Subgroups — #28355933

Order 13851.

926 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
2	#17987415	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987415	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
6	C₉×#10587023	C₃	1	3	yes	maximal, complemented
7	C₉×#10587022	C₃	1	3	yes	maximal, complemented
8	C₉×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987410	C₃	1	3	yes	maximal, complemented
10	#17987410	C₃	1	3	yes	maximal, complemented
11	#17987409	C₃	1	3	yes	maximal, complemented
12	#17987409	C₃	1	3	yes	maximal, complemented
13	C₉×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
14	#17987406	C₃	1	1	yes	maximal, characteristic
15	#17987406	C₃	1	1	yes	maximal, characteristic
16	#17987404	C₃	1	3	yes	maximal, complemented
17	#17987403	C₃	1	3	yes	maximal, complemented
18	#17987402	C₃	1	1	yes	maximal, characteristic, complemented
19	#17987402	C₃	1	1	yes	maximal, characteristic, complemented
20	C₃×#13095305	C₃	1	1	yes	maximal, characteristic, complemented
21	C₁₉⋊C₃×C₉⋊C₉	C₃	1	3	yes	maximal, complemented
22	C₉×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
23	#13095333	C₉	1	27	yes	complemented
24	C₃×#10587025	C₃²	1	1	yes	characteristic
25	C₃×#10587022	C₃²	1	3	yes	—
26	C₃×#10587023	C₃²	1	3	yes	—
27	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
28	C₃²×#4294	C₃²	1	1	yes	characteristic
29	C₃²×#4294	C₃²	1	1	yes	characteristic
30	C₃²×#4294	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
32	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
33	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
34	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
35	#13095307	C₃²	1	3	yes	—
36	#13095307	C₃²	1	3	yes	—
37	#13095306	C₃²	1	3	yes	—
38	#13095306	C₃²	1	3	yes	—
39	#13095305	C₃²	1	1	yes	characteristic
40	#13095305	C₃²	1	3	yes	—
41	#13095305	C₃²	1	3	yes	—
42	#13095305	C₃²	1	3	yes	—
43	#13095305	C₃²	1	3	yes	—
44	#13095305	C₃²	1	3	yes	—
45	#13095305	C₃²	1	1	yes	characteristic
46	#13095305	C₃²	1	1	yes	characteristic
47	#13095305	C₃²	1	1	yes	characteristic
48	#13095305	C₃²	1	1	yes	characteristic
49	#13095305	C₃²	1	1	yes	characteristic
50	#13095305	C₃²	1	1	yes	characteristic
51	#13095305	C₃²	1	1	yes	characteristic
52	#13095305	C₃²	1	1	yes	characteristic
53	#13095305	C₃²	1	3	yes	—
54	#13095305	C₃²	1	3	yes	—
55	#13095305	C₃²	1	3	yes	—
56	#13095305	C₃²	1	3	yes	—
57	#13095305	C₃²	1	3	yes	—
58	C₃²⋊#4294	C₃²	1	1	yes	characteristic
59	C₃²⋊#4294	C₃²	1	1	yes	characteristic
60	C₃²⋊#4294	C₃²	1	1	yes	characteristic
61	C₃²⋊#4294	C₃²	1	1	yes	characteristic
62	#13095303	C₃²	1	3	yes	—
63	#13095303	C₃²	1	3	yes	—
64	#13095303	C₃²	1	3	yes	—
65	#13095303	C₃²	1	3	yes	—
66	#13095303	C₃²	1	3	yes	—
67	#13095303	C₃²	1	3	yes	—
68	C₉×#4294	C₃²	1	3	yes	—
69	C₉×#4294	C₃²	1	3	yes	—
70	C₉×#4294	C₃²	1	3	yes	—
71	C₉×#4294	C₃²	1	3	yes	—
72	C₉×#4294	C₃²	1	3	yes	—
73	C₉×#4294	C₃²	1	3	yes	—
74	C₉×#4294	C₃²	1	3	yes	—
75	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
76	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
77	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
78	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
79	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₉²×C₁₉	C₃²	1	3	yes	—
82	#10624864	—	19	1	no	maximal, Sylow
83	C₃³×C₁₉	C₃³	1	1	yes	characteristic
84	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
85	C₁₉×3₋¹⁺²	C₃×C₉	1	9	yes	complemented
86	C₃×C₉×C₁₉	—	9	1	no	—
87	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
88	C₃×C₉×C₁₉	C₃³	1	3	yes	—
89	C₃×C₉×C₁₉	C₃³	1	3	yes	—
90	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
91	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
92	C₃×C₉×C₁₉	C₃³	1	3	yes	—
93	C₃²⋊(C₁₉⋊C₃)	C₃×C₉	1	9	yes	—
94	#10587025	C₃×C₉	1	9	yes	complemented
95	C₃×#4294	—	9	1	no	—
96	C₃×#4294	—	9	1	no	—
97	C₃×#4294	C₃×C₉	1	9	yes	complemented
98	C₃×#4294	C₃³	1	1	yes	characteristic
99	C₃×#4294	C₃³	1	1	yes	characteristic
100	C₃×#4294	C₃³	1	1	yes	characteristic
101	C₃×#4294	C₃³	1	1	yes	characteristic
102	C₃×#4294	C₃³	1	3	yes	—
103	C₃×#4294	C₃³	1	3	yes	—
104	C₃×#4294	C₃³	1	3	yes	—
105	C₃×#4294	C₃³	1	3	yes	—
106	C₃×#4294	C₃³	1	1	yes	characteristic
107	C₃×#4294	C₃³	1	1	yes	characteristic
108	C₃×#4294	C₃³	1	3	yes	—
109	C₃×#4294	C₃³	1	1	yes	characteristic
110	C₃×#4294	C₃³	1	1	yes	characteristic
111	C₃×#4294	C₃³	1	1	yes	characteristic
112	#10587023	C₃×C₉	1	27	yes	complemented
113	#10587022	C₃×C₉	1	27	yes	complemented
114	C₉×C₁₉⋊C₃	—	9	3	no	—
115	C₉×C₁₉⋊C₃	—	9	3	no	—
116	C₉×C₁₉⋊C₃	—	9	3	no	—
117	C₉×C₁₉⋊C₃	C₃×C₉	1	27	yes	complemented
118	C₃×#737	—	19	1	no	—
119	C₉×3₋¹⁺²	—	19	1	no	—
120	C₉×3₋¹⁺²	—	19	1	no	—
121	C₉×3₋¹⁺²	—	19	1	no	—
122	C₉×3₋¹⁺²	—	19	3	no	—
123	C₉×3₋¹⁺²	—	19	3	no	—
124	C₉×3₋¹⁺²	—	19	3	no	—
125	C₉×3₋¹⁺²	—	19	3	no	—
126	C₉×3₋¹⁺²	—	19	1	no	—
127	C₉×3₋¹⁺²	—	19	1	no	—
128	C₉×3₋¹⁺²	—	19	3	no	—
129	C₉×3₋¹⁺²	—	19	3	no	—
130	C₉×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
131	C₉×H(3,𝔽₃)	—	19	1	no	—
132	C₉×H(3,𝔽₃)	—	19	1	no	—
133	C₉×H(3,𝔽₃)	—	19	1	no	—
134	#6887	—	19	3	no	—
135	#6887	—	19	1	no	—
136	#6887	—	19	1	no	—
137	#6887	—	19	3	no	—
138	C₃×C₉⋊C₉	—	19	1	no	—
139	C₃×C₉⋊C₉	—	19	3	no	—
140	C₃²×C₁₉	—	3	1	no	—
141	C₃²×C₁₉	—	3	1	no	—
142	C₃²×C₁₉	—	3	1	no	—
143	C₃²×C₁₉	C₃²×C₉	1	3	yes	—
144	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
145	C₃×C₁₉⋊C₃	—	3	3	no	—
146	C₃×C₁₉⋊C₃	—	3	3	no	—
147	C₃×C₁₉⋊C₃	—	3	3	no	—
148	C₃×C₁₉⋊C₃	C₃²×C₉	1	9	yes	—
149	C₉×C₁₉	—	3	1	no	—
150	C₉×C₁₉	—	3	3	no	—
151	C₉×C₁₉	—	3	3	no	—
152	C₉×C₁₉	—	3	1	no	—
153	C₉×C₁₉	—	3	1	no	—
154	C₉×C₁₉	C₃²×C₉	1	9	yes	—
155	#4294	—	3	1	no	—
156	#4294	—	3	1	no	—
157	#4294	—	3	3	no	—
158	#4294	—	3	3	no	—
159	#4294	—	3	3	no	—
160	#4294	—	3	3	no	—
161	#4294	—	3	1	no	—
162	#4294	—	3	1	no	—
163	#4294	—	3	1	no	—
164	#4294	—	3	1	no	—
165	#4294	C₃²×C₉	1	3	yes	—
166	#4294	C₃²×C₉	1	9	yes	—
167	#4294	C₃²×C₉	1	3	yes	—
168	#4294	C₃²×C₉	1	3	yes	—
169	#737	—	19	27	no	—
170	C₃×3₋¹⁺²	—	19	1	no	—
171	C₃×3₋¹⁺²	—	19	3	no	—
172	C₃×3₋¹⁺²	—	19	3	no	—
173	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
174	C₃×H(3,𝔽₃)	—	19	1	no	—
175	C₃²×C₉	—	19	1	no	—
176	C₃²×C₉	—	19	3	no	—
177	C₃²×C₉	—	19	1	no	—
178	C₃²×C₉	—	19	1	no	—
179	C₃²×C₉	—	19	3	no	—
180	C₃²×C₉	—	19	3	no	—
181	C₃²×C₉	—	19	3	no	—
182	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
183	C₉⋊C₉	—	19	1	no	—
184	C₉⋊C₉	—	19	1	no	—
185	C₉⋊C₉	—	19	3	no	—
186	C₉⋊C₉	—	19	3	no	—
187	C₉⋊C₉	—	19	1	no	—
188	C₉⋊C₉	—	19	3	no	—
189	C₉⋊C₉	—	19	1	no	—
190	C₉⋊C₉	—	19	1	no	—
191	C₉⋊C₉	—	19	1	no	—
192	C₉⋊C₉	—	19	1	no	—
193	C₉⋊C₉	—	19	3	no	—
194	C₉⋊C₉	—	19	3	no	—
195	C₉⋊C₉	—	19	1	no	—
196	C₉⋊C₉	—	19	1	no	—
197	C₉⋊C₉	—	19	3	no	—
198	C₉⋊C₉	—	19	3	no	—
199	C₉⋊C₉	—	19	3	no	—
200	C₉⋊C₉	—	19	3	no	—
201	C₉⋊C₉	—	19	3	no	—
202	C₉⋊C₉	—	19	3	no	—
203	C₉⋊C₉	—	19	3	no	—
204	C₉⋊C₉	—	19	3	no	—
205	C₉⋊C₉	—	19	3	no	—
206	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
207	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
208	C₃²⋊C₉	—	19	1	no	—
209	C₃²⋊C₉	—	19	1	no	—
210	C₃²⋊C₉	—	19	1	no	—
211	C₃²⋊C₉	—	19	1	no	—
212	C₃²⋊C₉	—	19	3	no	—
213	C₃²⋊C₉	—	19	3	no	—
214	C₃²⋊C₉	—	19	3	no	—
215	C₃²⋊C₉	—	19	3	no	—
216	C₃²⋊C₉	—	19	3	no	—
217	C₃²⋊C₉	—	19	3	no	—
218	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
219	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
220	C₉²	—	19	3	no	—
221	C₉²	—	19	3	no	—
222	C₉²	—	19	3	no	—
223	C₉²	—	19	3	no	—
224	C₉²	—	19	3	no	—
225	C₉²	—	19	3	no	—
226	C₉²	—	19	3	no	—
227	C₉²	C₃×C₁₉⋊C₃	1	3	yes	—
228	C₃×C₁₉	—	3	3	no	—
229	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
230	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
231	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
232	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
233	C₁₉⋊C₃	—	3	9	no	—
234	C₃³	—	19	3	no	—
235	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
236	3₋¹⁺²	—	19	9	no	—
237	3₋¹⁺²	—	19	27	no	—
238	3₋¹⁺²	—	19	27	no	—
239	3₋¹⁺²	C₉×C₁₉⋊C₃	1	9	yes	complemented
240	H(3,𝔽₃)	—	19	9	no	—
241	C₃×C₉	—	9	1	no	—
242	C₃×C₉	—	19	9	no	—
243	C₃×C₉	—	19	3	no	—
244	C₃×C₉	—	19	1	no	—
245	C₃×C₉	—	19	1	no	—
246	C₃×C₉	—	19	1	no	—
247	C₃×C₉	—	19	27	no	—
248	C₃×C₉	—	19	1	no	—
249	C₃×C₉	—	19	1	no	—
250	C₃×C₉	—	19	3	no	—
251	C₃×C₉	—	19	1	no	—
252	C₃×C₉	—	19	1	no	—
253	C₃×C₉	—	19	3	no	—
254	C₃×C₉	—	19	3	no	—
255	C₃×C₉	—	19	3	no	—
256	C₃×C₉	—	19	1	no	—
257	C₃×C₉	—	19	1	no	—
258	C₃×C₉	—	171	3	no	—
259	C₃×C₉	—	171	3	no	—
260	C₃×C₉	—	171	3	no	—
261	C₃×C₉	—	171	1	no	—
262	C₃×C₉	—	171	1	no	—
263	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
264	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
265	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
266	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
267	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
268	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
269	C₁₉	#10624864	1	1	yes	characteristic, complemented, Sylow
270	C₃²	—	3	1	no	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	19	9	no	—
274	C₃²	—	57	3	no	—
275	C₃²	—	57	3	no	—
276	C₃²	—	57	3	no	—
277	C₃²	C₃×C₉×C₁₉⋊C₃	1	3	yes	—
278	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
279	C₉	—	3	3	no	—
280	C₉	—	3	3	no	—
281	C₉	—	3	1	no	—
282	C₉	—	3	1	no	—
283	C₉	—	3	1	no	—
284	C₉	—	19	3	no	—
285	C₉	—	19	3	no	—
286	C₉	—	19	3	no	—
287	C₉	—	19	9	no	—
288	C₉	—	57	1	no	—
289	C₉	—	57	3	no	—
290	C₉	—	57	1	no	—
291	C₉	—	57	1	no	—
292	C₉	—	57	1	no	—
293	C₉	—	57	3	no	—
294	C₉	—	57	1	no	—
295	C₉	—	57	1	no	—
296	C₉	—	57	3	no	—
297	C₉	—	57	3	no	—
298	C₉	C₃×C₉×C₁₉⋊C₃	1	9	yes	—
299	C₃	—	3	3	no	—
300	C₃	—	57	9	no	—
301	C₃	C₃²×C₉×C₁₉⋊C₃	1	1	yes	characteristic
302	C₃	#17987544	1	1	yes	characteristic
303	C₃	#17987545	1	1	yes	characteristic
304	C₃	#17987546	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃³×C₉
Fitting	C₉×C₁₉×3₋¹⁺²	C₃
socle, cosocle	C₃²×C₁₉	C₃⁴
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624864
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #17987415	C₃	semidirect
3. #17987415	C₃	semidirect
4. 3₋¹⁺²×#4294	C₃	semidirect
5. 3₋¹⁺²×#4294	C₃	semidirect
6. C₉×#10587023	C₃	semidirect
7. C₉×#10587022	C₃	semidirect
8. C₉×#10587025	C₃	semidirect
9. #17987410	C₃	semidirect
10. #17987410	C₃	semidirect
11. #17987409	C₃	semidirect
12. #17987409	C₃	semidirect
16. #17987404	C₃	semidirect
17. #17987403	C₃	semidirect
18. #17987402	C₃	semidirect
19. #17987402	C₃	semidirect
20. C₃×#13095305	C₃	semidirect
21. C₁₉⋊C₃×C₉⋊C₉	C₃	semidirect
22. C₉×C₁₉×3₋¹⁺²	C₃	semidirect
23. #13095333	C₉	semidirect
85. C₁₉×3₋¹⁺²	C₃×C₉	semidirect
94. #10587025	C₃×C₉	semidirect
97. C₃×#4294	C₃×C₉	semidirect
112. #10587023	C₃×C₉	semidirect
113. #10587022	C₃×C₉	semidirect
117. C₉×C₁₉⋊C₃	C₃×C₉	semidirect
130. C₉×3₋¹⁺²	C₁₉⋊C₃	semidirect
239. 3₋¹⁺²	C₉×C₁₉⋊C₃	semidirect
269. C₁₉	#10624864	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095333	3
#17987415	3
3₋¹⁺²×#4294	3
C₉×#10587023	3
C₉×#10587022	3
C₉×#10587025	3
#17987410	3
#17987409	3
C₉×C₃²⋊(C₁₉⋊C₃)	3
#17987406	3
#17987404	3
#17987403	3
#17987402	3
C₃×#13095305	3
C₁₉⋊C₃×C₉⋊C₉	3
C₉×C₁₉×3₋¹⁺²	3
#10624864	19