Subgroups — C₁₇×#10624547 #27136711

Order 12393.

2150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁵×C₁₇	C₃	1	1	yes	maximal, characteristic
2	C₃×C₁₇×C₃²⋊C₉	C₃	1	12	yes	maximal, complemented
3	C₃⁴×C₁₇	—	3	12	no	—
4	C₃⁴×C₁₇	—	3	24	no	—
5	C₃⁴×C₁₇	C₉	1	9	yes	complemented
6	C₃⁴×C₁₇	C₃²	1	4	yes	—
7	C₃²×C₉×C₁₇	C₃²	1	9	yes	complemented
8	C₁₇×C₃²⋊C₉	—	3	36	no	—
9	#10624547	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃³×C₁₇	—	3	27	no	—
11	C₃³×C₁₇	—	3	216	no	—
12	C₃³×C₁₇	—	3	108	no	—
13	C₃³×C₁₇	—	3	12	no	—
14	C₃³×C₁₇	—	3	24	no	—
15	C₃³×C₁₇	C₃×C₉	1	12	yes	—
16	C₃³×C₁₇	H(3,𝔽₃)	1	12	yes	—
17	C₃³×C₁₇	3₋¹⁺²	1	24	yes	—
18	C₃³×C₁₇	C₃³	1	1	yes	characteristic
19	C₃×C₉×C₁₇	—	3	36	no	—
20	C₃⁵	C₃×C₁₇	1	1	yes	characteristic
21	C₃×C₃²⋊C₉	C₃×C₁₇	1	12	yes	complemented
22	C₃²×C₁₇	—	3	243	no	—
23	C₃²×C₁₇	—	3	36	no	—
24	C₃²×C₁₇	—	3	12	no	—
25	C₃²×C₁₇	—	3	24	no	—
26	C₃²×C₁₇	—	3	72	no	—
27	C₃²×C₁₇	C₃²⋊C₉	1	36	yes	complemented
28	C₃²×C₁₇	C₃²×C₉	1	1	yes	characteristic
29	C₃²×C₁₇	C₃×H(3,𝔽₃)	1	4	yes	—
30	C₃²×C₁₇	C₃×3₋¹⁺²	1	8	yes	—
31	C₉×C₁₇	—	9	9	no	—
32	C₃⁴	—	3	12	no	—
33	C₃⁴	—	3	24	no	—
34	C₃⁴	C₉×C₁₇	1	9	yes	complemented
35	C₃⁴	C₃²×C₁₇	1	4	yes	—
36	C₃²×C₉	C₃²×C₁₇	1	9	yes	complemented
37	C₃²⋊C₉	—	3	36	no	—
38	C₃×C₁₇	—	3	36	no	—
39	C₃×C₁₇	C₃×C₃²⋊C₉	1	4	yes	—
40	C₃×C₁₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₁₇	#6891	1	8	yes	—
42	C₃³	—	3	27	no	—
43	C₃³	—	3	216	no	—
44	C₃³	—	3	108	no	—
45	C₃³	—	3	12	no	—
46	C₃³	—	3	24	no	—
47	C₃³	C₃×C₉×C₁₇	1	12	yes	—
48	C₃³	C₁₇×H(3,𝔽₃)	1	12	yes	—
49	C₃³	C₁₇×3₋¹⁺²	1	24	yes	—
50	C₃³	C₃³×C₁₇	1	1	yes	characteristic
51	C₃×C₉	—	3	36	no	—
52	C₁₇	#10624547	1	1	yes	characteristic, direct product factor, Sylow
53	C₃²	—	3	243	no	—
54	C₃²	—	3	36	no	—
55	C₃²	—	3	12	no	—
56	C₃²	—	3	24	no	—
57	C₃²	—	3	72	no	—
58	C₃²	C₁₇×C₃²⋊C₉	1	36	yes	complemented
59	C₃²	C₃²×C₉×C₁₇	1	1	yes	characteristic
60	C₃²	C₃×C₁₇×H(3,𝔽₃)	1	4	yes	—
61	C₃²	C₃×C₁₇×3₋¹⁺²	1	8	yes	—
62	C₉	—	9	9	no	—
63	C₃	—	3	36	no	—
64	C₃	C₃×C₁₇×C₃²⋊C₉	1	4	yes	—
65	C₃	C₁₇×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₃	C₁₇×#6891	1	8	yes	—

	subgroup	G/H
center, socle	C₃³×C₁₇	C₃³
commutator	C₃²	C₃²×C₉×C₁₇
Frattini, cosocle	C₃³	C₃³×C₁₇
prime radical	C₃⁵×C₁₇	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₇×C₃²⋊C₉	C₃	semidirect
5. C₃⁴×C₁₇	C₉	semidirect
7. C₃²×C₉×C₁₇	C₃²	semidirect
9. #10624547	C₁₇	direct (N × H)
21. C₃×C₃²⋊C₉	C₃×C₁₇	semidirect
27. C₃²×C₁₇	C₃²⋊C₉	semidirect
34. C₃⁴	C₉×C₁₇	semidirect
36. C₃²×C₉	C₃²×C₁₇	semidirect
52. C₁₇	#10624547	direct (N × H)
58. C₃²	C₁₇×C₃²⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃⁵×C₁₇	3
C₃×C₁₇×C₃²⋊C₉	3
#10624547	17

Subgroups — C17×#10624547 #27136711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10624547 #27136711