Subgroups — C₁₇×#10624477 #27136641

Order 12393.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×#732	C₃	1	2	yes	maximal, complemented
2	C₁₇×#6867	C₃	1	2	yes	maximal, complemented
3	C₃×C₁₇×H(3,𝔽₃)	—	3	2	no	—
4	C₃²×C₉×C₁₇	C₃²	1	1	yes	characteristic
5	C₁₇×#732	—	3	6	no	—
6	C₁₇×C₃²⋊C₉	—	3	2	no	—
7	#10624477	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₃³×C₁₇	—	9	2	no	—
9	C₃³×C₁₇	H(3,𝔽₃)	1	1	yes	characteristic
10	C₁₇×H(3,𝔽₃)	—	3	6	no	—
11	C₁₇×H(3,𝔽₃)	—	9	4	no	—
12	C₃×C₉×C₁₇	—	3	2	no	—
13	C₃×C₉×C₁₇	—	3	2	no	—
14	C₃×C₉×C₁₇	—	9	2	no	—
15	C₃×#732	C₃×C₁₇	1	2	yes	complemented
16	#6867	C₃×C₁₇	1	2	yes	complemented
17	C₃²×C₁₇	—	9	1	no	—
18	C₃²×C₁₇	—	9	6	no	—
19	C₃²×C₁₇	—	27	6	no	—
20	C₃²×C₁₇	AΣL(1,𝔽₂₇)	1	2	yes	—
21	C₃²×C₁₇	#732	1	2	yes	—
22	C₉×C₁₇	—	9	1	no	—
23	C₉×C₁₇	—	27	2	no	—
24	C₃×H(3,𝔽₃)	—	3	2	no	—
25	C₃²×C₉	C₃²×C₁₇	1	1	yes	characteristic
26	#732	—	3	6	no	—
27	C₃²⋊C₉	—	3	2	no	—
28	C₃×C₁₇	—	3	2	no	—
29	C₃×C₁₇	—	3	2	no	—
30	C₃×C₁₇	—	27	6	no	—
31	C₃×C₁₇	#6856	1	1	yes	characteristic
32	C₃³	—	9	2	no	—
33	C₃³	C₁₇×H(3,𝔽₃)	1	1	yes	characteristic
34	H(3,𝔽₃)	—	3	6	no	—
35	H(3,𝔽₃)	—	9	4	no	—
36	C₃×C₉	—	3	2	no	—
37	C₃×C₉	—	3	2	no	—
38	C₃×C₉	—	9	2	no	—
39	C₁₇	#10624477	1	1	yes	characteristic, direct product factor, Sylow
40	C₃²	—	9	1	no	—
41	C₃²	—	9	6	no	—
42	C₃²	—	27	6	no	—
43	C₃²	C₁₇×AΣL(1,𝔽₂₇)	1	2	yes	—
44	C₃²	C₁₇×#732	1	2	yes	—
45	C₉	—	9	1	no	—
46	C₉	—	27	2	no	—
47	C₃	—	3	2	no	—
48	C₃	—	3	2	no	—
49	C₃	—	27	6	no	—
50	C₃	C₁₇×#6856	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₁₇	#6856
commutator, Frattini, cosocle	C₃²×C₉	C₃²×C₁₇
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₇ ◁ C₃³×C₁₇ ◁ C₃²×C₉×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×#732	C₃	semidirect
2. C₁₇×#6867	C₃	semidirect
7. #10624477	C₁₇	direct (N × H)
15. C₃×#732	C₃×C₁₇	semidirect
16. #6867	C₃×C₁₇	semidirect
39. C₁₇	#10624477	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×#732	3
C₁₇×#6867	3
#10624477	17

Subgroups — C17×#10624477 #27136641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10624477 #27136641