Subgroups — C₉×#15688532 #41007514

Order 26973.

986 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
2	#22582548	C₃	1	2	yes	maximal, complemented
3	3₋¹⁺²×#66524	C₃	1	2	yes	maximal, complemented
4	C₉×#11758606	C₃	1	3	yes	maximal, complemented
5	C₉×#11758605	C₃	1	3	yes	maximal, complemented
6	C₉×#11758608	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582543	C₃	1	6	yes	maximal, complemented
8	#22582542	C₃	1	6	yes	maximal, complemented
9	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
10	#22582539	C₃	1	2	yes	maximal
11	#22582537	C₃	1	6	yes	maximal, complemented
12	#22582535	C₃	1	2	yes	maximal, complemented
13	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
14	C₉²×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
15	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
16	#15688532	C₉	1	27	yes	direct product factor
17	C₃×#11758608	C₃²	1	1	yes	characteristic
18	C₃×#11758605	C₃²	1	3	yes	—
19	C₃×#11758606	C₃²	1	3	yes	—
20	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
21	C₃²×#66524	C₃²	1	2	yes	—
22	C₃²×#66524	C₃²	1	1	yes	characteristic
23	C₃×C₉×C₃₇⋊C₃	C₃²	1	6	yes	—
24	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
25	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
26	#15688506	C₃²	1	6	yes	—
27	#15688505	C₃²	1	6	yes	—
28	#15688504	C₃²	1	6	yes	—
29	#15688504	C₃²	1	6	yes	—
30	#15688504	C₃²	1	6	yes	—
31	#15688504	C₃²	1	6	yes	—
32	#15688504	C₃²	1	6	yes	—
33	C₃²⋊#66524	C₃²	1	2	yes	—
34	C₃²⋊#66524	C₃²	1	2	yes	—
35	#15688502	C₃²	1	6	yes	—
36	#15688502	C₃²	1	6	yes	—
37	#15688502	C₃²	1	6	yes	—
38	C₉×#66524	C₃²	1	6	yes	—
39	C₉×#66524	C₃²	1	2	yes	—
40	C₉×#66524	C₃²	1	2	yes	—
41	C₉×#66524	C₃²	1	6	yes	—
42	C₉×#66524	C₃²	1	6	yes	—
43	C₉×#66524	C₃²	1	2	yes	—
44	C₉×#66524	C₃²	1	3	yes	—
45	C₉×#66524	C₃²	1	1	yes	characteristic
46	C₉×#66524	C₃²	1	1	yes	characteristic
47	C₉×#66524	C₃²	1	1	yes	characteristic
48	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
49	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
50	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
51	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
52	C₉²×C₃₇	C₃²	1	3	yes	—
53	C₃³×C₃₇	C₃³	1	1	yes	characteristic
54	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
55	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	complemented
56	C₃×C₉×C₃₇	—	3	3	no	—
57	C₃×C₉×C₃₇	C₃³	1	6	yes	—
58	C₃×C₉×C₃₇	C₃³	1	2	yes	—
59	C₃×C₉×C₃₇	C₃³	1	3	yes	—
60	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
61	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	9	yes	—
62	#11758608	C₃×C₉	1	9	yes	complemented
63	C₃×#66524	—	3	6	no	—
64	C₃×#66524	C₃×C₉	1	9	yes	complemented
65	C₃×#66524	C₃³	1	2	yes	—
66	C₃×#66524	C₃³	1	2	yes	—
67	C₃×#66524	C₃³	1	6	yes	—
68	C₃×#66524	C₃³	1	6	yes	—
69	C₃×#66524	C₃³	1	1	yes	characteristic
70	C₃×#66524	C₃³	1	3	yes	—
71	C₃×#66524	C₃³	1	1	yes	characteristic
72	C₃×#66524	C₃³	1	1	yes	characteristic
73	C₃×#66524	C₃³	1	2	yes	—
74	#11758606	C₃×C₉	1	27	yes	complemented
75	#11758605	C₃×C₉	1	27	yes	complemented
76	C₉×C₃₇⋊C₃	—	3	9	no	—
77	C₉×C₃₇⋊C₃	—	3	18	no	—
78	C₉×C₃₇⋊C₃	C₃×C₉	1	27	yes	complemented
79	C₉×#737	—	37	1	no	maximal, Sylow
80	C₃²×C₃₇	—	3	2	no	—
81	C₃²×C₃₇	—	3	1	no	—
82	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
83	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
84	C₃×C₃₇⋊C₃	—	3	6	no	—
85	C₃×C₃₇⋊C₃	—	3	3	no	—
86	C₃×C₃₇⋊C₃	C₃²×C₉	1	9	yes	—
87	C₉×C₃₇	—	3	6	no	—
88	C₉×C₃₇	—	3	2	no	—
89	C₉×C₃₇	C₃²×C₉	1	9	yes	—
90	C₉×C₃₇	#737	1	3	yes	complemented
91	#66524	—	3	6	no	—
92	#66524	—	3	6	no	—
93	#66524	—	3	2	no	—
94	#66524	—	3	2	no	—
95	#66524	C₃²×C₉	1	3	yes	—
96	#66524	C₃²×C₉	1	9	yes	—
97	#66524	C₃²×C₉	1	3	yes	—
98	#66524	C₃²×C₉	1	3	yes	—
99	#66524	#737	1	6	yes	complemented
100	C₃×#737	—	37	1	no	—
101	C₉×3₋¹⁺²	—	37	2	no	—
102	C₉×3₋¹⁺²	—	37	1	no	—
103	C₉×3₋¹⁺²	—	37	6	no	—
104	C₉×3₋¹⁺²	—	37	6	no	—
105	C₉×3₋¹⁺²	—	37	2	no	—
106	C₉×3₋¹⁺²	—	37	3	no	—
107	C₉×3₋¹⁺²	—	37	3	no	—
108	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
109	C₉×H(3,𝔽₃)	—	37	1	no	—
110	C₉×H(3,𝔽₃)	—	37	2	no	—
111	#6887	—	37	6	no	—
112	#6887	—	37	2	no	—
113	C₃×C₉²	—	37	1	no	—
114	C₃×C₉²	—	37	3	no	—
115	C₃×C₃₇	—	3	3	no	—
116	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
117	C₃×C₃₇	C₃×#737	1	2	yes	—
118	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
119	C₃₇⋊C₃	—	3	9	no	—
120	#737	—	37	27	no	—
121	C₃×3₋¹⁺²	—	37	1	no	—
122	C₃×3₋¹⁺²	—	37	3	no	—
123	C₃×3₋¹⁺²	—	37	3	no	—
124	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
125	C₃×H(3,𝔽₃)	—	37	1	no	—
126	C₃²×C₉	—	37	1	no	—
127	C₃²×C₉	—	37	3	no	—
128	C₃²×C₉	—	37	2	no	—
129	C₃²×C₉	—	37	6	no	—
130	C₃²×C₉	—	37	3	no	—
131	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
132	C₉⋊C₉	—	37	6	no	—
133	C₉⋊C₉	—	37	6	no	—
134	C₉⋊C₉	—	37	6	no	—
135	C₉⋊C₉	—	37	6	no	—
136	C₉⋊C₉	—	37	6	no	—
137	C₉⋊C₉	—	37	6	no	—
138	C₉⋊C₉	—	37	6	no	—
139	C₉⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
140	C₃²⋊C₉	—	37	2	no	—
141	C₃²⋊C₉	—	37	2	no	—
142	C₃²⋊C₉	—	37	6	no	—
143	C₃²⋊C₉	—	37	6	no	—
144	C₃²⋊C₉	—	37	6	no	—
145	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
146	C₉²	—	37	2	no	—
147	C₉²	—	37	6	no	—
148	C₉²	—	37	2	no	—
149	C₉²	—	37	2	no	—
150	C₉²	—	37	6	no	—
151	C₉²	—	37	6	no	—
152	C₉²	—	37	1	no	—
153	C₉²	—	37	3	no	—
154	C₉²	—	37	1	no	—
155	C₉²	—	37	1	no	—
156	C₉²	C₃×C₃₇⋊C₃	1	3	yes	—
157	C₃₇	C₉×#737	1	1	yes	characteristic, complemented, Sylow
158	C₃³	—	37	3	no	—
159	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
160	3₋¹⁺²	—	37	9	no	—
161	3₋¹⁺²	—	37	27	no	—
162	3₋¹⁺²	—	37	27	no	—
163	3₋¹⁺²	C₉×C₃₇⋊C₃	1	9	yes	complemented
164	H(3,𝔽₃)	—	37	9	no	—
165	C₃×C₉	—	3	3	no	—
166	C₃×C₉	—	37	9	no	—
167	C₃×C₉	—	37	1	no	—
168	C₃×C₉	—	37	2	no	—
169	C₃×C₉	—	37	3	no	—
170	C₃×C₉	—	37	1	no	—
171	C₃×C₉	—	37	1	no	—
172	C₃×C₉	—	37	27	no	—
173	C₃×C₉	—	37	2	no	—
174	C₃×C₉	—	37	2	no	—
175	C₃×C₉	—	37	6	no	—
176	C₃×C₉	—	37	6	no	—
177	C₃×C₉	—	111	18	no	—
178	C₃×C₉	—	111	6	no	—
179	C₃×C₉	—	111	9	no	—
180	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
181	C₃×C₉	C₃²×C₃₇⋊C₃	1	6	yes	—
182	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
183	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
184	C₃²	—	3	2	no	—
185	C₃²	—	3	1	no	—
186	C₃²	—	37	9	no	—
187	C₃²	—	111	6	no	—
188	C₃²	—	111	3	no	—
189	C₃²	C₃×C₉×C₃₇⋊C₃	1	3	yes	—
190	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
191	C₉	—	3	6	no	—
192	C₉	—	3	2	no	—
193	C₉	—	37	3	no	—
194	C₉	—	37	3	no	—
195	C₉	—	37	3	no	—
196	C₉	—	37	9	no	—
197	C₉	—	37	6	no	—
198	C₉	—	111	6	no	—
199	C₉	—	111	2	no	—
200	C₉	—	111	2	no	—
201	C₉	—	111	6	no	—
202	C₉	C₃×C₉×C₃₇⋊C₃	1	9	yes	—
203	C₉	#15688532	1	3	yes	direct product factor
204	C₃	—	3	3	no	—
205	C₃	—	111	9	no	—
206	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic
207	C₃	#737×C₃₇⋊C₃	1	2	yes	—
208	C₃	C₃×#15688532	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₉×#737
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #22582548	C₃	semidirect
3. 3₋¹⁺²×#66524	C₃	semidirect
4. C₉×#11758606	C₃	semidirect
5. C₉×#11758605	C₃	semidirect
6. C₉×#11758608	C₃	semidirect
7. #22582543	C₃	semidirect
8. #22582542	C₃	semidirect
11. #22582537	C₃	semidirect
12. #22582535	C₃	semidirect
13. C₃×C₉×#66524	C₃	semidirect
14. C₉²×C₃₇⋊C₃	C₃	semidirect
15. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
16. #15688532	C₉	direct (N × H)
55. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
62. #11758608	C₃×C₉	semidirect
64. C₃×#66524	C₃×C₉	semidirect
74. #11758606	C₃×C₉	semidirect
75. #11758605	C₃×C₉	semidirect
78. C₉×C₃₇⋊C₃	C₃×C₉	semidirect
90. C₉×C₃₇	#737	semidirect
99. #66524	#737	semidirect
108. C₉×3₋¹⁺²	C₃₇⋊C₃	semidirect
157. C₃₇	C₉×#737	semidirect
163. 3₋¹⁺²	C₉×C₃₇⋊C₃	semidirect
203. C₉	#15688532	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×#15688532	3
#22582548	3
3₋¹⁺²×#66524	3
C₉×#11758606	3
C₉×#11758605	3
C₉×#11758608	3
#22582543	3
#22582542	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582539	3
#22582537	3
#22582535	3
C₃×C₉×#66524	3
C₉²×C₃₇⋊C₃	3
C₉×3₋¹⁺²×C₃₇	3
C₉×#737	37

Subgroups — C9×#15688532 #41007514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#15688532 #41007514