Group #50304136 — subgroups

Subgroups — C₁₃⋊C₃×D₃₅⋊D₇ #50304136

Order 38220.

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

normal subgroups only

This group has 208 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×C₁₃⋊C₃×D₃₅	C₂	1	2	yes	maximal, complemented
2	C₅×C₁₃⋊C₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×D₃₅⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅×C₇²×C₁₃⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₇²×C₁₃⋊C₃	—	5	1	no	maximal, Hall
6	C₇×C₁₃×D₃₅	C₂×C₃	1	2	yes	complemented
7	C₅×C₁₃×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
8	D₅×D₇×C₁₃⋊C₃	—	7	2	no	maximal
9	C₁₃⋊C₃×C₇⋊D₇	D₅	1	1	yes	characteristic, complemented
10	C₇×D₇×C₁₃⋊C₃	—	5	2	no	—
11	C₅×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×D₃₅⋊D₇	—	13	1	no	maximal, Hall
13	C₁₃⋊C₃×D₃₅	D₇	1	2	yes	complemented
14	C₇×D₅×C₁₃⋊C₃	—	7	2	no	—
15	C₅×D₇×C₁₃⋊C₃	—	7	2	no	—
16	C₅×D₇×C₁₃⋊C₃	—	14	3	no	—
17	C₁₃×D₇²	—	5	1	no	Hall
18	C₇²×C₁₃⋊C₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
19	D₅×C₁₃×D₇	—	7	2	no	—
20	C₃×C₇×D₃₅	—	13	2	no	—
21	C₃×C₅×C₇⋊D₇	—	13	1	no	—
22	C₅×C₇×C₁₃⋊C₃	—	2	3	no	—
23	C₅×C₇×C₁₃⋊C₃	C₂×D₇	1	2	yes	complemented
24	C₁₃×C₇⋊D₇	C₃×D₅	1	1	yes	characteristic, complemented
25	C₇×C₁₃×D₇	—	5	2	no	—
26	C₂×D₇×C₁₃⋊C₃	—	35	2	no	—
27	D₃₅⋊D₇	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
28	C₁₃×D₃₅	C₃×D₇	1	2	yes	complemented
29	C₇×D₅×C₁₃	—	7	2	no	—
30	C₅×C₁₃×D₇	—	7	2	no	—
31	C₅×C₁₃×D₇	—	14	3	no	—
32	C₂×D₅×C₁₃⋊C₃	—	49	1	no	Hall
33	C₃×C₅×C₇²	—	13	1	no	Hall
34	C₇²×C₁₃	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Hall
35	C₃×D₇²	—	65	1	no	Hall
36	C₂×C₇×C₁₃⋊C₃	—	35	2	no	—
37	D₇×C₁₃⋊C₃	—	5	2	no	—
38	D₇×C₁₃⋊C₃	—	7	2	no	—
39	D₇×C₁₃⋊C₃	—	14	3	no	—
40	C₇×D₃₅	C₂×C₁₃⋊C₃	1	2	yes	complemented
41	C₅×C₇⋊D₇	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
42	C₅×C₇×C₁₃	—	2	3	no	—
43	C₅×C₇×C₁₃	C₂×C₃×D₇	1	2	yes	complemented
44	C₃×D₅×D₇	—	91	2	no	—
45	C₂×C₅×C₁₃⋊C₃	—	49	1	no	—
46	D₅×C₁₃⋊C₃	—	7	2	no	—
47	C₂×C₁₃×D₇	—	35	2	no	—
48	C₃×C₇⋊D₇	—	13	1	no	—
49	C₃×C₇×D₇	—	65	2	no	—
50	C₇×C₁₃⋊C₃	—	2	3	no	—
51	C₇×C₁₃⋊C₃	D₅×D₇	1	2	yes	complemented
52	C₂×D₅×C₁₃	—	49	1	no	Hall
53	C₅×C₇²	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
54	C₃×D₃₅	—	13	2	no	—
55	C₃×C₇×D₅	—	91	2	no	—
56	C₃×C₅×D₇	—	91	2	no	—
57	C₃×C₅×D₇	—	182	3	no	—
58	D₇²	—	5	1	no	Hall
59	C₅×C₁₃⋊C₃	D₇²	1	1	yes	characteristic, complemented, Hall
60	C₂×C₇×C₁₃	—	35	2	no	—
61	C₁₃×D₇	—	5	2	no	—
62	C₁₃×D₇	—	7	2	no	—
63	C₁₃×D₇	—	14	3	no	—
64	C₂²×C₁₃⋊C₃	—	245	1	no	Hall
65	C₃×C₇²	—	13	1	no	Hall
66	D₅×D₇	—	7	2	no	—
67	C₂×C₅×C₁₃	—	49	1	no	—
68	D₅×C₁₃	—	7	2	no	—
69	C₃×C₅×C₇	—	13	2	no	—
70	C₃×C₅×C₇	—	26	3	no	—
71	C₇⋊D₇	D₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented
72	C₇×D₇	—	5	2	no	—
73	C₇×C₁₃	—	2	3	no	—
74	C₇×C₁₃	C₃×D₅×D₇	1	2	yes	complemented
75	C₂×C₃×D₇	—	455	2	no	—
76	C₂×C₁₃⋊C₃	—	35	2	no	—
77	C₂×C₁₃⋊C₃	—	49	1	no	—
78	D₃₅	D₇×C₁₃⋊C₃	1	2	yes	complemented
79	C₅×D₇	—	7	2	no	—
80	C₅×D₇	—	14	3	no	—
81	C₇×D₅	—	7	2	no	—
82	C₅×C₁₃	C₃×D₇²	1	1	yes	characteristic, complemented, Hall
83	C₂×C₃×D₅	—	637	1	no	Hall
84	C₂²×C₁₃	—	245	1	no	Hall
85	C₇²	C₂×D₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃×C₇	—	455	2	no	—
87	C₃×D₇	—	65	2	no	—
88	C₃×D₇	—	91	2	no	—
89	C₃×D₇	—	182	3	no	—
90	C₁₃⋊C₃	D₃₅⋊D₇	1	1	yes	characteristic, direct product factor, Hall
91	C₅×C₇	—	2	3	no	—
92	C₅×C₇	C₂×D₇×C₁₃⋊C₃	1	2	yes	complemented
93	C₂×C₃×C₅	—	637	1	no	—
94	C₃×D₅	—	91	2	no	—
95	C₂×D₇	—	35	2	no	—
96	C₂×C₁₃	—	35	2	no	—
97	C₂×C₁₃	—	49	1	no	—
98	C₃×C₇	—	13	2	no	—
99	C₃×C₇	—	26	3	no	—
100	C₂×D₅	—	49	1	no	Hall
101	C₃×C₅	—	13	1	no	Hall
102	C₂×C₇	—	35	2	no	—
103	D₇	—	5	2	no	—
104	D₇	—	7	2	no	—
105	D₇	—	14	3	no	—
106	C₁₃	C₃×D₃₅⋊D₇	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₃	—	3185	1	no	Hall
108	C₂×C₅	—	49	1	no	—
109	D₅	—	7	2	no	—
110	C₇	—	2	3	no	—
111	C₇	D₅×D₇×C₁₃⋊C₃	1	2	yes	complemented
112	C₂×C₃	—	637	1	no	—
113	C₂×C₃	—	455	2	no	—
114	C₅	D₇²×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	245	1	no	Sylow
116	C₃	—	13	1	no	Sylow
117	C₂	—	35	2	no	—
118	C₂	—	49	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×C₁₃⋊C₃×D₃₅

C₂

yes

maximal, complemented

C₅×C₁₃⋊C₃×C₇⋊D₇

C₂

yes

maximal, characteristic, complemented

C₁₃×D₃₅⋊D₇

C₃

yes

maximal, characteristic, complemented, Hall

C₅×C₇²×C₁₃⋊C₃

C₂²

yes

characteristic, complemented, Hall

D₇²×C₁₃⋊C₃

—

maximal, Hall

C₇×C₁₃×D₃₅

C₂×C₃

yes

complemented

C₅×C₁₃×C₇⋊D₇

C₂×C₃

yes

characteristic, complemented

D₅×D₇×C₁₃⋊C₃

—

maximal

C₁₃⋊C₃×C₇⋊D₇

D₅

yes

characteristic, complemented

C₇×D₇×C₁₃⋊C₃

—

C₅×C₇²×C₁₃

C₂²×C₃

yes

characteristic, complemented, Hall

C₃×D₃₅⋊D₇

—

maximal, Hall

C₁₃⋊C₃×D₃₅

D₇

yes

complemented

C₇×D₅×C₁₃⋊C₃

—

C₅×D₇×C₁₃⋊C₃

—

C₅×D₇×C₁₃⋊C₃

—

C₁₃×D₇²

—

Hall

C₇²×C₁₃⋊C₃

C₂×D₅

yes

characteristic, complemented, Hall

D₅×C₁₃×D₇

—

C₃×C₇×D₃₅

—

C₃×C₅×C₇⋊D₇

—

C₅×C₇×C₁₃⋊C₃

—

C₅×C₇×C₁₃⋊C₃

C₂×D₇

yes

complemented

C₁₃×C₇⋊D₇

C₃×D₅

yes

characteristic, complemented

C₇×C₁₃×D₇

—

C₂×D₇×C₁₃⋊C₃

—

D₃₅⋊D₇

C₁₃⋊C₃

yes

characteristic, direct product factor, Hall

C₁₃×D₃₅

C₃×D₇

yes

complemented

C₇×D₅×C₁₃

—

C₅×C₁₃×D₇

—

C₅×C₁₃×D₇

—

C₂×D₅×C₁₃⋊C₃

—

Hall

C₃×C₅×C₇²

—

Hall

C₇²×C₁₃

C₂×C₃×D₅

yes

characteristic, complemented, Hall

C₃×D₇²

—

Hall

C₂×C₇×C₁₃⋊C₃

—

D₇×C₁₃⋊C₃

—

D₇×C₁₃⋊C₃

—

D₇×C₁₃⋊C₃

—

C₇×D₃₅

C₂×C₁₃⋊C₃

yes

complemented

C₅×C₇⋊D₇

C₂×C₁₃⋊C₃

yes

characteristic, complemented

C₅×C₇×C₁₃

—

C₅×C₇×C₁₃

C₂×C₃×D₇

yes

complemented

C₃×D₅×D₇

—

C₂×C₅×C₁₃⋊C₃

—

D₅×C₁₃⋊C₃

—

C₂×C₁₃×D₇

—

C₃×C₇⋊D₇

—

C₃×C₇×D₇

—

C₇×C₁₃⋊C₃

—

C₇×C₁₃⋊C₃

D₅×D₇

yes

complemented

C₂×D₅×C₁₃

—

Hall

C₅×C₇²

C₂²×C₁₃⋊C₃

yes

characteristic, complemented, Hall

C₃×D₃₅

—

C₃×C₇×D₅

—

C₃×C₅×D₇

—

C₃×C₅×D₇

—

182

—

D₇²

—

Hall

C₅×C₁₃⋊C₃

D₇²

yes

characteristic, complemented, Hall

C₂×C₇×C₁₃

—

C₁₃×D₇

—

C₁₃×D₇

—

C₁₃×D₇

—

C₂²×C₁₃⋊C₃

—

245

Hall

C₃×C₇²

—

Hall

D₅×D₇

—

C₂×C₅×C₁₃

—

D₅×C₁₃

—

C₃×C₅×C₇

—

C₃×C₅×C₇

—

C₇⋊D₇

D₅×C₁₃⋊C₃

yes

characteristic, complemented

C₇×D₇

—

C₇×C₁₃

—

C₇×C₁₃

C₃×D₅×D₇

yes

complemented

C₂×C₃×D₇

—

455

—

C₂×C₁₃⋊C₃

—

C₂×C₁₃⋊C₃

—

D₃₅

D₇×C₁₃⋊C₃

yes

complemented

C₅×D₇

—

C₅×D₇

—

C₇×D₅

—

C₅×C₁₃

C₃×D₇²

yes

characteristic, complemented, Hall

C₂×C₃×D₅

—

637

Hall

C₂²×C₁₃

—

245

Hall

C₇²

C₂×D₅×C₁₃⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

455

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

—

182

—

C₁₃⋊C₃

D₃₅⋊D₇

yes

characteristic, direct product factor, Hall

C₅×C₇

—

C₅×C₇

C₂×D₇×C₁₃⋊C₃

yes

complemented

C₂×C₃×C₅

—

637

—

C₃×D₅

—

C₂×D₇

—

C₂×C₁₃

—

C₂×C₁₃

—

C₃×C₇

—

C₃×C₇

—

100

C₂×D₅

—

Hall

101

C₃×C₅

—

Hall

102

C₂×C₇

—

103

D₇

—

104

D₇

—

105

D₇

—

106

C₁₃

C₃×D₃₅⋊D₇

yes

characteristic, complemented, Sylow

107

C₂²×C₃

—

3185

Hall

108

C₂×C₅

—

109

D₅

—

110

C₇

—

111

C₇

D₅×D₇×C₁₃⋊C₃

yes

complemented

112

C₂×C₃

—

637

—

113

C₂×C₃

—

455

—

114

C₅

D₇²×C₁₃⋊C₃

yes

characteristic, complemented, Sylow

115

C₂²

—

245

Sylow

116

C₃

—

Sylow

117

C₂

—

118

C₂

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇²×C₁₃	C₂²×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₅×C₇²×C₁₃

C₂²×C₃

LCS

G ▷ C₅×C₇²×C₁₃

UCS

C₁

N	H ≅ G/N	type
1. C₇×C₁₃⋊C₃×D₃₅	C₂	semidirect
2. C₅×C₁₃⋊C₃×C₇⋊D₇	C₂	semidirect
3. C₁₃×D₃₅⋊D₇	C₃	semidirect
4. C₅×C₇²×C₁₃⋊C₃	C₂²	semidirect
6. C₇×C₁₃×D₃₅	C₂×C₃	semidirect
7. C₅×C₁₃×C₇⋊D₇	C₂×C₃	semidirect
9. C₁₃⋊C₃×C₇⋊D₇	D₅	semidirect
11. C₅×C₇²×C₁₃	C₂²×C₃	semidirect
13. C₁₃⋊C₃×D₃₅	D₇	semidirect
18. C₇²×C₁₃⋊C₃	C₂×D₅	semidirect
23. C₅×C₇×C₁₃⋊C₃	C₂×D₇	semidirect
24. C₁₃×C₇⋊D₇	C₃×D₅	semidirect
27. D₃₅⋊D₇	C₁₃⋊C₃	direct (N × H)
28. C₁₃×D₃₅	C₃×D₇	semidirect
34. C₇²×C₁₃	C₂×C₃×D₅	semidirect
40. C₇×D₃₅	C₂×C₁₃⋊C₃	semidirect
41. C₅×C₇⋊D₇	C₂×C₁₃⋊C₃	semidirect
43. C₅×C₇×C₁₃	C₂×C₃×D₇	semidirect
51. C₇×C₁₃⋊C₃	D₅×D₇	semidirect
53. C₅×C₇²	C₂²×C₁₃⋊C₃	semidirect
59. C₅×C₁₃⋊C₃	D₇²	semidirect
71. C₇⋊D₇	D₅×C₁₃⋊C₃	semidirect
74. C₇×C₁₃	C₃×D₅×D₇	semidirect
78. D₃₅	D₇×C₁₃⋊C₃	semidirect
82. C₅×C₁₃	C₃×D₇²	semidirect
85. C₇²	C₂×D₅×C₁₃⋊C₃	semidirect
90. C₁₃⋊C₃	D₃₅⋊D₇	direct (N × H)
92. C₅×C₇	C₂×D₇×C₁₃⋊C₃	semidirect
106. C₁₃	C₃×D₃₅⋊D₇	semidirect
111. C₇	D₅×D₇×C₁₃⋊C₃	semidirect
114. C₅	D₇²×C₁₃⋊C₃	semidirect

H ≅ G/N

type

1. C₇×C₁₃⋊C₃×D₃₅

C₂

semidirect

2. C₅×C₁₃⋊C₃×C₇⋊D₇

C₂

semidirect

3. C₁₃×D₃₅⋊D₇

C₃

semidirect

4. C₅×C₇²×C₁₃⋊C₃

C₂²

semidirect

6. C₇×C₁₃×D₃₅

C₂×C₃

semidirect

7. C₅×C₁₃×C₇⋊D₇

C₂×C₃

semidirect

9. C₁₃⋊C₃×C₇⋊D₇

D₅

semidirect

11. C₅×C₇²×C₁₃

C₂²×C₃

semidirect

13. C₁₃⋊C₃×D₃₅

D₇

semidirect

18. C₇²×C₁₃⋊C₃

C₂×D₅

semidirect

23. C₅×C₇×C₁₃⋊C₃

C₂×D₇

semidirect

24. C₁₃×C₇⋊D₇

C₃×D₅

semidirect

27. D₃₅⋊D₇

C₁₃⋊C₃

direct (N × H)

28. C₁₃×D₃₅

C₃×D₇

semidirect

34. C₇²×C₁₃

C₂×C₃×D₅

semidirect

40. C₇×D₃₅

C₂×C₁₃⋊C₃

semidirect

41. C₅×C₇⋊D₇

C₂×C₁₃⋊C₃

semidirect

43. C₅×C₇×C₁₃

C₂×C₃×D₇

semidirect

51. C₇×C₁₃⋊C₃

D₅×D₇

semidirect

53. C₅×C₇²

C₂²×C₁₃⋊C₃

semidirect

59. C₅×C₁₃⋊C₃

D₇²

semidirect

71. C₇⋊D₇

D₅×C₁₃⋊C₃

semidirect

74. C₇×C₁₃

C₃×D₅×D₇

semidirect

78. D₃₅

D₇×C₁₃⋊C₃

semidirect

82. C₅×C₁₃

C₃×D₇²

semidirect

85. C₇²

C₂×D₅×C₁₃⋊C₃

semidirect

90. C₁₃⋊C₃

D₃₅⋊D₇

direct (N × H)

92. C₅×C₇

C₂×D₇×C₁₃⋊C₃

semidirect

106. C₁₃

C₃×D₃₅⋊D₇

semidirect

111. C₇

D₅×D₇×C₁₃⋊C₃

semidirect

114. C₅

D₇²×C₁₃⋊C₃

semidirect

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

subgroup	index
C₇×C₁₃⋊C₃×D₃₅	2
C₅×C₁₃⋊C₃×C₇⋊D₇	2
C₁₃×D₃₅⋊D₇	3
D₇²×C₁₃⋊C₃	5
D₅×D₇×C₁₃⋊C₃	7
C₃×D₃₅⋊D₇	13

subgroup

index

C₇×C₁₃⋊C₃×D₃₅

C₅×C₁₃⋊C₃×C₇⋊D₇

C₁₃×D₃₅⋊D₇

D₇²×C₁₃⋊C₃

D₅×D₇×C₁₃⋊C₃

C₃×D₃₅⋊D₇

Subgroups — C₁₃⋊C₃×D₃₅⋊D₇ #50304136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13⋊C3×D35⋊D7 #50304136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃⋊C₃×D₃₅⋊D₇ #50304136