Subgroups — C₁₇×#10624541 #27136705

Order 12393.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#6880	C₃	1	3	yes	maximal
2	C₁₇×#6864	C₃	1	1	yes	maximal, characteristic
3	C₁₇×#733	—	3	3	no	—
4	C₃×C₁₇×C₂₇	—	3	1	no	—
5	C₉²×C₁₇	C₃²	1	1	yes	characteristic
6	#10624541	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₁₇×3₋¹⁺²	—	9	3	no	—
8	C₃×C₉×C₁₇	—	3	1	no	—
9	C₃×C₉×C₁₇	H(3,𝔽₃)	1	1	yes	characteristic
10	C₁₇×C₂₇	—	3	3	no	—
11	#6880	C₃×C₁₇	1	3	yes	—
12	#6864	C₃×C₁₇	1	1	yes	characteristic
13	C₃²×C₁₇	#732	1	1	yes	characteristic
14	C₉×C₁₇	—	3	1	no	—
15	C₉×C₁₇	—	9	1	no	—
16	C₉×C₁₇	—	27	3	no	—
17	#733	—	3	3	no	—
18	C₃×C₂₇	—	3	1	no	—
19	C₉²	C₃²×C₁₇	1	1	yes	characteristic
20	C₃×C₁₇	—	3	1	no	—
21	C₃×C₁₇	#6879	1	1	yes	characteristic
22	3₋¹⁺²	—	9	3	no	—
23	C₃×C₉	—	3	1	no	—
24	C₃×C₉	C₁₇×H(3,𝔽₃)	1	1	yes	characteristic
25	C₂₇	—	3	3	no	—
26	C₁₇	#10624541	1	1	yes	characteristic, direct product factor, Sylow
27	C₃²	C₁₇×#732	1	1	yes	characteristic
28	C₉	—	3	1	no	—
29	C₉	—	9	1	no	—
30	C₉	—	27	3	no	—
31	C₃	—	3	1	no	—
32	C₃	C₁₇×#6879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₁₇	#6879
commutator, Frattini, cosocle	C₉²	C₃²×C₁₇
prime radical	C₃²×C₁₇	#732
LCS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₇ ◁ C₃²×C₁₇ ◁ C₃×C₉×C₁₇ ◁ C₉²×C₁₇ ◁ G

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

N	H ≅ G/N	type
6. #10624541	C₁₇	direct (N × H)
26. C₁₇	#10624541	direct (N × H)

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