Subgroups — C₉×#15688530 #41007510

Order 26973.

986 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
2	#22582548	C₃	1	16	yes	maximal, complemented
3	C₉×#11758608	C₃	1	8	yes	maximal, complemented
4	H(3,𝔽₃)×#66524	C₃	1	2	yes	maximal
5	#22582535	C₃	1	8	yes	maximal, complemented
6	C₃×C₉×#66524	C₃	1	4	yes	maximal, complemented
7	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
8	#15688530	C₉	1	27	yes	direct product factor
9	C₃×#11758608	C₃²	1	8	yes	—
10	C₃²×#66524	C₃²	1	8	yes	—
11	C₃²×#66524	C₃²	1	4	yes	—
12	#15688504	C₃²	1	48	yes	—
13	C₃²⋊#66524	C₃²	1	16	yes	—
14	C₉×#66524	C₃²	1	8	yes	—
15	C₉×#66524	C₃²	1	8	yes	—
16	C₉×#66524	C₃²	1	16	yes	—
17	C₉×#66524	C₃²	1	1	yes	characteristic
18	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	8	yes	—
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
23	C₃×C₉×C₃₇	—	3	12	no	—
24	C₃×C₉×C₃₇	C₃³	1	8	yes	—
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	#11758608	C₃×C₉	1	72	yes	complemented
27	C₃×#66524	—	3	24	no	—
28	C₃×#66524	C₃×C₉	1	36	yes	complemented
29	C₃×#66524	C₃³	1	16	yes	—
30	C₃×#66524	C₃³	1	1	yes	characteristic
31	C₃×#66524	C₃³	1	8	yes	—
32	C₃×#66524	C₃³	1	2	yes	—
33	C₉×#737	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	—	3	8	no	—
35	C₃²×C₃₇	—	3	4	no	—
36	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
37	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
38	C₉×C₃₇	—	3	8	no	—
39	C₉×C₃₇	#737	1	3	yes	complemented
40	#66524	—	3	16	no	—
41	#66524	C₃²×C₉	1	3	yes	—
42	#66524	C₃²×C₉	1	24	yes	—
43	#66524	#737	1	6	yes	complemented
44	C₃×#737	—	37	1	no	—
45	C₉×3₋¹⁺²	—	37	8	no	—
46	C₉×3₋¹⁺²	—	37	16	no	—
47	C₉×H(3,𝔽₃)	—	37	2	no	—
48	C₉×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
49	#6887	—	37	8	no	—
50	C₃×C₉²	—	37	4	no	—
51	C₃×C₃₇	—	3	12	no	—
52	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
53	C₃×C₃₇	C₃×#737	1	2	yes	—
54	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
55	#737	—	37	27	no	—
56	C₃×3₋¹⁺²	—	37	8	no	—
57	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
58	C₃²×C₉	—	37	4	no	—
59	C₃²×C₉	—	37	8	no	—
60	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
61	C₉⋊C₉	—	37	48	no	—
62	C₃²⋊C₉	—	37	16	no	—
63	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	8	yes	—
64	C₉²	—	37	8	no	—
65	C₉²	—	37	16	no	—
66	C₉²	—	37	1	no	—
67	C₉²	—	37	8	no	—
68	C₃₇	C₉×#737	1	1	yes	characteristic, complemented, Sylow
69	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
70	3₋¹⁺²	—	37	72	no	—
71	H(3,𝔽₃)	C₉×C₃₇⋊C₃	1	9	yes	—
72	C₃×C₉	—	3	12	no	—
73	C₃×C₉	—	37	36	no	—
74	C₃×C₉	—	37	1	no	—
75	C₃×C₉	—	37	2	no	—
76	C₃×C₉	—	37	8	no	—
77	C₃×C₉	—	37	16	no	—
78	C₃×C₉	—	111	24	no	—
79	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
80	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃²	—	3	8	no	—
82	C₃²	—	3	4	no	—
83	C₃²	C₃×C₉×C₃₇⋊C₃	1	12	yes	—
84	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
85	C₉	—	3	8	no	—
86	C₉	—	37	3	no	—
87	C₉	—	37	24	no	—
88	C₉	—	37	6	no	—
89	C₉	—	111	16	no	—
90	C₉	#15688530	1	3	yes	direct product factor
91	C₃	—	3	12	no	—
92	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic
93	C₃	#737×C₃₇⋊C₃	1	2	yes	—
94	C₃	C₃×#15688530	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₉×H(3,𝔽₃)×C₃₇	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₉×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₉×#737
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. #22582548	C₃	semidirect
3. C₉×#11758608	C₃	semidirect
5. #22582535	C₃	semidirect
6. C₃×C₉×#66524	C₃	semidirect
8. #15688530	C₉	direct (N × H)
26. #11758608	C₃×C₉	semidirect
28. C₃×#66524	C₃×C₉	semidirect
39. C₉×C₃₇	#737	semidirect
43. #66524	#737	semidirect
68. C₃₇	C₉×#737	semidirect
90. C₉	#15688530	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×#15688530	3
#22582548	3
C₉×#11758608	3
H(3,𝔽₃)×#66524	3
#22582535	3
C₃×C₉×#66524	3
C₉×H(3,𝔽₃)×C₃₇	3
C₉×#737	37

Subgroups — C9×#15688530 #41007510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#15688530 #41007510