Subgroups — C₇²×(C₃⋊D₃)⋊D₁₇ #44472956

Order 29988.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×D₅₁	C₂	1	2	yes	maximal, complemented
2	C₇²×C₁₇×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₇²×D₁₇	—	3	2	no	maximal
4	C₃²×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇²×D₅₁	D₃	1	2	yes	complemented
6	D₃×C₇²×C₁₇	—	3	2	no	—
7	D₃×C₇²×C₁₇	—	6	1	no	—
8	C₃×C₇²×D₁₇	—	3	2	no	—
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	—	2	1	no	—
12	C₃×C₇²×C₁₇	C₂×D₃	1	2	yes	complemented
13	C₇×C₁₇×C₃⋊D₃	C₂×C₇	1	8	yes	complemented
14	C₃×C₇×D₅₁	C₂×C₇	1	16	yes	complemented
15	D₃²×C₇²	—	17	1	no	maximal, Hall
16	C₂×C₇²×C₁₇	—	9	1	no	—
17	C₇²×D₁₇	—	3	2	no	—
18	D₃×C₇×D₁₇	—	3	16	no	—
19	C₃²×C₇×C₁₇	C₂²×C₇	1	8	yes	complemented
20	C₇²×C₃⋊D₃	D₁₇	1	1	yes	characteristic, complemented
21	C₃×D₃×C₇²	—	17	2	no	—
22	C₇²×C₁₇	D₃²	1	1	yes	characteristic, complemented, Hall
23	C₇×D₅₁	D₃×C₇	1	16	yes	complemented
24	D₃×C₇×C₁₇	—	3	16	no	—
25	D₃×C₇×C₁₇	—	6	8	no	—
26	C₃×C₇×D₁₇	—	3	16	no	—
27	(C₃⋊D₃)⋊D₁₇	C₇²	1	1	yes	characteristic, direct product factor, Hall
28	C₂×D₃×C₇²	—	51	2	no	—
29	C₂×C₇×D₁₇	—	9	8	no	—
30	C₃²×C₇²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
31	C₃×C₇×C₁₇	—	2	8	no	—
32	C₃×C₇×C₁₇	C₂×D₃×C₇	1	16	yes	complemented
33	C₁₇×C₃⋊D₃	C₂×C₇²	1	1	yes	characteristic, complemented
34	C₃×D₅₁	C₂×C₇²	1	2	yes	complemented
35	C₂×C₃×C₇²	—	51	2	no	—
36	D₃×C₇²	—	3	2	no	—
37	D₃×C₇²	—	6	1	no	—
38	D₃×C₇²	—	17	2	no	—
39	D₃²×C₇	—	17	8	no	—
40	C₂×C₇×C₁₇	—	9	8	no	—
41	C₇×D₁₇	—	3	16	no	—
42	D₃×D₁₇	—	3	2	no	—
43	C₂²×C₇²	—	153	1	no	Hall
44	C₃²×C₁₇	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇²	—	2	1	no	—
46	C₃×C₇²	D₃×D₁₇	1	2	yes	complemented
47	C₇×C₃⋊D₃	C₇×D₁₇	1	8	yes	complemented
48	C₃×D₃×C₇	—	17	16	no	—
49	C₇×C₁₇	D₃²×C₇	1	8	yes	complemented
50	D₅₁	D₃×C₇²	1	2	yes	complemented
51	C₃×D₁₇	—	3	2	no	—
52	D₃×C₁₇	—	3	2	no	—
53	D₃×C₁₇	—	6	1	no	—
54	C₂×C₇²	—	9	1	no	—
55	C₂×C₇²	—	51	2	no	—
56	C₂×D₃×C₇	—	51	16	no	—
57	C₂×D₁₇	—	9	1	no	Hall
58	C₃²×C₇	C₂×C₇×D₁₇	1	8	yes	complemented
59	C₃×C₁₇	—	2	1	no	—
60	C₃×C₁₇	C₂×D₃×C₇²	1	2	yes	complemented
61	C₇²	(C₃⋊D₃)⋊D₁₇	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃×C₇	—	51	16	no	—
63	D₃×C₇	—	3	16	no	—
64	D₃×C₇	—	6	8	no	—
65	D₃×C₇	—	17	16	no	—
66	D₃²	—	17	1	no	Hall
67	C₂×C₁₇	—	9	1	no	—
68	D₁₇	—	3	2	no	—
69	C₂²×C₇	—	153	8	no	—
70	C₃×C₇	—	2	8	no	—
71	C₃×C₇	D₃×C₇×D₁₇	1	16	yes	complemented
72	C₃⋊D₃	C₇²×D₁₇	1	1	yes	characteristic, complemented
73	C₃×D₃	—	17	2	no	—
74	C₁₇	D₃²×C₇²	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₇	—	9	8	no	—
76	C₂×C₇	—	51	16	no	—
77	C₂×D₃	—	51	2	no	—
78	C₃²	C₂×C₇²×D₁₇	1	1	yes	characteristic, complemented, Sylow
79	C₇	C₇×(C₃⋊D₃)⋊D₁₇	1	8	yes	direct product factor
80	C₂×C₃	—	51	2	no	—
81	D₃	—	3	2	no	—
82	D₃	—	6	1	no	—
83	D₃	—	17	2	no	—
84	C₂²	—	153	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	D₃×C₇²×D₁₇	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	51	2	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. C₃×C₇²×D₅₁	C₂	semidirect
2. C₇²×C₁₇×C₃⋊D₃	C₂	semidirect
4. C₃²×C₇²×C₁₇	C₂²	semidirect
5. C₇²×D₅₁	D₃	semidirect
9. C₇×(C₃⋊D₃)⋊D₁₇	C₇	direct (N × H)
12. C₃×C₇²×C₁₇	C₂×D₃	semidirect
13. C₇×C₁₇×C₃⋊D₃	C₂×C₇	semidirect
14. C₃×C₇×D₅₁	C₂×C₇	semidirect
19. C₃²×C₇×C₁₇	C₂²×C₇	semidirect
20. C₇²×C₃⋊D₃	D₁₇	semidirect
22. C₇²×C₁₇	D₃²	semidirect
23. C₇×D₅₁	D₃×C₇	semidirect
27. (C₃⋊D₃)⋊D₁₇	C₇²	direct (N × H)
30. C₃²×C₇²	C₂×D₁₇	semidirect
32. C₃×C₇×C₁₇	C₂×D₃×C₇	semidirect
33. C₁₇×C₃⋊D₃	C₂×C₇²	semidirect
34. C₃×D₅₁	C₂×C₇²	semidirect
44. C₃²×C₁₇	C₂²×C₇²	semidirect
46. C₃×C₇²	D₃×D₁₇	semidirect
47. C₇×C₃⋊D₃	C₇×D₁₇	semidirect
49. C₇×C₁₇	D₃²×C₇	semidirect
50. D₅₁	D₃×C₇²	semidirect
58. C₃²×C₇	C₂×C₇×D₁₇	semidirect
60. C₃×C₁₇	C₂×D₃×C₇²	semidirect
61. C₇²	(C₃⋊D₃)⋊D₁₇	direct (N × H)
71. C₃×C₇	D₃×C₇×D₁₇	semidirect
72. C₃⋊D₃	C₇²×D₁₇	semidirect
74. C₁₇	D₃²×C₇²	semidirect
78. C₃²	C₂×C₇²×D₁₇	semidirect
79. C₇	C₇×(C₃⋊D₃)⋊D₁₇	direct (N × H)
86. 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
C₃×C₇²×D₅₁	2
C₇²×C₁₇×C₃⋊D₃	2
D₃×C₇²×D₁₇	3
C₇×(C₃⋊D₃)⋊D₁₇	7
D₃²×C₇²	17

Subgroups — C72×(C3⋊D3)⋊D17 #44472956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×(C₃⋊D₃)⋊D₁₇ #44472956