Group #37561603 — subgroups

Subgroups — D₁₃₃⋊(C₃⋊D₁₅) #37561603

Order 23940.

240 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 65 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×D₁₃₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₃⋊D₂₈₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×C₃⋊D₁₀₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₃₃⋊D₁₅	—	3	4	no	maximal
5	C₃²×C₅×C₇×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₃₃⋊(C₃⋊D₃)	—	5	1	no	maximal, Hall
7	C₁₉×D₁₀₅	—	3	4	no	—
8	C₇×D₂₈₅	—	3	4	no	—
9	C₃×C₅×D₁₃₃	D₃	1	4	yes	complemented
10	D₁₉×C₃⋊D₁₅	—	7	1	no	maximal, Hall
11	D₃₅⋊D₁₉	—	9	1	no	Hall
12	C₁₉×C₃⋊D₂₁	—	5	1	no	—
13	C₇×C₃⋊D₅₇	—	5	1	no	—
14	C₃²×D₁₃₃	D₅	1	1	yes	characteristic, complemented
15	C₃×C₅×C₇×C₁₉	C₂×D₃	1	4	yes	complemented
16	C₃⋊D₂₈₅	D₇	1	1	yes	characteristic, complemented
17	C₁₉×C₃⋊D₁₅	—	7	1	no	—
18	C₃²×C₅×D₁₉	—	7	1	no	—
19	D₂₁⋊D₁₉	—	15	4	no	—
20	C₁₉×D₃₅	—	9	1	no	—
21	C₇×D₉₅	—	9	1	no	—
22	C₅×D₁₃₃	C₃⋊D₃	1	1	yes	characteristic, complemented
23	D₇×C₃⋊D₁₅	—	19	1	no	maximal, Hall
24	C₃²×C₇×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
25	D₁₅×D₁₉	—	21	4	no	—
26	C₃²×C₅×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented, Hall
27	C₁₉×D₂₁	—	15	4	no	—
28	C₇×D₅₇	—	15	4	no	—
29	C₃×D₁₃₃	D₁₅	1	4	yes	complemented
30	C₃⋊D₃×D₁₉	—	35	1	no	Hall
31	C₅×C₇×C₁₉	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
32	C₃⋊D₁₀₅	D₁₉	1	1	yes	characteristic, complemented
33	C₇×C₃⋊D₁₅	—	19	1	no	—
34	C₃²×C₅×D₇	—	19	1	no	—
35	D₂₈₅	—	3	4	no	—
36	C₁₉×D₁₅	—	21	4	no	—
37	C₃×C₅×D₁₉	—	7	4	no	—
38	D₇×D₁₉	—	45	1	no	Hall
39	D₇×D₁₅	—	57	4	no	—
40	C₃×C₇×C₁₉	C₂×D₁₅	1	4	yes	complemented
41	D₅×D₁₉	—	63	1	no	Hall
42	C₃⋊D₅₇	—	5	1	no	—
43	C₃⋊D₃×C₁₉	—	35	1	no	—
44	C₃²×D₁₉	—	7	1	no	—
45	C₃²×C₅×C₇	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
46	C₃×C₅×C₁₉	D₃×D₇	1	4	yes	complemented
47	D₁₃₃	C₃⋊D₁₅	1	1	yes	characteristic, complemented
48	C₇×D₁₉	—	45	1	no	—
49	D₇×C₁₉	—	45	1	no	—
50	D₇×C₃⋊D₃	—	95	1	no	Hall
51	D₃×D₁₉	—	105	4	no	—
52	D₁₀₅	—	3	4	no	—
53	C₇×D₁₅	—	57	4	no	—
54	C₃×C₅×D₇	—	19	4	no	—
55	D₉₅	—	9	1	no	—
56	C₅×D₁₉	—	7	1	no	—
57	D₅×C₁₉	—	63	1	no	—
58	C₂×C₃⋊D₁₅	—	133	1	no	Hall
59	C₃²×C₁₉	D₅×D₇	1	1	yes	characteristic, complemented, Hall
60	D₅×D₇	—	171	1	no	Hall
61	C₇×C₁₉	C₂×C₃⋊D₁₅	1	1	yes	characteristic, complemented, Hall
62	C₃⋊D₂₁	—	5	1	no	—
63	C₇×C₃⋊D₃	—	95	1	no	—
64	C₃²×D₇	—	19	1	no	—
65	D₅₇	—	15	4	no	—
66	C₃×D₁₉	—	7	4	no	—
67	D₃×C₁₉	—	105	4	no	—
68	C₃×C₅×C₇	D₃×D₁₉	1	4	yes	complemented
69	C₅×C₁₉	D₇×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
70	C₂×C₃²×C₅	—	133	1	no	—
71	C₃⋊D₁₅	—	7	1	no	—
72	C₃⋊D₁₅	—	19	1	no	—
73	D₃×D₇	—	285	4	no	—
74	C₂×D₁₉	—	315	1	no	Hall
75	D₃₅	—	9	1	no	—
76	C₅×D₇	—	19	1	no	—
77	C₇×D₅	—	171	1	no	—
78	C₃²×C₇	D₅×D₁₉	1	1	yes	characteristic, complemented, Hall
79	C₂×D₁₅	—	399	4	no	—
80	C₃×C₁₉	D₇×D₁₅	1	4	yes	complemented
81	C₃²×C₅	D₇×D₁₉	1	1	yes	characteristic, complemented, Hall
82	D₂₁	—	15	4	no	—
83	C₃×D₇	—	19	4	no	—
84	D₃×C₇	—	285	4	no	—
85	C₂×C₁₉	—	315	1	no	—
86	D₁₉	—	7	1	no	—
87	D₁₉	—	45	1	no	—
88	C₂×C₃⋊D₃	—	665	1	no	Hall
89	C₅×C₇	C₃⋊D₃×D₁₉	1	1	yes	characteristic, complemented, Hall
90	C₂×C₃×C₅	—	133	4	no	—
91	D₁₅	—	21	4	no	—
92	D₁₅	—	57	4	no	—
93	C₂×D₇	—	855	1	no	Hall
94	C₃×C₇	D₁₅×D₁₉	1	4	yes	complemented
95	C₂×D₅	—	1197	1	no	Hall
96	C₁₉	D₇×C₃⋊D₁₅	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₃²	—	133	1	no	—
98	C₃⋊D₃	—	35	1	no	—
99	C₃⋊D₃	—	95	1	no	—
100	C₃×C₅	D₂₁⋊D₁₉	1	4	yes	complemented
101	C₂×C₇	—	855	1	no	—
102	D₇	—	19	1	no	—
103	D₇	—	45	1	no	—
104	C₂×D₃	—	1995	4	no	—
105	C₂×C₅	—	133	1	no	—
106	D₅	—	63	1	no	—
107	D₅	—	171	1	no	—
108	C₃²	D₃₅⋊D₁₉	1	1	yes	characteristic, complemented, Sylow
109	C₇	D₁₉×C₃⋊D₁₅	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	133	4	no	—
111	D₃	—	285	4	no	—
112	D₃	—	105	4	no	—
113	C₅	D₁₃₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	5985	1	no	Sylow
115	C₃	D₁₃₃⋊D₁₅	1	4	yes	complemented
116	C₂	—	315	1	no	—
117	C₂	—	855	1	no	—
118	C₂	—	133	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₅×D₁₃₃

C₂

yes

maximal, characteristic, complemented

C₇×C₃⋊D₂₈₅

C₂

yes

maximal, characteristic, complemented

C₁₉×C₃⋊D₁₀₅

C₂

yes

maximal, characteristic, complemented

D₁₃₃⋊D₁₅

—

maximal

C₃²×C₅×C₇×C₁₉

C₂²

yes

characteristic, complemented, Hall

D₁₃₃⋊(C₃⋊D₃)

—

maximal, Hall

C₁₉×D₁₀₅

—

C₇×D₂₈₅

—

C₃×C₅×D₁₃₃

D₃

yes

complemented

D₁₉×C₃⋊D₁₅

—

maximal, Hall

D₃₅⋊D₁₉

—

Hall

C₁₉×C₃⋊D₂₁

—

C₇×C₃⋊D₅₇

—

C₃²×D₁₃₃

D₅

yes

characteristic, complemented

C₃×C₅×C₇×C₁₉

C₂×D₃

yes

complemented

C₃⋊D₂₈₅

D₇

yes

characteristic, complemented

C₁₉×C₃⋊D₁₅

—

C₃²×C₅×D₁₉

—

D₂₁⋊D₁₉

—

C₁₉×D₃₅

—

C₇×D₉₅

—

C₅×D₁₃₃

C₃⋊D₃

yes

characteristic, complemented

D₇×C₃⋊D₁₅

—

maximal, Hall

C₃²×C₇×C₁₉

C₂×D₅

yes

characteristic, complemented, Hall

D₁₅×D₁₉

—

C₃²×C₅×C₁₉

C₂×D₇

yes

characteristic, complemented, Hall

C₁₉×D₂₁

—

C₇×D₅₇

—

C₃×D₁₃₃

D₁₅

yes

complemented

C₃⋊D₃×D₁₉

—

Hall

C₅×C₇×C₁₉

C₂×C₃⋊D₃

yes

characteristic, complemented, Hall

C₃⋊D₁₀₅

D₁₉

yes

characteristic, complemented

C₇×C₃⋊D₁₅

—

C₃²×C₅×D₇

—

D₂₈₅

—

C₁₉×D₁₅

—

C₃×C₅×D₁₉

—

D₇×D₁₉

—

Hall

D₇×D₁₅

—

C₃×C₇×C₁₉

C₂×D₁₅

yes

complemented

D₅×D₁₉

—

Hall

C₃⋊D₅₇

—

C₃⋊D₃×C₁₉

—

C₃²×D₁₉

—

C₃²×C₅×C₇

C₂×D₁₉

yes

characteristic, complemented, Hall

C₃×C₅×C₁₉

D₃×D₇

yes

complemented

D₁₃₃

C₃⋊D₁₅

yes

characteristic, complemented

C₇×D₁₉

—

D₇×C₁₉

—

D₇×C₃⋊D₃

—

Hall

D₃×D₁₉

—

105

—

D₁₀₅

—

C₇×D₁₅

—

C₃×C₅×D₇

—

D₉₅

—

C₅×D₁₉

—

D₅×C₁₉

—

C₂×C₃⋊D₁₅

—

133

Hall

C₃²×C₁₉

D₅×D₇

yes

characteristic, complemented, Hall

D₅×D₇

—

171

Hall

C₇×C₁₉

C₂×C₃⋊D₁₅

yes

characteristic, complemented, Hall

C₃⋊D₂₁

—

C₇×C₃⋊D₃

—

C₃²×D₇

—

D₅₇

—

C₃×D₁₉

—

D₃×C₁₉

—

105

—

C₃×C₅×C₇

D₃×D₁₉

yes

complemented

C₅×C₁₉

D₇×C₃⋊D₃

yes

characteristic, complemented, Hall

C₂×C₃²×C₅

—

133

—

C₃⋊D₁₅

—

C₃⋊D₁₅

—

D₃×D₇

—

285

—

C₂×D₁₉

—

315

Hall

D₃₅

—

C₅×D₇

—

C₇×D₅

—

171

—

C₃²×C₇

D₅×D₁₉

yes

characteristic, complemented, Hall

C₂×D₁₅

—

399

—

C₃×C₁₉

D₇×D₁₅

yes

complemented

C₃²×C₅

D₇×D₁₉

yes

characteristic, complemented, Hall

D₂₁

—

C₃×D₇

—

D₃×C₇

—

285

—

C₂×C₁₉

—

315

—

D₁₉

—

D₁₉

—

C₂×C₃⋊D₃

—

665

Hall

C₅×C₇

C₃⋊D₃×D₁₉

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

133

—

D₁₅

—

D₁₅

—

C₂×D₇

—

855

Hall

C₃×C₇

D₁₅×D₁₉

yes

complemented

C₂×D₅

—

1197

Hall

C₁₉

D₇×C₃⋊D₁₅

yes

characteristic, complemented, Sylow

C₂×C₃²

—

133

—

C₃⋊D₃

—

C₃⋊D₃

—

100

C₃×C₅

D₂₁⋊D₁₉

yes

complemented

101

C₂×C₇

—

855

—

102

D₇

—

103

D₇

—

104

C₂×D₃

—

1995

—

105

C₂×C₅

—

133

—

106

D₅

—

107

D₅

—

171

—

108

C₃²

D₃₅⋊D₁₉

yes

characteristic, complemented, Sylow

109

C₇

D₁₉×C₃⋊D₁₅

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

133

—

111

D₃

—

285

—

112

D₃

—

105

—

113

C₅

D₁₃₃⋊(C₃⋊D₃)

yes

characteristic, complemented, Sylow

114

C₂²

—

5985

Sylow

115

C₃

D₁₃₃⋊D₁₅

yes

complemented

116

C₂

—

315

—

117

C₂

—

855

—

118

C₂

—

133

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅×C₇×C₁₉	C₂²
LCS	G ▷ C₃²×C₅×C₇×C₁₉
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃²×C₅×C₇×C₁₉

C₂²

LCS

G ▷ C₃²×C₅×C₇×C₁₉

UCS

C₁

N	H ≅ G/N	type
1. C₃²×C₅×D₁₃₃	C₂	semidirect
2. C₇×C₃⋊D₂₈₅	C₂	semidirect
3. C₁₉×C₃⋊D₁₀₅	C₂	semidirect
5. C₃²×C₅×C₇×C₁₉	C₂²	semidirect
9. C₃×C₅×D₁₃₃	D₃	semidirect
14. C₃²×D₁₃₃	D₅	semidirect
15. C₃×C₅×C₇×C₁₉	C₂×D₃	semidirect
16. C₃⋊D₂₈₅	D₇	semidirect
22. C₅×D₁₃₃	C₃⋊D₃	semidirect
24. C₃²×C₇×C₁₉	C₂×D₅	semidirect
26. C₃²×C₅×C₁₉	C₂×D₇	semidirect
29. C₃×D₁₃₃	D₁₅	semidirect
31. C₅×C₇×C₁₉	C₂×C₃⋊D₃	semidirect
32. C₃⋊D₁₀₅	D₁₉	semidirect
40. C₃×C₇×C₁₉	C₂×D₁₅	semidirect
45. C₃²×C₅×C₇	C₂×D₁₉	semidirect
46. C₃×C₅×C₁₉	D₃×D₇	semidirect
47. D₁₃₃	C₃⋊D₁₅	semidirect
59. C₃²×C₁₉	D₅×D₇	semidirect
61. C₇×C₁₉	C₂×C₃⋊D₁₅	semidirect
68. C₃×C₅×C₇	D₃×D₁₉	semidirect
69. C₅×C₁₉	D₇×C₃⋊D₃	semidirect
78. C₃²×C₇	D₅×D₁₉	semidirect
80. C₃×C₁₉	D₇×D₁₅	semidirect
81. C₃²×C₅	D₇×D₁₉	semidirect
89. C₅×C₇	C₃⋊D₃×D₁₉	semidirect
94. C₃×C₇	D₁₅×D₁₉	semidirect
96. C₁₉	D₇×C₃⋊D₁₅	semidirect
100. C₃×C₅	D₂₁⋊D₁₉	semidirect
108. C₃²	D₃₅⋊D₁₉	semidirect
109. C₇	D₁₉×C₃⋊D₁₅	semidirect
113. C₅	D₁₃₃⋊(C₃⋊D₃)	semidirect
115. C₃	D₁₃₃⋊D₁₅	semidirect

H ≅ G/N

type

1. C₃²×C₅×D₁₃₃

C₂

semidirect

2. C₇×C₃⋊D₂₈₅

C₂

semidirect

3. C₁₉×C₃⋊D₁₀₅

C₂

semidirect

5. C₃²×C₅×C₇×C₁₉

C₂²

semidirect

9. C₃×C₅×D₁₃₃

D₃

semidirect

14. C₃²×D₁₃₃

D₅

semidirect

15. C₃×C₅×C₇×C₁₉

C₂×D₃

semidirect

16. C₃⋊D₂₈₅

D₇

semidirect

22. C₅×D₁₃₃

C₃⋊D₃

semidirect

24. C₃²×C₇×C₁₉

C₂×D₅

semidirect

26. C₃²×C₅×C₁₉

C₂×D₇

semidirect

29. C₃×D₁₃₃

D₁₅

semidirect

31. C₅×C₇×C₁₉

C₂×C₃⋊D₃

semidirect

32. C₃⋊D₁₀₅

D₁₉

semidirect

40. C₃×C₇×C₁₉

C₂×D₁₅

semidirect

45. C₃²×C₅×C₇

C₂×D₁₉

semidirect

46. C₃×C₅×C₁₉

D₃×D₇

semidirect

47. D₁₃₃

C₃⋊D₁₅

semidirect

59. C₃²×C₁₉

D₅×D₇

semidirect

61. C₇×C₁₉

C₂×C₃⋊D₁₅

semidirect

68. C₃×C₅×C₇

D₃×D₁₉

semidirect

69. C₅×C₁₉

D₇×C₃⋊D₃

semidirect

78. C₃²×C₇

D₅×D₁₉

semidirect

80. C₃×C₁₉

D₇×D₁₅

semidirect

81. C₃²×C₅

D₇×D₁₉

semidirect

89. C₅×C₇

C₃⋊D₃×D₁₉

semidirect

94. C₃×C₇

D₁₅×D₁₉

semidirect

96. C₁₉

D₇×C₃⋊D₁₅

semidirect

100. C₃×C₅

D₂₁⋊D₁₉

semidirect

108. C₃²

D₃₅⋊D₁₉

semidirect

109. C₇

D₁₉×C₃⋊D₁₅

semidirect

113. C₅

D₁₃₃⋊(C₃⋊D₃)

semidirect

115. C₃

D₁₃₃⋊D₁₅

semidirect

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

subgroup	index
C₃²×C₅×D₁₃₃	2
C₇×C₃⋊D₂₈₅	2
C₁₉×C₃⋊D₁₀₅	2
D₁₃₃⋊D₁₅	3
D₁₃₃⋊(C₃⋊D₃)	5
D₁₉×C₃⋊D₁₅	7
D₇×C₃⋊D₁₅	19

subgroup

index

C₃²×C₅×D₁₃₃

C₇×C₃⋊D₂₈₅

C₁₉×C₃⋊D₁₀₅

D₁₃₃⋊D₁₅

D₁₃₃⋊(C₃⋊D₃)

D₁₉×C₃⋊D₁₅

D₇×C₃⋊D₁₅

Subgroups — D₁₃₃⋊(C₃⋊D₁₅) #37561603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D133⋊(C3⋊D15) #37561603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃₃⋊(C₃⋊D₁₅) #37561603