Subgroups — C₃×#10601080 #14304777

Order 1920.

460 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×C₅×#416	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3969	C₂	1	1	yes	maximal, characteristic
3	C₃×#65523	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65288	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64772	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64752	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601080	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	1	yes	characteristic
14	C₂×C₃×#681	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₃×C₅×#103	C₄	1	2	yes	complemented
17	C₃×C₅×#102	C₄	1	2	yes	complemented
18	C₃×#3969	—	2	2	no	—
19	C₃×C₂²⋊#168	—	2	2	no	—
20	C₃×C₈×Dic₅	—	2	2	no	—
21	C₃×C₄⋊#168	—	2	2	no	—
22	C₃×#1937	—	5	1	no	maximal, Hall
23	C₅×#416	C₂×C₃	1	1	yes	characteristic
24	C₂×#3969	C₂×C₃	1	1	yes	characteristic
25	#65523	C₂×C₃	1	1	yes	characteristic
26	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
27	#65288	C₂×C₃	1	1	yes	characteristic, complemented
28	#64772	C₂×C₃	1	1	yes	characteristic, complemented
29	#64752	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×C₅×#41	—	2	1	no	—
31	C₃×C₅×#41	—	2	1	no	—
32	C₃×C₅×#41	—	2	1	no	—
33	C₃×C₅×#41	—	2	1	no	—
34	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	—	2	1	no	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₅	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
45	C₃×C₄²×C₅	—	2	1	no	—
46	C₃×C₄×Dic₅	—	2	1	no	—
47	C₃×C₄×Dic₅	D₄	1	2	yes	—
48	C₃×#681	—	2	1	no	—
49	C₃×#681	—	2	1	no	—
50	C₃×#681	—	2	1	no	—
51	C₃×#681	—	2	1	no	—
52	C₂×C₃×#168	—	2	1	no	—
53	C₂×C₃×#168	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	C₂×C₃×#168	—	2	1	no	—
56	C₂×C₃×#168	D₄	1	2	yes	—
57	C₃×#424	—	5	1	no	—
58	C₂×C₃×#104	—	5	1	no	—
59	C₃×#416	D₅	1	1	yes	characteristic
60	C₃×#408	—	5	1	no	—
61	C₂×C₃×C₄×C₈	—	5	1	no	—
62	C₃×#341	—	5	1	no	—
63	C₃×#339	—	5	1	no	—
64	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
65	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₅×#117	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
68	C₂×#681	C₂²×C₃	1	1	yes	characteristic
69	C₂×#681	C₂²×C₃	1	1	yes	characteristic
70	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
71	C₅×#103	C₃×C₄	1	2	yes	complemented
72	C₅×#102	C₃×C₄	1	2	yes	complemented
73	#3969	—	2	2	no	—
74	C₂²⋊#168	—	2	2	no	—
75	C₈×Dic₅	—	2	2	no	—
76	C₄⋊#168	—	2	2	no	—
77	#1937	—	5	1	no	Sylow
78	C₂³×C₃×C₅	—	2	1	no	—
79	C₂³×C₃×C₅	#41	1	1	yes	characteristic
80	C₃×C₅×Q₈	—	2	1	no	—
81	C₃×C₅×Q₈	—	2	1	no	—
82	C₃×C₅×D₄	—	2	1	no	—
83	C₃×C₅×D₄	—	2	1	no	—
84	C₃×C₅×D₄	—	4	1	no	—
85	C₃×C₅×D₄	—	4	1	no	—
86	C₂×C₃×C₄×C₅	—	2	2	no	—
87	C₂×C₃×C₄×C₅	—	2	1	no	—
88	C₂×C₃×C₄×C₅	—	4	1	no	—
89	C₂×C₃×C₄×C₅	—	4	1	no	—
90	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
96	C₂×C₃×Dic₅	—	2	4	no	—
97	C₂×C₃×Dic₅	—	2	2	no	—
98	C₃×C₅×C₈	—	2	2	no	—
99	C₃×#168	—	2	2	no	—
100	C₃×#168	—	4	1	no	—
101	C₃×#168	—	4	1	no	—
102	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
103	C₂×C₃×M₁₆	—	5	1	no	—
104	C₂×C₃×M₁₆	—	5	1	no	—
105	C₂²×C₃×C₈	—	5	1	no	—
106	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
107	C₃×#117	C₂×D₅	1	1	yes	characteristic
108	C₂×C₃×C₄²	—	5	1	no	—
109	C₃×C₄⋊C₈	—	10	2	no	—
110	C₃×#104	—	10	2	no	—
111	C₃×#103	Dic₅	1	2	yes	complemented
112	C₃×#102	Dic₅	1	2	yes	complemented
113	C₃×C₂²⋊C₈	—	10	2	no	—
114	C₃×C₄×C₈	—	10	2	no	—
115	C₅×#41	—	2	1	no	—
116	C₅×#41	—	2	1	no	—
117	C₅×#41	—	2	1	no	—
118	C₅×#41	—	2	1	no	—
119	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
120	C₂×C₅×D₄	—	2	1	no	—
121	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₅	—	2	1	no	—
123	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
124	C₂²×Dic₅	—	2	1	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₂×C₅×C₈	—	2	1	no	—
127	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
128	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
129	C₅×C₂²⋊C₄	—	2	1	no	—
130	C₄²×C₅	—	2	1	no	—
131	C₄×Dic₅	—	2	1	no	—
132	C₄×Dic₅	C₃×D₄	1	2	yes	—
133	#681	—	2	1	no	—
134	#681	—	2	1	no	—
135	#681	—	2	1	no	—
136	#681	—	2	1	no	—
137	C₂×#168	—	2	1	no	—
138	C₂×#168	—	2	1	no	—
139	C₂×#168	—	2	1	no	—
140	C₂×#168	—	2	1	no	—
141	C₂×#168	C₃×D₄	1	2	yes	—
142	#424	—	5	1	no	—
143	C₂×#104	—	5	1	no	—
144	#416	C₃×D₅	1	1	yes	characteristic
145	#408	—	5	1	no	—
146	C₂×C₄×C₈	—	5	1	no	—
147	#341	—	5	1	no	—
148	#339	—	5	1	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	—	4	1	no	—
151	C₂²×C₃×C₅	—	4	2	no	—
152	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
153	C₂²×C₃×C₅	#123	1	1	yes	characteristic
154	C₂²×C₃×C₅	#126	1	1	yes	characteristic
155	C₃×C₄×C₅	—	4	1	no	—
156	C₃×C₄×C₅	—	4	2	no	—
157	C₃×C₄×C₅	#117	1	1	yes	characteristic
158	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
159	C₃×C₄×C₅	#121	1	1	yes	characteristic
160	C₃×C₄×C₅	#124	1	1	yes	characteristic
161	C₃×Dic₅	—	2	4	no	—
162	C₃×#41	—	2	1	no	—
163	C₃×#41	—	2	1	no	—
164	C₃×#41	—	2	1	no	—
165	C₃×#41	—	2	1	no	—
166	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
167	C₂×C₃×D₄	—	2	1	no	—
168	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
169	C₂²×C₃×C₄	—	2	1	no	—
170	C₂²×C₃×C₄	—	10	1	no	—
171	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
172	C₃×M₁₆	—	10	1	no	—
173	C₃×M₁₆	—	10	1	no	—
174	C₃×M₁₆	—	10	1	no	—
175	C₃×M₁₆	—	10	1	no	—
176	C₂×C₃×C₈	—	2	1	no	—
177	C₂×C₃×C₈	—	2	1	no	—
178	C₂×C₃×C₈	—	5	2	no	—
179	C₂×C₃×C₈	—	10	1	no	—
180	C₂×C₃×C₈	—	10	1	no	—
181	C₂×C₃×C₈	—	10	1	no	—
182	C₂×C₃×C₈	—	10	1	no	—
183	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
184	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
185	C₃×C₂²⋊C₄	—	2	1	no	—
186	C₃×C₄²	—	2	1	no	—
187	C₃×C₄²	—	5	2	no	—
188	C₃×C₄²	—	10	1	no	—
189	C₂³×C₅	—	2	1	no	—
190	C₂³×C₅	C₃×#41	1	1	yes	characteristic
191	C₅×Q₈	—	2	1	no	—
192	C₅×Q₈	—	2	1	no	—
193	C₅×D₄	—	2	1	no	—
194	C₅×D₄	—	2	1	no	—
195	C₅×D₄	—	4	1	no	—
196	C₅×D₄	—	4	1	no	—
197	C₂×C₄×C₅	—	2	2	no	—
198	C₂×C₄×C₅	—	2	1	no	—
199	C₂×C₄×C₅	—	4	1	no	—
200	C₂×C₄×C₅	—	4	1	no	—
201	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
205	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
206	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
207	C₂×Dic₅	—	2	4	no	—
208	C₂×Dic₅	—	2	2	no	—
209	C₅×C₈	—	2	2	no	—
210	#168	—	2	2	no	—
211	#168	—	4	1	no	—
212	#168	—	4	1	no	—
213	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
214	C₂×M₁₆	—	5	1	no	—
215	C₂×M₁₆	—	5	1	no	—
216	C₂²×C₈	—	5	1	no	—
217	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
218	#117	C₂×C₃×D₅	1	1	yes	characteristic
219	C₂×C₄²	—	5	1	no	—
220	C₄⋊C₈	—	10	2	no	—
221	#104	—	10	2	no	—
222	#103	C₃×Dic₅	1	2	yes	complemented
223	#102	C₃×Dic₅	1	2	yes	complemented
224	C₂²⋊C₈	—	10	2	no	—
225	C₄×C₈	—	10	2	no	—
226	C₂×C₃×C₅	—	2	1	no	—
227	C₂×C₃×C₅	—	2	1	no	—
228	C₂×C₃×C₅	—	8	1	no	—
229	C₂×C₃×C₅	#388	1	1	yes	characteristic
230	C₂×C₃×C₅	#443	1	2	yes	—
231	C₂³×C₃	—	2	1	no	—
232	C₂³×C₃	#711	1	1	yes	characteristic
233	C₃×Q₈	—	2	1	no	—
234	C₃×Q₈	—	2	1	no	—
235	C₃×D₄	—	2	1	no	—
236	C₃×D₄	—	2	1	no	—
237	C₃×D₄	—	4	1	no	—
238	C₃×D₄	—	4	1	no	—
239	C₂×C₃×C₄	—	2	2	no	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	—	4	1	no	—
242	C₂×C₃×C₄	—	4	1	no	—
243	C₂×C₃×C₄	—	10	4	no	—
244	C₂×C₃×C₄	—	10	2	no	—
245	C₂×C₃×C₄	#709	1	1	yes	characteristic
246	C₂×C₃×C₄	#709	1	1	yes	characteristic
247	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
248	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
249	C₂×C₃×C₄	#711	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
251	C₃×C₈	—	2	2	no	—
252	C₃×C₈	—	10	2	no	—
253	C₃×C₈	—	20	1	no	—
254	C₃×C₈	—	20	1	no	—
255	C₂²×C₅	—	2	2	no	—
256	C₂²×C₅	—	4	1	no	—
257	C₂²×C₅	—	4	2	no	—
258	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
259	C₂²×C₅	C₃×#123	1	1	yes	characteristic
260	C₂²×C₅	C₃×#126	1	1	yes	characteristic
261	C₄×C₅	—	4	1	no	—
262	C₄×C₅	—	4	2	no	—
263	C₄×C₅	C₃×#117	1	1	yes	characteristic
264	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
265	C₄×C₅	C₃×#121	1	1	yes	characteristic
266	C₄×C₅	C₃×#124	1	1	yes	characteristic
267	Dic₅	—	2	4	no	—
268	#41	—	2	1	no	—
269	#41	—	2	1	no	—
270	#41	—	2	1	no	—
271	#41	—	2	1	no	—
272	C₂×Q₈	C₂×C₃×Dic₅	1	1	yes	characteristic
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	C₂×C₃×Dic₅	1	1	yes	characteristic
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	10	1	no	—
277	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
278	M₁₆	—	10	1	no	—
279	M₁₆	—	10	1	no	—
280	M₁₆	—	10	1	no	—
281	M₁₆	—	10	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	5	2	no	—
285	C₂×C₈	—	10	1	no	—
286	C₂×C₈	—	10	1	no	—
287	C₂×C₈	—	10	1	no	—
288	C₂×C₈	—	10	1	no	—
289	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
290	C₄⋊C₄	C₂×C₃×Dic₅	1	2	yes	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	5	2	no	—
294	C₄²	—	10	1	no	—
295	C₃×C₅	#1937	1	1	yes	characteristic, complemented, Hall
296	C₂²×C₃	—	2	2	no	—
297	C₂²×C₃	—	4	2	no	—
298	C₂²×C₃	—	4	1	no	—
299	C₂²×C₃	#4025	1	1	yes	characteristic
300	C₂²×C₃	#4029	1	1	yes	characteristic
301	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
302	C₃×C₄	—	4	2	no	—
303	C₃×C₄	—	4	1	no	—
304	C₃×C₄	—	10	4	no	—
305	C₃×C₄	#4030	1	1	yes	characteristic
306	C₃×C₄	#4031	1	1	yes	characteristic
307	C₃×C₄	#4072	1	1	yes	characteristic
308	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	2	1	no	—
311	C₂×C₅	—	8	1	no	—
312	C₂×C₅	C₃×#388	1	1	yes	characteristic
313	C₂×C₅	C₃×#443	1	2	yes	—
314	C₂³	—	2	1	no	—
315	C₂³	C₃×#711	1	1	yes	characteristic
316	Q₈	—	2	1	no	—
317	Q₈	—	2	1	no	—
318	D₄	—	2	1	no	—
319	D₄	—	2	1	no	—
320	D₄	—	4	1	no	—
321	D₄	—	4	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	10	4	no	—
327	C₂×C₄	—	10	2	no	—
328	C₂×C₄	C₃×#709	1	1	yes	characteristic
329	C₂×C₄	C₃×#709	1	1	yes	characteristic
330	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
331	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
332	C₂×C₄	C₃×#711	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	10	2	no	—
336	C₈	—	20	1	no	—
337	C₈	—	20	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	8	1	no	—
341	C₂×C₃	#65240	1	1	yes	characteristic
342	C₂×C₃	#65485	1	2	yes	—
343	C₅	C₃×#1937	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	C₃×#4025	1	1	yes	characteristic
348	C₂²	C₃×#4029	1	1	yes	characteristic
349	C₂²	C₃×D₄×Dic₅	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	10	4	no	—
353	C₄	C₃×#4030	1	1	yes	characteristic
354	C₄	C₃×#4031	1	1	yes	characteristic
355	C₄	C₃×#4072	1	1	yes	characteristic
356	C₄	C₃×D₄×Dic₅	1	1	yes	characteristic
357	C₃	#10601080	1	1	yes	characteristic, direct product factor, Sylow
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	1	no	—
361	C₂	C₃×#65240	1	1	yes	characteristic
362	C₂	C₃×#65485	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#416	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#416	D₅
hypocenter	C₅	C₃×#1937
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#416

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×C₈×Dic₅	C₂	semidirect
5. C₃×#65288	C₂	semidirect
6. C₃×#64772	C₂	semidirect
7. C₃×#64752	C₂	semidirect
8. #10601080	C₃	direct (N × H)
16. C₃×C₅×#103	C₄	semidirect
17. C₃×C₅×#102	C₄	semidirect
26. C₂×C₈×Dic₅	C₂×C₃	semidirect
27. #65288	C₂×C₃	semidirect
28. #64772	C₂×C₃	semidirect
29. #64752	C₂×C₃	semidirect
71. C₅×#103	C₃×C₄	semidirect
72. C₅×#102	C₃×C₄	semidirect
111. C₃×#103	Dic₅	semidirect
112. C₃×#102	Dic₅	semidirect
222. #103	C₃×Dic₅	semidirect
223. #102	C₃×Dic₅	semidirect
295. C₃×C₅	#1937	semidirect
343. C₅	C₃×#1937	semidirect
357. C₃	#10601080	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×C₅×#416	2
C₂×C₃×#3969	2
C₃×#65523	2
C₂×C₃×C₈×Dic₅	2
C₃×#65288	2
C₃×#64772	2
C₃×#64752	2
#10601080	3
C₃×#1937	5

Subgroups — C3×#10601080 #14304777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601080 #14304777