Subgroups — #23748578

Order 9477.

980 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15691696	C₃	1	3	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691681	C₃	1	2	yes	maximal, complemented
4	#15691679	C₃	1	2	yes	maximal, complemented
5	#15691677	C₃	1	3	yes	maximal, complemented
6	#15691676	C₃	1	1	yes	maximal, characteristic, complemented
7	#15691674	C₃	1	3	yes	maximal, complemented
8	#15691672	C₃	1	3	yes	maximal, complemented
9	#15691671	C₃	1	3	yes	maximal, complemented
10	#15691667	C₃	1	2	yes	maximal, complemented
11	#15691663	C₃	1	1	yes	maximal, characteristic, complemented
12	#15691659	C₃	1	3	yes	maximal, complemented
13	#15691652	C₃	1	2	yes	maximal, complemented
14	#15691650	C₃	1	3	yes	maximal, complemented
15	#15691649	C₃	1	1	yes	maximal, characteristic, complemented
16	#15691648	C₃	1	3	yes	maximal, complemented
17	C₁₃⋊C₃×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
18	C₁₃×#6913	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃³×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
20	#11760454	—	3	6	no	—
21	#11760454	—	3	6	no	—
22	#11760454	C₃²	1	6	yes	complemented
23	#11760454	C₃²	1	6	yes	complemented
24	#11760453	—	3	6	no	—
25	#11760453	C₃²	1	6	yes	complemented
26	#11760452	C₃²	1	3	yes	complemented
27	#11760452	C₃²	1	3	yes	complemented
28	C₃×#66851	C₃²	1	1	yes	characteristic, complemented
29	C₃×#66851	C₃²	1	1	yes	characteristic, complemented
30	C₃×#66848	—	3	3	no	—
31	C₃×#66848	C₃²	1	1	yes	characteristic
32	C₃×#66848	C₃²	1	1	yes	characteristic
33	C₃×#66848	C₃²	1	1	yes	characteristic
34	C₃×#66849	C₃²	1	1	yes	characteristic, complemented
35	C₃×#66849	C₃²	1	1	yes	characteristic, complemented
36	C₃×#66849	C₃²	1	1	yes	characteristic, complemented
37	3₋¹⁺²×C₁₃⋊C₃	—	3	6	no	—
38	3₋¹⁺²×C₁₃⋊C₃	—	3	3	no	—
39	3₋¹⁺²×C₁₃⋊C₃	—	3	6	no	—
40	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	6	yes	—
41	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	3	yes	—
42	C₃×C₉×C₁₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
43	C₃×C₉×C₁₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
44	C₃×C₉×C₁₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
45	#11760443	C₃²	1	2	yes	complemented
46	#11760443	C₃²	1	2	yes	complemented
47	#11760442	C₃²	1	6	yes	complemented
48	#11760442	C₃²	1	6	yes	complemented
49	#11760441	C₃²	1	2	yes	complemented
50	#11760440	C₃²	1	3	yes	complemented
51	#11760440	C₃²	1	3	yes	complemented
52	#11760439	C₃²	1	6	yes	complemented
53	#11760439	C₃²	1	6	yes	complemented
54	#11760438	C₃²	1	6	yes	complemented
55	#11760437	C₃²	1	3	yes	complemented
56	#11760437	C₃²	1	3	yes	complemented
57	#11760436	C₃²	1	6	yes	complemented
58	#11760436	C₃²	1	6	yes	complemented
59	#11760435	C₃²	1	6	yes	complemented
60	#11760434	C₃²	1	3	yes	complemented
61	#11760434	C₃²	1	3	yes	complemented
62	C₁₃×#737	C₃²	1	3	yes	complemented
63	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic, complemented
64	C₁₃×#733	C₃²	1	2	yes	complemented
65	C₁₃×#732	C₃²	1	1	yes	characteristic, complemented
66	C₁₃×#731	C₃²	1	3	yes	complemented
67	C₁₃×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	complemented
68	#10624934	—	13	1	no	maximal, Sylow
69	C₃³×C₁₃	C₃³	1	1	yes	characteristic
70	C₃²×C₁₃⋊C₃	—	3	3	no	—
71	C₃²×C₁₃⋊C₃	—	3	3	no	—
72	C₃²×C₁₃⋊C₃	—	9	3	no	—
73	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
74	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
75	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
76	C₁₃×3₋¹⁺²	—	3	3	no	—
77	C₁₃×3₋¹⁺²	—	3	6	no	—
78	C₁₃×3₋¹⁺²	C₃³	1	6	yes	—
79	C₁₃×H(3,𝔽₃)	C₃³	1	3	yes	—
80	C₃×C₉×C₁₃	—	3	3	no	—
81	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
82	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
83	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
84	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
85	C₃²⋊(C₁₃⋊C₃)	C₃³	1	6	yes	—
86	#66851	—	3	2	no	—
87	#66851	—	3	3	no	—
88	#66851	—	3	1	no	—
89	#66851	—	3	6	no	—
90	#66851	—	3	6	no	—
91	#66851	C₃³	1	6	yes	—
92	#66851	C₃³	1	6	yes	—
93	C₃×C₁₃⋊C₉	—	3	3	no	—
94	C₃×C₁₃⋊C₉	—	3	3	no	—
95	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
96	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
97	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
98	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
99	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
100	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
101	#66849	—	3	6	no	—
102	#66849	—	3	3	no	—
103	#66849	—	3	6	no	—
104	#66849	—	3	2	no	—
105	#66849	—	3	3	no	—
106	#66849	—	3	1	no	—
107	#66849	—	3	6	no	—
108	#66849	—	3	6	no	—
109	#66848	—	3	3	no	—
110	#66848	—	3	3	no	—
111	#66848	—	3	6	no	—
112	#66848	—	3	6	no	—
113	#66848	—	3	6	no	—
114	#66848	—	3	6	no	—
115	#66848	H(3,𝔽₃)	1	6	yes	—
116	#66848	H(3,𝔽₃)	1	3	yes	—
117	C₉×C₁₃⋊C₃	—	3	6	no	—
118	C₉×C₁₃⋊C₃	—	3	3	no	—
119	C₉×C₁₃⋊C₃	—	3	6	no	—
120	C₉×C₁₃⋊C₃	—	3	2	no	—
121	C₉×C₁₃⋊C₃	—	3	3	no	—
122	C₉×C₁₃⋊C₃	—	3	1	no	—
123	C₉×C₁₃⋊C₃	—	3	6	no	—
124	C₉×C₁₃⋊C₃	—	3	6	no	—
125	3₋¹⁺⁴	—	13	3	no	—
126	C₃²×3₋¹⁺²	—	13	1	no	—
127	#6913	—	13	3	no	—
128	#6913	—	13	2	no	—
129	#6913	—	13	2	no	—
130	#6913	—	13	1	no	—
131	#6913	—	13	3	no	—
132	#6913	—	13	3	no	—
133	#6913	—	13	3	no	—
134	#6913	C₁₃⋊C₃	1	1	yes	characteristic, complemented
135	#6912	—	13	2	no	—
136	#6911	—	13	1	no	—
137	#6910	—	13	3	no	—
138	#6908	—	13	3	no	—
139	#6908	—	13	2	no	—
140	#6908	—	13	1	no	—
141	#6908	—	13	3	no	—
142	C₃×AΣL(1,𝔽₂₇)	—	13	3	no	—
143	C₃²×C₁₃	—	3	3	no	—
144	C₃²×C₁₃	—	3	1	no	—
145	C₃²×C₁₃	—	9	1	no	—
146	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
147	C₃×C₁₃⋊C₃	—	3	1	no	—
148	C₃×C₁₃⋊C₃	—	3	3	no	—
149	C₃×C₁₃⋊C₃	—	3	6	no	—
150	C₃×C₁₃⋊C₃	—	3	1	no	—
151	C₃×C₁₃⋊C₃	—	9	1	no	—
152	C₃×C₁₃⋊C₃	—	9	9	no	—
153	C₃×C₁₃⋊C₃	—	9	1	no	—
154	C₃×C₁₃⋊C₃	—	27	3	no	—
155	C₃×C₁₃⋊C₃	C₃×H(3,𝔽₃)	1	3	yes	—
156	C₉×C₁₃	—	3	6	no	—
157	C₉×C₁₃	—	3	1	no	—
158	C₉×C₁₃	—	3	1	no	—
159	C₉×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
160	C₁₃⋊C₉	—	3	6	no	—
161	C₁₃⋊C₉	—	3	6	no	—
162	C₁₃⋊C₉	—	3	1	no	—
163	C₁₃⋊C₉	—	3	1	no	—
164	C₁₃⋊C₉	—	3	1	no	—
165	C₁₃⋊C₉	—	3	1	no	—
166	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	2	yes	—
167	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
168	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
169	C₃⁴	—	13	1	no	—
170	#737	—	13	6	no	—
171	#737	—	13	6	no	—
172	#737	—	13	3	no	—
173	#737	—	13	3	no	—
174	#737	—	13	6	no	—
175	#737	—	39	6	no	—
176	#737	—	39	6	no	—
177	#737	—	39	6	no	—
178	#737	C₃×C₁₃⋊C₃	1	3	yes	complemented
179	C₃×3₋¹⁺²	—	13	1	no	—
180	C₃×3₋¹⁺²	—	13	1	no	—
181	C₃×3₋¹⁺²	—	13	1	no	—
182	C₃×3₋¹⁺²	—	13	1	no	—
183	C₃×3₋¹⁺²	—	13	1	no	—
184	C₃×3₋¹⁺²	—	13	1	no	—
185	C₃×3₋¹⁺²	—	13	1	no	—
186	C₃×3₋¹⁺²	—	13	1	no	—
187	C₃×3₋¹⁺²	—	13	6	no	—
188	C₃×3₋¹⁺²	—	39	3	no	—
189	C₃×3₋¹⁺²	—	39	6	no	—
190	C₃×3₋¹⁺²	—	39	3	no	—
191	C₃×3₋¹⁺²	—	39	6	no	—
192	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
193	C₃×H(3,𝔽₃)	—	13	3	no	—
194	C₃²×C₉	—	13	1	no	—
195	C₃²×C₉	—	13	1	no	—
196	C₃²×C₉	—	13	1	no	—
197	#733	—	13	6	no	—
198	#733	—	13	6	no	—
199	#733	—	13	2	no	—
200	#733	—	13	2	no	—
201	#733	C₃×C₁₃⋊C₃	1	2	yes	complemented
202	#732	—	13	3	no	—
203	#732	—	13	3	no	—
204	#732	—	13	2	no	—
205	#732	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
206	#731	—	13	6	no	—
207	#731	—	13	6	no	—
208	#731	—	13	3	no	—
209	#731	—	13	3	no	—
210	#731	—	13	6	no	—
211	#731	C₃×C₁₃⋊C₃	1	3	yes	complemented
212	AΣL(1,𝔽₂₇)	—	13	6	no	—
213	AΣL(1,𝔽₂₇)	—	13	6	no	—
214	AΣL(1,𝔽₂₇)	—	13	6	no	—
215	AΣL(1,𝔽₂₇)	—	13	3	no	—
216	AΣL(1,𝔽₂₇)	—	13	3	no	—
217	AΣL(1,𝔽₂₇)	C₃×C₁₃⋊C₃	1	3	yes	complemented
218	C₃×C₁₃	—	3	1	no	—
219	C₃×C₁₃	—	9	1	no	—
220	C₃×C₁₃	—	9	3	no	—
221	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
222	C₁₃⋊C₃	—	3	3	no	—
223	C₁₃⋊C₃	—	3	3	no	—
224	C₁₃⋊C₃	—	9	1	no	—
225	C₁₃⋊C₃	—	9	6	no	—
226	C₃³	—	13	1	no	—
227	C₃³	—	13	1	no	—
228	C₃³	—	13	1	no	—
229	C₃³	—	39	3	no	—
230	C₃³	—	39	3	no	—
231	C₃³	—	117	3	no	—
232	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
233	3₋¹⁺²	—	3	3	no	—
234	3₋¹⁺²	—	3	6	no	—
235	3₋¹⁺²	—	13	6	no	—
236	3₋¹⁺²	—	13	3	no	—
237	3₋¹⁺²	—	13	6	no	—
238	3₋¹⁺²	—	13	6	no	—
239	3₋¹⁺²	—	39	6	no	—
240	3₋¹⁺²	—	39	6	no	—
241	3₋¹⁺²	—	39	2	no	—
242	3₋¹⁺²	—	39	1	no	—
243	3₋¹⁺²	—	39	3	no	—
244	3₋¹⁺²	—	39	6	no	—
245	3₋¹⁺²	—	39	6	no	—
246	3₋¹⁺²	—	39	6	no	—
247	3₋¹⁺²	—	39	3	no	—
248	3₋¹⁺²	—	39	3	no	—
249	3₋¹⁺²	—	39	6	no	—
250	3₋¹⁺²	—	39	2	no	—
251	3₋¹⁺²	—	39	3	no	—
252	3₋¹⁺²	—	39	1	no	—
253	3₋¹⁺²	—	39	6	no	—
254	3₋¹⁺²	—	39	6	no	—
255	3₋¹⁺²	—	39	6	no	—
256	3₋¹⁺²	—	39	3	no	—
257	3₋¹⁺²	—	39	6	no	—
258	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	6	yes	—
259	H(3,𝔽₃)	—	13	6	no	—
260	H(3,𝔽₃)	—	39	6	no	—
261	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	3	yes	—
262	C₃×C₉	—	3	3	no	—
263	C₃×C₉	—	13	1	no	—
264	C₃×C₉	—	13	1	no	—
265	C₃×C₉	—	13	1	no	—
266	C₃×C₉	—	13	1	no	—
267	C₃×C₉	—	13	1	no	—
268	C₃×C₉	—	13	1	no	—
269	C₃×C₉	—	39	3	no	—
270	C₃×C₉	—	39	2	no	—
271	C₃×C₉	—	39	1	no	—
272	C₃×C₉	—	39	6	no	—
273	C₃×C₉	—	39	6	no	—
274	C₃×C₉	—	39	6	no	—
275	C₃×C₉	—	39	3	no	—
276	C₃×C₉	—	39	3	no	—
277	C₃×C₉	—	39	3	no	—
278	C₃×C₉	—	39	6	no	—
279	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
280	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
281	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
282	C₁₃	#10624934	1	1	yes	characteristic, complemented, Sylow
283	C₃²	—	3	3	no	—
284	C₃²	—	3	1	no	—
285	C₃²	—	9	1	no	—
286	C₃²	—	13	3	no	—
287	C₃²	—	39	3	no	—
288	C₃²	—	39	1	no	—
289	C₃²	—	39	6	no	—
290	C₃²	—	39	1	no	—
291	C₃²	—	351	3	no	—
292	C₃²	—	117	1	no	—
293	C₃²	—	117	1	no	—
294	C₃²	—	117	9	no	—
295	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
296	C₉	—	3	6	no	—
297	C₉	—	3	1	no	—
298	C₉	—	3	1	no	—
299	C₉	—	13	2	no	—
300	C₉	—	13	3	no	—
301	C₉	—	13	1	no	—
302	C₉	—	39	1	no	—
303	C₉	—	39	1	no	—
304	C₉	—	39	1	no	—
305	C₉	—	39	1	no	—
306	C₉	—	39	6	no	—
307	C₉	—	39	6	no	—
308	C₉	H(3,𝔽₃)×C₁₃⋊C₃	1	3	yes	—
309	C₃	—	3	1	no	—
310	C₃	—	9	1	no	—
311	C₃	—	9	3	no	—
312	C₃	—	39	3	no	—
313	C₃	—	39	3	no	—
314	C₃	—	117	6	no	—
315	C₃	—	117	1	no	—
316	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₁₃×#6913	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	#6913	C₁₃⋊C₃
hypocenter	C₁₃	#10624934
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6913

Complements

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

N	H ≅ G/N	type
1. #15691696	C₃	semidirect
2. C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
3. #15691681	C₃	semidirect
4. #15691679	C₃	semidirect
5. #15691677	C₃	semidirect
6. #15691676	C₃	semidirect
7. #15691674	C₃	semidirect
8. #15691672	C₃	semidirect
9. #15691671	C₃	semidirect
10. #15691667	C₃	semidirect
11. #15691663	C₃	semidirect
12. #15691659	C₃	semidirect
13. #15691652	C₃	semidirect
14. #15691650	C₃	semidirect
15. #15691649	C₃	semidirect
16. #15691648	C₃	semidirect
17. C₁₃⋊C₃×AΣL(1,𝔽₂₇)	C₃	semidirect
18. C₁₃×#6913	C₃	semidirect
22. #11760454	C₃²	semidirect
23. #11760454	C₃²	semidirect
25. #11760453	C₃²	semidirect
26. #11760452	C₃²	semidirect
27. #11760452	C₃²	semidirect
28. C₃×#66851	C₃²	semidirect
29. C₃×#66851	C₃²	semidirect
34. C₃×#66849	C₃²	semidirect
35. C₃×#66849	C₃²	semidirect
36. C₃×#66849	C₃²	semidirect
42. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
43. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
44. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
45. #11760443	C₃²	semidirect
46. #11760443	C₃²	semidirect
47. #11760442	C₃²	semidirect
48. #11760442	C₃²	semidirect
49. #11760441	C₃²	semidirect
50. #11760440	C₃²	semidirect
51. #11760440	C₃²	semidirect
52. #11760439	C₃²	semidirect
53. #11760439	C₃²	semidirect
54. #11760438	C₃²	semidirect
55. #11760437	C₃²	semidirect
56. #11760437	C₃²	semidirect
57. #11760436	C₃²	semidirect
58. #11760436	C₃²	semidirect
59. #11760435	C₃²	semidirect
60. #11760434	C₃²	semidirect
61. #11760434	C₃²	semidirect
62. C₁₃×#737	C₃²	semidirect
63. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
64. C₁₃×#733	C₃²	semidirect
65. C₁₃×#732	C₃²	semidirect
66. C₁₃×#731	C₃²	semidirect
67. C₁₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
134. #6913	C₁₃⋊C₃	semidirect
178. #737	C₃×C₁₃⋊C₃	semidirect
192. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
201. #733	C₃×C₁₃⋊C₃	semidirect
205. #732	C₃×C₁₃⋊C₃	semidirect
211. #731	C₃×C₁₃⋊C₃	semidirect
217. AΣL(1,𝔽₂₇)	C₃×C₁₃⋊C₃	semidirect
282. C₁₃	#10624934	semidirect