Subgroups — C₃×C₁₇×#6855 #27136707

Order 12393.

1052 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₁₇	C₃	1	4	yes	maximal
2	C₁₇×#6855	C₃	1	9	yes	maximal, direct product factor
3	C₃⁴×C₁₇	C₃²	1	1	yes	characteristic
4	C₃²×C₉×C₁₇	—	3	12	no	—
5	C₃²×C₉×C₁₇	—	3	24	no	—
6	C₃²×C₉×C₁₇	C₉	1	36	yes	complemented
7	C₃²×C₉×C₁₇	C₃²	1	12	yes	—
8	C₃×#6855	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃³×C₁₇	C₃×C₉	1	12	yes	—
10	C₃³×C₁₇	H(3,𝔽₃)	1	3	yes	—
11	C₃³×C₁₇	3₋¹⁺²	1	24	yes	—
12	C₃³×C₁₇	C₃³	1	1	yes	characteristic
13	C₃×C₉×C₁₇	—	3	108	no	—
14	C₃×C₉×C₁₇	—	3	12	no	—
15	C₃×C₉×C₁₇	—	3	24	no	—
16	C₃×C₉×C₁₇	C₃×C₉	1	36	yes	complemented
17	C₃³×C₉	C₃×C₁₇	1	4	yes	—
18	#6855	C₃×C₁₇	1	9	yes	direct product factor
19	C₃²×C₁₇	C₉²	1	9	yes	—
20	C₃²×C₁₇	C₃²⋊C₉	1	36	yes	—
21	C₃²×C₁₇	C₉⋊C₉	1	72	yes	—
22	C₃²×C₁₇	C₃²×C₉	1	4	yes	—
23	C₃²×C₁₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²×C₁₇	C₃×3₋¹⁺²	1	8	yes	—
25	C₉×C₁₇	—	3	36	no	—
26	C₃⁴	C₃²×C₁₇	1	1	yes	characteristic
27	C₃²×C₉	—	3	12	no	—
28	C₃²×C₉	—	3	24	no	—
29	C₃²×C₉	C₉×C₁₇	1	36	yes	complemented
30	C₃²×C₉	C₃²×C₁₇	1	12	yes	—
31	C₃×C₁₇	#6855	1	27	yes	direct product factor
32	C₃×C₁₇	C₃×C₉²	1	1	yes	characteristic
33	C₃×C₁₇	C₃×C₃²⋊C₉	1	4	yes	—
34	C₃×C₁₇	C₃×C₉⋊C₉	1	8	yes	—
35	C₃³	C₃×C₉×C₁₇	1	12	yes	—
36	C₃³	C₁₇×H(3,𝔽₃)	1	3	yes	—
37	C₃³	C₁₇×3₋¹⁺²	1	24	yes	—
38	C₃³	C₃³×C₁₇	1	1	yes	characteristic
39	C₃×C₉	—	3	108	no	—
40	C₃×C₉	—	3	12	no	—
41	C₃×C₉	—	3	24	no	—
42	C₃×C₉	C₃×C₉×C₁₇	1	36	yes	complemented
43	C₁₇	C₃×#6855	1	1	yes	characteristic, direct product factor, Sylow
44	C₃²	C₉²×C₁₇	1	9	yes	—
45	C₃²	C₁₇×C₃²⋊C₉	1	36	yes	—
46	C₃²	C₁₇×C₉⋊C₉	1	72	yes	—
47	C₃²	C₃²×C₉×C₁₇	1	4	yes	—
48	C₃²	C₃×C₁₇×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃²	C₃×C₁₇×3₋¹⁺²	1	8	yes	—
50	C₉	—	3	36	no	—
51	C₃	C₁₇×#6855	1	27	yes	direct product factor
52	C₃	C₃×C₉²×C₁₇	1	1	yes	characteristic
53	C₃	C₃×C₁₇×C₃²⋊C₉	1	4	yes	—
54	C₃	C₃×C₁₇×C₉⋊C₉	1	8	yes	—

	subgroup	G/H
center, socle, prime radical	C₃⁴×C₁₇	C₃²
commutator	C₃	C₃×C₉²×C₁₇
Frattini, cosocle	C₃³	C₃³×C₁₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃⁴×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#6855	C₃	direct (N × H)
6. C₃²×C₉×C₁₇	C₉	semidirect
8. C₃×#6855	C₁₇	direct (N × H)
16. C₃×C₉×C₁₇	C₃×C₉	semidirect
18. #6855	C₃×C₁₇	direct (N × H)
29. C₃²×C₉	C₉×C₁₇	semidirect
31. C₃×C₁₇	#6855	direct (N × H)
42. C₃×C₉	C₃×C₉×C₁₇	semidirect
43. C₁₇	C₃×#6855	direct (N × H)
51. C₃	C₁₇×#6855	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₉×C₁₇	3
C₁₇×#6855	3
C₃×#6855	17

Subgroups — C3×C17×#6855 #27136707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₇×#6855 #27136707