Subgroups — C₇×C₉×C₁₇×Dic₇ #44472513

Order 29988.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×C₉×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₁₇×Dic₇	C₃	1	1	yes	maximal, characteristic
3	C₇²×C₉×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic
5	C₉×C₁₇×Dic₇	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₄×C₇×C₉×C₁₇	—	7	1	no	maximal
7	C₇×C₁₇×Dic₇	C₉	1	1	yes	characteristic, direct product factor, Hall
8	C₃×C₇²×C₁₇	C₃×C₄	1	1	yes	characteristic
9	C₂×C₇×C₉×C₁₇	—	2	3	no	—
10	C₂×C₇×C₉×C₁₇	D₇	1	1	yes	characteristic
11	C₂×C₇×C₉×C₁₇	C₂×C₇	1	1	yes	characteristic
12	C₇×C₉×Dic₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₇²×C₁₇	C₂×C₉	1	1	yes	characteristic
14	C₃×C₄×C₇×C₁₇	—	7	1	no	—
15	C₃×C₁₇×Dic₇	C₃×C₇	1	1	yes	characteristic
16	C₇×C₉×C₁₇	—	2	3	no	—
17	C₇×C₉×C₁₇	Dic₇	1	1	yes	characteristic, direct product factor
18	C₇×C₉×C₁₇	C₄×C₇	1	1	yes	characteristic, complemented
19	C₂×C₇²×C₉	C₂×C₁₇	1	1	yes	characteristic
20	C₇²×C₁₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₇×C₁₇	—	2	3	no	—
22	C₂×C₃×C₇×C₁₇	C₃×D₇	1	1	yes	characteristic
23	C₂×C₃×C₇×C₁₇	C₂×C₃×C₇	1	1	yes	characteristic
24	C₄×C₉×C₁₇	—	7	1	no	Hall
25	C₃×C₇×Dic₇	C₃×C₁₇	1	1	yes	characteristic
26	C₄×C₇×C₁₇	—	7	1	no	—
27	C₁₇×Dic₇	C₇×C₉	1	1	yes	characteristic, direct product factor
28	C₇²×C₉	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
29	C₃×C₇×C₁₇	—	2	3	no	—
30	C₃×C₇×C₁₇	C₃×Dic₇	1	1	yes	characteristic
31	C₃×C₇×C₁₇	C₃×C₄×C₇	1	1	yes	characteristic
32	C₂×C₉×C₁₇	C₇×D₇	1	1	yes	characteristic
33	C₂×C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic
34	C₄×C₇×C₉	—	7	1	no	—
35	C₉×Dic₇	C₇×C₁₇	1	1	yes	characteristic, direct product factor
36	C₂×C₇×C₁₇	—	2	3	no	—
37	C₂×C₇×C₁₇	C₉×D₇	1	1	yes	characteristic
38	C₂×C₇×C₁₇	C₂×C₇×C₉	1	1	yes	characteristic
39	C₃×C₄×C₁₇	—	7	1	no	—
40	C₇×Dic₇	C₉×C₁₇	1	1	yes	characteristic, direct product factor, Hall
41	C₉×C₁₇	C₇×Dic₇	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₇²	C₃×C₄×C₁₇	1	1	yes	characteristic
43	C₂×C₇×C₉	—	2	3	no	—
44	C₂×C₇×C₉	D₇×C₁₇	1	1	yes	characteristic
45	C₂×C₇×C₉	C₂×C₇×C₁₇	1	1	yes	characteristic
46	C₇×C₁₇	—	2	3	no	—
47	C₇×C₁₇	C₉×Dic₇	1	1	yes	characteristic, direct product factor
48	C₇×C₁₇	C₄×C₇×C₉	1	1	yes	characteristic, complemented
49	C₂×C₃×C₁₇	C₃×C₇×D₇	1	1	yes	characteristic
50	C₂×C₇²	C₂×C₉×C₁₇	1	1	yes	characteristic
51	C₃×C₄×C₇	—	7	1	no	—
52	C₃×Dic₇	C₃×C₇×C₁₇	1	1	yes	characteristic
53	C₄×C₁₇	—	7	1	no	Hall
54	C₇×C₉	—	2	3	no	—
55	C₇×C₉	C₁₇×Dic₇	1	1	yes	characteristic, direct product factor
56	C₇×C₉	C₄×C₇×C₁₇	1	1	yes	characteristic, complemented
57	C₃×C₁₇	C₃×C₇×Dic₇	1	1	yes	characteristic
58	C₇²	C₄×C₉×C₁₇	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃×C₇	—	2	3	no	—
60	C₂×C₃×C₇	C₃×D₇×C₁₇	1	1	yes	characteristic
61	C₂×C₃×C₇	C₂×C₃×C₇×C₁₇	1	1	yes	characteristic
62	C₄×C₉	—	7	1	no	Hall
63	C₂×C₁₇	C₇×C₉×D₇	1	1	yes	characteristic
64	C₄×C₇	—	7	1	no	—
65	Dic₇	C₇×C₉×C₁₇	1	1	yes	characteristic, direct product factor
66	C₃×C₇	—	2	3	no	—
67	C₃×C₇	C₃×C₁₇×Dic₇	1	1	yes	characteristic
68	C₃×C₇	C₃×C₄×C₇×C₁₇	1	1	yes	characteristic
69	C₂×C₉	C₇×D₇×C₁₇	1	1	yes	characteristic
70	C₁₇	C₇×C₉×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₇	—	2	3	no	—
72	C₂×C₇	C₂×C₇×C₉×C₁₇	1	1	yes	characteristic
73	C₂×C₇	C₉×D₇×C₁₇	1	1	yes	characteristic
74	C₃×C₄	—	7	1	no	—
75	C₉	C₇×C₁₇×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
76	C₇	—	2	3	no	—
77	C₇	C₄×C₇×C₉×C₁₇	1	1	yes	characteristic, complemented
78	C₇	C₉×C₁₇×Dic₇	1	1	yes	characteristic, direct product factor
79	C₂×C₃	C₃×C₇×D₇×C₁₇	1	1	yes	characteristic
80	C₄	—	7	1	no	Sylow
81	C₃	C₃×C₇×C₁₇×Dic₇	1	1	yes	characteristic
82	C₂	C₇×C₉×D₇×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₇×C₉×C₁₇	D₇
commutator, hypocenter	C₇	C₄×C₇×C₉×C₁₇
Frattini	C₂×C₃	C₃×C₇×D₇×C₁₇
Fitting	C₂×C₇²×C₉×C₁₇	C₂
socle, prime radical	C₂×C₃×C₇²×C₁₇	C₂×C₃
cosocle	C₂×C₃×C₇	C₂×C₃×C₇×C₁₇
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₇×C₉×C₁₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₇²×C₉×C₁₇	C₄	semidirect
5. C₉×C₁₇×Dic₇	C₇	direct (N × H)
7. C₇×C₁₇×Dic₇	C₉	direct (N × H)
12. C₇×C₉×Dic₇	C₁₇	direct (N × H)
17. C₇×C₉×C₁₇	Dic₇	direct (N × H)
18. C₇×C₉×C₁₇	C₄×C₇	semidirect
20. C₇²×C₁₇	C₄×C₉	semidirect
27. C₁₇×Dic₇	C₇×C₉	direct (N × H)
28. C₇²×C₉	C₄×C₁₇	semidirect
35. C₉×Dic₇	C₇×C₁₇	direct (N × H)
40. C₇×Dic₇	C₉×C₁₇	direct (N × H)
41. C₉×C₁₇	C₇×Dic₇	direct (N × H)
47. C₇×C₁₇	C₉×Dic₇	direct (N × H)
48. C₇×C₁₇	C₄×C₇×C₉	semidirect
55. C₇×C₉	C₁₇×Dic₇	direct (N × H)
56. C₇×C₉	C₄×C₇×C₁₇	semidirect
58. C₇²	C₄×C₉×C₁₇	semidirect
65. Dic₇	C₇×C₉×C₁₇	direct (N × H)
70. C₁₇	C₇×C₉×Dic₇	direct (N × H)
75. C₉	C₇×C₁₇×Dic₇	direct (N × H)
77. C₇	C₄×C₇×C₉×C₁₇	semidirect
78. C₇	C₉×C₁₇×Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇²×C₉×C₁₇	2
C₃×C₇×C₁₇×Dic₇	3
C₉×C₁₇×Dic₇	7
C₄×C₇×C₉×C₁₇	7
C₇×C₉×Dic₇	17

Subgroups — C7×C9×C17×Dic7 #44472513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₉×C₁₇×Dic₇ #44472513