Group #37524621 — subgroups

Subgroups — C₂×C₇×C₉×D₉₁ #37524621

Order 22932.

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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×C₉×D₉₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₇²×C₉×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×D₉₁	C₃	1	1	yes	maximal, characteristic
4	C₇²×C₉×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇×D₉₁	C₂×C₃	1	2	yes	—
6	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
7	C₂×C₉×D₉₁	C₇	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₇×C₉×D₁₃	—	7	1	no	maximal
9	C₂×C₇×D₉₁	C₉	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic
11	C₂×C₇×C₉×D₇	—	13	1	no	maximal, Hall
12	C₂×C₇×C₉×C₁₃	—	2	3	no	—
13	C₂×C₇×C₉×C₁₃	D₇	1	1	yes	characteristic, complemented
14	C₂×C₇×C₉×C₁₃	C₂×C₇	1	1	yes	characteristic, complemented
15	C₉×D₉₁	C₂×C₇	1	2	yes	direct product factor
16	C₇×C₉×D₁₃	—	7	2	no	—
17	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic, complemented
18	C₇×D₉₁	C₂×C₉	1	2	yes	direct product factor
19	C₂×C₃×D₉₁	C₃×C₇	1	1	yes	characteristic
20	C₂×C₃×C₇×D₁₃	—	7	1	no	—
21	C₂×C₇²×C₉	D₁₃	1	1	yes	characteristic, complemented
22	C₇×C₉×D₇	—	13	2	no	—
23	C₇×C₉×C₁₃	—	2	3	no	—
24	C₇×C₉×C₁₃	C₂×D₇	1	1	yes	characteristic, complemented
25	C₇×C₉×C₁₃	C₂²×C₇	1	1	yes	characteristic, complemented
26	C₇²×C₁₃	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₇×D₇	—	13	1	no	—
28	C₂×C₃×C₇×C₁₃	—	2	3	no	—
29	C₂×C₃×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic
30	C₂×C₃×C₇×C₁₃	C₂×C₃×C₇	1	1	yes	characteristic
31	C₃×D₉₁	C₂×C₃×C₇	1	2	yes	—
32	C₃×C₇×D₁₃	—	7	2	no	—
33	C₂×C₉×D₁₃	—	7	1	no	Hall
34	C₇²×C₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
35	C₂×D₉₁	C₇×C₉	1	1	yes	characteristic, direct product factor
36	C₂×C₇×D₁₃	—	7	1	no	—
37	C₂×C₃×C₇²	C₃×D₁₃	1	1	yes	characteristic
38	C₃×C₇×D₇	—	13	2	no	—
39	C₃×C₇×C₁₃	—	2	3	no	—
40	C₃×C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic
41	C₃×C₇×C₁₃	C₂²×C₃×C₇	1	1	yes	characteristic
42	C₂²×C₇×C₉	—	91	1	no	—
43	C₂×C₉×D₇	—	13	1	no	—
44	C₂×C₉×C₁₃	C₇×D₇	1	1	yes	characteristic, complemented
45	C₉×D₁₃	—	7	2	no	—
46	C₂×C₇×D₇	—	13	1	no	Hall
47	C₂×C₇×C₁₃	—	2	3	no	—
48	C₂×C₇×C₁₃	C₉×D₇	1	1	yes	characteristic, complemented
49	C₂×C₇×C₁₃	C₂×C₇×C₉	1	1	yes	characteristic, complemented
50	D₉₁	C₂×C₇×C₉	1	2	yes	direct product factor
51	C₇×D₁₃	—	7	2	no	—
52	C₂×C₃×D₁₃	—	7	1	no	—
53	C₃×C₇²	C₂×C₃×D₁₃	1	1	yes	characteristic
54	C₂×C₇×C₉	—	2	3	no	—
55	C₂×C₇×C₉	—	91	2	no	—
56	C₂×C₇×C₉	C₇×D₁₃	1	1	yes	characteristic, complemented
57	C₂×C₇×C₉	D₉₁	1	1	yes	characteristic, direct product factor
58	C₉×D₇	—	13	2	no	—
59	C₉×C₁₃	C₂×C₇×D₇	1	1	yes	characteristic, complemented, Hall
60	C₂×C₇²	C₉×D₁₃	1	1	yes	characteristic, complemented
61	C₇×D₇	—	13	2	no	—
62	C₇×C₁₃	—	2	3	no	—
63	C₇×C₁₃	C₂×C₉×D₇	1	1	yes	characteristic, complemented
64	C₇×C₁₃	C₂²×C₇×C₉	1	1	yes	characteristic, complemented
65	C₂²×C₃×C₇	—	91	1	no	—
66	C₂×C₃×D₇	—	13	1	no	—
67	C₂×C₃×C₁₃	C₃×C₇×D₇	1	1	yes	characteristic
68	C₃×D₁₃	—	7	2	no	—
69	C₇×C₉	—	2	3	no	—
70	C₇×C₉	C₂×C₇×D₁₃	1	1	yes	characteristic, complemented
71	C₇×C₉	C₂×D₉₁	1	1	yes	characteristic, direct product factor
72	C₂×D₁₃	—	7	1	no	Hall
73	C₇²	C₂×C₉×D₁₃	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃×C₇	—	2	3	no	—
75	C₂×C₃×C₇	—	91	2	no	—
76	C₂×C₃×C₇	C₃×C₇×D₁₃	1	1	yes	characteristic
77	C₂×C₃×C₇	C₃×D₉₁	1	1	yes	characteristic
78	C₃×D₇	—	13	2	no	—
79	C₃×C₁₃	C₂×C₃×C₇×D₇	1	1	yes	characteristic
80	C₂²×C₉	—	91	1	no	Hall
81	C₂²×C₇	—	91	1	no	—
82	C₂×D₇	—	13	1	no	—
83	C₂×C₁₃	C₇×C₉×D₇	1	1	yes	characteristic, complemented
84	D₁₃	—	7	2	no	—
85	C₃×C₇	—	2	3	no	—
86	C₃×C₇	C₂×C₃×C₇×D₁₃	1	1	yes	characteristic
87	C₃×C₇	C₂×C₃×D₉₁	1	1	yes	characteristic
88	C₂×C₉	—	91	2	no	—
89	C₂×C₉	C₇×D₉₁	1	1	yes	characteristic, direct product factor
90	C₂×C₇	—	2	3	no	—
91	C₂×C₇	—	91	2	no	—
92	C₂×C₇	C₇×C₉×D₁₃	1	1	yes	characteristic, complemented
93	C₂×C₇	C₉×D₉₁	1	1	yes	characteristic, direct product factor
94	D₇	—	13	2	no	—
95	C₁₃	C₂×C₇×C₉×D₇	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₃	—	91	1	no	—
97	C₉	C₂×C₇×D₉₁	1	1	yes	characteristic, direct product factor, Sylow
98	C₇	—	2	3	no	—
99	C₇	C₂×C₇×C₉×D₁₃	1	1	yes	characteristic, complemented
100	C₇	C₂×C₉×D₉₁	1	1	yes	characteristic, direct product factor
101	C₂×C₃	—	91	2	no	—
102	C₂×C₃	C₃×C₇×D₉₁	1	1	yes	characteristic
103	C₂²	—	91	1	no	Sylow
104	C₃	C₂×C₃×C₇×D₉₁	1	1	yes	characteristic
105	C₂	—	91	2	no	—
106	C₂	C₇×C₉×D₉₁	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×C₉×D₉₁

C₂

yes

maximal, direct product factor

C₂×C₇²×C₉×C₁₃

C₂

yes

maximal, characteristic, complemented

C₂×C₃×C₇×D₉₁

C₃

yes

maximal, characteristic

C₇²×C₉×C₁₃

C₂²

yes

characteristic, complemented, Hall

C₃×C₇×D₉₁

C₂×C₃

yes

—

C₂×C₃×C₇²×C₁₃

C₂×C₃

yes

characteristic

C₂×C₉×D₉₁

C₇

yes

maximal, characteristic, direct product factor

C₂×C₇×C₉×D₁₃

—

maximal

C₂×C₇×D₉₁

C₉

yes

characteristic, direct product factor, Hall

C₃×C₇²×C₁₃

C₂²×C₃

yes

characteristic

C₂×C₇×C₉×D₇

—

maximal, Hall

C₂×C₇×C₉×C₁₃

—

C₂×C₇×C₉×C₁₃

D₇

yes

characteristic, complemented

C₂×C₇×C₉×C₁₃

C₂×C₇

yes

characteristic, complemented

C₉×D₉₁

C₂×C₇

yes

direct product factor

C₇×C₉×D₁₃

—

C₂×C₇²×C₁₃

C₂×C₉

yes

characteristic, complemented

C₇×D₉₁

C₂×C₉

yes

direct product factor

C₂×C₃×D₉₁

C₃×C₇

yes

characteristic

C₂×C₃×C₇×D₁₃

—

C₂×C₇²×C₉

D₁₃

yes

characteristic, complemented

C₇×C₉×D₇

—

C₇×C₉×C₁₃

—

C₇×C₉×C₁₃

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₃

C₂²×C₇

yes

characteristic, complemented

C₇²×C₁₃

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×D₇

—

C₂×C₃×C₇×C₁₃

—

C₂×C₃×C₇×C₁₃

C₃×D₇

yes

characteristic

C₂×C₃×C₇×C₁₃

C₂×C₃×C₇

yes

characteristic

C₃×D₉₁

C₂×C₃×C₇

yes

—

C₃×C₇×D₁₃

—

C₂×C₉×D₁₃

—

Hall

C₇²×C₉

C₂×D₁₃

yes

characteristic, complemented, Hall

C₂×D₉₁

C₇×C₉

yes

characteristic, direct product factor

C₂×C₇×D₁₃

—

C₂×C₃×C₇²

C₃×D₁₃

yes

characteristic

C₃×C₇×D₇

—

C₃×C₇×C₁₃

—

C₃×C₇×C₁₃

C₂×C₃×D₇

yes

characteristic

C₃×C₇×C₁₃

C₂²×C₃×C₇

yes

characteristic

C₂²×C₇×C₉

—

C₂×C₉×D₇

—

C₂×C₉×C₁₃

C₇×D₇

yes

characteristic, complemented

C₉×D₁₃

—

C₂×C₇×D₇

—

Hall

C₂×C₇×C₁₃

—

C₂×C₇×C₁₃

C₉×D₇

yes

characteristic, complemented

C₂×C₇×C₁₃

C₂×C₇×C₉

yes

characteristic, complemented

D₉₁

C₂×C₇×C₉

yes

direct product factor

C₇×D₁₃

—

C₂×C₃×D₁₃

—

C₃×C₇²

C₂×C₃×D₁₃

yes

characteristic

C₂×C₇×C₉

—

C₂×C₇×C₉

—

C₂×C₇×C₉

C₇×D₁₃

yes

characteristic, complemented

C₂×C₇×C₉

D₉₁

yes

characteristic, direct product factor

C₉×D₇

—

C₉×C₁₃

C₂×C₇×D₇

yes

characteristic, complemented, Hall

C₂×C₇²

C₉×D₁₃

yes

characteristic, complemented

C₇×D₇

—

C₇×C₁₃

—

C₇×C₁₃

C₂×C₉×D₇

yes

characteristic, complemented

C₇×C₁₃

C₂²×C₇×C₉

yes

characteristic, complemented

C₂²×C₃×C₇

—

C₂×C₃×D₇

—

C₂×C₃×C₁₃

C₃×C₇×D₇

yes

characteristic

C₃×D₁₃

—

C₇×C₉

—

C₇×C₉

C₂×C₇×D₁₃

yes

characteristic, complemented

C₇×C₉

C₂×D₉₁

yes

characteristic, direct product factor

C₂×D₁₃

—

Hall

C₇²

C₂×C₉×D₁₃

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

C₃×C₇×D₁₃

yes

characteristic

C₂×C₃×C₇

C₃×D₉₁

yes

characteristic

C₃×D₇

—

C₃×C₁₃

C₂×C₃×C₇×D₇

yes

characteristic

C₂²×C₉

—

Hall

C₂²×C₇

—

C₂×D₇

—

C₂×C₁₃

C₇×C₉×D₇

yes

characteristic, complemented

D₁₃

—

C₃×C₇

—

C₃×C₇

C₂×C₃×C₇×D₁₃

yes

characteristic

C₃×C₇

C₂×C₃×D₉₁

yes

characteristic

C₂×C₉

—

C₂×C₉

C₇×D₉₁

yes

characteristic, direct product factor

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₇×C₉×D₁₃

yes

characteristic, complemented

C₂×C₇

C₉×D₉₁

yes

characteristic, direct product factor

D₇

—

C₁₃

C₂×C₇×C₉×D₇

yes

characteristic, complemented, Sylow

C₂²×C₃

—

C₉

C₂×C₇×D₉₁

yes

characteristic, direct product factor, Sylow

C₇

—

C₇

C₂×C₇×C₉×D₁₃

yes

characteristic, complemented

100

C₇

C₂×C₉×D₉₁

yes

characteristic, direct product factor

101

C₂×C₃

—

102

C₂×C₃

C₃×C₇×D₉₁

yes

characteristic

103

C₂²

—

Sylow

104

C₃

C₂×C₃×C₇×D₉₁

yes

characteristic

105

C₂

—

106

C₂

C₇×C₉×D₉₁

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇×C₉	D₉₁
commutator, hypocenter	C₇×C₁₃	C₂²×C₇×C₉
Frattini	C₃	C₂×C₃×C₇×D₉₁
Fitting	C₂×C₇²×C₉×C₁₃	C₂
socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₃×C₇×D₉₁	C₃
cosocle	C₃×C₇×C₁₃	C₂²×C₃×C₇
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₂×C₇×C₉

subgroup

G/H

center, hypercenter

C₂×C₇×C₉

D₉₁

commutator, hypocenter

C₇×C₁₃

C₂²×C₇×C₉

Frattini

C₃

C₂×C₃×C₇×D₉₁

Fitting

C₂×C₇²×C₉×C₁₃

C₂

socle

C₂×C₃×C₇²×C₁₃

C₂×C₃

prime radical

C₂×C₃×C₇×D₉₁

C₃

cosocle

C₃×C₇×C₁₃

C₂²×C₃×C₇

LCS

G ▷ C₇×C₁₃

UCS

C₁ ◁ C₂×C₇×C₉

N	H ≅ G/N	type
1. C₇×C₉×D₉₁	C₂	direct (N × H)
2. C₂×C₇²×C₉×C₁₃	C₂	semidirect
4. C₇²×C₉×C₁₃	C₂²	semidirect
7. C₂×C₉×D₉₁	C₇	direct (N × H)
9. C₂×C₇×D₉₁	C₉	direct (N × H)
13. C₂×C₇×C₉×C₁₃	D₇	semidirect
14. C₂×C₇×C₉×C₁₃	C₂×C₇	semidirect
15. C₉×D₉₁	C₂×C₇	direct (N × H)
17. C₂×C₇²×C₁₃	C₂×C₉	semidirect
18. C₇×D₉₁	C₂×C₉	direct (N × H)
21. C₂×C₇²×C₉	D₁₃	semidirect
24. C₇×C₉×C₁₃	C₂×D₇	semidirect
25. C₇×C₉×C₁₃	C₂²×C₇	semidirect
26. C₇²×C₁₃	C₂²×C₉	semidirect
34. C₇²×C₉	C₂×D₁₃	semidirect
35. C₂×D₉₁	C₇×C₉	direct (N × H)
44. C₂×C₉×C₁₃	C₇×D₇	semidirect
48. C₂×C₇×C₁₃	C₉×D₇	semidirect
49. C₂×C₇×C₁₃	C₂×C₇×C₉	semidirect
50. D₉₁	C₂×C₇×C₉	direct (N × H)
56. C₂×C₇×C₉	C₇×D₁₃	semidirect
57. C₂×C₇×C₉	D₉₁	direct (N × H)
59. C₉×C₁₃	C₂×C₇×D₇	semidirect
60. C₂×C₇²	C₉×D₁₃	semidirect
63. C₇×C₁₃	C₂×C₉×D₇	semidirect
64. C₇×C₁₃	C₂²×C₇×C₉	semidirect
70. C₇×C₉	C₂×C₇×D₁₃	semidirect
71. C₇×C₉	C₂×D₉₁	direct (N × H)
73. C₇²	C₂×C₉×D₁₃	semidirect
83. C₂×C₁₃	C₇×C₉×D₇	semidirect
89. C₂×C₉	C₇×D₉₁	direct (N × H)
92. C₂×C₇	C₇×C₉×D₁₃	semidirect
93. C₂×C₇	C₉×D₉₁	direct (N × H)
95. C₁₃	C₂×C₇×C₉×D₇	semidirect
97. C₉	C₂×C₇×D₉₁	direct (N × H)
99. C₇	C₂×C₇×C₉×D₁₃	semidirect
100. C₇	C₂×C₉×D₉₁	direct (N × H)
106. C₂	C₇×C₉×D₉₁	direct (N × H)

H ≅ G/N

type

1. C₇×C₉×D₉₁

C₂

direct (N × H)

2. C₂×C₇²×C₉×C₁₃

C₂

semidirect

4. C₇²×C₉×C₁₃

C₂²

semidirect

7. C₂×C₉×D₉₁

C₇

direct (N × H)

9. C₂×C₇×D₉₁

C₉

direct (N × H)

13. C₂×C₇×C₉×C₁₃

D₇

semidirect

14. C₂×C₇×C₉×C₁₃

C₂×C₇

semidirect

15. C₉×D₉₁

C₂×C₇

direct (N × H)

17. C₂×C₇²×C₁₃

C₂×C₉

semidirect

18. C₇×D₉₁

C₂×C₉

direct (N × H)

21. C₂×C₇²×C₉

D₁₃

semidirect

24. C₇×C₉×C₁₃

C₂×D₇

semidirect

25. C₇×C₉×C₁₃

C₂²×C₇

semidirect

26. C₇²×C₁₃

C₂²×C₉

semidirect

34. C₇²×C₉

C₂×D₁₃

semidirect

35. C₂×D₉₁

C₇×C₉

direct (N × H)

44. C₂×C₉×C₁₃

C₇×D₇

semidirect

48. C₂×C₇×C₁₃

C₉×D₇

semidirect

49. C₂×C₇×C₁₃

C₂×C₇×C₉

semidirect

50. D₉₁

C₂×C₇×C₉

direct (N × H)

56. C₂×C₇×C₉

C₇×D₁₃

semidirect

57. C₂×C₇×C₉

D₉₁

direct (N × H)

59. C₉×C₁₃

C₂×C₇×D₇

semidirect

60. C₂×C₇²

C₉×D₁₃

semidirect

63. C₇×C₁₃

C₂×C₉×D₇

semidirect

64. C₇×C₁₃

C₂²×C₇×C₉

semidirect

70. C₇×C₉

C₂×C₇×D₁₃

semidirect

71. C₇×C₉

C₂×D₉₁

direct (N × H)

73. C₇²

C₂×C₉×D₁₃

semidirect

83. C₂×C₁₃

C₇×C₉×D₇

semidirect

89. C₂×C₉

C₇×D₉₁

direct (N × H)

92. C₂×C₇

C₇×C₉×D₁₃

semidirect

93. C₂×C₇

C₉×D₉₁

direct (N × H)

95. C₁₃

C₂×C₇×C₉×D₇

semidirect

97. C₉

C₂×C₇×D₉₁

direct (N × H)

99. C₇

C₂×C₇×C₉×D₁₃

semidirect

100. C₇

C₂×C₉×D₉₁

direct (N × H)

106. C₂

C₇×C₉×D₉₁

direct (N × H)

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

subgroup	index
C₇×C₉×D₉₁	2
C₂×C₇²×C₉×C₁₃	2
C₂×C₃×C₇×D₉₁	3
C₂×C₉×D₉₁	7
C₂×C₇×C₉×D₁₃	7
C₂×C₇×C₉×D₇	13

subgroup

index

C₇×C₉×D₉₁

C₂×C₇²×C₉×C₁₃

C₂×C₃×C₇×D₉₁

C₂×C₉×D₉₁

C₂×C₇×C₉×D₁₃

C₂×C₇×C₉×D₇

Subgroups — C₂×C₇×C₉×D₉₁ #37524621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C7×C9×D91 #37524621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₉×D₉₁ #37524621