95	C₃×D₇	—	266	2	no	—
96	C₃×D₇	—	266	6	no	—
97	C₃×D₇	—	266	3	no	—
98	C₃×D₇	—	14	3	no	—
99	C₃×D₇	—	57	2	no	—
100	C₃×D₇	—	133	2	no	—
101	D₃×C₇	—	7	2	no	—
102	C₂×C₁₉	—	21	2	no	—
103	C₂×C₁₉	—	49	1	no	—
104	C₂×C₃×D₃	—	931	1	no	Hall
105	C₂×D₇	—	21	2	no	—
106	C₃×C₇	—	2	3	no	—
107	C₃×C₇	—	19	2	no	—
108	C₃×C₇	—	38	3	no	—
109	C₃×C₇	—	38	6	no	—
110	C₃×C₇	—	38	2	no	—
111	C₃×C₇	C₂×D₇×C₁₉⋊C₃	1	2	yes	complemented
112	C₁₉	C₃×D₂₁⋊D₇	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₃²	—	931	1	no	—
114	C₃×D₃	—	133	2	no	—
115	C₂×C₇	—	21	2	no	—
116	D₇	—	3	2	no	—
117	D₇	—	7	2	no	—
118	D₇	—	14	3	no	—
119	C₂²×C₃	—	2793	1	no	—
120	C₂×D₃	—	49	1	no	—
121	C₃²	—	19	1	no	Sylow
122	C₇	—	2	3	no	—
123	C₇	D₃×D₇×C₁₉⋊C₃	1	2	yes	complemented
124	C₂×C₃	—	49	1	no	—
125	C₂×C₃	—	1862	1	no	—
126	C₂×C₃	—	399	2	no	—
127	C₂×C₃	—	931	1	no	—
128	D₃	—	7	2	no	—
129	C₂²	—	147	1	no	Sylow
130	C₃	—	19	1	no	—
131	C₃	—	38	1	no	—
132	C₃	D₇²×C₁₉⋊C₃	1	1	yes	characteristic, complemented
133	C₂	—	21	2	no	—
134	C₂	—	49	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×D₂₁×C₁₉⋊C₃

C₂

yes

maximal, complemented

C₃×C₁₉⋊C₃×C₇⋊D₇

C₂

yes

maximal, characteristic, complemented

C₁₉×D₂₁⋊D₇

C₃

yes

maximal, characteristic, complemented

D₇²×C₁₉⋊C₃

—

maximal

C₃×C₇²×C₁₉⋊C₃

C₂²

yes

characteristic, complemented, Hall

C₇×C₁₉×D₂₁

C₂×C₃

yes

complemented

C₃×C₁₉×C₇⋊D₇

C₂×C₃

yes

characteristic, complemented

C₁₉⋊C₃×C₇⋊D₇

—

C₁₉⋊C₃×C₇⋊D₇

D₃

yes

characteristic, complemented

C₇×D₇×C₁₉⋊C₃

—

D₃×D₇×C₁₉⋊C₃

—

maximal

D₇²×C₁₉

—

Hall

C₇²×C₁₉⋊C₃

—

C₇²×C₁₉⋊C₃

C₂×D₃

yes

characteristic, complemented

C₃×C₇²×C₁₉

C₂²×C₃

yes

characteristic, complemented

D₂₁×C₁₉⋊C₃

D₇

yes

complemented

D₃×C₇×C₁₉⋊C₃

—

C₃×D₇×C₁₉⋊C₃

—

C₃×D₇×C₁₉⋊C₃

—

C₁₉×C₇⋊D₇

C₃×D₃

yes

characteristic, complemented

C₇×D₇×C₁₉

—

C₃×D₂₁⋊D₇

—

maximal, Hall

D₃×D₇×C₁₉

—

C₂×D₇×C₁₉⋊C₃

—

C₃×C₇×C₁₉⋊C₃

—

C₃×C₇×C₁₉⋊C₃

C₂×D₇

yes

complemented

C₇²×C₁₉

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₃×C₇×D₂₁

—

C₃²×C₇⋊D₇

—

C₁₉×D₂₁

C₃×D₇

yes

complemented

D₃×C₇×C₁₉

—

C₃×D₇×C₁₉

—

C₃×D₇×C₁₉

—

C₂×C₇×C₁₉⋊C₃

—

D₇×C₁₉⋊C₃

—

D₇×C₁₉⋊C₃

—

D₇×C₁₉⋊C₃

—

D₇×C₁₉⋊C₃

—

D₇×C₁₉⋊C₃

—

C₂×D₃×C₁₉⋊C₃

—

Hall

D₂₁⋊D₇

C₁₉⋊C₃

yes

characteristic, direct product factor

C₃×D₇²

—

C₂×D₇×C₁₉

—

C₃²×C₇²

—

Hall

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

C₂×C₃×D₇

yes

complemented

C₇×C₁₉⋊C₃

—

C₇×C₁₉⋊C₃

—

C₇×C₁₉⋊C₃

—

C₇×C₁₉⋊C₃

D₃×D₇

yes

complemented

C₂×C₃×C₁₉⋊C₃

—

D₃×C₁₉⋊C₃

—

C₇×D₂₁

C₂×C₁₉⋊C₃

yes

complemented

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

C₂×C₁₉⋊C₃

yes

characteristic, complemented

C₃×C₇×D₇

—

C₂×C₇×C₁₉

—

D₇×C₁₉

—

D₇×C₁₉

—

D₇×C₁₉

—

C₃×D₃×D₇

—

133

—

C₂×D₃×C₁₉

—

C₂²×C₁₉⋊C₃

—

147

—

D₇²

—

Hall

C₃×C₁₉⋊C₃

D₇²

yes

characteristic, complemented, Hall

C₃×C₇²

—

C₃×C₇²

—

C₃×C₇²

C₂²×C₁₉⋊C₃

yes

characteristic, complemented

C₇×C₁₉

—

C₇×C₁₉

C₃×D₃×D₇

yes

complemented

C₃×D₂₁

—

C₃×D₃×C₇

—

133

—

C₃²×D₇

—

266

—

C₃²×D₇

—

133

—

C₂×C₃×C₁₉

—

D₃×C₁₉

—

C₂×C₁₉⋊C₃

—

C₂×C₁₉⋊C₃

—

C₂×C₁₉⋊C₃

—

C₇⋊D₇

D₃×C₁₉⋊C₃

yes

characteristic, complemented

C₇×D₇

—

C₂×C₃×D₇

—

399

—

D₃×D₇

—

C₂²×C₁₉

—

147

Hall

C₃²×C₇

—

C₃²×C₇

—

C₃×C₁₉

C₃×D₇²

yes

characteristic, complemented

C₁₉⋊C₃

—

C₁₉⋊C₃

D₂₁⋊D₇

yes

characteristic, direct product factor

C₇²

C₂×D₃×C₁₉⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

399

—

D₂₁

D₇×C₁₉⋊C₃

yes

complemented

C₃×D₇

—

C₃×D₇

—

266

—

C₃×D₇

—

266

—

C₃×D₇

—

266

—

C₃×D₇

—

C₃×D₇

—

100

C₃×D₇

—

133

—

101

D₃×C₇

—

102

C₂×C₁₉

—

103

C₂×C₁₉

—

104

C₂×C₃×D₃

—

931

Hall

105

C₂×D₇

—

106

C₃×C₇

—

107

C₃×C₇

—

108

C₃×C₇

—

109

C₃×C₇

—

110

C₃×C₇

—

111

C₃×C₇

C₂×D₇×C₁₉⋊C₃

yes

complemented

112

C₁₉

C₃×D₂₁⋊D₇

yes

characteristic, complemented, Sylow

113

C₂×C₃²

—

931

—

114

C₃×D₃

—

133

—

115

C₂×C₇

—

116

D₇

—

117

D₇

—

118

D₇

—

119

C₂²×C₃

—

2793

—

120

C₂×D₃

—

121

C₃²

—

Sylow

122

C₇

—

123

C₇

D₃×D₇×C₁₉⋊C₃

yes

complemented

124

C₂×C₃

—

125

C₂×C₃

—

1862

—

126

C₂×C₃

—

399

—

127

C₂×C₃

—

931

—

128

D₃

—

129

C₂²

—

147

Sylow

130

C₃

—

131

C₃

—

132

C₃

D₇²×C₁₉⋊C₃

yes

characteristic, complemented

133

C₂

—

134

C₂

—

	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. C₇×D₂₁×C₁₉⋊C₃	C₂	semidirect
2. C₃×C₁₉⋊C₃×C₇⋊D₇	C₂	semidirect
3. C₁₉×D₂₁⋊D₇	C₃	semidirect
5. C₃×C₇²×C₁₉⋊C₃	C₂²	semidirect
6. C₇×C₁₉×D₂₁	C₂×C₃	semidirect
7. C₃×C₁₉×C₇⋊D₇	C₂×C₃	semidirect
9. C₁₉⋊C₃×C₇⋊D₇	D₃	semidirect
14. C₇²×C₁₉⋊C₃	C₂×D₃	semidirect
15. C₃×C₇²×C₁₉	C₂²×C₃	semidirect
16. D₂₁×C₁₉⋊C₃	D₇	semidirect
20. C₁₉×C₇⋊D₇	C₃×D₃	semidirect
26. C₃×C₇×C₁₉⋊C₃	C₂×D₇	semidirect
27. C₇²×C₁₉	C₂×C₃×D₃	semidirect
30. C₁₉×D₂₁	C₃×D₇	semidirect
41. D₂₁⋊D₇	C₁₉⋊C₃	direct (N × H)
46. C₃×C₇×C₁₉	C₂×C₃×D₇	semidirect
50. C₇×C₁₉⋊C₃	D₃×D₇	semidirect
53. C₇×D₂₁	C₂×C₁₉⋊C₃	semidirect
56. C₃×C₇⋊D₇	C₂×C₁₉⋊C₃	semidirect
66. C₃×C₁₉⋊C₃	D₇²	semidirect
69. C₃×C₇²	C₂²×C₁₉⋊C₃	semidirect
71. C₇×C₁₉	C₃×D₃×D₇	semidirect
81. C₇⋊D₇	D₃×C₁₉⋊C₃	semidirect
88. C₃×C₁₉	C₃×D₇²	semidirect
90. C₁₉⋊C₃	D₂₁⋊D₇	direct (N × H)
91. C₇²	C₂×D₃×C₁₉⋊C₃	semidirect
93. D₂₁	D₇×C₁₉⋊C₃	semidirect
111. C₃×C₇	C₂×D₇×C₁₉⋊C₃	semidirect
112. C₁₉	C₃×D₂₁⋊D₇	semidirect
123. C₇	D₃×D₇×C₁₉⋊C₃	semidirect
132. C₃	D₇²×C₁₉⋊C₃	semidirect

H ≅ G/N

type

1. C₇×D₂₁×C₁₉⋊C₃

C₂

semidirect

2. C₃×C₁₉⋊C₃×C₇⋊D₇

C₂

semidirect

3. C₁₉×D₂₁⋊D₇

C₃

semidirect

5. C₃×C₇²×C₁₉⋊C₃

C₂²

semidirect

6. C₇×C₁₉×D₂₁

C₂×C₃

semidirect

7. C₃×C₁₉×C₇⋊D₇

C₂×C₃

semidirect

9. C₁₉⋊C₃×C₇⋊D₇

D₃

semidirect

14. C₇²×C₁₉⋊C₃

C₂×D₃

semidirect

15. C₃×C₇²×C₁₉

C₂²×C₃

semidirect

16. D₂₁×C₁₉⋊C₃

D₇

semidirect

20. C₁₉×C₇⋊D₇

C₃×D₃

semidirect

26. C₃×C₇×C₁₉⋊C₃

C₂×D₇

semidirect

27. C₇²×C₁₉

C₂×C₃×D₃

semidirect

30. C₁₉×D₂₁

C₃×D₇

semidirect

41. D₂₁⋊D₇

C₁₉⋊C₃

direct (N × H)

46. C₃×C₇×C₁₉

C₂×C₃×D₇

semidirect

50. C₇×C₁₉⋊C₃

D₃×D₇

semidirect

53. C₇×D₂₁

C₂×C₁₉⋊C₃

semidirect

56. C₃×C₇⋊D₇

C₂×C₁₉⋊C₃

semidirect

66. C₃×C₁₉⋊C₃

D₇²

semidirect

69. C₃×C₇²

C₂²×C₁₉⋊C₃

semidirect

71. C₇×C₁₉

C₃×D₃×D₇

semidirect

81. C₇⋊D₇

D₃×C₁₉⋊C₃

semidirect

88. C₃×C₁₉

C₃×D₇²

semidirect

90. C₁₉⋊C₃

D₂₁⋊D₇

direct (N × H)

91. C₇²

C₂×D₃×C₁₉⋊C₃

semidirect

93. D₂₁

D₇×C₁₉⋊C₃

semidirect

111. C₃×C₇

C₂×D₇×C₁₉⋊C₃

semidirect

112. C₁₉

C₃×D₂₁⋊D₇

semidirect

123. C₇

D₃×D₇×C₁₉⋊C₃

semidirect

132. C₃

D₇²×C₁₉⋊C₃

semidirect

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

subgroup	index
C₇×D₂₁×C₁₉⋊C₃	2
C₃×C₁₉⋊C₃×C₇⋊D₇	2
C₁₉×D₂₁⋊D₇	3
D₇²×C₁₉⋊C₃	3
D₃×D₇×C₁₉⋊C₃	7
C₃×D₂₁⋊D₇	19

subgroup

index

C₇×D₂₁×C₁₉⋊C₃

C₃×C₁₉⋊C₃×C₇⋊D₇

C₁₉×D₂₁⋊D₇

D₇²×C₁₉⋊C₃

D₃×D₇×C₁₉⋊C₃

C₃×D₂₁⋊D₇

Subgroups — C₁₉⋊C₃×D₂₁⋊D₇ #45692413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19⋊C3×D21⋊D7 #45692413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉⋊C₃×D₂₁⋊D₇ #45692413