Subgroups — C₁₇×#10624444 #27136608

Order 12393.

316 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₁₇	C₃	1	1	yes	maximal, characteristic
2	C₁₇×C₃²⋊C₂₇	C₃	1	3	yes	maximal, complemented
3	C₃⁴×C₁₇	C₉	1	1	yes	characteristic
4	C₃²×C₉×C₁₇	—	3	1	no	—
5	C₃²×C₉×C₁₇	—	3	2	no	—
6	C₃²×C₉×C₁₇	—	3	9	no	—
7	C₃²×C₉×C₁₇	C₉	1	2	yes	—
8	C₃²×C₉×C₁₇	C₃²	1	1	yes	characteristic
9	C₃×C₁₇×C₂₇	—	3	3	no	—
10	#10624444	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃³×C₁₇	—	3	1	no	—
12	C₃³×C₁₇	—	3	2	no	—
13	C₃³×C₁₇	—	3	3	no	—
14	C₃³×C₁₇	—	3	6	no	—
15	C₃³×C₁₇	C₂₇	1	3	yes	complemented
16	C₃³×C₁₇	C₃×C₉	1	1	yes	characteristic
17	C₃×C₉×C₁₇	—	3	18	no	—
18	C₃×C₉×C₁₇	—	3	2	no	—
19	C₃×C₉×C₁₇	—	3	9	no	—
20	C₃×C₉×C₁₇	—	3	3	no	—
21	C₃×C₉×C₁₇	—	3	6	no	—
22	C₃×C₉×C₁₇	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₉×C₁₇	3₋¹⁺²	1	2	yes	—
24	C₁₇×C₂₇	—	9	3	no	—
25	C₃³×C₉	C₃×C₁₇	1	1	yes	characteristic
26	C₃²⋊C₂₇	C₃×C₁₇	1	3	yes	complemented
27	C₃²×C₁₇	—	3	18	no	—
28	C₃²×C₁₇	—	3	3	no	—
29	C₃²×C₁₇	—	3	3	no	—
30	C₃²×C₁₇	—	3	1	no	—
31	C₃²×C₁₇	—	3	6	no	—
32	C₃²×C₁₇	—	3	2	no	—
33	C₃²×C₁₇	—	3	9	no	—
34	C₃²×C₁₇	C₃²⋊C₉	1	1	yes	characteristic
35	C₃²×C₁₇	C₃×C₂₇	1	1	yes	characteristic
36	C₃²×C₁₇	C₂₇⋊C₃	1	2	yes	—
37	C₉×C₁₇	—	3	6	no	—
38	C₉×C₁₇	—	3	2	no	—
39	C₉×C₁₇	AΣL(1,𝔽₂₇)	1	3	yes	—
40	C₃⁴	C₉×C₁₇	1	1	yes	characteristic
41	C₃²×C₉	—	3	1	no	—
42	C₃²×C₉	—	3	2	no	—
43	C₃²×C₉	—	3	9	no	—
44	C₃²×C₉	C₉×C₁₇	1	2	yes	—
45	C₃²×C₉	C₃²×C₁₇	1	1	yes	characteristic
46	C₃×C₂₇	—	3	3	no	—
47	C₃×C₁₇	—	3	9	no	—
48	C₃×C₁₇	—	3	1	no	—
49	C₃×C₁₇	—	3	2	no	—
50	C₃×C₁₇	C₃²⋊C₂₇	1	1	yes	characteristic
51	C₃×C₁₇	#6866	1	1	yes	characteristic
52	C₃×C₁₇	#6869	1	2	yes	—
53	C₃³	—	3	3	no	—
54	C₃³	—	3	6	no	—
55	C₃³	—	3	1	no	—
56	C₃³	—	3	2	no	—
57	C₃³	C₁₇×C₂₇	1	3	yes	complemented
58	C₃³	C₃×C₉×C₁₇	1	1	yes	characteristic
59	C₃×C₉	—	3	18	no	—
60	C₃×C₉	—	3	2	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	C₁₇×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃×C₉	C₁₇×3₋¹⁺²	1	2	yes	—
66	C₂₇	—	9	3	no	—
67	C₁₇	#10624444	1	1	yes	characteristic, direct product factor, Sylow
68	C₃²	—	3	9	no	—
69	C₃²	—	3	18	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	6	no	—
74	C₃²	—	3	2	no	—
75	C₃²	C₁₇×C₃²⋊C₉	1	1	yes	characteristic
76	C₃²	C₃×C₁₇×C₂₇	1	1	yes	characteristic
77	C₃²	C₁₇×C₂₇⋊C₃	1	2	yes	—
78	C₉	—	3	6	no	—
79	C₉	—	3	2	no	—
80	C₉	C₁₇×AΣL(1,𝔽₂₇)	1	3	yes	—
81	C₃	—	3	9	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	2	no	—
84	C₃	C₁₇×C₃²⋊C₂₇	1	1	yes	characteristic
85	C₃	C₁₇×#6866	1	1	yes	characteristic
86	C₃	C₁₇×#6869	1	2	yes	—

	subgroup	G/H
center	C₃×C₉×C₁₇	H(3,𝔽₃)
commutator	C₃²	C₃×C₁₇×C₂₇
Frattini, cosocle	C₃²×C₉	C₃²×C₁₇
socle	C₃²×C₁₇	C₃²⋊C₉
prime radical	C₃⁴×C₁₇	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₁₇ ◁ C₃²×C₉×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×C₃²⋊C₂₇	C₃	semidirect
10. #10624444	C₁₇	direct (N × H)
15. C₃³×C₁₇	C₂₇	semidirect
26. C₃²⋊C₂₇	C₃×C₁₇	semidirect
57. C₃³	C₁₇×C₂₇	semidirect
67. C₁₇	#10624444	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×C₃²⋊C₂₇	3
#10624444	17

Subgroups — C17×#10624444 #27136608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10624444 #27136608