Subgroups — C₃×#10613444 #14437996

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66050	C₂	1	1	yes	maximal, characteristic
2	C₃×#66022	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65358	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65061	C₂	1	4	yes	maximal, complemented
7	C₃×#65054	C₂	1	4	yes	maximal, complemented
8	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#64992	C₂	1	1	yes	maximal, characteristic, complemented
10	#10613444	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×#711	C₂²	1	2	yes	—
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₃×C₅×#125	C₂²	1	1	yes	characteristic
14	C₃×#4084	—	2	2	no	—
15	C₃×#4082	C₂²	1	2	yes	—
16	C₃×D₄×Dic₅	—	2	2	no	—
17	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
18	C₃×#4042	—	2	2	no	—
19	C₃×#4038	C₂²	1	2	yes	—
20	C₃×#4034	C₂²	1	2	yes	—
21	C₃×#4033	—	2	2	no	—
22	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
23	C₃×#4016	C₂²	1	1	yes	characteristic
24	C₃×#4015	C₂²	1	1	yes	characteristic
25	C₃×C₅×#102	C₂²	1	4	yes	—
26	C₃×C₅×#97	C₂²	1	1	yes	characteristic
27	C₃×#3964	C₂²	1	4	yes	—
28	C₃×#3948	C₂²	1	4	yes	—
29	C₃×#3947	C₂²	1	2	yes	—
30	C₃×#3942	C₂²	1	4	yes	—
31	C₃×#3935	C₂²	1	1	yes	characteristic
32	C₃×#3135	—	5	1	no	maximal, Hall
33	#66050	C₂×C₃	1	1	yes	characteristic
34	#66022	C₂×C₃	1	1	yes	characteristic, complemented
35	C₅×#490	C₂×C₃	1	1	yes	characteristic, complemented
36	#65360	C₂×C₃	1	1	yes	characteristic, complemented
37	#65358	C₂×C₃	1	1	yes	characteristic, complemented
38	#65061	C₂×C₃	1	4	yes	complemented
39	#65054	C₂×C₃	1	4	yes	complemented
40	D₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
41	#64992	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₃×C₅×D₄	—	2	2	no	—
43	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
44	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
45	C₂²×C₃×Dic₅	—	2	2	no	—
46	C₃×#711	—	2	8	no	—
47	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
48	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×Dic₁₀	—	2	2	no	—
50	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
51	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
52	C₃×C₅×C₄⋊C₄	—	2	2	no	—
53	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
54	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₃×C₂²⋊Dic₅	—	2	4	no	—
56	C₃×#685	—	2	2	no	—
57	C₃×C₄⋊Dic₅	—	2	2	no	—
58	C₃×C₄⋊Dic₅	C₂³	1	2	yes	—
59	C₃×#683	—	2	2	no	—
60	C₃×#683	—	2	2	no	—
61	C₃×C₄×Dic₅	—	2	2	no	—
62	C₃×C₄×Dic₅	D₄	1	2	yes	—
63	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
64	C₂×C₃×#168	C₂³	1	2	yes	—
65	C₃×C₈×D₅	—	2	4	no	—
66	C₃×#533	—	5	1	no	—
67	C₃×#532	—	5	1	no	—
68	C₃×#491	—	5	1	no	—
69	C₃×#491	—	5	1	no	—
70	C₃×#490	—	5	1	no	—
71	C₃×#490	D₅	1	1	yes	characteristic, complemented
72	C₃×#489	—	5	4	no	—
73	C₃×#416	—	5	4	no	—
74	C₂×C₃×#97	—	5	1	no	—
75	C₂×#711	C₂²×C₃	1	2	yes	—
76	C₅×#127	C₂²×C₃	1	1	yes	characteristic
77	C₅×#125	C₂²×C₃	1	1	yes	characteristic
78	#4084	—	2	2	no	—
79	#4082	C₂²×C₃	1	2	yes	—
80	D₄×Dic₅	—	2	2	no	—
81	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
82	#4042	—	2	2	no	—
83	#4038	C₂²×C₃	1	2	yes	—
84	#4034	C₂²×C₃	1	2	yes	—
85	#4033	—	2	2	no	—
86	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
87	#4016	C₂²×C₃	1	1	yes	characteristic
88	#4015	C₂²×C₃	1	1	yes	characteristic
89	C₅×#102	C₂²×C₃	1	4	yes	—
90	C₅×#97	C₂²×C₃	1	1	yes	characteristic
91	#3964	C₂²×C₃	1	4	yes	—
92	#3948	C₂²×C₃	1	4	yes	—
93	#3947	C₂²×C₃	1	2	yes	—
94	#3942	C₂²×C₃	1	4	yes	—
95	#3935	C₂²×C₃	1	1	yes	characteristic
96	#3135	—	5	1	no	Sylow
97	C₂³×C₃×C₅	—	2	2	no	—
98	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
99	C₃×C₅×D₄	—	2	4	no	—
100	C₃×C₅×D₄	—	4	4	no	—
101	C₂×C₃×C₄×C₅	—	2	2	no	—
102	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
103	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₃×C₂²⋊D₅	—	4	4	no	—
105	C₂×C₃×Dic₅	—	2	2	no	—
106	C₂×C₃×Dic₅	—	2	2	no	—
107	C₂×C₃×Dic₅	—	4	4	no	—
108	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
109	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
110	C₃×C₄×D₅	—	4	2	no	—
111	C₃×C₄×D₅	#41	1	4	yes	—
112	C₃×Dic₁₀	—	2	4	no	—
113	C₃×Dic₁₀	—	4	4	no	—
114	C₃×C₅×C₈	—	2	2	no	—
115	C₃×#168	—	2	2	no	—
116	C₂×C₃×#41	—	5	2	no	—
117	C₂²×C₃×C₈	—	5	2	no	—
118	C₃×C₄⋊Q₈	—	5	2	no	—
119	C₃×C₄⋊Q₈	—	5	1	no	—
120	C₃×#127	C₂×D₅	1	1	yes	characteristic
121	C₃×#125	—	5	1	no	—
122	C₃×#125	C₂×D₅	1	1	yes	characteristic
123	C₃×#124	—	5	2	no	—
124	C₃×C₂²⋊Q₈	—	10	2	no	—
125	C₃×#121	—	10	2	no	—
126	C₃×C₄×Q₈	—	10	2	no	—
127	C₃×C₄×D₄	—	10	2	no	—
128	C₃×#117	—	5	2	no	—
129	C₂×C₃×C₄²	—	5	1	no	—
130	C₃×#103	—	5	4	no	—
131	C₃×#103	—	5	4	no	—
132	C₃×#102	—	5	4	no	—
133	C₃×#102	C₂×D₅	1	4	yes	—
134	C₃×#97	—	5	2	no	—
135	C₃×#97	—	5	1	no	—
136	C₃×#97	C₂×D₅	1	1	yes	characteristic
137	C₂×C₅×D₄	—	2	2	no	—
138	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
139	C₂×C₂²⋊D₅	—	2	2	no	—
140	C₂²×Dic₅	—	2	2	no	—
141	#711	—	2	8	no	—
142	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
143	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
144	C₂×Dic₁₀	—	2	2	no	—
145	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
146	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
147	C₅×C₄⋊C₄	—	2	2	no	—
148	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
149	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
150	C₂²⋊Dic₅	—	2	4	no	—
151	#685	—	2	2	no	—
152	C₄⋊Dic₅	—	2	2	no	—
153	C₄⋊Dic₅	C₂³×C₃	1	2	yes	—
154	#683	—	2	2	no	—
155	#683	—	2	2	no	—
156	C₄×Dic₅	—	2	2	no	—
157	C₄×Dic₅	C₃×D₄	1	2	yes	—
158	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
159	C₂×#168	C₂³×C₃	1	2	yes	—
160	C₈×D₅	—	2	4	no	—
161	#533	—	5	1	no	—
162	#532	—	5	1	no	—
163	#491	—	5	1	no	—
164	#491	—	5	1	no	—
165	#490	—	5	1	no	—
166	#490	C₃×D₅	1	1	yes	characteristic, complemented
167	#489	—	5	4	no	—
168	#416	—	5	4	no	—
169	C₂×#97	—	5	1	no	—
170	C₂²×C₃×C₅	—	4	2	no	—
171	C₂²×C₃×C₅	—	4	4	no	—
172	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₃×D₅	—	2	2	no	—
174	C₂×C₃×D₅	#124	1	2	yes	—
175	C₃×C₄×C₅	—	2	2	no	—
176	C₃×C₄×C₅	—	4	2	no	—
177	C₃×C₄×C₅	C₂×#41	1	2	yes	—
178	C₃×Dic₅	—	2	2	no	—
179	C₃×Dic₅	—	4	2	no	—
180	C₃×Dic₅	—	4	2	no	—
181	C₃×Dic₅	#124	1	2	yes	—
182	C₃×#41	—	10	8	no	—
183	C₂×C₃×Q₈	—	5	2	no	—
184	C₂×C₃×Q₈	—	10	2	no	—
185	C₂×C₃×D₄	—	2	2	no	—
186	C₂×C₃×D₄	—	10	2	no	—
187	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
188	C₂²×C₃×C₄	—	5	1	no	—
189	C₂²×C₃×C₄	—	5	2	no	—
190	C₂²×C₃×C₄	—	10	2	no	—
191	C₂×C₃×C₈	—	5	2	no	—
192	C₂×C₃×C₈	—	10	4	no	—
193	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
194	C₃×C₄⋊C₄	—	2	2	no	—
195	C₃×C₄⋊C₄	—	5	2	no	—
196	C₃×C₄⋊C₄	—	10	2	no	—
197	C₃×C₄⋊C₄	—	10	2	no	—
198	C₃×C₄⋊C₄	—	10	2	no	—
199	C₃×C₄⋊C₄	C₂²×D₅	1	2	yes	—
200	C₃×C₂²⋊C₄	—	10	2	no	—
201	C₃×C₂²⋊C₄	—	10	4	no	—
202	C₃×C₄²	—	5	2	no	—
203	C₃×C₄²	—	5	1	no	—
204	C₃×C₄²	—	10	2	no	—
205	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
206	C₂³×C₅	—	2	2	no	—
207	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
208	C₅×D₄	—	2	4	no	—
209	C₅×D₄	—	4	4	no	—
210	C₂×C₄×C₅	—	2	2	no	—
211	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
213	C₂²⋊D₅	—	4	4	no	—
214	C₂×Dic₅	—	2	2	no	—
215	C₂×Dic₅	—	2	2	no	—
216	C₂×Dic₅	—	4	4	no	—
217	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
218	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
219	C₄×D₅	—	4	2	no	—
220	C₄×D₅	C₃×#41	1	4	yes	—
221	Dic₁₀	—	2	4	no	—
222	Dic₁₀	—	4	4	no	—
223	C₅×C₈	—	2	2	no	—
224	#168	—	2	2	no	—
225	C₂×#41	—	5	2	no	—
226	C₂²×C₈	—	5	2	no	—
227	C₄⋊Q₈	—	5	2	no	—
228	C₄⋊Q₈	—	5	1	no	—
229	#127	C₂×C₃×D₅	1	1	yes	characteristic
230	#125	—	5	1	no	—
231	#125	C₂×C₃×D₅	1	1	yes	characteristic
232	#124	—	5	2	no	—
233	C₂²⋊Q₈	—	10	2	no	—
234	#121	—	10	2	no	—
235	C₄×Q₈	—	10	2	no	—
236	C₄×D₄	—	10	2	no	—
237	#117	—	5	2	no	—
238	C₂×C₄²	—	5	1	no	—
239	#103	—	5	4	no	—
240	#103	—	5	4	no	—
241	#102	—	5	4	no	—
242	#102	C₂×C₃×D₅	1	4	yes	—
243	#97	—	5	2	no	—
244	#97	—	5	1	no	—
245	#97	C₂×C₃×D₅	1	1	yes	characteristic
246	C₂×C₃×C₅	—	8	2	no	—
247	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
248	C₂×C₃×C₅	#575	1	2	yes	—
249	C₃×D₅	—	2	2	no	—
250	C₂³×C₃	—	2	2	no	—
251	C₂³×C₃	—	5	1	no	—
252	C₃×Q₈	—	10	4	no	—
253	C₃×Q₈	—	20	4	no	—
254	C₃×D₄	—	2	4	no	—
255	C₃×D₄	—	4	4	no	—
256	C₃×D₄	—	20	4	no	—
257	C₂×C₃×C₄	—	2	2	no	—
258	C₂×C₃×C₄	—	5	1	no	—
259	C₂×C₃×C₄	—	5	4	no	—
260	C₂×C₃×C₄	—	5	2	no	—
261	C₂×C₃×C₄	—	10	2	no	—
262	C₂×C₃×C₄	—	10	2	no	—
263	C₂×C₃×C₄	—	20	2	no	—
264	C₂×C₃×C₄	—	20	4	no	—
265	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
266	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
267	C₃×C₈	—	2	2	no	—
268	C₃×C₈	—	10	2	no	—
269	C₂²×C₅	—	4	4	no	—
270	C₂²×C₅	—	4	2	no	—
271	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
272	C₂×D₅	—	2	2	no	—
273	C₂×D₅	C₃×#124	1	2	yes	—
274	C₄×C₅	—	2	2	no	—
275	C₄×C₅	—	4	2	no	—
276	C₄×C₅	C₂×C₃×#41	1	2	yes	—
277	Dic₅	—	2	2	no	—
278	Dic₅	—	4	2	no	—
279	Dic₅	—	4	2	no	—
280	Dic₅	C₃×#124	1	2	yes	—
281	#41	—	10	8	no	—
282	C₂×Q₈	—	5	2	no	—
283	C₂×Q₈	—	10	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	10	2	no	—
286	C₂×D₄	C₂²×C₃×D₅	1	2	yes	—
287	C₂²×C₄	—	5	1	no	—
288	C₂²×C₄	—	5	2	no	—
289	C₂²×C₄	—	10	2	no	—
290	C₂×C₈	—	5	2	no	—
291	C₂×C₈	—	10	4	no	—
292	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	5	2	no	—
295	C₄⋊C₄	—	10	2	no	—
296	C₄⋊C₄	—	10	2	no	—
297	C₄⋊C₄	—	10	2	no	—
298	C₄⋊C₄	C₂²×C₃×D₅	1	2	yes	—
299	C₂²⋊C₄	—	10	2	no	—
300	C₂²⋊C₄	—	10	4	no	—
301	C₄²	—	5	2	no	—
302	C₄²	—	5	1	no	—
303	C₄²	—	10	2	no	—
304	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
305	C₃×C₅	#3135	1	1	yes	characteristic, complemented, Hall
306	C₂²×C₃	—	4	4	no	—
307	C₂²×C₃	—	4	2	no	—
308	C₂²×C₃	—	5	2	no	—
309	C₂²×C₃	—	10	2	no	—
310	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
311	C₃×C₄	—	2	2	no	—
312	C₃×C₄	—	4	2	no	—
313	C₃×C₄	—	5	2	no	—
314	C₃×C₄	—	10	2	no	—
315	C₃×C₄	—	20	2	no	—
316	C₃×C₄	—	20	2	no	—
317	C₃×C₄	D₅×#41	1	2	yes	—
318	C₂×C₅	—	8	2	no	—
319	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
320	C₂×C₅	C₃×#575	1	2	yes	—
321	D₅	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	—	5	1	no	—
324	Q₈	—	10	4	no	—
325	Q₈	—	20	4	no	—
326	D₄	—	2	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	20	4	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	5	1	no	—
331	C₂×C₄	—	5	4	no	—
332	C₂×C₄	—	5	2	no	—
333	C₂×C₄	—	10	2	no	—
334	C₂×C₄	—	10	2	no	—
335	C₂×C₄	—	20	2	no	—
336	C₂×C₄	—	20	4	no	—
337	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
338	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	10	2	no	—
341	C₂×C₃	—	8	2	no	—
342	C₂×C₃	—	10	2	no	—
343	C₂×C₃	D₅×#124	1	1	yes	characteristic
344	C₂×C₃	#66107	1	2	yes	—
345	C₅	C₃×#3135	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	4	4	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	5	2	no	—
349	C₂²	—	10	2	no	—
350	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	5	2	no	—
354	C₄	—	10	2	no	—
355	C₄	—	20	2	no	—
356	C₄	—	20	2	no	—
357	C₄	C₃×D₅×#41	1	2	yes	—
358	C₃	#10613444	1	1	yes	characteristic, direct product factor, Sylow
359	C₂	—	8	2	no	—
360	C₂	—	10	2	no	—
361	C₂	C₃×D₅×#124	1	1	yes	characteristic
362	C₂	C₃×#66107	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#490	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#66050	C₂
hypercenter	C₃×#490	D₅
hypocenter	C₅	C₃×#3135
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#490

Complements

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

N	H ≅ G/N	type
2. C₃×#66022	C₂	semidirect
3. C₃×C₅×#490	C₂	semidirect
4. C₃×#65360	C₂	semidirect
5. C₃×#65358	C₂	semidirect
6. C₃×#65061	C₂	semidirect
7. C₃×#65054	C₂	semidirect
8. C₃×D₅×#97	C₂	semidirect
9. C₃×#64992	C₂	semidirect
10. #10613444	C₃	direct (N × H)
34. #66022	C₂×C₃	semidirect
35. C₅×#490	C₂×C₃	semidirect
36. #65360	C₂×C₃	semidirect
37. #65358	C₂×C₃	semidirect
38. #65061	C₂×C₃	semidirect
39. #65054	C₂×C₃	semidirect
40. D₅×#97	C₂×C₃	semidirect
41. #64992	C₂×C₃	semidirect
71. C₃×#490	D₅	semidirect
166. #490	C₃×D₅	semidirect
305. C₃×C₅	#3135	semidirect
345. C₅	C₃×#3135	semidirect
358. C₃	#10613444	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#66050	2
C₃×#66022	2
C₃×C₅×#490	2
C₃×#65360	2
C₃×#65358	2
C₃×#65061	2
C₃×#65054	2
C₃×D₅×#97	2
C₃×#64992	2
#10613444	3
C₃×#3135	5

Subgroups — C3×#10613444 #14437996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613444 #14437996