Group #44472577 — subgroups

Subgroups — C₂×D₇×C₁₇×C₇⋊C₉ #44472577

Order 29988.

128 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 70 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×C₁₇×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×C₁₇×C₇⋊C₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₇×D₇×C₁₇	C₃	1	1	yes	maximal, characteristic
4	C₇×C₁₇×C₇⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇×D₇×C₁₇	C₂×C₃	1	2	yes	—
6	C₂×C₃×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic
7	C₂×C₉×D₇×C₁₇	—	7	1	no	maximal
8	C₂²×C₁₇×C₇⋊C₉	—	7	1	no	maximal
9	C₂×C₇×D₇×C₁₇	C₉	1	1	yes	characteristic, complemented, Hall
10	C₃×C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic
11	C₉×D₇×C₁₇	—	7	2	no	—
12	C₂×C₁₇×C₇⋊C₉	—	7	2	no	—
13	C₂×C₁₇×C₇⋊C₉	D₇	1	1	yes	characteristic, direct product factor
14	C₂×C₇×C₉×C₁₇	—	7	1	no	—
15	C₂×D₇×C₇⋊C₉	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₇²×C₁₇	C₂×C₉	1	1	yes	characteristic, complemented
17	C₇×D₇×C₁₇	C₂×C₉	1	2	yes	complemented
18	C₂²×C₃×C₇×C₁₇	—	7	1	no	—
19	C₂×C₃×D₇×C₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic
20	C₇×C₉×C₁₇	—	7	1	no	—
21	C₁₇×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, direct product factor
22	C₂×C₇×C₇⋊C₉	C₂×C₁₇	1	1	yes	characteristic, complemented
23	D₇×C₇⋊C₉	C₂×C₁₇	1	2	yes	direct product factor
24	C₇²×C₁₇	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₇×C₁₇	—	6	1	no	—
26	C₂×C₃×C₇×C₁₇	—	7	2	no	—
27	C₂×C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₂×C₃×C₇×C₁₇	C₃×D₇	1	1	yes	characteristic
29	C₃×D₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	2	yes	—
30	C₂²×C₉×C₁₇	—	49	1	no	Hall
31	C₂×C₃×C₇×D₇	C₃×C₁₇	1	1	yes	characteristic
32	C₂²×C₇×C₁₇	—	7	1	no	—
33	C₂×D₇×C₁₇	C₇⋊C₉	1	1	yes	characteristic, direct product factor
34	C₇×C₇⋊C₉	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
35	C₃×C₇×C₁₇	—	6	1	no	—
36	C₃×C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₃×C₇×C₁₇	C₂×C₃×D₇	1	1	yes	characteristic
38	C₂×C₉×C₁₇	—	7	1	no	—
39	C₂×C₉×C₁₇	—	49	2	no	—
40	C₂×C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic
41	C₃×C₇×D₇	C₂×C₃×C₁₇	1	2	yes	—
42	C₂×C₉×D₇	—	7	1	no	—
43	C₂²×C₇⋊C₉	—	7	1	no	—
44	C₂×C₇×C₁₇	—	6	1	no	—
45	C₂×C₇×C₁₇	—	7	2	no	—
46	C₂×C₇×C₁₇	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
47	C₂×C₇×C₁₇	C₉×D₇	1	1	yes	characteristic, complemented
48	D₇×C₁₇	C₂×C₇⋊C₉	1	2	yes	direct product factor
49	C₂²×C₃×C₁₇	—	7	1	no	—
50	C₂×C₇×D₇	C₉×C₁₇	1	1	yes	characteristic, complemented, Hall
51	C₉×C₁₇	—	7	1	no	Hall
52	C₃×C₇²	C₂²×C₃×C₁₇	1	1	yes	characteristic
53	C₂×C₇×C₉	—	7	1	no	—
54	C₉×D₇	—	7	2	no	—
55	C₂×C₇⋊C₉	—	7	2	no	—
56	C₂×C₇⋊C₉	D₇×C₁₇	1	1	yes	characteristic, direct product factor
57	C₇×C₁₇	—	6	1	no	—
58	C₇×C₁₇	C₂²×C₇⋊C₉	1	1	yes	characteristic, complemented
59	C₇×C₁₇	C₂×C₉×D₇	1	1	yes	characteristic, complemented
60	C₂×C₃×C₁₇	—	7	2	no	—
61	C₂×C₃×C₁₇	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
62	C₂×C₇²	C₂×C₉×C₁₇	1	1	yes	characteristic, complemented
63	C₇×D₇	C₂×C₉×C₁₇	1	2	yes	complemented
64	C₂²×C₃×C₇	—	7	1	no	—
65	C₂×C₃×D₇	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₂²×C₁₇	—	7	1	no	Hall
67	C₇×C₉	—	7	1	no	—
68	C₇⋊C₉	C₂×D₇×C₁₇	1	1	yes	characteristic, direct product factor
69	C₃×C₁₇	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
70	C₇²	C₂²×C₉×C₁₇	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃×C₇	—	6	1	no	—
72	C₂×C₃×C₇	—	7	2	no	—
73	C₂×C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
74	C₂×C₃×C₇	C₃×D₇×C₁₇	1	1	yes	characteristic
75	C₃×D₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	2	yes	—
76	C₂²×C₉	—	49	1	no	Hall
77	C₂×C₁₇	—	7	2	no	—
78	C₂×C₁₇	D₇×C₇⋊C₉	1	1	yes	characteristic, direct product factor
79	C₂²×C₇	—	7	1	no	—
80	C₂×D₇	C₁₇×C₇⋊C₉	1	1	yes	characteristic, direct product factor
81	C₃×C₇	—	6	1	no	—
82	C₃×C₇	C₂²×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
83	C₃×C₇	C₂×C₃×D₇×C₁₇	1	1	yes	characteristic
84	C₂×C₉	—	7	1	no	—
85	C₂×C₉	—	49	2	no	—
86	C₁₇	C₂×D₇×C₇⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×C₇	—	6	1	no	—
88	C₂×C₇	—	7	2	no	—
89	C₂×C₇	C₂×C₁₇×C₇⋊C₉	1	1	yes	characteristic, complemented
90	C₂×C₇	C₉×D₇×C₁₇	1	1	yes	characteristic, complemented
91	D₇	C₂×C₁₇×C₇⋊C₉	1	2	yes	direct product factor
92	C₂²×C₃	—	7	1	no	—
93	C₉	—	7	1	no	Sylow
94	C₇	—	6	1	no	—
95	C₇	C₂²×C₁₇×C₇⋊C₉	1	1	yes	characteristic, complemented
96	C₇	C₂×C₉×D₇×C₁₇	1	1	yes	characteristic, complemented
97	C₂×C₃	—	7	2	no	—
98	C₂×C₃	D₇×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
99	C₂²	—	7	1	no	Sylow
100	C₃	C₂×D₇×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
101	C₂	—	7	2	no	—
102	C₂	D₇×C₁₇×C₇⋊C₉	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₁₇×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

D₇×C₁₇×C₇⋊C₉

C₂

yes

maximal, direct product factor

C₂×C₃×C₇×D₇×C₁₇

C₃

yes

maximal, characteristic

C₇×C₁₇×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₇×D₇×C₁₇

C₂×C₃

yes

—

C₂×C₃×C₇²×C₁₇

C₂×C₃

yes

characteristic

C₂×C₉×D₇×C₁₇

—

maximal

C₂²×C₁₇×C₇⋊C₉

—

maximal

C₂×C₇×D₇×C₁₇

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₇

C₂²×C₃

yes

characteristic

C₉×D₇×C₁₇

—

C₂×C₁₇×C₇⋊C₉

—

C₂×C₁₇×C₇⋊C₉

D₇

yes

characteristic, direct product factor

C₂×C₇×C₉×C₁₇

—

C₂×D₇×C₇⋊C₉

C₁₇

yes

maximal, characteristic, direct product factor, Hall

C₂×C₇²×C₁₇

C₂×C₉

yes

characteristic, complemented

C₇×D₇×C₁₇

C₂×C₉

yes

complemented

C₂²×C₃×C₇×C₁₇

—

C₂×C₃×D₇×C₁₇

ΓL(1,𝔽₈)

yes

characteristic

C₇×C₉×C₁₇

—

C₁₇×C₇⋊C₉

C₂×D₇

yes

characteristic, direct product factor

C₂×C₇×C₇⋊C₉

C₂×C₁₇

yes

characteristic, complemented

D₇×C₇⋊C₉

C₂×C₁₇

yes

direct product factor

C₇²×C₁₇

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

C₂×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₃×C₇×C₁₇

C₃×D₇

yes

characteristic

C₃×D₇×C₁₇

C₂×ΓL(1,𝔽₈)

yes

—

C₂²×C₉×C₁₇

—

Hall

C₂×C₃×C₇×D₇

C₃×C₁₇

yes

characteristic

C₂²×C₇×C₁₇

—

C₂×D₇×C₁₇

C₇⋊C₉

yes

characteristic, direct product factor

C₇×C₇⋊C₉

C₂²×C₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₃×C₇×C₁₇

C₂×C₃×D₇

yes

characteristic

C₂×C₉×C₁₇

—

C₂×C₉×C₁₇

—

C₂×C₃×C₇²

C₂×C₃×C₁₇

yes

characteristic

C₃×C₇×D₇

C₂×C₃×C₁₇

yes

—

C₂×C₉×D₇

—

C₂²×C₇⋊C₉

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

C₂×C₇⋊C₉

yes

characteristic, complemented

C₂×C₇×C₁₇

C₉×D₇

yes

characteristic, complemented

D₇×C₁₇

C₂×C₇⋊C₉

yes

direct product factor

C₂²×C₃×C₁₇

—

C₂×C₇×D₇

C₉×C₁₇

yes

characteristic, complemented, Hall

C₉×C₁₇

—

Hall

C₃×C₇²

C₂²×C₃×C₁₇

yes

characteristic

C₂×C₇×C₉

—

C₉×D₇

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

D₇×C₁₇

yes

characteristic, direct product factor

C₇×C₁₇

—

C₇×C₁₇

C₂²×C₇⋊C₉

yes

characteristic, complemented

C₇×C₁₇

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₃×C₁₇

—

C₂×C₃×C₁₇

D₇×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₇²

C₂×C₉×C₁₇

yes

characteristic, complemented

C₇×D₇

C₂×C₉×C₁₇

yes

complemented

C₂²×C₃×C₇

—

C₂×C₃×D₇

C₁₇×ΓL(1,𝔽₈)

yes

characteristic

C₂²×C₁₇

—

Hall

C₇×C₉

—

C₇⋊C₉

C₂×D₇×C₁₇

yes

characteristic, direct product factor

C₃×C₁₇

C₂×D₇×ΓL(1,𝔽₈)

yes

characteristic

C₇²

C₂²×C₉×C₁₇

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

C₂×C₁₇×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₃×C₇

C₃×D₇×C₁₇

yes

characteristic

C₃×D₇

C₂×C₁₇×ΓL(1,𝔽₈)

yes

—

C₂²×C₉

—

Hall

C₂×C₁₇

—

C₂×C₁₇

D₇×C₇⋊C₉

yes

characteristic, direct product factor

C₂²×C₇

—

C₂×D₇

C₁₇×C₇⋊C₉

yes

characteristic, direct product factor

C₃×C₇

—

C₃×C₇

C₂²×C₁₇×ΓL(1,𝔽₈)

yes

characteristic

C₃×C₇

C₂×C₃×D₇×C₁₇

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₁₇

C₂×D₇×C₇⋊C₉

yes

characteristic, direct product factor, Sylow

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂×C₁₇×C₇⋊C₉

yes

characteristic, complemented

C₂×C₇

C₉×D₇×C₁₇

yes

characteristic, complemented

D₇

C₂×C₁₇×C₇⋊C₉

yes

direct product factor

C₂²×C₃

—

C₉

—

Sylow

C₇

—

C₇

C₂²×C₁₇×C₇⋊C₉

yes

characteristic, complemented

C₇

C₂×C₉×D₇×C₁₇

yes

characteristic, complemented

C₂×C₃

—

C₂×C₃

D₇×C₁₇×ΓL(1,𝔽₈)

yes

characteristic

C₂²

—

Sylow

100

C₃

C₂×D₇×C₁₇×ΓL(1,𝔽₈)

yes

characteristic

101

C₂

—

102

C₂

D₇×C₁₇×C₇⋊C₉

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₇	D₇×ΓL(1,𝔽₈)
commutator, hypocenter	C₇²	C₂²×C₉×C₁₇
Frattini	C₃	C₂×D₇×C₁₇×ΓL(1,𝔽₈)
Fitting, socle	C₂×C₃×C₇²×C₁₇	C₂×C₃
prime radical	C₂×C₃×C₇×D₇×C₁₇	C₃
cosocle	C₃×C₇²	C₂²×C₃×C₁₇
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₁₇

subgroup

G/H

center, hypercenter

C₂×C₃×C₁₇

D₇×ΓL(1,𝔽₈)

commutator, hypocenter

C₇²

C₂²×C₉×C₁₇

Frattini

C₃

C₂×D₇×C₁₇×ΓL(1,𝔽₈)

Fitting, socle

C₂×C₃×C₇²×C₁₇

C₂×C₃

prime radical

C₂×C₃×C₇×D₇×C₁₇

C₃

cosocle

C₃×C₇²

C₂²×C₃×C₁₇

LCS

G ▷ C₇²

UCS

C₁ ◁ C₂×C₃×C₁₇

N	H ≅ G/N	type
1. C₂×C₇×C₁₇×C₇⋊C₉	C₂	semidirect
2. D₇×C₁₇×C₇⋊C₉	C₂	direct (N × H)
4. C₇×C₁₇×C₇⋊C₉	C₂²	semidirect
9. C₂×C₇×D₇×C₁₇	C₉	semidirect
13. C₂×C₁₇×C₇⋊C₉	D₇	direct (N × H)
15. C₂×D₇×C₇⋊C₉	C₁₇	direct (N × H)
16. C₂×C₇²×C₁₇	C₂×C₉	semidirect
17. C₇×D₇×C₁₇	C₂×C₉	semidirect
21. C₁₇×C₇⋊C₉	C₂×D₇	direct (N × H)
22. C₂×C₇×C₇⋊C₉	C₂×C₁₇	semidirect
23. D₇×C₇⋊C₉	C₂×C₁₇	direct (N × H)
24. C₇²×C₁₇	C₂²×C₉	semidirect
33. C₂×D₇×C₁₇	C₇⋊C₉	direct (N × H)
34. C₇×C₇⋊C₉	C₂²×C₁₇	semidirect
46. C₂×C₇×C₁₇	C₂×C₇⋊C₉	semidirect
47. C₂×C₇×C₁₇	C₉×D₇	semidirect
48. D₇×C₁₇	C₂×C₇⋊C₉	direct (N × H)
50. C₂×C₇×D₇	C₉×C₁₇	semidirect
56. C₂×C₇⋊C₉	D₇×C₁₇	direct (N × H)
58. C₇×C₁₇	C₂²×C₇⋊C₉	semidirect
59. C₇×C₁₇	C₂×C₉×D₇	semidirect
62. C₂×C₇²	C₂×C₉×C₁₇	semidirect
63. C₇×D₇	C₂×C₉×C₁₇	semidirect
68. C₇⋊C₉	C₂×D₇×C₁₇	direct (N × H)
70. C₇²	C₂²×C₉×C₁₇	semidirect
78. C₂×C₁₇	D₇×C₇⋊C₉	direct (N × H)
80. C₂×D₇	C₁₇×C₇⋊C₉	direct (N × H)
86. C₁₇	C₂×D₇×C₇⋊C₉	direct (N × H)
89. C₂×C₇	C₂×C₁₇×C₇⋊C₉	semidirect
90. C₂×C₇	C₉×D₇×C₁₇	semidirect
91. D₇	C₂×C₁₇×C₇⋊C₉	direct (N × H)
95. C₇	C₂²×C₁₇×C₇⋊C₉	semidirect
96. C₇	C₂×C₉×D₇×C₁₇	semidirect
102. C₂	D₇×C₁₇×C₇⋊C₉	direct (N × H)

H ≅ G/N

type

1. C₂×C₇×C₁₇×C₇⋊C₉

C₂

semidirect

2. D₇×C₁₇×C₇⋊C₉

C₂

direct (N × H)

4. C₇×C₁₇×C₇⋊C₉

C₂²

semidirect

9. C₂×C₇×D₇×C₁₇

C₉

semidirect

13. C₂×C₁₇×C₇⋊C₉

D₇

direct (N × H)

15. C₂×D₇×C₇⋊C₉

C₁₇

direct (N × H)

16. C₂×C₇²×C₁₇

C₂×C₉

semidirect

17. C₇×D₇×C₁₇

C₂×C₉

semidirect

21. C₁₇×C₇⋊C₉

C₂×D₇

direct (N × H)

22. C₂×C₇×C₇⋊C₉

C₂×C₁₇

semidirect

23. D₇×C₇⋊C₉

C₂×C₁₇

direct (N × H)

24. C₇²×C₁₇

C₂²×C₉

semidirect

33. C₂×D₇×C₁₇

C₇⋊C₉

direct (N × H)

34. C₇×C₇⋊C₉

C₂²×C₁₇

semidirect

46. C₂×C₇×C₁₇

C₂×C₇⋊C₉

semidirect

47. C₂×C₇×C₁₇

C₉×D₇

semidirect

48. D₇×C₁₇

C₂×C₇⋊C₉

direct (N × H)

50. C₂×C₇×D₇

C₉×C₁₇

semidirect

56. C₂×C₇⋊C₉

D₇×C₁₇

direct (N × H)

58. C₇×C₁₇

C₂²×C₇⋊C₉

semidirect

59. C₇×C₁₇

C₂×C₉×D₇

semidirect

62. C₂×C₇²

C₂×C₉×C₁₇

semidirect

63. C₇×D₇

C₂×C₉×C₁₇

semidirect

68. C₇⋊C₉

C₂×D₇×C₁₇

direct (N × H)

70. C₇²

C₂²×C₉×C₁₇

semidirect

78. C₂×C₁₇

D₇×C₇⋊C₉

direct (N × H)

80. C₂×D₇

C₁₇×C₇⋊C₉

direct (N × H)

86. C₁₇

C₂×D₇×C₇⋊C₉

direct (N × H)

89. C₂×C₇

C₂×C₁₇×C₇⋊C₉

semidirect

90. C₂×C₇

C₉×D₇×C₁₇

semidirect

91. D₇

C₂×C₁₇×C₇⋊C₉

direct (N × H)

95. C₇

C₂²×C₁₇×C₇⋊C₉

semidirect

96. C₇

C₂×C₉×D₇×C₁₇

semidirect

102. C₂

D₇×C₁₇×C₇⋊C₉

direct (N × H)

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

subgroup	index
C₂×C₇×C₁₇×C₇⋊C₉	2
D₇×C₁₇×C₇⋊C₉	2
C₂×C₃×C₇×D₇×C₁₇	3
C₂×C₉×D₇×C₁₇	7
C₂²×C₁₇×C₇⋊C₉	7
C₂×D₇×C₇⋊C₉	17

subgroup

index

C₂×C₇×C₁₇×C₇⋊C₉

D₇×C₁₇×C₇⋊C₉

C₂×C₃×C₇×D₇×C₁₇

C₂×C₉×D₇×C₁₇

C₂²×C₁₇×C₇⋊C₉

C₂×D₇×C₇⋊C₉

Subgroups — C₂×D₇×C₁₇×C₇⋊C₉ #44472577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D7×C17×C7⋊C9 #44472577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇×C₁₇×C₇⋊C₉ #44472577