Subgroups — C₁₇×#10624480 #27136644

Order 12393.

156 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×#6868	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₇×H(3,𝔽₃)	—	3	2	no	—
5	C₃²×C₉×C₁₇	—	3	1	no	—
6	C₃²×C₉×C₁₇	C₃²	1	1	yes	characteristic
7	C₁₇×#732	—	3	6	no	—
8	C₁₇×C₃²⋊C₉	—	3	1	no	—
9	#10624480	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃³×C₁₇	—	9	2	no	—
11	C₃³×C₁₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₁₇×H(3,𝔽₃)	—	3	6	no	—
13	C₁₇×H(3,𝔽₃)	—	9	2	no	—
14	C₁₇×H(3,𝔽₃)	—	9	2	no	—
15	C₃×C₉×C₁₇	—	3	3	no	—
16	C₃×C₉×C₁₇	—	3	2	no	—
17	C₃×C₉×C₁₇	—	3	1	no	—
18	C₃×C₉×C₁₇	—	3	1	no	—
19	C₃×C₉×C₁₇	—	9	2	no	—
20	C₃×C₉×C₁₇	—	9	1	no	—
21	C₃×C₉×C₁₇	—	9	1	no	—
22	C₃×#732	C₃×C₁₇	1	2	yes	complemented
23	#6868	C₃×C₁₇	1	1	yes	characteristic, complemented
24	#6855	C₃×C₁₇	1	1	yes	characteristic, complemented
25	C₃²×C₁₇	—	3	1	no	—
26	C₃²×C₁₇	—	3	2	no	—
27	C₃²×C₁₇	—	9	6	no	—
28	C₃²×C₁₇	—	27	2	no	—
29	C₃²×C₁₇	—	27	4	no	—
30	C₃²×C₁₇	AΣL(1,𝔽₂₇)	1	2	yes	—
31	C₃²×C₁₇	#732	1	1	yes	characteristic
32	C₃²×C₁₇	#732	1	1	yes	characteristic
33	C₉×C₁₇	—	9	3	no	—
34	C₉×C₁₇	—	9	1	no	—
35	C₉×C₁₇	—	27	1	no	—
36	C₃×H(3,𝔽₃)	—	3	2	no	—
37	C₃²×C₉	—	3	1	no	—
38	C₃²×C₉	C₃²×C₁₇	1	1	yes	characteristic
39	#732	—	3	6	no	—
40	C₃²⋊C₉	—	3	1	no	—
41	C₃×C₁₇	—	3	1	no	—
42	C₃×C₁₇	—	3	2	no	—
43	C₃×C₁₇	—	27	6	no	—
44	C₃×C₁₇	#6856	1	1	yes	characteristic
45	C₃×C₁₇	#6879	1	1	yes	characteristic
46	C₃×C₁₇	#6881	1	2	yes	—
47	C₃³	—	9	2	no	—
48	C₃³	C₁₇×H(3,𝔽₃)	1	1	yes	characteristic
49	H(3,𝔽₃)	—	3	6	no	—
50	H(3,𝔽₃)	—	9	2	no	—
51	H(3,𝔽₃)	—	9	2	no	—
52	C₃×C₉	—	3	3	no	—
53	C₃×C₉	—	3	2	no	—
54	C₃×C₉	—	3	1	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	9	2	no	—
57	C₃×C₉	—	9	1	no	—
58	C₃×C₉	—	9	1	no	—
59	C₁₇	#10624480	1	1	yes	characteristic, direct product factor, Sylow
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	9	6	no	—
63	C₃²	—	27	2	no	—
64	C₃²	—	27	4	no	—
65	C₃²	C₁₇×AΣL(1,𝔽₂₇)	1	2	yes	—
66	C₃²	C₁₇×#732	1	1	yes	characteristic
67	C₃²	C₁₇×#732	1	1	yes	characteristic
68	C₉	—	9	3	no	—
69	C₉	—	9	1	no	—
70	C₉	—	27	1	no	—
71	C₃	—	3	1	no	—
72	C₃	—	3	2	no	—
73	C₃	—	27	6	no	—
74	C₃	C₁₇×#6856	1	1	yes	characteristic
75	C₃	C₁₇×#6879	1	1	yes	characteristic
76	C₃	C₁₇×#6881	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₇	#732
commutator, Frattini, cosocle	C₃²×C₉	C₃²×C₁₇
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×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×#6868	C₃	semidirect
3. C₁₇×#6855	C₃	semidirect
9. #10624480	C₁₇	direct (N × H)
22. C₃×#732	C₃×C₁₇	semidirect
23. #6868	C₃×C₁₇	semidirect
24. #6855	C₃×C₁₇	semidirect
59. C₁₇	#10624480	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×#6868	3
C₁₇×#6855	3
#10624480	17

Subgroups — C17×#10624480 #27136644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10624480 #27136644