Subgroups — #28356015

Order 13851.

662 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987465	C₃	1	2	yes	maximal, complemented
2	#17987464	C₃	1	2	yes	maximal, complemented
3	#17987460	C₃	1	2	yes	maximal, complemented
4	#17987459	C₃	1	2	yes	maximal, complemented
5	#17987457	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987456	C₃	1	2	yes	maximal, complemented
7	#17987455	C₃	1	2	yes	maximal, complemented
8	#17987452	C₃	1	2	yes	maximal, complemented
9	#17987449	C₃	1	2	yes	maximal, complemented
10	#17987446	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987444	C₃	1	2	yes	maximal, complemented
12	#17987437	C₃	1	2	yes	maximal, complemented
13	#17987436	C₃	1	2	yes	maximal, complemented
14	#17987430	C₃	1	1	yes	maximal, characteristic
15	#17987424	C₃	1	2	yes	maximal
16	C₉×#10587023	C₃	1	2	yes	maximal, complemented
17	C₉×#10587023	C₃	1	1	yes	maximal, characteristic, complemented
18	C₉×#10587022	C₃	1	2	yes	maximal, complemented
19	C₉×#10587022	C₃	1	1	yes	maximal, characteristic, complemented
20	C₉×#10587025	C₃	1	2	yes	maximal, complemented
21	C₉×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
22	#17987403	C₃	1	2	yes	maximal, complemented
23	C₉²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
24	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
25	C₃×#10587025	C₃²	1	2	yes	—
26	C₃×#10587022	C₃²	1	2	yes	—
27	C₃×#10587022	C₃²	1	1	yes	characteristic
28	C₃×#10587023	C₃²	1	2	yes	—
29	C₃×#10587023	C₃²	1	1	yes	characteristic
30	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
33	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
34	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
35	#13095307	C₃²	1	2	yes	—
36	#13095307	C₃²	1	2	yes	—
37	#13095307	C₃²	1	1	yes	characteristic
38	#13095307	C₃²	1	1	yes	characteristic
39	#13095306	C₃²	1	2	yes	—
40	#13095306	C₃²	1	2	yes	—
41	#13095306	C₃²	1	1	yes	characteristic
42	#13095306	C₃²	1	1	yes	characteristic
43	#13095305	C₃²	1	2	yes	—
44	#13095305	C₃²	1	2	yes	—
45	#13095305	C₃²	1	2	yes	—
46	#13095305	C₃²	1	2	yes	—
47	#13095305	C₃²	1	2	yes	—
48	#13095305	C₃²	1	2	yes	—
49	#13095305	C₃²	1	2	yes	—
50	#13095305	C₃²	1	2	yes	—
51	#13095305	C₃²	1	2	yes	—
52	#13095305	C₃²	1	2	yes	—
53	#13095305	C₃²	1	2	yes	—
54	#13095305	C₃²	1	2	yes	—
55	#13095305	C₃²	1	2	yes	—
56	#13095305	C₃²	1	2	yes	—
57	#13095305	C₃²	1	2	yes	—
58	#13095305	C₃²	1	2	yes	—
59	#13095305	C₃²	1	2	yes	—
60	#13095305	C₃²	1	2	yes	—
61	#13095305	C₃²	1	2	yes	—
62	#13095305	C₃²	1	2	yes	—
63	#13095305	C₃²	1	2	yes	—
64	#13095305	C₃²	1	2	yes	—
65	#13095305	C₃²	1	2	yes	—
66	#13095305	C₃²	1	2	yes	—
67	C₃²⋊#4294	C₃²	1	1	yes	characteristic
68	C₃²⋊#4294	C₃²	1	1	yes	characteristic
69	C₃²⋊#4294	C₃²	1	2	yes	—
70	C₃²⋊#4294	C₃²	1	2	yes	—
71	#13095303	C₃²	1	2	yes	—
72	#13095303	C₃²	1	2	yes	—
73	#13095303	C₃²	1	2	yes	—
74	#13095303	C₃²	1	2	yes	—
75	#13095303	C₃²	1	2	yes	—
76	#13095303	C₃²	1	2	yes	—
77	#13095303	C₃²	1	2	yes	—
78	#13095303	C₃²	1	2	yes	—
79	#13095303	C₃²	1	2	yes	—
80	#13095303	C₃²	1	2	yes	—
81	#13095303	C₃²	1	2	yes	—
82	#13095303	C₃²	1	2	yes	—
83	C₉×#4294	C₃²	1	2	yes	—
84	C₉×#4294	C₃²	1	1	yes	characteristic
85	C₉×#4294	C₃²	1	2	yes	—
86	C₉×#4294	C₃²	1	1	yes	characteristic
87	C₉×#4294	C₃²	1	2	yes	—
88	C₉×#4294	C₃²	1	2	yes	—
89	C₉×#4294	C₃²	1	2	yes	—
90	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
91	C₁₉×C₉⋊C₉	C₃²	1	2	yes	—
92	C₁₉×C₉⋊C₉	C₃²	1	2	yes	—
93	C₁₉×C₉⋊C₉	C₃²	1	2	yes	—
94	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
95	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
96	C₉²×C₁₉	C₃²	1	1	yes	characteristic
97	C₉²×C₁₉	C₃²	1	1	yes	characteristic
98	C₉²×C₁₉	C₃²	1	1	yes	characteristic
99	#10624877	—	19	1	no	maximal, Sylow
100	C₃³×C₁₉	C₃³	1	1	yes	characteristic
101	C₃²×C₁₉⋊C₃	C₃³	1	2	yes	—
102	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
103	C₃×C₉×C₁₉	—	3	3	no	—
104	C₃×C₉×C₁₉	C₃³	1	2	yes	—
105	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
106	C₃×C₉×C₁₉	C₃³	1	2	yes	—
107	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
108	C₃×C₉×C₁₉	C₃³	1	2	yes	—
109	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
110	C₃×C₉×C₁₉	C₃³	1	2	yes	—
111	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
112	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
113	#10587025	—	3	6	no	—
114	C₃×#4294	C₃³	1	2	yes	—
115	C₃×#4294	C₃³	1	2	yes	—
116	C₃×#4294	C₃³	1	1	yes	characteristic
117	C₃×#4294	C₃³	1	1	yes	characteristic
118	C₃×#4294	C₃³	1	2	yes	—
119	C₃×#4294	C₃³	1	2	yes	—
120	C₃×#4294	C₃³	1	2	yes	—
121	C₃×#4294	C₃³	1	2	yes	—
122	C₃×#4294	C₃³	1	2	yes	—
123	C₃×#4294	C₃³	1	2	yes	—
124	C₃×#4294	C₃³	1	2	yes	—
125	C₃×#4294	C₃³	1	2	yes	—
126	C₃×#4294	C₃³	1	2	yes	—
127	#10587023	—	3	3	no	—
128	#10587023	—	3	6	no	—
129	#10587022	—	3	3	no	—
130	#10587022	—	3	6	no	—
131	C₉×C₁₉⋊C₃	—	3	3	no	—
132	C₉×C₁₉⋊C₃	—	3	3	no	—
133	C₉×C₁₉⋊C₃	—	3	6	no	—
134	C₉×C₁₉⋊C₃	—	3	6	no	—
135	#6899	—	19	2	no	—
136	#6899	—	19	2	no	—
137	#6899	—	19	2	no	—
138	#6898	—	19	2	no	—
139	#6898	—	19	1	no	—
140	#6897	—	19	2	no	—
141	#6897	—	19	2	no	—
142	#6897	—	19	2	no	—
143	#6896	—	19	2	no	—
144	#6896	—	19	1	no	—
145	#6895	—	19	2	no	—
146	#6894	—	19	2	no	—
147	#6894	—	19	2	no	—
148	#6893	—	19	1	no	—
149	C₉⋊H(3,𝔽₃)	—	19	2	no	—
150	C₉×3₋¹⁺²	—	19	2	no	—
151	C₉×3₋¹⁺²	—	19	2	no	—
152	C₉×3₋¹⁺²	—	19	1	no	—
153	C₉×3₋¹⁺²	—	19	2	no	—
154	C₉×3₋¹⁺²	—	19	1	no	—
155	C₉×H(3,𝔽₃)	—	19	1	no	—
156	#6887	—	19	2	no	—
157	#6887	C₁₉⋊C₃	1	1	yes	characteristic, complemented
158	C₃×C₉²	—	19	1	no	—
159	C₃²×C₁₉	—	3	2	no	—
160	C₃²×C₁₉	—	3	1	no	—
161	C₃²×C₁₉	—	3	1	no	—
162	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
163	C₃×C₁₉⋊C₃	—	3	2	no	—
164	C₃×C₁₉⋊C₃	—	3	1	no	—
165	C₃×C₁₉⋊C₃	—	3	2	no	—
166	C₃×C₁₉⋊C₃	—	3	2	no	—
167	C₃×C₁₉⋊C₃	—	3	2	no	—
168	C₃×C₁₉⋊C₃	—	3	2	no	—
169	C₃×C₁₉⋊C₃	#737	1	3	yes	—
170	C₉×C₁₉	—	3	1	no	—
171	C₉×C₁₉	—	3	2	no	—
172	C₉×C₁₉	—	3	1	no	—
173	C₉×C₁₉	—	3	2	no	—
174	C₉×C₁₉	—	3	1	no	—
175	C₉×C₁₉	—	3	2	no	—
176	C₉×C₁₉	—	3	2	no	—
177	C₉×C₁₉	#737	1	3	yes	—
178	#4294	—	3	2	no	—
179	#4294	—	3	2	no	—
180	#4294	—	3	1	no	—
181	#4294	—	3	1	no	—
182	#4294	—	3	2	no	—
183	#4294	—	3	2	no	—
184	#4294	—	3	2	no	—
185	#4294	—	3	2	no	—
186	#4294	—	3	2	no	—
187	#4294	—	3	2	no	—
188	#4294	—	3	2	no	—
189	#4294	—	3	2	no	—
190	#4294	#737	1	6	yes	—
191	C₃×3₋¹⁺²	—	19	2	no	—
192	C₃×3₋¹⁺²	—	19	2	no	—
193	C₃×3₋¹⁺²	—	19	1	no	—
194	C₃×3₋¹⁺²	—	19	2	no	—
195	C₃×3₋¹⁺²	—	19	1	no	—
196	C₃×H(3,𝔽₃)	—	19	1	no	—
197	C₃²×C₉	—	19	2	no	—
198	C₃²×C₉	—	19	1	no	—
199	C₃²×C₉	—	19	1	no	—
200	C₃²×C₉	—	19	2	no	—
201	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
202	C₉⋊C₉	—	19	2	no	—
203	C₉⋊C₉	—	19	2	no	—
204	C₉⋊C₉	—	19	2	no	—
205	C₉⋊C₉	—	19	2	no	—
206	C₉⋊C₉	—	19	2	no	—
207	C₉⋊C₉	—	19	2	no	—
208	C₉⋊C₉	—	19	2	no	—
209	C₉⋊C₉	—	19	2	no	—
210	C₉⋊C₉	—	19	2	no	—
211	C₉⋊C₉	—	19	2	no	—
212	C₉⋊C₉	—	19	2	no	—
213	C₉⋊C₉	—	19	2	no	—
214	C₉⋊C₉	—	19	2	no	—
215	C₉⋊C₉	—	19	2	no	—
216	C₉⋊C₉	—	19	2	no	—
217	C₉⋊C₉	—	19	2	no	—
218	C₉⋊C₉	—	19	2	no	—
219	C₉⋊C₉	—	19	2	no	—
220	C₉⋊C₉	—	19	2	no	—
221	C₉⋊C₉	—	19	2	no	—
222	C₉⋊C₉	—	19	2	no	—
223	C₉⋊C₉	—	19	2	no	—
224	C₉⋊C₉	—	19	2	no	—
225	C₉⋊C₉	—	19	2	no	—
226	C₉⋊C₉	—	19	2	no	—
227	C₉⋊C₉	—	19	2	no	—
228	C₉⋊C₉	—	19	1	no	—
229	C₉⋊C₉	—	19	1	no	—
230	C₉⋊C₉	—	19	2	no	—
231	C₉⋊C₉	—	19	2	no	—
232	C₉⋊C₉	—	19	1	no	—
233	C₉⋊C₉	—	19	1	no	—
234	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
235	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
236	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
237	C₃²⋊C₉	—	19	1	no	—
238	C₃²⋊C₉	—	19	1	no	—
239	C₃²⋊C₉	—	19	2	no	—
240	C₃²⋊C₉	—	19	2	no	—
241	C₃²⋊C₉	—	19	2	no	—
242	C₃²⋊C₉	—	19	2	no	—
243	C₃²⋊C₉	—	19	2	no	—
244	C₃²⋊C₉	—	19	2	no	—
245	C₃²⋊C₉	—	19	2	no	—
246	C₃²⋊C₉	—	19	2	no	—
247	C₃²⋊C₉	—	19	2	no	—
248	C₃²⋊C₉	—	19	2	no	—
249	C₃²⋊C₉	—	19	2	no	—
250	C₃²⋊C₉	—	19	2	no	—
251	C₃²⋊C₉	—	19	2	no	—
252	C₃²⋊C₉	—	19	2	no	—
253	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
254	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
255	C₉²	—	19	1	no	—
256	C₉²	—	19	2	no	—
257	C₉²	—	19	1	no	—
258	C₉²	—	19	2	no	—
259	C₉²	—	19	2	no	—
260	C₉²	—	19	2	no	—
261	C₉²	—	19	2	no	—
262	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
263	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
264	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
265	C₃×C₁₉	—	9	1	no	—
266	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
267	C₃×C₁₉	C₃×#737	1	2	yes	—
268	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
269	C₁₉⋊C₃	—	3	3	no	—
270	C₁₉⋊C₃	—	9	2	no	—
271	C₃³	—	19	1	no	—
272	C₃³	—	19	2	no	—
273	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
274	3₋¹⁺²	—	57	6	no	—
275	3₋¹⁺²	—	57	6	no	—
276	3₋¹⁺²	—	57	3	no	—
277	3₋¹⁺²	—	57	3	no	—
278	3₋¹⁺²	—	57	6	no	—
279	H(3,𝔽₃)	—	57	3	no	—
280	C₃×C₉	—	3	3	no	—
281	C₃×C₉	—	19	2	no	—
282	C₃×C₉	—	19	2	no	—
283	C₃×C₉	—	19	2	no	—
284	C₃×C₉	—	19	1	no	—
285	C₃×C₉	—	19	1	no	—
286	C₃×C₉	—	19	2	no	—
287	C₃×C₉	—	19	2	no	—
288	C₃×C₉	—	19	2	no	—
289	C₃×C₉	—	19	2	no	—
290	C₃×C₉	—	19	2	no	—
291	C₃×C₉	—	19	2	no	—
292	C₃×C₉	—	19	2	no	—
293	C₃×C₉	—	19	2	no	—
294	C₃×C₉	—	57	6	no	—
295	C₃×C₉	—	57	3	no	—
296	C₃×C₉	—	57	3	no	—
297	C₃×C₉	—	57	6	no	—
298	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
299	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
300	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
301	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
302	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
303	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
304	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
305	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
306	C₁₉	#10624877	1	1	yes	characteristic, complemented, Sylow
307	C₃²	—	3	1	no	—
308	C₃²	—	3	2	no	—
309	C₃²	—	3	1	no	—
310	C₃²	—	19	3	no	—
311	C₃²	—	57	2	no	—
312	C₃²	—	57	1	no	—
313	C₃²	—	57	2	no	—
314	C₃²	—	57	2	no	—
315	C₃²	—	57	2	no	—
316	C₃²	—	57	2	no	—
317	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
318	C₉	—	3	2	no	—
319	C₉	—	3	1	no	—
320	C₉	—	3	2	no	—
321	C₉	—	3	2	no	—
322	C₉	—	3	1	no	—
323	C₉	—	3	1	no	—
324	C₉	—	3	2	no	—
325	C₉	—	19	6	no	—
326	C₉	—	57	2	no	—
327	C₉	—	57	2	no	—
328	C₉	—	57	2	no	—
329	C₉	—	57	2	no	—
330	C₉	—	57	2	no	—
331	C₉	—	57	2	no	—
332	C₉	—	57	2	no	—
333	C₉	—	57	2	no	—
334	C₉	—	57	1	no	—
335	C₉	—	57	1	no	—
336	C₉	—	57	2	no	—
337	C₉	—	57	2	no	—
338	C₉	#13095332	1	3	yes	—
339	C₃	—	9	1	no	—
340	C₃	—	57	3	no	—
341	C₃	—	171	2	no	—
342	C₃	C₁₉⋊C₃×#737	1	2	yes	—
343	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
344	C₃	C₃×#13095332	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₁₉×#6887	C₃
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	#6887	C₁₉⋊C₃
hypocenter	C₁₉	#10624877
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #17987465	C₃	semidirect
2. #17987464	C₃	semidirect
3. #17987460	C₃	semidirect
4. #17987459	C₃	semidirect
5. #17987457	C₃	semidirect
6. #17987456	C₃	semidirect
7. #17987455	C₃	semidirect
8. #17987452	C₃	semidirect
9. #17987449	C₃	semidirect
10. #17987446	C₃	semidirect
11. #17987444	C₃	semidirect
12. #17987437	C₃	semidirect
13. #17987436	C₃	semidirect
16. C₉×#10587023	C₃	semidirect
17. C₉×#10587023	C₃	semidirect
18. C₉×#10587022	C₃	semidirect
19. C₉×#10587022	C₃	semidirect
20. C₉×#10587025	C₃	semidirect
22. #17987403	C₃	semidirect
23. C₉²×C₁₉⋊C₃	C₃	semidirect
24. C₁₉×#6887	C₃	semidirect
157. #6887	C₁₉⋊C₃	semidirect
306. C₁₉	#10624877	semidirect

Maximal subgroups

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

subgroup	index
#17987465	3
#17987464	3
#17987460	3
#17987459	3
#17987457	3
#17987456	3
#17987455	3
#17987452	3
#17987449	3
#17987446	3
#17987444	3
#17987437	3
#17987436	3
#17987430	3
#17987424	3
C₉×#10587023	3
C₉×#10587022	3
C₉×#10587025	3
C₉×C₃²⋊(C₁₉⋊C₃)	3
#17987403	3
C₉²×C₁₉⋊C₃	3
C₁₉×#6887	3
#10624877	19