Group #37524554 — subgroups

Subgroups — C₂×C₇⋊C₉×D₉₁ #37524554

Order 22932.

128 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×C₇×C₁₃×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇⋊C₉×D₉₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₇×D₉₁	C₃	1	1	yes	maximal, characteristic
4	C₇×C₁₃×C₇⋊C₉	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	—	7	1	no	maximal
8	C₂×D₁₃×C₇⋊C₉	—	7	1	no	maximal
9	C₂×C₇×D₉₁	C₉	1	1	yes	characteristic, complemented, Hall
10	C₃×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic
11	C₂×D₇×C₇⋊C₉	—	13	1	no	maximal, Hall
12	C₂×C₇×C₉×C₁₃	—	7	1	no	—
13	C₉×D₉₁	—	7	2	no	—
14	C₂×C₁₃×C₇⋊C₉	D₇	1	1	yes	characteristic, complemented
15	D₁₃×C₇⋊C₉	—	7	2	no	—
16	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic, complemented
17	C₇×D₉₁	C₂×C₉	1	2	yes	complemented
18	C₂×C₃×D₉₁	ΓL(1,𝔽₈)	1	1	yes	characteristic
19	C₂×C₃×C₇×D₁₃	—	7	1	no	—
20	C₂×C₇×C₇⋊C₉	D₁₃	1	1	yes	characteristic, complemented
21	D₇×C₇⋊C₉	—	13	2	no	—
22	C₇×C₉×C₁₃	—	7	1	no	—
23	C₁₃×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
24	C₇²×C₁₃	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₇×D₇	—	13	1	no	—
26	C₂×C₃×C₇×C₁₃	—	6	1	no	—
27	C₂×C₃×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₂×C₃×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic
29	C₃×D₉₁	C₂×ΓL(1,𝔽₈)	1	2	yes	—
30	C₃×C₇×D₁₃	—	7	2	no	—
31	C₂×C₉×D₁₃	—	49	1	no	Hall
32	C₇×C₇⋊C₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
33	C₂×D₉₁	C₇⋊C₉	1	1	yes	characteristic, direct product factor
34	C₂×C₇×D₁₃	—	7	1	no	—
35	C₂×C₃×C₇²	C₃×D₁₃	1	1	yes	characteristic
36	C₃×C₇×D₇	—	13	2	no	—
37	C₃×C₇×C₁₃	—	6	1	no	—
38	C₃×C₇×C₁₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
39	C₃×C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic
40	C₂×C₉×D₇	—	91	1	no	—
41	C₂²×C₇⋊C₉	—	91	1	no	—
42	C₂×C₉×C₁₃	—	7	1	no	—
43	C₉×D₁₃	—	49	2	no	—
44	C₂×C₇×D₇	—	13	1	no	Hall
45	C₂×C₇×C₁₃	—	6	1	no	—
46	C₂×C₇×C₁₃	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
47	C₂×C₇×C₁₃	C₉×D₇	1	1	yes	characteristic, complemented
48	D₉₁	C₂×C₇⋊C₉	1	2	yes	direct product factor
49	C₇×D₁₃	—	7	2	no	—
50	C₂×C₃×D₁₃	—	7	1	no	—
51	C₃×C₇²	C₂×C₃×D₁₃	1	1	yes	characteristic
52	C₂×C₇×C₉	—	7	1	no	—
53	C₉×D₇	—	91	2	no	—
54	C₂×C₇⋊C₉	—	91	2	no	—
55	C₂×C₇⋊C₉	D₉₁	1	1	yes	characteristic, direct product factor
56	C₉×C₁₃	—	7	1	no	Hall
57	C₂×C₇²	C₉×D₁₃	1	1	yes	characteristic, complemented
58	C₇×D₇	—	13	2	no	—
59	C₇×C₁₃	—	6	1	no	—
60	C₇×C₁₃	C₂²×C₇⋊C₉	1	1	yes	characteristic, complemented
61	C₇×C₁₃	C₂×C₉×D₇	1	1	yes	characteristic, complemented
62	C₂²×C₃×C₇	—	91	1	no	—
63	C₂×C₃×D₇	—	13	1	no	—
64	C₂×C₃×C₁₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
65	C₃×D₁₃	—	7	2	no	—
66	C₇×C₉	—	7	1	no	—
67	C₇⋊C₉	C₂×D₉₁	1	1	yes	characteristic, direct product factor
68	C₂×D₁₃	—	7	1	no	Hall
69	C₇²	C₂×C₉×D₁₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃×C₇	—	6	1	no	—
71	C₂×C₃×C₇	—	91	2	no	—
72	C₂×C₃×C₇	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic
73	C₂×C₃×C₇	C₃×D₉₁	1	1	yes	characteristic
74	C₃×D₇	—	13	2	no	—
75	C₃×C₁₃	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
76	C₂²×C₉	—	637	1	no	Hall
77	C₂²×C₇	—	91	1	no	—
78	C₂×D₇	—	13	1	no	—
79	C₂×C₁₃	D₇×C₇⋊C₉	1	1	yes	characteristic, complemented
80	D₁₃	—	7	2	no	—
81	C₃×C₇	—	6	1	no	—
82	C₃×C₇	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic
83	C₃×C₇	C₂×C₃×D₉₁	1	1	yes	characteristic
84	C₂×C₉	—	7	1	no	—
85	C₂×C₉	—	637	2	no	—
86	C₂×C₇	—	6	1	no	—
87	C₂×C₇	—	91	2	no	—
88	C₂×C₇	D₁₃×C₇⋊C₉	1	1	yes	characteristic, complemented
89	C₂×C₇	C₉×D₉₁	1	1	yes	characteristic, complemented
90	D₇	—	13	2	no	—
91	C₁₃	C₂×D₇×C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₃	—	91	1	no	—
93	C₉	—	7	1	no	Sylow
94	C₇	—	6	1	no	—
95	C₇	C₂×D₁₃×C₇⋊C₉	1	1	yes	characteristic, complemented
96	C₇	C₂×C₉×D₉₁	1	1	yes	characteristic, complemented
97	C₂×C₃	—	91	2	no	—
98	C₂×C₃	ΓL(1,𝔽₈)×D₉₁	1	1	yes	characteristic
99	C₂²	—	91	1	no	Sylow
100	C₃	C₂×ΓL(1,𝔽₈)×D₉₁	1	1	yes	characteristic
101	C₂	—	91	2	no	—
102	C₂	C₇⋊C₉×D₉₁	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₁₃×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

C₇⋊C₉×D₉₁

C₂

yes

maximal, direct product factor

C₂×C₃×C₇×D₉₁

C₃

yes

maximal, characteristic

C₇×C₁₃×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₇×D₉₁

C₂×C₃

yes

—

C₂×C₃×C₇²×C₁₃

C₂×C₃

yes

characteristic

C₂×C₉×D₉₁

—

maximal

C₂×D₁₃×C₇⋊C₉

—

maximal

C₂×C₇×D₉₁

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₃

C₂²×C₃

yes

characteristic

C₂×D₇×C₇⋊C₉

—

maximal, Hall

C₂×C₇×C₉×C₁₃

—

C₉×D₉₁

—

C₂×C₁₃×C₇⋊C₉

D₇

yes

characteristic, complemented

D₁₃×C₇⋊C₉

—

C₂×C₇²×C₁₃

C₂×C₉

yes

characteristic, complemented

C₇×D₉₁

C₂×C₉

yes

complemented

C₂×C₃×D₉₁

ΓL(1,𝔽₈)

yes

characteristic

C₂×C₃×C₇×D₁₃

—

C₂×C₇×C₇⋊C₉

D₁₃

yes

characteristic, complemented

D₇×C₇⋊C₉

—

C₇×C₉×C₁₃

—

C₁₃×C₇⋊C₉

C₂×D₇

yes

characteristic, complemented

C₇²×C₁₃

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×D₇

—

C₂×C₃×C₇×C₁₃

—

C₂×C₃×C₇×C₁₃

C₂×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₃×C₇×C₁₃

C₃×D₇

yes

characteristic

C₃×D₉₁

C₂×ΓL(1,𝔽₈)

yes

—

C₃×C₇×D₁₃

—

C₂×C₉×D₁₃

—

Hall

C₇×C₇⋊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

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₃×C₇×C₁₃

C₂×C₃×D₇

yes

characteristic

C₂×C₉×D₇

—

C₂²×C₇⋊C₉

—

C₂×C₉×C₁₃

—

C₉×D₁₃

—

C₂×C₇×D₇

—

Hall

C₂×C₇×C₁₃

—

C₂×C₇×C₁₃

C₂×C₇⋊C₉

yes

characteristic, complemented

C₂×C₇×C₁₃

C₉×D₇

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

—

C₉×D₇

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

D₉₁

yes

characteristic, direct product factor

C₉×C₁₃

—

Hall

C₂×C₇²

C₉×D₁₃

yes

characteristic, complemented

C₇×D₇

—

C₇×C₁₃

—

C₇×C₁₃

C₂²×C₇⋊C₉

yes

characteristic, complemented

C₇×C₁₃

C₂×C₉×D₇

yes

characteristic, complemented

C₂²×C₃×C₇

—

C₂×C₃×D₇

—

C₂×C₃×C₁₃

D₇×ΓL(1,𝔽₈)

yes

characteristic

C₃×D₁₃

—

C₇×C₉

—

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

ΓL(1,𝔽₈)×D₁₃

yes

characteristic

C₂×C₃×C₇

C₃×D₉₁

yes

characteristic

C₃×D₇

—

C₃×C₁₃

C₂×D₇×ΓL(1,𝔽₈)

yes

characteristic

C₂²×C₉

—

637

Hall

C₂²×C₇

—

C₂×D₇

—

C₂×C₁₃

D₇×C₇⋊C₉

yes

characteristic, complemented

D₁₃

—

C₃×C₇

—

C₃×C₇

C₂×ΓL(1,𝔽₈)×D₁₃

yes

characteristic

C₃×C₇

C₂×C₃×D₉₁

yes

characteristic

C₂×C₉

—

C₂×C₉

—

637

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

D₁₃×C₇⋊C₉

yes

characteristic, complemented

C₂×C₇

C₉×D₉₁

yes

characteristic, complemented

D₇

—

C₁₃

C₂×D₇×C₇⋊C₉

yes

characteristic, complemented, Sylow

C₂²×C₃

—

C₉

—

Sylow

C₇

—

C₇

C₂×D₁₃×C₇⋊C₉

yes

characteristic, complemented

C₇

C₂×C₉×D₉₁

yes

characteristic, complemented

C₂×C₃

—

C₂×C₃

ΓL(1,𝔽₈)×D₉₁

yes

characteristic

C₂²

—

Sylow

100

C₃

C₂×ΓL(1,𝔽₈)×D₉₁

yes

characteristic

101

C₂

—

102

C₂

C₇⋊C₉×D₉₁

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)×D₉₁
commutator, hypocenter	C₇²×C₁₃	C₂²×C₉
Frattini	C₃	C₂×ΓL(1,𝔽₈)×D₉₁
Fitting, socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₃×C₇×D₉₁	C₃
cosocle	C₃×C₇²×C₁₃	C₂²×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₃

subgroup

G/H

center, hypercenter

C₂×C₃

ΓL(1,𝔽₈)×D₉₁

commutator, hypocenter

C₇²×C₁₃

C₂²×C₉

Frattini

C₃

C₂×ΓL(1,𝔽₈)×D₉₁

Fitting, socle

C₂×C₃×C₇²×C₁₃

C₂×C₃

prime radical

C₂×C₃×C₇×D₉₁

C₃

cosocle

C₃×C₇²×C₁₃

C₂²×C₃

LCS

G ▷ C₇²×C₁₃

UCS

C₁ ◁ C₂×C₃

N	H ≅ G/N	type
1. C₂×C₇×C₁₃×C₇⋊C₉	C₂	semidirect
2. C₇⋊C₉×D₉₁	C₂	direct (N × H)
4. C₇×C₁₃×C₇⋊C₉	C₂²	semidirect
9. C₂×C₇×D₉₁	C₉	semidirect
14. C₂×C₁₃×C₇⋊C₉	D₇	semidirect
16. C₂×C₇²×C₁₃	C₂×C₉	semidirect
17. C₇×D₉₁	C₂×C₉	semidirect
20. C₂×C₇×C₇⋊C₉	D₁₃	semidirect
23. C₁₃×C₇⋊C₉	C₂×D₇	semidirect
24. C₇²×C₁₃	C₂²×C₉	semidirect
32. C₇×C₇⋊C₉	C₂×D₁₃	semidirect
33. C₂×D₉₁	C₇⋊C₉	direct (N × H)
46. C₂×C₇×C₁₃	C₂×C₇⋊C₉	semidirect
47. C₂×C₇×C₁₃	C₉×D₇	semidirect
48. D₉₁	C₂×C₇⋊C₉	direct (N × H)
55. C₂×C₇⋊C₉	D₉₁	direct (N × H)
57. C₂×C₇²	C₉×D₁₃	semidirect
60. C₇×C₁₃	C₂²×C₇⋊C₉	semidirect
61. C₇×C₁₃	C₂×C₉×D₇	semidirect
67. C₇⋊C₉	C₂×D₉₁	direct (N × H)
69. C₇²	C₂×C₉×D₁₃	semidirect
79. C₂×C₁₃	D₇×C₇⋊C₉	semidirect
88. C₂×C₇	D₁₃×C₇⋊C₉	semidirect
89. C₂×C₇	C₉×D₉₁	semidirect
91. C₁₃	C₂×D₇×C₇⋊C₉	semidirect
95. C₇	C₂×D₁₃×C₇⋊C₉	semidirect
96. C₇	C₂×C₉×D₉₁	semidirect
102. C₂	C₇⋊C₉×D₉₁	direct (N × H)

H ≅ G/N

type

1. C₂×C₇×C₁₃×C₇⋊C₉

C₂

semidirect

2. C₇⋊C₉×D₉₁

C₂

direct (N × H)

4. C₇×C₁₃×C₇⋊C₉

C₂²

semidirect

9. C₂×C₇×D₉₁

C₉

semidirect

14. C₂×C₁₃×C₇⋊C₉

D₇

semidirect

16. C₂×C₇²×C₁₃

C₂×C₉

semidirect

17. C₇×D₉₁

C₂×C₉

semidirect

20. C₂×C₇×C₇⋊C₉

D₁₃

semidirect

23. C₁₃×C₇⋊C₉

C₂×D₇

semidirect

24. C₇²×C₁₃

C₂²×C₉

semidirect

32. C₇×C₇⋊C₉

C₂×D₁₃

semidirect

33. C₂×D₉₁

C₇⋊C₉

direct (N × H)

46. C₂×C₇×C₁₃

C₂×C₇⋊C₉

semidirect

47. C₂×C₇×C₁₃

C₉×D₇

semidirect

48. D₉₁

C₂×C₇⋊C₉

direct (N × H)

55. C₂×C₇⋊C₉

D₉₁

direct (N × H)

57. C₂×C₇²

C₉×D₁₃

semidirect

60. C₇×C₁₃

C₂²×C₇⋊C₉

semidirect

61. C₇×C₁₃

C₂×C₉×D₇

semidirect

67. C₇⋊C₉

C₂×D₉₁

direct (N × H)

69. C₇²

C₂×C₉×D₁₃

semidirect

79. C₂×C₁₃

D₇×C₇⋊C₉

semidirect

88. C₂×C₇

D₁₃×C₇⋊C₉

semidirect

89. C₂×C₇

C₉×D₉₁

semidirect

91. C₁₃

C₂×D₇×C₇⋊C₉

semidirect

95. C₇

C₂×D₁₃×C₇⋊C₉

semidirect

96. C₇

C₂×C₉×D₉₁

semidirect

102. 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
C₂×C₇×C₁₃×C₇⋊C₉	2
C₇⋊C₉×D₉₁	2
C₂×C₃×C₇×D₉₁	3
C₂×C₉×D₉₁	7
C₂×D₁₃×C₇⋊C₉	7
C₂×D₇×C₇⋊C₉	13

subgroup

index

C₂×C₇×C₁₃×C₇⋊C₉

C₇⋊C₉×D₉₁

C₂×C₃×C₇×D₉₁

C₂×C₉×D₉₁

C₂×D₁₃×C₇⋊C₉

C₂×D₇×C₇⋊C₉

Subgroups — C₂×C₇⋊C₉×D₉₁ #37524554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C7⋊C9×D91 #37524554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇⋊C₉×D₉₁ #37524554