Subgroups — #28355986

Order 13851.

824 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#17987444	C₃	1	3	yes	maximal, complemented
3	#17987444	C₃	1	3	yes	maximal, complemented
4	#17987441	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987439	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987438	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987437	C₃	1	3	yes	maximal, complemented
9	#17987436	C₃	1	3	yes	maximal, complemented
10	#17987435	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987434	C₃	1	3	yes	maximal, complemented
12	#17987433	C₃	1	3	yes	maximal, complemented
13	#17987420	C₃	1	1	yes	maximal, characteristic
14	#17987419	C₃	1	1	yes	maximal, characteristic
15	#17987418	C₃	1	1	yes	maximal, characteristic
16	#17987415	C₃	1	1	yes	maximal, characteristic, complemented
17	#17987415	C₃	1	1	yes	maximal, characteristic, complemented
18	3₋¹⁺²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
19	#17987410	C₃	1	3	yes	maximal, complemented
20	#17987409	C₃	1	3	yes	maximal, complemented
21	#17987404	C₃	1	3	yes	maximal, complemented
22	C₁₉×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×#10587025	C₃²	1	1	yes	characteristic
25	C₃×#10587025	C₃²	1	1	yes	characteristic
26	C₃×#10587022	C₃²	1	3	yes	—
27	C₃×#10587023	C₃²	1	3	yes	—
28	C₃²×#4294	C₃²	1	1	yes	characteristic
29	C₃²×#4294	C₃²	1	1	yes	characteristic
30	C₃×C₉×C₁₉⋊C₃	—	3	9	no	—
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
34	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
35	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
36	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
38	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
39	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
40	#13095307	C₃²	1	3	yes	complemented
41	#13095307	C₃²	1	3	yes	complemented
42	#13095307	C₃²	1	3	yes	complemented
43	#13095306	C₃²	1	3	yes	complemented
44	#13095306	C₃²	1	3	yes	complemented
45	#13095306	C₃²	1	3	yes	complemented
46	#13095305	C₃²	1	3	yes	complemented
47	#13095305	C₃²	1	3	yes	complemented
48	#13095305	C₃²	1	3	yes	complemented
49	#13095305	C₃²	1	3	yes	complemented
50	#13095305	C₃²	1	3	yes	complemented
51	#13095305	C₃²	1	3	yes	complemented
52	#13095305	C₃²	1	3	yes	complemented
53	#13095305	C₃²	1	3	yes	complemented
54	#13095305	C₃²	1	3	yes	complemented
55	#13095305	C₃²	1	3	yes	complemented
56	#13095305	C₃²	1	3	yes	complemented
57	#13095305	C₃²	1	3	yes	complemented
58	#13095305	C₃²	1	3	yes	complemented
59	#13095305	C₃²	1	3	yes	complemented
60	#13095305	C₃²	1	3	yes	complemented
61	C₃²⋊#4294	C₃²	1	1	yes	characteristic
62	C₃²⋊#4294	C₃²	1	1	yes	characteristic
63	C₃²⋊#4294	C₃²	1	1	yes	characteristic
64	C₃²⋊#4294	C₃²	1	1	yes	characteristic
65	#13095303	C₃²	1	3	yes	—
66	#13095303	C₃²	1	3	yes	—
67	#13095303	C₃²	1	3	yes	—
68	#13095303	C₃²	1	3	yes	—
69	#13095303	C₃²	1	3	yes	—
70	#13095303	C₃²	1	3	yes	—
71	#13095303	C₃²	1	3	yes	—
72	C₉×#4294	C₃²	1	3	yes	complemented
73	C₉×#4294	C₃²	1	3	yes	complemented
74	C₉×#4294	C₃²	1	3	yes	complemented
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	complemented
78	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
79	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
80	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
82	#10624872	—	19	1	no	maximal, Sylow
83	C₃³×C₁₉	C₃³	1	1	yes	characteristic
84	C₃²×C₁₉⋊C₃	—	3	3	no	—
85	C₃²×C₁₉⋊C₃	—	3	3	no	—
86	C₃²×C₁₉⋊C₃	—	3	3	no	—
87	C₃²×C₁₉⋊C₃	—	3	3	no	—
88	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
89	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
90	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
91	C₁₉×3₋¹⁺²	—	3	3	no	—
92	C₃×C₉×C₁₉	—	3	3	no	—
93	C₃×C₉×C₁₉	C₃³	1	3	yes	—
94	C₃×C₉×C₁₉	C₃³	1	3	yes	—
95	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
96	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
97	C₃×C₉×C₁₉	C₃³	1	3	yes	—
98	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
99	#10587025	—	3	3	no	—
100	#10587025	—	3	3	no	—
101	C₃×#4294	—	3	3	no	—
102	C₃×#4294	—	3	3	no	—
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	3	yes	—
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	C₃×#4294	C₃³	1	1	yes	characteristic
113	C₃×#4294	C₃³	1	1	yes	characteristic
114	C₃×#4294	C₃³	1	1	yes	characteristic
115	#10587023	—	3	9	no	—
116	#10587022	—	3	9	no	—
117	C₉×C₁₉⋊C₃	—	3	3	no	—
118	C₉×C₁₉⋊C₃	—	3	3	no	—
119	C₉×C₁₉⋊C₃	—	3	3	no	—
120	C₉×C₁₉⋊C₃	—	3	3	no	—
121	C₉×C₁₉⋊C₃	—	3	3	no	—
122	C₉×C₁₉⋊C₃	—	3	3	no	—
123	C₉×C₁₉⋊C₃	—	3	3	no	—
124	C₉×C₁₉⋊C₃	—	3	3	no	—
125	C₉×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	complemented
126	C₃³×C₉	—	19	1	no	—
127	#6895	—	19	3	no	—
128	#6895	—	19	3	no	—
129	#6894	—	19	1	no	—
130	#6894	—	19	1	no	—
131	#6894	—	19	1	no	—
132	#6894	—	19	3	no	—
133	#6894	—	19	3	no	—
134	#6894	—	19	3	no	—
135	#6894	—	19	3	no	—
136	#6894	—	19	1	no	—
137	#6894	—	19	1	no	—
138	#6894	C₁₉⋊C₃	1	1	yes	characteristic, complemented
139	#6891	—	19	1	no	—
140	#6891	—	19	1	no	—
141	#6891	—	19	1	no	—
142	C₉×3₋¹⁺²	—	19	1	no	—
143	C₉×3₋¹⁺²	—	19	3	no	—
144	C₉×3₋¹⁺²	—	19	3	no	—
145	C₉×3₋¹⁺²	—	19	1	no	—
146	C₉×3₋¹⁺²	—	19	1	no	—
147	#6887	—	19	3	no	—
148	C₃²×C₁₉	—	3	1	no	—
149	C₃²×C₁₉	—	3	1	no	—
150	C₃²×C₁₉	—	3	1	no	—
151	C₃²×C₁₉	#737	1	3	yes	—
152	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
153	C₃×C₁₉⋊C₃	—	3	27	no	—
154	C₃×C₁₉⋊C₃	—	3	1	no	—
155	C₃×C₁₉⋊C₃	—	3	1	no	—
156	C₃×C₁₉⋊C₃	—	3	1	no	—
157	C₃×C₁₉⋊C₃	—	3	1	no	—
158	C₃×C₁₉⋊C₃	—	3	1	no	—
159	C₃×C₁₉⋊C₃	—	3	1	no	—
160	C₃×C₁₉⋊C₃	—	3	1	no	—
161	C₃×C₁₉⋊C₃	—	3	1	no	—
162	C₃×C₁₉⋊C₃	—	3	1	no	—
163	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
164	C₃×C₁₉⋊C₃	#737	1	3	yes	—
165	C₃×C₁₉⋊C₃	#737	1	3	yes	—
166	C₉×C₁₉	—	3	1	no	—
167	C₉×C₁₉	—	3	1	no	—
168	C₉×C₁₉	—	3	3	no	—
169	C₉×C₁₉	—	3	3	no	—
170	C₉×C₁₉	—	3	3	no	—
171	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
172	#4294	—	3	1	no	—
173	#4294	—	3	1	no	—
174	#4294	—	3	1	no	—
175	#4294	—	3	1	no	—
176	#4294	—	3	3	no	—
177	#4294	—	3	3	no	—
178	#4294	—	3	3	no	—
179	#4294	—	3	3	no	—
180	#4294	—	3	3	no	—
181	#4294	—	3	3	no	—
182	#4294	C₃×3₋¹⁺²	1	3	yes	—
183	#4294	C₃×3₋¹⁺²	1	3	yes	—
184	C₃⁴	—	19	1	no	—
185	C₃×3₋¹⁺²	—	19	1	no	—
186	C₃×3₋¹⁺²	—	19	1	no	—
187	C₃×3₋¹⁺²	—	19	3	no	—
188	C₃×3₋¹⁺²	—	19	3	no	—
189	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
190	C₃²×C₉	—	19	1	no	—
191	C₃²×C₉	—	19	1	no	—
192	C₃²×C₉	—	19	1	no	—
193	C₃²×C₉	—	19	1	no	—
194	C₃²×C₉	—	19	1	no	—
195	C₃²×C₉	—	19	1	no	—
196	C₃²×C₉	—	19	1	no	—
197	C₃²×C₉	—	19	1	no	—
198	C₃²×C₉	—	19	1	no	—
199	C₃²×C₉	—	19	1	no	—
200	C₃²×C₉	—	19	1	no	—
201	C₃²×C₉	—	57	9	no	—
202	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
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	C₉⋊C₉	—	19	3	no	—
215	C₉⋊C₉	—	19	3	no	—
216	C₉⋊C₉	—	19	3	no	—
217	C₉⋊C₉	—	19	3	no	—
218	C₉⋊C₉	—	19	3	no	—
219	C₉⋊C₉	—	19	3	no	—
220	C₉⋊C₉	—	19	3	no	—
221	C₉⋊C₉	—	19	3	no	—
222	C₉⋊C₉	—	19	3	no	—
223	C₉⋊C₉	—	19	3	no	—
224	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
225	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
226	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
227	C₃²⋊C₉	—	19	1	no	—
228	C₃²⋊C₉	—	19	1	no	—
229	C₃²⋊C₉	—	19	1	no	—
230	C₃²⋊C₉	—	19	1	no	—
231	C₃²⋊C₉	—	19	3	no	—
232	C₃²⋊C₉	—	19	3	no	—
233	C₃²⋊C₉	—	19	3	no	—
234	C₃²⋊C₉	—	19	3	no	—
235	C₃²⋊C₉	—	19	3	no	—
236	C₃²⋊C₉	—	19	3	no	—
237	C₃²⋊C₉	—	19	3	no	—
238	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
239	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
240	C₉²	—	19	3	no	—
241	C₉²	—	19	3	no	—
242	C₉²	—	19	3	no	—
243	C₃×C₁₉	—	3	3	no	—
244	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
245	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
246	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
247	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
248	C₁₉⋊C₃	—	3	3	no	—
249	C₁₉⋊C₃	—	3	3	no	—
250	C₁₉⋊C₃	—	3	3	no	—
251	C₃³	—	19	1	no	—
252	C₃³	—	19	1	no	—
253	C₃³	—	19	1	no	—
254	C₃³	—	57	3	no	—
255	C₃³	—	57	3	no	—
256	C₃³	—	57	3	no	—
257	C₃³	—	57	3	no	—
258	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
259	3₋¹⁺²	—	3	3	no	—
260	3₋¹⁺²	—	57	9	no	—
261	3₋¹⁺²	—	57	9	no	—
262	3₋¹⁺²	—	57	3	no	—
263	3₋¹⁺²	—	57	3	no	—
264	C₃×C₉	—	3	3	no	—
265	C₃×C₉	—	19	1	no	—
266	C₃×C₉	—	19	1	no	—
267	C₃×C₉	—	19	9	no	—
268	C₃×C₉	—	19	3	no	—
269	C₃×C₉	—	19	3	no	—
270	C₃×C₉	—	19	3	no	—
271	C₃×C₉	—	19	3	no	—
272	C₃×C₉	—	19	3	no	—
273	C₃×C₉	—	19	3	no	—
274	C₃×C₉	—	19	1	no	—
275	C₃×C₉	—	19	1	no	—
276	C₃×C₉	—	19	1	no	—
277	C₃×C₉	—	19	1	no	—
278	C₃×C₉	—	57	3	no	—
279	C₃×C₉	—	57	3	no	—
280	C₃×C₉	—	57	3	no	—
281	C₃×C₉	—	57	3	no	—
282	C₃×C₉	—	57	3	no	—
283	C₃×C₉	—	57	3	no	—
284	C₃×C₉	—	57	3	no	—
285	C₃×C₉	—	57	3	no	—
286	C₃×C₉	—	57	3	no	—
287	C₃×C₉	—	57	3	no	—
288	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
289	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
290	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
291	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
292	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
293	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
294	C₁₉	#10624872	1	1	yes	characteristic, complemented, Sylow
295	C₃²	—	3	1	no	—
296	C₃²	—	3	1	no	—
297	C₃²	—	3	1	no	—
298	C₃²	—	19	3	no	—
299	C₃²	—	19	3	no	—
300	C₃²	—	19	3	no	—
301	C₃²	—	57	27	no	—
302	C₃²	—	57	1	no	—
303	C₃²	—	57	1	no	—
304	C₃²	—	57	1	no	—
305	C₃²	—	57	1	no	—
306	C₃²	—	57	1	no	—
307	C₃²	—	57	1	no	—
308	C₃²	—	57	1	no	—
309	C₃²	—	57	1	no	—
310	C₃²	—	57	1	no	—
311	C₃²	#13095333	1	3	yes	—
312	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
313	C₉	—	3	1	no	—
314	C₉	—	3	1	no	—
315	C₉	—	3	3	no	—
316	C₉	—	3	3	no	—
317	C₉	—	3	3	no	—
318	C₉	—	19	3	no	—
319	C₉	—	19	3	no	—
320	C₉	—	57	1	no	—
321	C₉	—	57	1	no	—
322	C₉	—	57	1	no	—
323	C₉	—	57	1	no	—
324	C₉	—	57	3	no	—
325	C₉	—	57	3	no	—
326	C₉	—	57	3	no	—
327	C₉	—	57	3	no	—
328	C₉	—	57	3	no	—
329	C₉	—	57	3	no	—
330	C₉	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
331	C₃	—	3	3	no	—
332	C₃	—	57	3	no	—
333	C₃	—	57	3	no	—
334	C₃	—	57	3	no	—
335	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
336	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
337	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
338	C₃	C₃×#13095333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6894	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6894	C₁₉⋊C₃
hypocenter	C₁₉	#10624872
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6894

Complements

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

N	H ≅ G/N	type
2. #17987444	C₃	semidirect
3. #17987444	C₃	semidirect
4. #17987441	C₃	semidirect
5. #17987440	C₃	semidirect
6. #17987439	C₃	semidirect
7. #17987438	C₃	semidirect
8. #17987437	C₃	semidirect
9. #17987436	C₃	semidirect
10. #17987435	C₃	semidirect
11. #17987434	C₃	semidirect
12. #17987433	C₃	semidirect
16. #17987415	C₃	semidirect
17. #17987415	C₃	semidirect
18. 3₋¹⁺²×#4294	C₃	semidirect
19. #17987410	C₃	semidirect
20. #17987409	C₃	semidirect
21. #17987404	C₃	semidirect
22. C₁₉×#6894	C₃	semidirect
40. #13095307	C₃²	semidirect
41. #13095307	C₃²	semidirect
42. #13095307	C₃²	semidirect
43. #13095306	C₃²	semidirect
44. #13095306	C₃²	semidirect
45. #13095306	C₃²	semidirect
46. #13095305	C₃²	semidirect
47. #13095305	C₃²	semidirect
48. #13095305	C₃²	semidirect
49. #13095305	C₃²	semidirect
50. #13095305	C₃²	semidirect
51. #13095305	C₃²	semidirect
52. #13095305	C₃²	semidirect
53. #13095305	C₃²	semidirect
54. #13095305	C₃²	semidirect
55. #13095305	C₃²	semidirect
56. #13095305	C₃²	semidirect
57. #13095305	C₃²	semidirect
58. #13095305	C₃²	semidirect
59. #13095305	C₃²	semidirect
60. #13095305	C₃²	semidirect
72. C₉×#4294	C₃²	semidirect
73. C₉×#4294	C₃²	semidirect
74. C₉×#4294	C₃²	semidirect
77. C₁₉×C₉⋊C₉	C₃²	semidirect
78. C₁₉×C₉⋊C₉	C₃²	semidirect
79. C₁₉×C₉⋊C₉	C₃²	semidirect
125. C₉×C₁₉⋊C₃	3₋¹⁺²	semidirect
138. #6894	C₁₉⋊C₃	semidirect
224. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
225. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
226. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
294. C₁₉	#10624872	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉×C₁₉⋊C₃	3
#17987444	3
#17987441	3
#17987440	3
#17987439	3
#17987438	3
#17987437	3
#17987436	3
#17987435	3
#17987434	3
#17987433	3
#17987420	3
#17987419	3
#17987418	3
#17987415	3
3₋¹⁺²×#4294	3
#17987410	3
#17987409	3
#17987404	3
C₁₉×#6894	3
#10624872	19