Subgroups — C₇²×C₃⋊D₃×D₁₇ #44472982

Order 29988.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇²×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇²×C₁₇×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×C₃⋊D₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₇²×D₁₇	—	3	4	no	maximal
5	C₃²×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇²×D₅₁	—	3	4	no	—
7	D₃×C₇²×C₁₇	—	3	4	no	—
8	C₃×C₇²×D₁₇	D₃	1	4	yes	complemented
9	C₇×C₃⋊D₃×D₁₇	C₇	1	8	yes	maximal, direct product factor
10	C₂×C₇²×D₁₇	—	9	1	no	Hall
11	C₃×C₇²×C₁₇	C₂×D₃	1	4	yes	complemented
12	C₇×C₃⋊D₅₁	C₂×C₇	1	8	yes	complemented
13	C₇×C₁₇×C₃⋊D₃	C₂×C₇	1	8	yes	complemented
14	C₃²×C₇×D₁₇	C₂×C₇	1	8	yes	complemented
15	C₂×C₇²×C₃⋊D₃	—	17	1	no	maximal, Hall
16	C₂×C₇²×C₁₇	—	9	1	no	—
17	C₇²×D₁₇	—	9	1	no	—
18	C₇²×D₁₇	C₃⋊D₃	1	1	yes	characteristic, direct product factor
19	D₃×C₇×D₁₇	—	3	32	no	—
20	C₃²×C₇×C₁₇	C₂²×C₇	1	8	yes	complemented
21	C₂×C₃²×C₇²	—	17	1	no	—
22	C₇²×C₃⋊D₃	—	17	1	no	—
23	C₇²×C₃⋊D₃	D₁₇	1	1	yes	characteristic, direct product factor
24	C₇²×C₁₇	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
25	C₇×D₅₁	—	3	32	no	—
26	D₃×C₇×C₁₇	—	3	32	no	—
27	C₃×C₇×D₁₇	D₃×C₇	1	32	yes	complemented
28	C₃⋊D₃×D₁₇	C₇²	1	1	yes	characteristic, direct product factor, Hall
29	C₂×D₃×C₇²	—	51	4	no	—
30	C₂×C₇×D₁₇	—	9	8	no	—
31	C₃²×C₇²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇×C₁₇	C₂×D₃×C₇	1	32	yes	complemented
33	C₃⋊D₅₁	C₂×C₇²	1	1	yes	characteristic, complemented
34	C₁₇×C₃⋊D₃	C₂×C₇²	1	1	yes	characteristic, complemented
35	C₃²×D₁₇	C₂×C₇²	1	1	yes	characteristic, complemented
36	C₂×C₃×C₇²	—	17	4	no	—
37	D₃×C₇²	—	3	4	no	—
38	D₃×C₇²	—	51	4	no	—
39	C₂×C₇×C₃⋊D₃	—	17	8	no	—
40	C₂×C₇×C₁₇	—	9	8	no	—
41	C₇×D₁₇	—	9	8	no	—
42	C₇×D₁₇	C₇×C₃⋊D₃	1	8	yes	direct product factor
43	D₃×D₁₇	—	3	4	no	—
44	C₂²×C₇²	—	153	1	no	Hall
45	C₃²×C₁₇	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
46	C₃×C₇²	D₃×D₁₇	1	4	yes	complemented
47	C₂×C₃²×C₇	—	17	8	no	—
48	C₇×C₃⋊D₃	—	17	8	no	—
49	C₇×C₃⋊D₃	C₇×D₁₇	1	8	yes	direct product factor
50	C₇×C₁₇	C₂×C₇×C₃⋊D₃	1	8	yes	complemented
51	D₅₁	—	3	4	no	—
52	C₃×D₁₇	D₃×C₇²	1	4	yes	complemented
53	D₃×C₁₇	—	3	4	no	—
54	C₂×C₇²	—	9	1	no	—
55	C₂×C₇²	—	17	1	no	—
56	C₂×C₇²	—	153	1	no	—
57	C₂×D₃×C₇	—	51	32	no	—
58	C₂×D₁₇	—	9	1	no	Hall
59	C₃²×C₇	C₂×C₇×D₁₇	1	8	yes	complemented
60	C₃×C₁₇	C₂×D₃×C₇²	1	4	yes	complemented
61	C₇²	C₃⋊D₃×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃×C₇	—	17	32	no	—
63	D₃×C₇	—	3	32	no	—
64	D₃×C₇	—	51	32	no	—
65	C₂×C₃⋊D₃	—	17	1	no	Hall
66	C₂×C₁₇	—	9	1	no	—
67	D₁₇	—	9	1	no	—
68	D₁₇	C₇²×C₃⋊D₃	1	1	yes	characteristic, direct product factor
69	C₂²×C₇	—	153	8	no	—
70	C₃×C₇	D₃×C₇×D₁₇	1	32	yes	complemented
71	C₂×C₃²	—	17	1	no	—
72	C₃⋊D₃	—	17	1	no	—
73	C₃⋊D₃	C₇²×D₁₇	1	1	yes	characteristic, direct product factor
74	C₁₇	C₂×C₇²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₇	—	9	8	no	—
76	C₂×C₇	—	17	8	no	—
77	C₂×C₇	—	153	8	no	—
78	C₂×D₃	—	51	4	no	—
79	C₃²	C₂×C₇²×D₁₇	1	1	yes	characteristic, complemented, Sylow
80	C₇	C₇×C₃⋊D₃×D₁₇	1	8	yes	direct product factor
81	C₂×C₃	—	17	4	no	—
82	D₃	—	3	4	no	—
83	D₃	—	51	4	no	—
84	C₂²	—	153	1	no	Sylow
85	C₃	D₃×C₇²×D₁₇	1	4	yes	complemented
86	C₂	—	9	1	no	—
87	C₂	—	17	1	no	—
88	C₂	—	153	1	no	—

	subgroup	G/H
center, hypercenter	C₇²	C₃⋊D₃×D₁₇
commutator, cosocle, hypocenter	C₃²×C₁₇	C₂²×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂²
LCS	G ▷ C₃²×C₁₇
UCS	C₁ ◁ C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇²×D₁₇	C₂	semidirect
2. C₇²×C₁₇×C₃⋊D₃	C₂	semidirect
3. C₇²×C₃⋊D₅₁	C₂	semidirect
5. C₃²×C₇²×C₁₇	C₂²	semidirect
8. C₃×C₇²×D₁₇	D₃	semidirect
9. C₇×C₃⋊D₃×D₁₇	C₇	direct (N × H)
11. C₃×C₇²×C₁₇	C₂×D₃	semidirect
12. C₇×C₃⋊D₅₁	C₂×C₇	semidirect
13. C₇×C₁₇×C₃⋊D₃	C₂×C₇	semidirect
14. C₃²×C₇×D₁₇	C₂×C₇	semidirect
18. C₇²×D₁₇	C₃⋊D₃	direct (N × H)
20. C₃²×C₇×C₁₇	C₂²×C₇	semidirect
23. C₇²×C₃⋊D₃	D₁₇	direct (N × H)
24. C₇²×C₁₇	C₂×C₃⋊D₃	semidirect
27. C₃×C₇×D₁₇	D₃×C₇	semidirect
28. C₃⋊D₃×D₁₇	C₇²	direct (N × H)
31. C₃²×C₇²	C₂×D₁₇	semidirect
32. C₃×C₇×C₁₇	C₂×D₃×C₇	semidirect
33. C₃⋊D₅₁	C₂×C₇²	semidirect
34. C₁₇×C₃⋊D₃	C₂×C₇²	semidirect
35. C₃²×D₁₇	C₂×C₇²	semidirect
42. C₇×D₁₇	C₇×C₃⋊D₃	direct (N × H)
45. C₃²×C₁₇	C₂²×C₇²	semidirect
46. C₃×C₇²	D₃×D₁₇	semidirect
49. C₇×C₃⋊D₃	C₇×D₁₇	direct (N × H)
50. C₇×C₁₇	C₂×C₇×C₃⋊D₃	semidirect
52. C₃×D₁₇	D₃×C₇²	semidirect
59. C₃²×C₇	C₂×C₇×D₁₇	semidirect
60. C₃×C₁₇	C₂×D₃×C₇²	semidirect
61. C₇²	C₃⋊D₃×D₁₇	direct (N × H)
68. D₁₇	C₇²×C₃⋊D₃	direct (N × H)
70. C₃×C₇	D₃×C₇×D₁₇	semidirect
73. C₃⋊D₃	C₇²×D₁₇	direct (N × H)
74. C₁₇	C₂×C₇²×C₃⋊D₃	semidirect
79. C₃²	C₂×C₇²×D₁₇	semidirect
80. C₇	C₇×C₃⋊D₃×D₁₇	direct (N × H)
85. C₃	D₃×C₇²×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₇²×D₁₇	2
C₇²×C₁₇×C₃⋊D₃	2
C₇²×C₃⋊D₅₁	2
D₃×C₇²×D₁₇	3
C₇×C₃⋊D₃×D₁₇	7
C₂×C₇²×C₃⋊D₃	17

Subgroups — C72×C3⋊D3×D17 #44472982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₃⋊D₃×D₁₇ #44472982