Group #41031280 — subgroups

Subgroups — C₃×D₇×D₃₂₅ #41031280

Order 27300.

120 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 44 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	C₃×D₂₂₇₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇×D₃₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₃×D₇×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₃₂₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₇×C₁₃×C₂₅	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×D₇×D₆₅	—	5	1	no	maximal
7	D₂₂₇₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₃₂₅	C₂×C₃	1	1	yes	characteristic, complemented
9	C₁₃×D₇×C₂₅	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₃₂₅	—	7	1	no	maximal, Hall
11	C₃×D₄₅₅	—	5	1	no	—
12	C₃×C₇×D₆₅	—	5	1	no	—
13	C₃×C₅×C₁₃×D₇	D₅	1	1	yes	characteristic
14	C₇×C₁₃×C₂₅	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
15	C₃×D₇×D₂₅	—	13	1	no	maximal, Hall
16	C₂×C₃×C₁₃×C₂₅	—	7	1	no	—
17	C₃×D₃₂₅	—	7	1	no	—
18	C₃×D₃₂₅	D₇	1	1	yes	characteristic, direct product factor
19	D₇×D₆₅	—	5	1	no	—
20	C₃×C₅×C₇×C₁₃	C₂×D₅	1	1	yes	characteristic
21	C₂×D₃₂₅	—	7	1	no	Hall
22	C₃×D₇×D₁₃	—	25	1	no	Hall
23	C₃×D₁₇₅	—	13	1	no	—
24	C₃×C₇×D₂₅	—	13	1	no	—
25	C₃×D₇×C₂₅	D₁₃	1	1	yes	characteristic, complemented
26	C₃×C₁₃×C₂₅	C₂×D₇	1	1	yes	characteristic, complemented, Hall
27	D₄₅₅	—	5	1	no	—
28	C₇×D₆₅	—	5	1	no	—
29	C₅×C₁₃×D₇	C₃×D₅	1	1	yes	characteristic
30	C₂×C₃×D₆₅	—	35	1	no	—
31	D₇×D₂₅	—	13	1	no	Hall
32	C₂×C₁₃×C₂₅	—	7	1	no	—
33	D₃₂₅	—	7	1	no	—
34	D₃₂₅	C₃×D₇	1	1	yes	characteristic, direct product factor
35	C₃×D₉₁	—	25	1	no	—
36	C₃×C₁₃×D₇	D₂₅	1	1	yes	characteristic, complemented
37	C₃×C₇×D₁₃	—	25	1	no	—
38	C₃×C₇×C₂₅	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
39	C₅×C₇×C₁₃	C₂×C₃×D₅	1	1	yes	characteristic
40	C₃×D₅×D₇	—	65	1	no	—
41	C₂×C₃×C₅×C₁₃	—	7	1	no	—
42	C₃×D₆₅	—	5	1	no	—
43	C₃×D₆₅	—	35	1	no	—
44	D₇×D₁₃	—	25	1	no	Hall
45	D₁₇₅	—	13	1	no	—
46	D₇×C₂₅	C₃×D₁₃	1	1	yes	characteristic, complemented
47	C₇×D₂₅	—	13	1	no	—
48	C₁₃×C₂₅	C₂×C₃×D₇	1	1	yes	characteristic, complemented, Hall
49	C₂×C₃×D₂₅	—	91	1	no	Hall
50	C₃×C₇×C₁₃	C₂×D₂₅	1	1	yes	characteristic, complemented, Hall
51	C₂×D₆₅	—	35	1	no	—
52	C₃×D₃₅	—	65	1	no	—
53	C₃×C₇×D₅	—	65	1	no	—
54	C₃×C₅×D₇	D₆₅	1	1	yes	characteristic
55	C₃×C₅×C₁₃	D₅×D₇	1	1	yes	characteristic
56	D₉₁	—	25	1	no	—
57	C₇×D₁₃	—	25	1	no	—
58	C₁₃×D₇	C₃×D₂₅	1	1	yes	characteristic, complemented
59	C₇×C₂₅	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
60	C₂×C₃×D₁₃	—	175	1	no	Hall
61	C₂×C₃×C₂₅	—	7	1	no	—
62	C₃×D₂₅	—	13	1	no	—
63	C₃×D₂₅	—	91	1	no	—
64	D₅×D₇	—	65	1	no	—
65	C₂×C₅×C₁₃	—	7	1	no	—
66	D₆₅	—	5	1	no	—
67	D₆₅	—	35	1	no	—
68	C₃×C₅×C₇	C₂×D₆₅	1	1	yes	characteristic
69	C₂×D₂₅	—	91	1	no	Hall
70	C₇×C₁₃	C₂×C₃×D₂₅	1	1	yes	characteristic, complemented, Hall
71	C₂×C₃×D₇	—	325	1	no	Hall
72	C₂×C₃×C₁₃	—	7	1	no	—
73	C₃×D₁₃	—	25	1	no	—
74	C₃×D₁₃	—	175	1	no	—
75	C₃×C₂₅	D₇×D₁₃	1	1	yes	characteristic, complemented, Hall
76	D₃₅	—	65	1	no	—
77	C₅×D₇	C₃×D₆₅	1	1	yes	characteristic
78	C₇×D₅	—	65	1	no	—
79	C₅×C₁₃	C₃×D₅×D₇	1	1	yes	characteristic
80	C₂×C₃×D₅	—	455	1	no	—
81	C₂×D₁₃	—	175	1	no	Hall
82	C₂×C₂₅	—	7	1	no	—
83	D₂₅	—	13	1	no	—
84	D₂₅	—	91	1	no	—
85	C₂×C₃×C₇	—	325	1	no	—
86	C₃×D₇	—	325	1	no	—
87	C₃×D₇	D₃₂₅	1	1	yes	characteristic, direct product factor
88	C₃×C₁₃	D₇×D₂₅	1	1	yes	characteristic, complemented, Hall
89	C₅×C₇	C₂×C₃×D₆₅	1	1	yes	characteristic
90	C₂×C₃×C₅	—	7	1	no	—
91	C₃×D₅	—	65	1	no	—
92	C₃×D₅	—	455	1	no	—
93	C₂×D₇	—	325	1	no	Hall
94	C₂×C₁₃	—	7	1	no	—
95	D₁₃	—	25	1	no	—
96	D₁₃	—	175	1	no	—
97	C₂₅	C₃×D₇×D₁₃	1	1	yes	characteristic, complemented, Sylow
98	C₃×C₇	C₂×D₃₂₅	1	1	yes	characteristic, complemented, Hall
99	C₂×D₅	—	455	1	no	—
100	C₃×C₅	D₇×D₆₅	1	1	yes	characteristic
101	C₂×C₇	—	325	1	no	—
102	D₇	—	325	1	no	—
103	D₇	C₃×D₃₂₅	1	1	yes	characteristic, direct product factor
104	C₁₃	C₃×D₇×D₂₅	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₃	—	2275	1	no	Hall
106	C₂×C₅	—	7	1	no	—
107	D₅	—	65	1	no	—
108	D₅	—	455	1	no	—
109	C₇	C₂×C₃×D₃₂₅	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	7	1	no	—
111	C₂×C₃	—	325	1	no	—
112	C₂×C₃	—	2275	1	no	—
113	C₅	C₃×D₇×D₆₅	1	1	yes	characteristic
114	C₂²	—	2275	1	no	Sylow
115	C₃	D₇×D₃₂₅	1	1	yes	characteristic, direct product factor, Sylow
116	C₂	—	7	1	no	—
117	C₂	—	325	1	no	—
118	C₂	—	2275	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×D₂₂₇₅

C₂

yes

maximal, characteristic, complemented

C₃×C₇×D₃₂₅

C₂

yes

maximal, characteristic, complemented

C₃×C₁₃×D₇×C₂₅

C₂

yes

maximal, characteristic, complemented

D₇×D₃₂₅

C₃

yes

maximal, characteristic, direct product factor, Hall

C₃×C₇×C₁₃×C₂₅

C₂²

yes

characteristic, complemented, Hall

C₃×D₇×D₆₅

—

maximal

D₂₂₇₅

C₂×C₃

yes

characteristic, complemented

C₇×D₃₂₅

C₂×C₃

yes

characteristic, complemented

C₁₃×D₇×C₂₅

C₂×C₃

yes

characteristic, complemented

C₂×C₃×D₃₂₅

—

maximal, Hall

C₃×D₄₅₅

—

C₃×C₇×D₆₅

—

C₃×C₅×C₁₃×D₇

D₅

yes

characteristic

C₇×C₁₃×C₂₅

C₂²×C₃

yes

characteristic, complemented, Hall

C₃×D₇×D₂₅

—

maximal, Hall

C₂×C₃×C₁₃×C₂₅

—

C₃×D₃₂₅

—

C₃×D₃₂₅

D₇

yes

characteristic, direct product factor

D₇×D₆₅

—

C₃×C₅×C₇×C₁₃

C₂×D₅

yes

characteristic

C₂×D₃₂₅

—

Hall

C₃×D₇×D₁₃

—

Hall

C₃×D₁₇₅

—

C₃×C₇×D₂₅

—

C₃×D₇×C₂₅

D₁₃

yes

characteristic, complemented

C₃×C₁₃×C₂₅

C₂×D₇

yes

characteristic, complemented, Hall

D₄₅₅

—

C₇×D₆₅

—

C₅×C₁₃×D₇

C₃×D₅

yes

characteristic

C₂×C₃×D₆₅

—

D₇×D₂₅

—

Hall

C₂×C₁₃×C₂₅

—

D₃₂₅

—

D₃₂₅

C₃×D₇

yes

characteristic, direct product factor

C₃×D₉₁

—

C₃×C₁₃×D₇

D₂₅

yes

characteristic, complemented

C₃×C₇×D₁₃

—

C₃×C₇×C₂₅

C₂×D₁₃

yes

characteristic, complemented, Hall

C₅×C₇×C₁₃

C₂×C₃×D₅

yes

characteristic

C₃×D₅×D₇

—

C₂×C₃×C₅×C₁₃

—

C₃×D₆₅

—

C₃×D₆₅

—

D₇×D₁₃

—

Hall

D₁₇₅

—

D₇×C₂₅

C₃×D₁₃

yes

characteristic, complemented

C₇×D₂₅

—

C₁₃×C₂₅

C₂×C₃×D₇

yes

characteristic, complemented, Hall

C₂×C₃×D₂₅

—

Hall

C₃×C₇×C₁₃

C₂×D₂₅

yes

characteristic, complemented, Hall

C₂×D₆₅

—

C₃×D₃₅

—

C₃×C₇×D₅

—

C₃×C₅×D₇

D₆₅

yes

characteristic

C₃×C₅×C₁₃

D₅×D₇

yes

characteristic

D₉₁

—

C₇×D₁₃

—

C₁₃×D₇

C₃×D₂₅

yes

characteristic, complemented

C₇×C₂₅

C₂×C₃×D₁₃

yes

characteristic, complemented, Hall

C₂×C₃×D₁₃

—

175

Hall

C₂×C₃×C₂₅

—

C₃×D₂₅

—

C₃×D₂₅

—

D₅×D₇

—

C₂×C₅×C₁₃

—

D₆₅

—

D₆₅

—

C₃×C₅×C₇

C₂×D₆₅

yes

characteristic

C₂×D₂₅

—

Hall

C₇×C₁₃

C₂×C₃×D₂₅

yes

characteristic, complemented, Hall

C₂×C₃×D₇

—

325

Hall

C₂×C₃×C₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

175

—

C₃×C₂₅

D₇×D₁₃

yes

characteristic, complemented, Hall

D₃₅

—

C₅×D₇

C₃×D₆₅

yes

characteristic

C₇×D₅

—

C₅×C₁₃

C₃×D₅×D₇

yes

characteristic

C₂×C₃×D₅

—

455

—

C₂×D₁₃

—

175

Hall

C₂×C₂₅

—

D₂₅

—

D₂₅

—

C₂×C₃×C₇

—

325

—

C₃×D₇

—

325

—

C₃×D₇

D₃₂₅

yes

characteristic, direct product factor

C₃×C₁₃

D₇×D₂₅

yes

characteristic, complemented, Hall

C₅×C₇

C₂×C₃×D₆₅

yes

characteristic

C₂×C₃×C₅

—

C₃×D₅

—

C₃×D₅

—

455

—

C₂×D₇

—

325

Hall

C₂×C₁₃

—

D₁₃

—

D₁₃

—

175

—

C₂₅

C₃×D₇×D₁₃

yes

characteristic, complemented, Sylow

C₃×C₇

C₂×D₃₂₅

yes

characteristic, complemented, Hall

C₂×D₅

—

455

—

100

C₃×C₅

D₇×D₆₅

yes

characteristic

101

C₂×C₇

—

325

—

102

D₇

—

325

—

103

D₇

C₃×D₃₂₅

yes

characteristic, direct product factor

104

C₁₃

C₃×D₇×D₂₅

yes

characteristic, complemented, Sylow

105

C₂²×C₃

—

2275

Hall

106

C₂×C₅

—

107

D₅

—

108

D₅

—

455

—

109

C₇

C₂×C₃×D₃₂₅

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

C₂×C₃

—

325

—

112

C₂×C₃

—

2275

—

113

C₅

C₃×D₇×D₆₅

yes

characteristic

114

C₂²

—

2275

Sylow

115

C₃

D₇×D₃₂₅

yes

characteristic, direct product factor, Sylow

116

C₂

—

117

C₂

—

325

—

118

C₂

—

2275

—

	subgroup	G/H
center, hypercenter	C₃	D₇×D₃₂₅
commutator, cosocle, hypocenter	C₇×C₁₃×C₂₅	C₂²×C₃
Frattini	C₅	C₃×D₇×D₆₅
Fitting	C₃×C₇×C₁₃×C₂₅	C₂²
socle	C₃×C₅×C₇×C₁₃	C₂×D₅
LCS	G ▷ C₇×C₁₃×C₂₅
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

D₇×D₃₂₅

commutator, cosocle, hypocenter

C₇×C₁₃×C₂₅

C₂²×C₃

Frattini

C₅

C₃×D₇×D₆₅

Fitting

C₃×C₇×C₁₃×C₂₅

C₂²

socle

C₃×C₅×C₇×C₁₃

C₂×D₅

LCS

G ▷ C₇×C₁₃×C₂₅

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. C₃×D₂₂₇₅	C₂	semidirect
2. C₃×C₇×D₃₂₅	C₂	semidirect
3. C₃×C₁₃×D₇×C₂₅	C₂	semidirect
4. D₇×D₃₂₅	C₃	direct (N × H)
5. C₃×C₇×C₁₃×C₂₅	C₂²	semidirect
7. D₂₂₇₅	C₂×C₃	semidirect
8. C₇×D₃₂₅	C₂×C₃	semidirect
9. C₁₃×D₇×C₂₅	C₂×C₃	semidirect
14. C₇×C₁₃×C₂₅	C₂²×C₃	semidirect
18. C₃×D₃₂₅	D₇	direct (N × H)
25. C₃×D₇×C₂₅	D₁₃	semidirect
26. C₃×C₁₃×C₂₅	C₂×D₇	semidirect
34. D₃₂₅	C₃×D₇	direct (N × H)
36. C₃×C₁₃×D₇	D₂₅	semidirect
38. C₃×C₇×C₂₅	C₂×D₁₃	semidirect
46. D₇×C₂₅	C₃×D₁₃	semidirect
48. C₁₃×C₂₅	C₂×C₃×D₇	semidirect
50. C₃×C₇×C₁₃	C₂×D₂₅	semidirect
58. C₁₃×D₇	C₃×D₂₅	semidirect
59. C₇×C₂₅	C₂×C₃×D₁₃	semidirect
70. C₇×C₁₃	C₂×C₃×D₂₅	semidirect
75. C₃×C₂₅	D₇×D₁₃	semidirect
87. C₃×D₇	D₃₂₅	direct (N × H)
88. C₃×C₁₃	D₇×D₂₅	semidirect
97. C₂₅	C₃×D₇×D₁₃	semidirect
98. C₃×C₇	C₂×D₃₂₅	semidirect
103. D₇	C₃×D₃₂₅	direct (N × H)
104. C₁₃	C₃×D₇×D₂₅	semidirect
109. C₇	C₂×C₃×D₃₂₅	semidirect
115. C₃	D₇×D₃₂₅	direct (N × H)

H ≅ G/N

type

1. C₃×D₂₂₇₅

C₂

semidirect

2. C₃×C₇×D₃₂₅

C₂

semidirect

3. C₃×C₁₃×D₇×C₂₅

C₂

semidirect

4. D₇×D₃₂₅

C₃

direct (N × H)

5. C₃×C₇×C₁₃×C₂₅

C₂²

semidirect

7. D₂₂₇₅

C₂×C₃

semidirect

8. C₇×D₃₂₅

C₂×C₃

semidirect

9. C₁₃×D₇×C₂₅

C₂×C₃

semidirect

14. C₇×C₁₃×C₂₅

C₂²×C₃

semidirect

18. C₃×D₃₂₅

D₇

direct (N × H)

25. C₃×D₇×C₂₅

D₁₃

semidirect

26. C₃×C₁₃×C₂₅

C₂×D₇

semidirect

34. D₃₂₅

C₃×D₇

direct (N × H)

36. C₃×C₁₃×D₇

D₂₅

semidirect

38. C₃×C₇×C₂₅

C₂×D₁₃

semidirect

46. D₇×C₂₅

C₃×D₁₃

semidirect

48. C₁₃×C₂₅

C₂×C₃×D₇

semidirect

50. C₃×C₇×C₁₃

C₂×D₂₅

semidirect

58. C₁₃×D₇

C₃×D₂₅

semidirect

59. C₇×C₂₅

C₂×C₃×D₁₃

semidirect

70. C₇×C₁₃

C₂×C₃×D₂₅

semidirect

75. C₃×C₂₅

D₇×D₁₃

semidirect

87. C₃×D₇

D₃₂₅

direct (N × H)

88. C₃×C₁₃

D₇×D₂₅

semidirect

97. C₂₅

C₃×D₇×D₁₃

semidirect

98. C₃×C₇

C₂×D₃₂₅

semidirect

103. D₇

C₃×D₃₂₅

direct (N × H)

104. C₁₃

C₃×D₇×D₂₅

semidirect

109. C₇

C₂×C₃×D₃₂₅

semidirect

115. C₃

D₇×D₃₂₅

direct (N × H)

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

subgroup	index
C₃×D₂₂₇₅	2
C₃×C₇×D₃₂₅	2
C₃×C₁₃×D₇×C₂₅	2
D₇×D₃₂₅	3
C₃×D₇×D₆₅	5
C₂×C₃×D₃₂₅	7
C₃×D₇×D₂₅	13

subgroup

index

C₃×D₂₂₇₅

C₃×C₇×D₃₂₅

C₃×C₁₃×D₇×C₂₅

D₇×D₃₂₅

C₃×D₇×D₆₅

C₂×C₃×D₃₂₅

C₃×D₇×D₂₅

Subgroups — C₃×D₇×D₃₂₅ #41031280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×D7×D325 #41031280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₇×D₃₂₅ #41031280