Subgroups — C₁₇×#10624483 #27136647

Order 12393.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×#733	C₃	1	2	yes	maximal
2	C₁₇×#6868	C₃	1	1	yes	maximal, characteristic
3	C₁₇×#6855	C₃	1	1	yes	maximal, characteristic
4	C₃×C₁₇×3₋¹⁺²	—	3	2	no	—
5	C₃²×C₉×C₁₇	—	3	1	no	—
6	C₃²×C₉×C₁₇	C₃²	1	1	yes	characteristic
7	C₁₇×#733	—	3	6	no	—
8	C₁₇×C₃²⋊C₉	—	3	1	no	—
9	#10624483	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃³×C₁₇	H(3,𝔽₃)	1	1	yes	characteristic
11	C₁₇×3₋¹⁺²	—	3	6	no	—
12	C₁₇×3₋¹⁺²	—	9	2	no	—
13	C₁₇×3₋¹⁺²	—	9	2	no	—
14	C₃×C₉×C₁₇	—	3	3	no	—
15	C₃×C₉×C₁₇	—	3	2	no	—
16	C₃×C₉×C₁₇	—	3	1	no	—
17	C₃×C₉×C₁₇	—	3	1	no	—
18	C₃×C₉×C₁₇	—	9	2	no	—
19	C₃×C₉×C₁₇	—	9	1	no	—
20	C₃×C₉×C₁₇	—	9	2	no	—
21	C₃×C₉×C₁₇	—	9	1	no	—
22	C₃×#733	C₃×C₁₇	1	2	yes	—
23	#6868	C₃×C₁₇	1	1	yes	characteristic
24	#6855	C₃×C₁₇	1	1	yes	characteristic
25	C₃²×C₁₇	—	3	1	no	—
26	C₃²×C₁₇	—	3	2	no	—
27	C₃²×C₁₇	AΣL(1,𝔽₂₇)	1	2	yes	—
28	C₃²×C₁₇	#732	1	1	yes	characteristic
29	C₃²×C₁₇	#732	1	1	yes	characteristic
30	C₉×C₁₇	—	9	3	no	—
31	C₉×C₁₇	—	9	1	no	—
32	C₉×C₁₇	—	9	6	no	—
33	C₉×C₁₇	—	27	1	no	—
34	C₃×3₋¹⁺²	—	3	2	no	—
35	C₃²×C₉	—	3	1	no	—
36	C₃²×C₉	C₃²×C₁₇	1	1	yes	characteristic
37	#733	—	3	6	no	—
38	C₃²⋊C₉	—	3	1	no	—
39	C₃×C₁₇	—	3	1	no	—
40	C₃×C₁₇	—	3	2	no	—
41	C₃×C₁₇	#6856	1	1	yes	characteristic
42	C₃×C₁₇	#6878	1	1	yes	characteristic
43	C₃×C₁₇	#6882	1	2	yes	—
44	C₃³	C₁₇×H(3,𝔽₃)	1	1	yes	characteristic
45	3₋¹⁺²	—	3	6	no	—
46	3₋¹⁺²	—	9	2	no	—
47	3₋¹⁺²	—	9	2	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	9	2	no	—
53	C₃×C₉	—	9	1	no	—
54	C₃×C₉	—	9	2	no	—
55	C₃×C₉	—	9	1	no	—
56	C₁₇	#10624483	1	1	yes	characteristic, direct product factor, Sylow
57	C₃²	—	3	1	no	—
58	C₃²	—	3	2	no	—
59	C₃²	C₁₇×AΣL(1,𝔽₂₇)	1	2	yes	—
60	C₃²	C₁₇×#732	1	1	yes	characteristic
61	C₃²	C₁₇×#732	1	1	yes	characteristic
62	C₉	—	9	3	no	—
63	C₉	—	9	1	no	—
64	C₉	—	9	6	no	—
65	C₉	—	27	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	C₃	C₁₇×#6856	1	1	yes	characteristic
69	C₃	C₁₇×#6878	1	1	yes	characteristic
70	C₃	C₁₇×#6882	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₇	#732
commutator, Frattini, cosocle	C₃²×C₉	C₃²×C₁₇
prime radical	C₃³×C₁₇	H(3,𝔽₃)
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₇ ◁ C₃³×C₁₇ ◁ C₃²×C₉×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. #10624483	C₁₇	direct (N × H)
56. C₁₇	#10624483	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₁₇×#733	3
C₁₇×#6868	3
C₁₇×#6855	3
#10624483	17

Subgroups — C17×#10624483 #27136647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10624483 #27136647