Subgroups — #14554428

Order 2187.

771 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6913	C₃	1	1	yes	maximal, characteristic
2	#10624887	C₃	1	3	yes	maximal, complemented
3	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624791	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624791	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624791	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624791	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624790	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624790	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624789	C₃	1	3	yes	maximal, complemented
11	#10624789	C₃	1	3	yes	maximal, complemented
12	#10624788	C₃	1	3	yes	maximal, complemented
13	#10624788	C₃	1	3	yes	maximal, complemented
14	#10624787	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624787	C₃	1	1	yes	maximal, characteristic, complemented
16	#10624786	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624785	C₃	1	3	yes	maximal, complemented
18	#10624784	C₃	1	3	yes	maximal, complemented
19	#10624700	C₃	1	3	yes	maximal, complemented
20	#10624697	C₃	1	3	yes	maximal, complemented
21	#10624692	C₃	1	1	yes	maximal, characteristic, complemented
22	#10624692	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#737	C₃²	1	3	yes	—
24	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
25	C₃³×C₉	C₃²	1	1	yes	characteristic
26	#6913	—	3	9	no	—
27	C₃×#733	C₃²	1	1	yes	characteristic
28	C₃×#733	C₃²	1	1	yes	characteristic
29	C₃×#732	C₃²	1	1	yes	characteristic
30	C₃×#731	C₃²	1	3	yes	—
31	C₃×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	—
32	#6900	—	3	3	no	—
33	#6899	—	3	3	no	—
34	#6899	—	3	3	no	—
35	#6896	—	3	3	no	—
36	#6896	—	3	3	no	—
37	#6896	—	3	3	no	—
38	#6894	—	3	3	no	—
39	#6894	—	3	3	no	—
40	#6894	—	3	3	no	—
41	#6894	C₃²	1	3	yes	—
42	#6894	C₃²	1	3	yes	—
43	#6894	C₃²	1	3	yes	—
44	#6894	C₃²	1	3	yes	—
45	#6894	C₃²	1	3	yes	—
46	#6894	C₃²	1	3	yes	—
47	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
48	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
49	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
50	C₉×3₋¹⁺²	—	3	3	no	—
51	C₉×3₋¹⁺²	—	3	3	no	—
52	C₉×3₋¹⁺²	—	3	3	no	—
53	#6887	C₃²	1	3	yes	—
54	#6887	C₃²	1	3	yes	—
55	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
56	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
57	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
58	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
59	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
60	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
61	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
62	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
63	C₃×C₉²	C₃²	1	1	yes	characteristic
64	C₃×C₉²	C₃²	1	1	yes	characteristic
65	C₃×C₉²	C₃²	1	1	yes	characteristic
66	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
67	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
68	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
69	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
70	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
71	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
72	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
73	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
74	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
75	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
76	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
77	3₋¹⁺²⋊C₉	C₃²	1	3	yes	—
78	H(3,𝔽₃)⋊C₉	C₃²	1	3	yes	—
79	H(3,𝔽₃)⋊C₉	C₃²	1	3	yes	—
80	H(3,𝔽₃)⋊C₉	C₃²	1	3	yes	—
81	H(3,𝔽₃)⋊C₉	C₃²	1	3	yes	—
82	H(3,𝔽₃)⋊C₉	C₃²	1	3	yes	—
83	H(3,𝔽₃)⋊C₉	C₃²	1	3	yes	—
84	#6868	C₃²	1	1	yes	characteristic
85	#6868	C₃²	1	1	yes	characteristic
86	#6868	C₃²	1	1	yes	characteristic
87	#6868	C₃²	1	1	yes	characteristic
88	#6868	C₃²	1	1	yes	characteristic
89	#6868	C₃²	1	1	yes	characteristic
90	#6867	C₃²	1	3	yes	—
91	#6867	C₃²	1	3	yes	—
92	#6866	C₃²	1	3	yes	—
93	#6866	C₃²	1	3	yes	—
94	C₃⁴	C₃³	1	1	yes	characteristic
95	#737	—	3	9	no	—
96	#737	—	3	18	no	—
97	C₃×3₋¹⁺²	—	3	3	no	—
98	C₃×3₋¹⁺²	—	3	3	no	—
99	C₃×3₋¹⁺²	—	3	3	no	—
100	C₃×3₋¹⁺²	—	3	3	no	—
101	C₃×3₋¹⁺²	—	3	3	no	—
102	C₃×3₋¹⁺²	—	3	3	no	—
103	C₃×3₋¹⁺²	—	3	6	no	—
104	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
105	C₃×3₋¹⁺²	C₃³	1	3	yes	—
106	C₃×3₋¹⁺²	C₃³	1	3	yes	—
107	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
108	C₃²×C₉	—	3	3	no	—
109	C₃²×C₉	—	3	3	no	—
110	C₃²×C₉	—	9	3	no	—
111	C₃²×C₉	C₃³	1	1	yes	characteristic
112	C₃²×C₉	C₃³	1	1	yes	characteristic
113	C₃²×C₉	C₃³	1	1	yes	characteristic
114	C₃²×C₉	C₃³	1	1	yes	characteristic
115	C₃²×C₉	C₃³	1	1	yes	characteristic
116	C₃²×C₉	C₃³	1	1	yes	characteristic
117	C₃²×C₉	C₃³	1	1	yes	characteristic
118	C₃²×C₉	C₃³	1	1	yes	characteristic
119	C₃²×C₉	C₃³	1	1	yes	characteristic
120	C₃²×C₉	C₃³	1	1	yes	characteristic
121	C₃²×C₉	C₃³	1	1	yes	characteristic
122	C₃²×C₉	C₃³	1	1	yes	characteristic
123	#733	—	3	3	no	—
124	#733	—	3	3	no	—
125	#732	—	3	3	no	—
126	#731	—	3	9	no	—
127	AΣL(1,𝔽₂₇)	—	3	9	no	—
128	C₉⋊C₉	—	3	3	no	—
129	C₉⋊C₉	—	3	3	no	—
130	C₉⋊C₉	—	3	1	no	—
131	C₉⋊C₉	—	3	1	no	—
132	C₉⋊C₉	—	3	3	no	—
133	C₉⋊C₉	—	3	1	no	—
134	C₉⋊C₉	—	3	3	no	—
135	C₉⋊C₉	—	3	3	no	—
136	C₉⋊C₉	—	3	3	no	—
137	C₉⋊C₉	—	3	3	no	—
138	C₉⋊C₉	—	3	3	no	—
139	C₉⋊C₉	—	3	3	no	—
140	C₉⋊C₉	—	3	3	no	—
141	C₉⋊C₉	—	3	3	no	—
142	C₉⋊C₉	—	3	3	no	—
143	C₉⋊C₉	—	3	3	no	—
144	C₉⋊C₉	—	3	3	no	—
145	C₉⋊C₉	—	3	3	no	—
146	C₉⋊C₉	—	3	3	no	—
147	C₉⋊C₉	—	3	3	no	—
148	C₉⋊C₉	—	3	3	no	—
149	C₉⋊C₉	—	3	3	no	—
150	C₉⋊C₉	—	3	3	no	—
151	C₉⋊C₉	—	3	3	no	—
152	C₉⋊C₉	—	3	3	no	—
153	C₉⋊C₉	—	3	3	no	—
154	C₉⋊C₉	—	3	3	no	—
155	C₉⋊C₉	—	3	3	no	—
156	C₉⋊C₉	—	3	3	no	—
157	C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
158	C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
159	C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
160	C₃²⋊C₉	—	3	1	no	—
161	C₃²⋊C₉	—	3	1	no	—
162	C₃²⋊C₉	—	3	3	no	—
163	C₃²⋊C₉	—	3	1	no	—
164	C₃²⋊C₉	—	3	3	no	—
165	C₃²⋊C₉	—	3	3	no	—
166	C₃²⋊C₉	—	3	3	no	—
167	C₃²⋊C₉	—	3	3	no	—
168	C₃²⋊C₉	—	3	3	no	—
169	C₃²⋊C₉	—	3	3	no	—
170	C₃²⋊C₉	C₃³	1	3	yes	—
171	C₃²⋊C₉	C₃³	1	3	yes	—
172	C₃²⋊C₉	C₃³	1	3	yes	—
173	C₃²⋊C₉	C₃³	1	3	yes	—
174	C₃²⋊C₉	C₃³	1	3	yes	—
175	C₃²⋊C₉	C₃³	1	3	yes	—
176	C₉²	—	3	1	no	—
177	C₉²	—	3	1	no	—
178	C₉²	—	3	3	no	—
179	C₉²	—	3	1	no	—
180	C₉²	—	3	3	no	—
181	C₉²	—	3	3	no	—
182	C₉²	—	3	3	no	—
183	C₃³	—	3	1	no	—
184	C₃³	—	3	2	no	—
185	C₃³	—	3	3	no	—
186	C₃³	—	9	1	no	—
187	C₃³	—	9	1	no	—
188	C₃³	—	9	1	no	—
189	C₃³	C₃×3₋¹⁺²	1	3	yes	—
190	C₃³	C₃⁴	1	1	yes	characteristic
191	3₋¹⁺²	—	3	27	no	—
192	3₋¹⁺²	—	3	3	no	—
193	3₋¹⁺²	—	3	3	no	—
194	3₋¹⁺²	—	3	6	no	—
195	3₋¹⁺²	—	3	6	no	—
196	3₋¹⁺²	—	9	9	no	—
197	3₋¹⁺²	—	9	9	no	—
198	3₋¹⁺²	—	27	3	no	—
199	3₋¹⁺²	—	27	3	no	—
200	3₋¹⁺²	—	27	3	no	—
201	H(3,𝔽₃)	—	3	3	no	—
202	H(3,𝔽₃)	—	3	6	no	—
203	C₃×C₉	—	3	1	no	—
204	C₃×C₉	—	3	1	no	—
205	C₃×C₉	—	3	1	no	—
206	C₃×C₉	—	3	2	no	—
207	C₃×C₉	—	3	2	no	—
208	C₃×C₉	—	3	3	no	—
209	C₃×C₉	—	3	1	no	—
210	C₃×C₉	—	3	1	no	—
211	C₃×C₉	—	3	1	no	—
212	C₃×C₉	—	3	1	no	—
213	C₃×C₉	—	3	3	no	—
214	C₃×C₉	—	3	3	no	—
215	C₃×C₉	—	3	3	no	—
216	C₃×C₉	—	3	3	no	—
217	C₃×C₉	—	3	3	no	—
218	C₃×C₉	—	3	3	no	—
219	C₃×C₉	—	3	1	no	—
220	C₃×C₉	—	3	3	no	—
221	C₃×C₉	—	3	3	no	—
222	C₃×C₉	—	9	1	no	—
223	C₃×C₉	—	9	1	no	—
224	C₃×C₉	—	9	1	no	—
225	C₃×C₉	—	9	9	no	—
226	C₃×C₉	—	9	9	no	—
227	C₃×C₉	—	9	1	no	—
228	C₃×C₉	—	9	1	no	—
229	C₃×C₉	—	9	1	no	—
230	C₃×C₉	—	9	1	no	—
231	C₃×C₉	—	9	1	no	—
232	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
233	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
234	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
235	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
236	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
237	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
238	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
239	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
240	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
241	C₃×C₉	#737	1	6	yes	—
242	C₃²	—	3	1	no	—
243	C₃²	—	3	1	no	—
244	C₃²	—	3	1	no	—
245	C₃²	—	3	3	no	—
246	C₃²	—	9	3	no	—
247	C₃²	—	9	6	no	—
248	C₃²	—	9	1	no	—
249	C₃²	—	9	1	no	—
250	C₃²	—	9	1	no	—
251	C₃²	—	9	3	no	—
252	C₃²	—	9	3	no	—
253	C₃²	—	9	3	no	—
254	C₃²	—	9	3	no	—
255	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
256	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
257	C₃²	C₃×#737	1	2	yes	—
258	C₉	—	3	3	no	—
259	C₉	—	3	3	no	—
260	C₉	—	3	3	no	—
261	C₉	—	3	3	no	—
262	C₉	—	3	3	no	—
263	C₉	—	3	1	no	—
264	C₉	—	3	1	no	—
265	C₉	—	3	1	no	—
266	C₉	—	9	3	no	—
267	C₉	—	9	3	no	—
268	C₉	—	9	1	no	—
269	C₉	—	9	3	no	—
270	C₉	—	9	3	no	—
271	C₉	—	9	1	no	—
272	C₉	—	9	1	no	—
273	C₉	—	9	3	no	—
274	C₉	—	9	3	no	—
275	C₉	—	9	1	no	—
276	C₉	—	9	1	no	—
277	C₉	—	9	3	no	—
278	C₉	—	9	3	no	—
279	C₉	C₉⋊H(3,𝔽₃)	1	9	yes	—
280	C₃	—	3	1	no	—
281	C₃	—	3	2	no	—
282	C₃	—	9	3	no	—
283	C₃	—	27	3	no	—
284	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
285	C₃	#10624934	1	1	yes	characteristic
286	C₃	#10624935	1	1	yes	characteristic
287	C₃	#10624936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃×#6913	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃×C₉⋊C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624887	C₃	semidirect
3. C₃×C₉×3₋¹⁺²	C₃	semidirect
4. #10624791	C₃	semidirect
5. #10624791	C₃	semidirect
6. #10624791	C₃	semidirect
7. #10624791	C₃	semidirect
8. #10624790	C₃	semidirect
9. #10624790	C₃	semidirect
10. #10624789	C₃	semidirect
11. #10624789	C₃	semidirect
12. #10624788	C₃	semidirect
13. #10624788	C₃	semidirect
14. #10624787	C₃	semidirect
15. #10624787	C₃	semidirect
16. #10624786	C₃	semidirect
17. #10624785	C₃	semidirect
18. #10624784	C₃	semidirect
19. #10624700	C₃	semidirect
20. #10624697	C₃	semidirect
21. #10624692	C₃	semidirect
22. #10624692	C₃	semidirect
104. C₃×3₋¹⁺²	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6913	3
#10624887	3
C₃×C₉×3₋¹⁺²	3
#10624791	3
#10624790	3
#10624789	3
#10624788	3
#10624787	3
#10624786	3
#10624785	3
#10624784	3
#10624700	3
#10624697	3
#10624692	3