Group #49141651 — subgroups

Subgroups — D₇×C₃⋊D₄₃₅ #49141651

Order 36540.

240 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 76 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×C₅×D₇×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₃⋊D₄₃₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊D₃₀₄₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₄₃₅	—	3	4	no	maximal
5	C₃²×C₅×C₇×C₂₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₇×C₃⋊D₈₇	—	5	1	no	maximal, Hall
7	D₃₀₄₅	—	3	4	no	—
8	C₇×D₄₃₅	—	3	4	no	—
9	C₃×C₅×D₇×C₂₉	D₃	1	4	yes	complemented
10	C₂×C₃⋊D₄₃₅	—	7	1	no	maximal, Hall
11	D₇×D₁₄₅	—	9	1	no	Hall
12	C₃⋊D₆₀₉	—	5	1	no	—
13	C₇×C₃⋊D₈₇	—	5	1	no	—
14	C₃²×D₇×C₂₉	D₅	1	1	yes	characteristic, complemented
15	C₃×C₅×C₇×C₂₉	C₂×D₃	1	4	yes	complemented
16	C₃⋊D₄₃₅	—	7	1	no	—
17	C₃⋊D₄₃₅	D₇	1	1	yes	characteristic, direct product factor
18	C₂×C₃²×C₅×C₂₉	—	7	1	no	—
19	D₇×D₈₇	—	15	4	no	—
20	D₁₀₁₅	—	9	1	no	—
21	C₇×D₁₄₅	—	9	1	no	—
22	C₅×D₇×C₂₉	C₃⋊D₃	1	1	yes	characteristic, complemented
23	C₃²×C₇×C₂₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
24	C₂×D₄₃₅	—	21	4	no	—
25	C₃²×C₅×C₂₉	C₂×D₇	1	1	yes	characteristic, complemented, Hall
26	D₇×C₃⋊D₁₅	—	29	1	no	maximal, Hall
27	D₆₀₉	—	15	4	no	—
28	C₇×D₈₇	—	15	4	no	—
29	C₃×D₇×C₂₉	D₁₅	1	4	yes	complemented
30	C₂×C₃⋊D₈₇	—	35	1	no	Hall
31	C₅×C₇×C₂₉	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×C₅×C₂₉	—	7	4	no	—
33	D₄₃₅	—	3	4	no	—
34	D₄₃₅	—	21	4	no	—
35	D₇×D₂₉	—	45	1	no	Hall
36	C₃⋊D₁₀₅	—	29	1	no	—
37	C₇×C₃⋊D₁₅	—	29	1	no	—
38	C₃²×C₅×D₇	D₂₉	1	1	yes	characteristic, complemented
39	C₃×C₇×C₂₉	C₂×D₁₅	1	4	yes	complemented
40	C₂×D₁₄₅	—	63	1	no	Hall
41	C₂×C₃²×C₂₉	—	7	1	no	—
42	C₃⋊D₈₇	—	5	1	no	—
43	C₃⋊D₈₇	—	35	1	no	—
44	C₃×C₅×C₂₉	D₃×D₇	1	4	yes	complemented
45	D₇×D₁₅	—	87	4	no	—
46	D₂₀₃	—	45	1	no	—
47	C₇×D₂₉	—	45	1	no	—
48	D₇×C₂₉	C₃⋊D₁₅	1	1	yes	characteristic, complemented
49	C₂×D₈₇	—	105	4	no	—
50	C₃²×C₅×C₇	C₂×D₂₉	1	1	yes	characteristic, complemented, Hall
51	C₂×C₅×C₂₉	—	7	1	no	—
52	D₁₄₅	—	9	1	no	—
53	D₁₄₅	—	63	1	no	—
54	C₃²×C₂₉	D₅×D₇	1	1	yes	characteristic, complemented, Hall
55	D₇×C₃⋊D₃	—	145	1	no	Hall
56	D₁₀₅	—	87	4	no	—
57	C₇×D₁₅	—	87	4	no	—
58	C₃×C₅×D₇	D₈₇	1	4	yes	complemented
59	C₇×C₂₉	C₂×C₃⋊D₁₅	1	1	yes	characteristic, complemented, Hall
60	C₂×C₃⋊D₁₅	—	203	1	no	Hall
61	C₂×C₃×C₂₉	—	7	4	no	—
62	D₈₇	—	15	4	no	—
63	D₈₇	—	105	4	no	—
64	C₅×C₂₉	D₇×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
65	D₅×D₇	—	261	1	no	Hall
66	C₃⋊D₂₁	—	145	1	no	—
67	C₇×C₃⋊D₃	—	145	1	no	—
68	C₃²×D₇	D₁₄₅	1	1	yes	characteristic, complemented
69	C₂×D₂₉	—	315	1	no	Hall
70	C₃×C₅×C₇	C₂×D₈₇	1	4	yes	complemented
71	C₂×C₃²×C₅	—	7	1	no	—
72	C₃⋊D₁₅	—	29	1	no	—
73	C₃⋊D₁₅	—	203	1	no	—
74	C₃×C₂₉	D₇×D₁₅	1	4	yes	complemented
75	D₃×D₇	—	435	4	no	—
76	D₃₅	—	261	1	no	—
77	C₅×D₇	C₃⋊D₈₇	1	1	yes	characteristic, complemented
78	C₇×D₅	—	261	1	no	—
79	C₃²×C₇	C₂×D₁₄₅	1	1	yes	characteristic, complemented, Hall
80	C₂×D₁₅	—	609	4	no	—
81	C₂×C₂₉	—	7	1	no	—
82	D₂₉	—	45	1	no	—
83	D₂₉	—	315	1	no	—
84	C₃²×C₅	D₇×D₂₉	1	1	yes	characteristic, complemented, Hall
85	D₂₁	—	435	4	no	—
86	C₃×D₇	D₄₃₅	1	4	yes	complemented
87	D₃×C₇	—	435	4	no	—
88	C₂×C₃⋊D₃	—	1015	1	no	Hall
89	C₅×C₇	C₂×C₃⋊D₈₇	1	1	yes	characteristic, complemented, Hall
90	C₂×C₃×C₅	—	7	4	no	—
91	D₁₅	—	87	4	no	—
92	D₁₅	—	609	4	no	—
93	C₂₉	D₇×C₃⋊D₁₅	1	1	yes	characteristic, complemented, Sylow
94	C₂×D₇	—	1305	1	no	Hall
95	C₃×C₇	C₂×D₄₃₅	1	4	yes	complemented
96	C₂×D₅	—	1827	1	no	Hall
97	C₂×C₃²	—	7	1	no	—
98	C₃⋊D₃	—	145	1	no	—
99	C₃⋊D₃	—	1015	1	no	—
100	C₃×C₅	D₇×D₈₇	1	4	yes	complemented
101	C₂×C₇	—	1305	1	no	—
102	D₇	—	1305	1	no	—
103	D₇	C₃⋊D₄₃₅	1	1	yes	characteristic, direct product factor
104	C₂×D₃	—	3045	4	no	—
105	C₂×C₅	—	7	1	no	—
106	D₅	—	261	1	no	—
107	D₅	—	1827	1	no	—
108	C₃²	D₇×D₁₄₅	1	1	yes	characteristic, complemented, Sylow
109	C₇	C₂×C₃⋊D₄₃₅	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	7	4	no	—
111	D₃	—	435	4	no	—
112	D₃	—	3045	4	no	—
113	C₅	D₇×C₃⋊D₈₇	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	9135	1	no	Sylow
115	C₃	D₇×D₄₃₅	1	4	yes	complemented
116	C₂	—	7	1	no	—
117	C₂	—	1305	1	no	—
118	C₂	—	9135	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₅×D₇×C₂₉

C₂

yes

maximal, characteristic, complemented

C₇×C₃⋊D₄₃₅

C₂

yes

maximal, characteristic, complemented

C₃⋊D₃₀₄₅

C₂

yes

maximal, characteristic, complemented

D₇×D₄₃₅

—

maximal

C₃²×C₅×C₇×C₂₉

C₂²

yes

characteristic, complemented, Hall

D₇×C₃⋊D₈₇

—

maximal, Hall

D₃₀₄₅

—

C₇×D₄₃₅

—

C₃×C₅×D₇×C₂₉

D₃

yes

complemented

C₂×C₃⋊D₄₃₅

—

maximal, Hall

D₇×D₁₄₅

—

Hall

C₃⋊D₆₀₉

—

C₇×C₃⋊D₈₇

—

C₃²×D₇×C₂₉

D₅

yes

characteristic, complemented

C₃×C₅×C₇×C₂₉

C₂×D₃

yes

complemented

C₃⋊D₄₃₅

—

C₃⋊D₄₃₅

D₇

yes

characteristic, direct product factor

C₂×C₃²×C₅×C₂₉

—

D₇×D₈₇

—

D₁₀₁₅

—

C₇×D₁₄₅

—

C₅×D₇×C₂₉

C₃⋊D₃

yes

characteristic, complemented

C₃²×C₇×C₂₉

C₂×D₅

yes

characteristic, complemented, Hall

C₂×D₄₃₅

—

C₃²×C₅×C₂₉

C₂×D₇

yes

characteristic, complemented, Hall

D₇×C₃⋊D₁₅

—

maximal, Hall

D₆₀₉

—

C₇×D₈₇

—

C₃×D₇×C₂₉

D₁₅

yes

complemented

C₂×C₃⋊D₈₇

—

Hall

C₅×C₇×C₂₉

C₂×C₃⋊D₃

yes

characteristic, complemented, Hall

C₂×C₃×C₅×C₂₉

—

D₄₃₅

—

D₄₃₅

—

D₇×D₂₉

—

Hall

C₃⋊D₁₀₅

—

C₇×C₃⋊D₁₅

—

C₃²×C₅×D₇

D₂₉

yes

characteristic, complemented

C₃×C₇×C₂₉

C₂×D₁₅

yes

complemented

C₂×D₁₄₅

—

Hall

C₂×C₃²×C₂₉

—

C₃⋊D₈₇

—

C₃⋊D₈₇

—

C₃×C₅×C₂₉

D₃×D₇

yes

complemented

D₇×D₁₅

—

D₂₀₃

—

C₇×D₂₉

—

D₇×C₂₉

C₃⋊D₁₅

yes

characteristic, complemented

C₂×D₈₇

—

105

—

C₃²×C₅×C₇

C₂×D₂₉

yes

characteristic, complemented, Hall

C₂×C₅×C₂₉

—

D₁₄₅

—

D₁₄₅

—

C₃²×C₂₉

D₅×D₇

yes

characteristic, complemented, Hall

D₇×C₃⋊D₃

—

145

Hall

D₁₀₅

—

C₇×D₁₅

—

C₃×C₅×D₇

D₈₇

yes

complemented

C₇×C₂₉

C₂×C₃⋊D₁₅

yes

characteristic, complemented, Hall

C₂×C₃⋊D₁₅

—

203

Hall

C₂×C₃×C₂₉

—

D₈₇

—

D₈₇

—

105

—

C₅×C₂₉

D₇×C₃⋊D₃

yes

characteristic, complemented, Hall

D₅×D₇

—

261

Hall

C₃⋊D₂₁

—

145

—

C₇×C₃⋊D₃

—

145

—

C₃²×D₇

D₁₄₅

yes

characteristic, complemented

C₂×D₂₉

—

315

Hall

C₃×C₅×C₇

C₂×D₈₇

yes

complemented

C₂×C₃²×C₅

—

C₃⋊D₁₅

—

C₃⋊D₁₅

—

203

—

C₃×C₂₉

D₇×D₁₅

yes

complemented

D₃×D₇

—

435

—

D₃₅

—

261

—

C₅×D₇

C₃⋊D₈₇

yes

characteristic, complemented

C₇×D₅

—

261

—

C₃²×C₇

C₂×D₁₄₅

yes

characteristic, complemented, Hall

C₂×D₁₅

—

609

—

C₂×C₂₉

—

D₂₉

—

D₂₉

—

315

—

C₃²×C₅

D₇×D₂₉

yes

characteristic, complemented, Hall

D₂₁

—

435

—

C₃×D₇

D₄₃₅

yes

complemented

D₃×C₇

—

435

—

C₂×C₃⋊D₃

—

1015

Hall

C₅×C₇

C₂×C₃⋊D₈₇

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

D₁₅

—

D₁₅

—

609

—

C₂₉

D₇×C₃⋊D₁₅

yes

characteristic, complemented, Sylow

C₂×D₇

—

1305

Hall

C₃×C₇

C₂×D₄₃₅

yes

complemented

C₂×D₅

—

1827

Hall

C₂×C₃²

—

C₃⋊D₃

—

145

—

C₃⋊D₃

—

1015

—

100

C₃×C₅

D₇×D₈₇

yes

complemented

101

C₂×C₇

—

1305

—

102

D₇

—

1305

—

103

D₇

C₃⋊D₄₃₅

yes

characteristic, direct product factor

104

C₂×D₃

—

3045

—

105

C₂×C₅

—

106

D₅

—

261

—

107

D₅

—

1827

—

108

C₃²

D₇×D₁₄₅

yes

characteristic, complemented, Sylow

109

C₇

C₂×C₃⋊D₄₃₅

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

D₃

—

435

—

112

D₃

—

3045

—

113

C₅

D₇×C₃⋊D₈₇

yes

characteristic, complemented, Sylow

114

C₂²

—

9135

Sylow

115

C₃

D₇×D₄₃₅

yes

complemented

116

C₂

—

117

C₂

—

1305

—

118

C₂

—

9135

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅×C₇×C₂₉	C₂²
LCS	G ▷ C₃²×C₅×C₇×C₂₉
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃²×C₅×C₇×C₂₉

C₂²

LCS

G ▷ C₃²×C₅×C₇×C₂₉

UCS

C₁

N	H ≅ G/N	type
1. C₃²×C₅×D₇×C₂₉	C₂	semidirect
2. C₇×C₃⋊D₄₃₅	C₂	semidirect
3. C₃⋊D₃₀₄₅	C₂	semidirect
5. C₃²×C₅×C₇×C₂₉	C₂²	semidirect
9. C₃×C₅×D₇×C₂₉	D₃	semidirect
14. C₃²×D₇×C₂₉	D₅	semidirect
15. C₃×C₅×C₇×C₂₉	C₂×D₃	semidirect
17. C₃⋊D₄₃₅	D₇	direct (N × H)
22. C₅×D₇×C₂₉	C₃⋊D₃	semidirect
23. C₃²×C₇×C₂₉	C₂×D₅	semidirect
25. C₃²×C₅×C₂₉	C₂×D₇	semidirect
29. C₃×D₇×C₂₉	D₁₅	semidirect
31. C₅×C₇×C₂₉	C₂×C₃⋊D₃	semidirect
38. C₃²×C₅×D₇	D₂₉	semidirect
39. C₃×C₇×C₂₉	C₂×D₁₅	semidirect
44. C₃×C₅×C₂₉	D₃×D₇	semidirect
48. D₇×C₂₉	C₃⋊D₁₅	semidirect
50. C₃²×C₅×C₇	C₂×D₂₉	semidirect
54. C₃²×C₂₉	D₅×D₇	semidirect
58. C₃×C₅×D₇	D₈₇	semidirect
59. C₇×C₂₉	C₂×C₃⋊D₁₅	semidirect
64. C₅×C₂₉	D₇×C₃⋊D₃	semidirect
68. C₃²×D₇	D₁₄₅	semidirect
70. C₃×C₅×C₇	C₂×D₈₇	semidirect
74. C₃×C₂₉	D₇×D₁₅	semidirect
77. C₅×D₇	C₃⋊D₈₇	semidirect
79. C₃²×C₇	C₂×D₁₄₅	semidirect
84. C₃²×C₅	D₇×D₂₉	semidirect
86. C₃×D₇	D₄₃₅	semidirect
89. C₅×C₇	C₂×C₃⋊D₈₇	semidirect
93. C₂₉	D₇×C₃⋊D₁₅	semidirect
95. C₃×C₇	C₂×D₄₃₅	semidirect
100. C₃×C₅	D₇×D₈₇	semidirect
103. D₇	C₃⋊D₄₃₅	direct (N × H)
108. C₃²	D₇×D₁₄₅	semidirect
109. C₇	C₂×C₃⋊D₄₃₅	semidirect
113. C₅	D₇×C₃⋊D₈₇	semidirect
115. C₃	D₇×D₄₃₅	semidirect

H ≅ G/N

type

1. C₃²×C₅×D₇×C₂₉

C₂

semidirect

2. C₇×C₃⋊D₄₃₅

C₂

semidirect

3. C₃⋊D₃₀₄₅

C₂

semidirect

5. C₃²×C₅×C₇×C₂₉

C₂²

semidirect

9. C₃×C₅×D₇×C₂₉

D₃

semidirect

14. C₃²×D₇×C₂₉

D₅

semidirect

15. C₃×C₅×C₇×C₂₉

C₂×D₃

semidirect

17. C₃⋊D₄₃₅

D₇

direct (N × H)

22. C₅×D₇×C₂₉

C₃⋊D₃

semidirect

23. C₃²×C₇×C₂₉

C₂×D₅

semidirect

25. C₃²×C₅×C₂₉

C₂×D₇

semidirect

29. C₃×D₇×C₂₉

D₁₅

semidirect

31. C₅×C₇×C₂₉

C₂×C₃⋊D₃

semidirect

38. C₃²×C₅×D₇

D₂₉

semidirect

39. C₃×C₇×C₂₉

C₂×D₁₅

semidirect

44. C₃×C₅×C₂₉

D₃×D₇

semidirect

48. D₇×C₂₉

C₃⋊D₁₅

semidirect

50. C₃²×C₅×C₇

C₂×D₂₉

semidirect

54. C₃²×C₂₉

D₅×D₇

semidirect

58. C₃×C₅×D₇

D₈₇

semidirect

59. C₇×C₂₉

C₂×C₃⋊D₁₅

semidirect

64. C₅×C₂₉

D₇×C₃⋊D₃

semidirect

68. C₃²×D₇

D₁₄₅

semidirect

70. C₃×C₅×C₇

C₂×D₈₇

semidirect

74. C₃×C₂₉

D₇×D₁₅

semidirect

77. C₅×D₇

C₃⋊D₈₇

semidirect

79. C₃²×C₇

C₂×D₁₄₅

semidirect

84. C₃²×C₅

D₇×D₂₉

semidirect

86. C₃×D₇

D₄₃₅

semidirect

89. C₅×C₇

C₂×C₃⋊D₈₇

semidirect

93. C₂₉

D₇×C₃⋊D₁₅

semidirect

95. C₃×C₇

C₂×D₄₃₅

semidirect

100. C₃×C₅

D₇×D₈₇

semidirect

103. D₇

C₃⋊D₄₃₅

direct (N × H)

108. C₃²

D₇×D₁₄₅

semidirect

109. C₇

C₂×C₃⋊D₄₃₅

semidirect

113. C₅

D₇×C₃⋊D₈₇

semidirect

115. C₃

D₇×D₄₃₅

semidirect

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

subgroup	index
C₃²×C₅×D₇×C₂₉	2
C₇×C₃⋊D₄₃₅	2
C₃⋊D₃₀₄₅	2
D₇×D₄₃₅	3
D₇×C₃⋊D₈₇	5
C₂×C₃⋊D₄₃₅	7
D₇×C₃⋊D₁₅	29

subgroup

index

C₃²×C₅×D₇×C₂₉

C₇×C₃⋊D₄₃₅

C₃⋊D₃₀₄₅

D₇×D₄₃₅

D₇×C₃⋊D₈₇

C₂×C₃⋊D₄₃₅

D₇×C₃⋊D₁₅

Subgroups — D₇×C₃⋊D₄₃₅ #49141651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×C3⋊D435 #49141651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₃⋊D₄₃₅ #49141651