Subgroups — #28356088

Order 13851.

698 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×#13095331	C₃	1	1	yes	maximal, characteristic
2	#17987468	C₃	1	3	yes	maximal, complemented
3	#17987460	C₃	1	3	yes	maximal, complemented
4	#17987460	C₃	1	3	yes	maximal, complemented
5	#17987447	C₃	1	3	yes	maximal, complemented
6	#17987446	C₃	1	3	yes	maximal, complemented
7	#17987446	C₃	1	3	yes	maximal, complemented
8	#17987441	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987441	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987441	C₃	1	3	yes	maximal, complemented
11	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
13	#17987440	C₃	1	3	yes	maximal, complemented
14	#17987439	C₃	1	1	yes	maximal, characteristic, complemented
15	#17987438	C₃	1	1	yes	maximal, characteristic, complemented
16	#17987435	C₃	1	3	yes	maximal, complemented
17	#17987423	C₃	1	1	yes	maximal, characteristic
18	#17987423	C₃	1	1	yes	maximal, characteristic
19	#17987402	C₃	1	1	yes	maximal, characteristic, complemented
20	#17987402	C₃	1	1	yes	maximal, characteristic, complemented
21	C₃×#13095305	C₃	1	1	yes	maximal, characteristic, complemented
22	C₁₉×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
23	#13095331	—	3	9	no	—
24	C₃×#10587025	C₃²	1	1	yes	characteristic
25	C₃×#10587025	C₃²	1	1	yes	characteristic
26	C₃×#10587025	C₃²	1	3	yes	—
27	C₃×#10587025	C₃²	1	3	yes	—
28	C₃×#10587025	C₃²	1	3	yes	—
29	C₃×#10587025	C₃²	1	3	yes	—
30	C₃²×#4294	C₃²	1	3	yes	—
31	C₃²×#4294	C₃²	1	1	yes	characteristic
32	C₃²×#4294	C₃²	1	1	yes	characteristic
33	C₃²×#4294	C₃²	1	1	yes	characteristic
34	#13095307	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	#13095306	C₃²	1	3	yes	—
40	#13095305	C₃²	1	1	yes	characteristic
41	#13095305	C₃²	1	1	yes	characteristic
42	#13095305	C₃²	1	1	yes	characteristic
43	#13095305	C₃²	1	1	yes	characteristic
44	#13095305	C₃²	1	1	yes	characteristic
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	3	yes	—
50	#13095305	C₃²	1	3	yes	—
51	#13095305	C₃²	1	3	yes	—
52	#13095305	C₃²	1	3	yes	—
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	#13095305	C₃²	1	3	yes	—
59	#13095305	C₃²	1	3	yes	—
60	#13095305	C₃²	1	3	yes	—
61	#13095305	C₃²	1	3	yes	—
62	#13095305	C₃²	1	3	yes	—
63	#13095305	C₃²	1	3	yes	—
64	#13095305	C₃²	1	3	yes	—
65	C₃²⋊#4294	C₃²	1	1	yes	characteristic
66	C₃²⋊#4294	C₃²	1	1	yes	characteristic
67	C₃²⋊#4294	C₃²	1	1	yes	characteristic
68	C₃²⋊#4294	C₃²	1	1	yes	characteristic
69	C₃²⋊#4294	C₃²	1	3	yes	—
70	C₃²⋊#4294	C₃²	1	3	yes	—
71	C₃²⋊#4294	C₃²	1	3	yes	—
72	C₃²⋊#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	3	yes	—
76	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
77	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
78	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
79	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
81	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
82	#10624887	—	19	1	no	maximal, Sylow
83	C₃³×C₁₉	C₃³	1	3	yes	—
84	C₃³×C₁₉	C₃³	1	1	yes	characteristic
85	C₁₉×3₋¹⁺²	—	9	3	no	—
86	C₁₉×H(3,𝔽₃)	—	3	3	no	—
87	C₃×C₉×C₁₉	—	9	1	no	—
88	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
89	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
90	C₃×C₉×C₁₉	C₃³	1	3	yes	—
91	C₃×C₉×C₁₉	C₃³	1	3	yes	—
92	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
93	#10587025	—	3	3	no	—
94	#10587025	—	3	3	no	—
95	#10587025	—	3	9	no	—
96	#10587025	—	3	9	no	—
97	#10587025	—	9	3	no	—
98	#10587025	—	9	3	no	—
99	C₃×#4294	—	3	9	no	—
100	C₃×#4294	—	9	1	no	—
101	C₃×#4294	—	9	1	no	—
102	C₃×#4294	3₋¹⁺²	1	9	yes	complemented
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	3	yes	—
107	C₃×#4294	C₃³	1	1	yes	characteristic
108	C₃×#4294	C₃³	1	1	yes	characteristic
109	C₃×#4294	C₃³	1	1	yes	characteristic
110	C₃×#4294	C₃³	1	1	yes	characteristic
111	C₃×#4294	C₃³	1	3	yes	—
112	C₃×#4294	C₃³	1	1	yes	characteristic
113	C₃×#4294	C₃³	1	3	yes	—
114	C₃×#4294	C₃³	1	1	yes	characteristic
115	C₃×#4294	C₃³	1	1	yes	characteristic
116	C₃×#4294	C₃³	1	1	yes	characteristic
117	C₃×#4294	C₃³	1	1	yes	characteristic
118	C₃×#737	—	19	1	no	—
119	#6900	—	19	3	no	—
120	#6899	—	19	3	no	—
121	#6899	—	19	3	no	—
122	#6896	—	19	3	no	—
123	#6896	—	19	3	no	—
124	#6896	—	19	3	no	—
125	#6894	—	19	3	no	—
126	#6894	—	19	1	no	—
127	#6894	—	19	1	no	—
128	#6894	—	19	1	no	—
129	#6894	—	19	1	no	—
130	#6894	—	19	3	no	—
131	#6894	—	19	1	no	—
132	#6894	—	19	1	no	—
133	#6894	—	19	3	no	—
134	C₉⋊H(3,𝔽₃)	—	19	1	no	—
135	C₉⋊H(3,𝔽₃)	—	19	1	no	—
136	C₉⋊H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
137	#6887	—	19	1	no	—
138	#6887	—	19	1	no	—
139	C₃×C₉⋊C₉	—	19	1	no	—
140	C₃²×C₁₉	—	3	1	no	—
141	C₃²×C₁₉	—	3	1	no	—
142	C₃²×C₁₉	—	3	1	no	—
143	C₃²×C₁₉	—	3	3	no	—
144	C₃²×C₁₉	—	3	3	no	—
145	C₃²×C₁₉	—	3	3	no	—
146	C₃²×C₁₉	—	3	3	no	—
147	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
148	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
149	C₉×C₁₉	—	3	1	no	—
150	C₉×C₁₉	—	3	1	no	—
151	C₉×C₁₉	—	3	1	no	—
152	C₉×C₁₉	—	3	3	no	—
153	C₉×C₁₉	—	3	3	no	—
154	#4294	—	3	3	no	—
155	#4294	—	3	3	no	—
156	#4294	—	3	1	no	—
157	#4294	—	3	1	no	—
158	#4294	—	3	1	no	—
159	#4294	—	3	1	no	—
160	#4294	—	3	1	no	—
161	#4294	—	3	1	no	—
162	#4294	—	3	3	no	—
163	#4294	—	3	3	no	—
164	#4294	—	3	3	no	—
165	#4294	—	3	3	no	—
166	#4294	C₃×3₋¹⁺²	1	3	yes	—
167	#4294	C₃×3₋¹⁺²	1	3	yes	—
168	#4294	C₃×3₋¹⁺²	1	3	yes	—
169	#737	—	57	9	no	—
170	C₃×3₋¹⁺²	—	19	1	no	—
171	C₃×3₋¹⁺²	—	19	1	no	—
172	C₃×3₋¹⁺²	—	19	3	no	—
173	C₃×3₋¹⁺²	—	19	3	no	—
174	C₃×3₋¹⁺²	—	19	3	no	—
175	C₃×3₋¹⁺²	—	19	3	no	—
176	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
177	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
178	C₃²×C₉	—	19	1	no	—
179	C₃²×C₉	—	19	3	no	—
180	C₃²×C₉	—	19	1	no	—
181	C₃²×C₉	—	19	1	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	1	no	—
186	C₉⋊C₉	—	19	1	no	—
187	C₉⋊C₉	—	19	1	no	—
188	C₉⋊C₉	—	19	1	no	—
189	C₉⋊C₉	—	19	1	no	—
190	C₉⋊C₉	—	19	1	no	—
191	C₉⋊C₉	—	19	1	no	—
192	C₉⋊C₉	—	19	3	no	—
193	C₉⋊C₉	—	19	3	no	—
194	C₉⋊C₉	—	19	3	no	—
195	C₉⋊C₉	—	19	3	no	—
196	C₉⋊C₉	—	19	3	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	—	19	3	no	—
207	C₉⋊C₉	—	19	3	no	—
208	C₉⋊C₉	—	19	3	no	—
209	C₉⋊C₉	—	19	3	no	—
210	C₉⋊C₉	—	19	3	no	—
211	C₉⋊C₉	—	19	3	no	—
212	C₉⋊C₉	—	19	3	no	—
213	C₉⋊C₉	—	19	3	no	—
214	C₃²⋊C₉	—	19	1	no	—
215	C₃²⋊C₉	—	19	1	no	—
216	C₃²⋊C₉	—	19	1	no	—
217	C₃²⋊C₉	—	19	1	no	—
218	C₃²⋊C₉	—	19	3	no	—
219	C₃²⋊C₉	—	19	3	no	—
220	C₃²⋊C₉	—	19	3	no	—
221	C₃²⋊C₉	—	19	3	no	—
222	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
223	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
224	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
225	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
226	C₉²	—	19	3	no	—
227	C₉²	—	19	3	no	—
228	C₃×C₁₉	—	3	3	no	—
229	C₃×C₁₉	—	9	3	no	—
230	C₃×C₁₉	C₃²×3₋¹⁺²	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	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
234	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
235	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
236	3₋¹⁺²	—	9	3	no	—
237	3₋¹⁺²	—	57	9	no	—
238	3₋¹⁺²	—	57	9	no	—
239	3₋¹⁺²	—	57	3	no	—
240	3₋¹⁺²	—	57	3	no	—
241	3₋¹⁺²	—	171	3	no	—
242	3₋¹⁺²	—	171	3	no	—
243	H(3,𝔽₃)	—	3	3	no	—
244	C₃×C₉	—	9	1	no	—
245	C₃×C₉	—	19	9	no	—
246	C₃×C₉	—	19	1	no	—
247	C₃×C₉	—	19	1	no	—
248	C₃×C₉	—	19	1	no	—
249	C₃×C₉	—	19	3	no	—
250	C₃×C₉	—	19	3	no	—
251	C₃×C₉	—	19	3	no	—
252	C₃×C₉	—	19	3	no	—
253	C₃×C₉	—	19	1	no	—
254	C₃×C₉	—	19	1	no	—
255	C₃×C₉	—	19	1	no	—
256	C₃×C₉	—	19	1	no	—
257	C₃×C₉	—	19	3	no	—
258	C₃×C₉	—	19	3	no	—
259	C₃×C₉	—	19	1	no	—
260	C₃×C₉	—	19	1	no	—
261	C₃×C₉	—	57	9	no	—
262	C₃×C₉	—	171	1	no	—
263	C₃×C₉	—	171	1	no	—
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	1	yes	characteristic
269	C₁₉	#10624887	1	1	yes	characteristic, complemented, Sylow
270	C₃²	—	3	1	no	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	3	3	no	—
274	C₃²	—	3	3	no	—
275	C₃²	—	3	3	no	—
276	C₃²	—	3	3	no	—
277	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
278	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
279	C₉	—	3	1	no	—
280	C₉	—	3	1	no	—
281	C₉	—	3	1	no	—
282	C₉	—	3	3	no	—
283	C₉	—	3	3	no	—
284	C₉	—	19	3	no	—
285	C₉	—	19	3	no	—
286	C₉	—	19	3	no	—
287	C₉	—	57	3	no	—
288	C₉	—	57	3	no	—
289	C₉	—	57	1	no	—
290	C₉	—	57	1	no	—
291	C₉	—	57	1	no	—
292	C₉	—	57	3	no	—
293	C₉	—	57	3	no	—
294	C₉	—	57	1	no	—
295	C₉	—	57	1	no	—
296	C₉	—	57	1	no	—
297	C₉	—	57	3	no	—
298	C₉	—	57	3	no	—
299	C₃	—	3	3	no	—
300	C₃	—	9	3	no	—
301	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
302	C₃	#17987544	1	1	yes	characteristic
303	C₃	#17987545	1	1	yes	characteristic
304	C₃	#17987547	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×C₉⋊H(3,𝔽₃)	C₃
prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃²
hypercenter	C₉⋊H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624887
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₉⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. #17987468	C₃	semidirect
3. #17987460	C₃	semidirect
4. #17987460	C₃	semidirect
5. #17987447	C₃	semidirect
6. #17987446	C₃	semidirect
7. #17987446	C₃	semidirect
8. #17987441	C₃	semidirect
9. #17987441	C₃	semidirect
10. #17987441	C₃	semidirect
11. #17987440	C₃	semidirect
12. #17987440	C₃	semidirect
13. #17987440	C₃	semidirect
14. #17987439	C₃	semidirect
15. #17987438	C₃	semidirect
16. #17987435	C₃	semidirect
19. #17987402	C₃	semidirect
20. #17987402	C₃	semidirect
21. C₃×#13095305	C₃	semidirect
102. C₃×#4294	3₋¹⁺²	semidirect
269. C₁₉	#10624887	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095331	3
#17987468	3
#17987460	3
#17987447	3
#17987446	3
#17987441	3
#17987440	3
#17987439	3
#17987438	3
#17987435	3
#17987423	3
#17987402	3
C₃×#13095305	3
C₁₉×C₉⋊H(3,𝔽₃)	3
#10624887	19