Subgroups — C₁₇×#13183685 #44472993

Order 29988.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₇×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₁₇×C₃⋊D₂₁	C₂	1	2	yes	maximal, complemented
3	C₁₇×D₂₁⋊D₇	—	3	4	no	maximal
4	C₃²×C₇²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×C₁₇×D₂₁	—	3	8	no	—
6	C₃×C₁₇×C₇⋊D₇	D₃	1	4	yes	complemented
7	D₇×C₁₇×C₃⋊D₃	—	7	2	no	maximal
8	D₇²×C₁₇	—	9	1	no	Hall
9	C₃×C₇²×C₁₇	C₂×D₃	1	4	yes	complemented
10	C₁₇×C₃⋊D₂₁	D₇	1	2	yes	complemented
11	C₇×C₁₇×C₃⋊D₃	—	7	2	no	—
12	C₃²×D₇×C₁₇	—	7	2	no	—
13	C₃²×D₇×C₁₇	—	14	3	no	—
14	#13183685	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₁₇×C₇⋊D₇	C₃⋊D₃	1	1	yes	characteristic, complemented
16	C₇×D₇×C₁₇	—	9	2	no	—
17	D₃×D₇×C₁₇	—	21	8	no	—
18	C₃²×C₇×C₁₇	—	2	3	no	—
19	C₃²×C₇×C₁₇	C₂×D₇	1	2	yes	complemented
20	C₇×C₃⋊D₂₁	C₂×C₁₇	1	2	yes	complemented
21	C₃²×C₇⋊D₇	C₂×C₁₇	1	1	yes	characteristic, complemented
22	C₇²×C₁₇	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
23	C₁₇×D₂₁	—	3	8	no	—
24	D₃×C₇×C₁₇	—	21	8	no	—
25	C₃×D₇×C₁₇	—	7	8	no	—
26	C₃×D₇×C₁₇	—	14	12	no	—
27	C₂×C₁₇×C₃⋊D₃	—	49	1	no	Hall
28	D₂₁⋊D₇	—	3	4	no	—
29	C₂×D₇×C₁₇	—	63	2	no	—
30	C₃²×C₇²	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
31	C₃×C₇×C₁₇	—	2	12	no	—
32	C₃×C₇×C₁₇	D₃×D₇	1	8	yes	complemented
33	C₂×C₃²×C₁₇	—	49	1	no	—
34	C₁₇×C₃⋊D₃	—	7	2	no	—
35	C₇×D₂₁	—	3	8	no	—
36	C₃×C₇⋊D₇	D₃×C₁₇	1	4	yes	complemented
37	D₇×C₃⋊D₃	—	7	2	no	—
38	C₂×C₇×C₁₇	—	63	2	no	—
39	D₇×C₁₇	—	7	2	no	—
40	D₇×C₁₇	—	9	2	no	—
41	D₇×C₁₇	—	14	3	no	—
42	C₂×D₃×C₁₇	—	147	4	no	—
43	D₇²	—	9	1	no	Hall
44	C₃²×C₁₇	D₇²	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇²	C₂×D₃×C₁₇	1	4	yes	complemented
46	C₃⋊D₂₁	D₇×C₁₇	1	2	yes	complemented
47	C₇×C₃⋊D₃	—	7	2	no	—
48	C₃²×D₇	—	7	2	no	—
49	C₃²×D₇	—	14	3	no	—
50	C₇×C₁₇	—	2	3	no	—
51	C₇×C₁₇	D₇×C₃⋊D₃	1	2	yes	complemented
52	C₂×C₃×C₁₇	—	49	4	no	—
53	D₃×C₁₇	—	21	8	no	—
54	C₇⋊D₇	C₁₇×C₃⋊D₃	1	1	yes	characteristic, complemented
55	C₇×D₇	—	9	2	no	—
56	D₃×D₇	—	21	8	no	—
57	C₂²×C₁₇	—	441	1	no	Hall
58	C₃²×C₇	—	2	3	no	—
59	C₃²×C₇	C₂×D₇×C₁₇	1	2	yes	complemented
60	C₃×C₁₇	D₂₁⋊D₇	1	4	yes	complemented
61	C₇²	C₂×C₁₇×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
62	D₂₁	—	3	8	no	—
63	C₃×D₇	—	7	8	no	—
64	C₃×D₇	—	14	12	no	—
65	D₃×C₇	—	21	8	no	—
66	C₂×C₃⋊D₃	—	49	1	no	Hall
67	C₂×C₁₇	—	49	1	no	—
68	C₂×C₁₇	—	63	2	no	—
69	C₂×D₇	—	63	2	no	—
70	C₃×C₇	—	2	12	no	—
71	C₃×C₇	D₃×D₇×C₁₇	1	8	yes	complemented
72	C₂×C₃²	—	49	1	no	—
73	C₃⋊D₃	—	7	2	no	—
74	C₁₇	#13183685	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₇	—	63	2	no	—
76	D₇	—	7	2	no	—
77	D₇	—	9	2	no	—
78	D₇	—	14	3	no	—
79	C₂×D₃	—	147	4	no	—
80	C₃²	D₇²×C₁₇	1	1	yes	characteristic, complemented, Sylow
81	C₇	—	2	3	no	—
82	C₇	D₇×C₁₇×C₃⋊D₃	1	2	yes	complemented
83	C₂×C₃	—	49	4	no	—
84	D₃	—	21	8	no	—
85	C₂²	—	441	1	no	Sylow
86	C₃	C₁₇×D₂₁⋊D₇	1	4	yes	complemented
87	C₂	—	49	1	no	—
88	C₂	—	63	2	no	—

	subgroup	G/H
center, hypercenter	C₁₇	#13183685
commutator, cosocle, hypocenter	C₃²×C₇²	C₂²×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂²
LCS	G ▷ C₃²×C₇²
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₁₇×C₇⋊D₇	C₂	semidirect
2. C₇×C₁₇×C₃⋊D₂₁	C₂	semidirect
4. C₃²×C₇²×C₁₇	C₂²	semidirect
6. C₃×C₁₇×C₇⋊D₇	D₃	semidirect
9. C₃×C₇²×C₁₇	C₂×D₃	semidirect
10. C₁₇×C₃⋊D₂₁	D₇	semidirect
14. #13183685	C₁₇	direct (N × H)
15. C₁₇×C₇⋊D₇	C₃⋊D₃	semidirect
19. C₃²×C₇×C₁₇	C₂×D₇	semidirect
20. C₇×C₃⋊D₂₁	C₂×C₁₇	semidirect
21. C₃²×C₇⋊D₇	C₂×C₁₇	semidirect
22. C₇²×C₁₇	C₂×C₃⋊D₃	semidirect
30. C₃²×C₇²	C₂²×C₁₇	semidirect
32. C₃×C₇×C₁₇	D₃×D₇	semidirect
36. C₃×C₇⋊D₇	D₃×C₁₇	semidirect
44. C₃²×C₁₇	D₇²	semidirect
45. C₃×C₇²	C₂×D₃×C₁₇	semidirect
46. C₃⋊D₂₁	D₇×C₁₇	semidirect
51. C₇×C₁₇	D₇×C₃⋊D₃	semidirect
54. C₇⋊D₇	C₁₇×C₃⋊D₃	semidirect
59. C₃²×C₇	C₂×D₇×C₁₇	semidirect
60. C₃×C₁₇	D₂₁⋊D₇	semidirect
61. C₇²	C₂×C₁₇×C₃⋊D₃	semidirect
71. C₃×C₇	D₃×D₇×C₁₇	semidirect
74. C₁₇	#13183685	direct (N × H)
80. C₃²	D₇²×C₁₇	semidirect
82. C₇	D₇×C₁₇×C₃⋊D₃	semidirect
86. C₃	C₁₇×D₂₁⋊D₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₁₇×C₇⋊D₇	2
C₇×C₁₇×C₃⋊D₂₁	2
C₁₇×D₂₁⋊D₇	3
D₇×C₁₇×C₃⋊D₃	7
#13183685	17

Subgroups — C17×#13183685 #44472993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13183685 #44472993