Subgroups — C₃×#10600654 #14284121

Order 1920.

432 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×#65813	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64685	C₂	1	2	yes	maximal, complemented
5	C₃×#64684	C₂	1	2	yes	maximal, complemented
6	#10600654	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×#4076	C₄	1	1	yes	characteristic
10	C₃×#4075	—	2	1	no	—
11	C₃×C₄×C₂²⋊D₅	—	2	1	no	—
12	C₃×#4070	C₄	1	1	yes	characteristic
13	C₃×#4022	—	2	1	no	—
14	C₃×#4021	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₂₀	C₄	1	1	yes	characteristic
16	C₃×#4018	—	2	1	no	—
17	C₃×#4016	C₂²	1	1	yes	characteristic
18	C₃×C₄×Dic₁₀	C₄	1	1	yes	characteristic
19	C₃×C₅×#97	C₂²	1	2	yes	—
20	C₃×C₄×C₅×C₈	—	2	1	no	—
21	C₃×C₂²⋊#168	—	2	1	no	—
22	C₃×C₄⋊#168	—	2	1	no	—
23	C₃×#3935	C₂²	1	2	yes	—
24	C₃×#1529	—	5	1	no	maximal, Hall
25	#65813	C₂×C₃	1	1	yes	characteristic
26	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	#65215	C₂×C₃	1	1	yes	characteristic, complemented
28	#64685	C₂×C₃	1	2	yes	complemented
29	#64684	C₂×C₃	1	2	yes	complemented
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
34	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	—	2	1	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	4	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊Dic₅	—	2	1	no	—
44	C₃×#685	—	2	1	no	—
45	C₃×#685	—	2	1	no	—
46	C₃×#685	—	2	1	no	—
47	C₃×#685	—	2	1	no	—
48	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
50	C₃×#683	—	2	1	no	—
51	C₃×#683	—	2	1	no	—
52	C₃×#683	—	2	1	no	—
53	C₃×#683	—	2	1	no	—
54	C₃×C₄×Dic₅	—	2	1	no	—
55	C₂×C₃×#168	—	2	2	no	—
56	C₃×#529	—	5	1	no	—
57	C₃×#432	—	5	1	no	—
58	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
59	C₃×#330	—	5	2	no	—
60	C₃×#329	—	5	2	no	—
61	C₂×C₅×M₁₆	—	2	1	no	—
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	#4076	C₃×C₄	1	1	yes	characteristic
64	#4075	—	2	1	no	—
65	C₄×C₂²⋊D₅	—	2	1	no	—
66	#4070	C₃×C₄	1	1	yes	characteristic
67	#4022	—	2	1	no	—
68	#4021	C₂²×C₃	1	1	yes	characteristic
69	C₄×D₂₀	C₃×C₄	1	1	yes	characteristic
70	#4018	—	2	1	no	—
71	#4016	C₂²×C₃	1	1	yes	characteristic
72	C₄×Dic₁₀	C₃×C₄	1	1	yes	characteristic
73	C₅×#97	C₂²×C₃	1	2	yes	—
74	C₄×C₅×C₈	—	2	1	no	—
75	C₂²⋊#168	—	2	1	no	—
76	C₄⋊#168	—	2	1	no	—
77	#3935	C₂²×C₃	1	2	yes	—
78	#1529	—	5	1	no	Sylow
79	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₃×D₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₃×C₂²⋊D₅	—	4	2	no	—
91	C₂×C₃×Dic₅	—	2	1	no	—
92	C₂×C₃×Dic₅	—	2	1	no	—
93	C₂×C₃×Dic₅	—	2	1	no	—
94	C₃×D₂₀	—	4	1	no	—
95	C₃×C₄×D₅	—	4	2	no	—
96	C₃×Dic₁₀	—	4	1	no	—
97	C₃×C₅×C₈	—	2	4	no	—
98	C₃×#168	—	4	2	no	—
99	C₂×C₃×M₁₆	—	2	1	no	—
100	C₃×#126	—	10	1	no	—
101	C₃×#125	—	5	1	no	—
102	C₃×#124	—	5	1	no	—
103	C₃×#123	—	10	1	no	—
104	C₃×C₂²⋊Q₈	—	5	1	no	—
105	C₃×#121	—	5	1	no	—
106	C₃×C₄×Q₈	—	5	1	no	—
107	C₃×C₄×D₄	—	5	1	no	—
108	C₃×C₄×D₄	—	10	1	no	—
109	C₃×#117	—	10	1	no	—
110	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
111	C₃×C₄⋊C₈	—	10	1	no	—
112	C₃×C₂²⋊C₈	—	10	1	no	—
113	C₃×#97	—	5	2	no	—
114	C₃×#97	C₂×D₅	1	2	yes	—
115	C₃×C₄×C₈	—	2	1	no	—
116	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
119	C₂×C₂²⋊D₅	—	2	1	no	—
120	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×C₄×D₅	—	2	1	no	—
122	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
123	C₅×M₁₆	—	2	4	no	—
124	C₂×C₅×C₈	—	2	2	no	—
125	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
126	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
127	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
129	C₂²⋊Dic₅	—	2	1	no	—
130	#685	—	2	1	no	—
131	#685	—	2	1	no	—
132	#685	—	2	1	no	—
133	#685	—	2	1	no	—
134	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
135	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
136	#683	—	2	1	no	—
137	#683	—	2	1	no	—
138	#683	—	2	1	no	—
139	#683	—	2	1	no	—
140	C₄×Dic₅	—	2	1	no	—
141	C₂×#168	—	2	2	no	—
142	#529	—	5	1	no	—
143	#432	—	5	1	no	—
144	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
145	#330	—	5	2	no	—
146	#329	—	5	2	no	—
147	C₂²×C₃×C₅	—	2	1	no	—
148	C₂²×C₃×C₅	—	2	1	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₃×D₅	—	4	1	no	—
152	C₂×C₃×D₅	—	4	1	no	—
153	C₂×C₃×D₅	—	4	1	no	—
154	C₃×C₄×C₅	—	2	1	no	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	—	2	1	no	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	#104	1	2	yes	—
160	C₃×Dic₅	—	4	1	no	—
161	C₃×Dic₅	—	4	1	no	—
162	C₃×Dic₅	—	4	1	no	—
163	C₂×C₃×Q₈	—	5	1	no	—
164	C₂×C₃×D₄	—	5	1	no	—
165	C₂×C₃×D₄	—	10	1	no	—
166	C₂²×C₃×C₄	—	10	1	no	—
167	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
168	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
169	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
170	C₃×M₁₆	—	2	4	no	—
171	C₂×C₃×C₈	—	2	2	no	—
172	C₂×C₃×C₈	—	10	2	no	—
173	C₃×C₄⋊C₄	—	5	1	no	—
174	C₃×C₄⋊C₄	—	5	1	no	—
175	C₃×C₄⋊C₄	—	10	1	no	—
176	C₃×C₄⋊C₄	—	10	1	no	—
177	C₃×C₄⋊C₄	—	10	1	no	—
178	C₃×C₄⋊C₄	—	10	1	no	—
179	C₃×C₂²⋊C₄	—	10	1	no	—
180	C₃×C₂²⋊C₄	—	10	1	no	—
181	C₃×C₂²⋊C₄	—	10	1	no	—
182	C₃×C₂²⋊C₄	—	10	1	no	—
183	C₃×C₂²⋊C₄	—	10	1	no	—
184	C₃×C₄²	—	10	1	no	—
185	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
186	C₃×C₄²	D₂₀	1	1	yes	characteristic
187	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
188	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
189	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×D₅	—	2	1	no	—
191	C₂×C₄×C₅	—	2	2	no	—
192	C₂×C₄×C₅	—	4	1	no	—
193	C₂×C₄×C₅	—	4	1	no	—
194	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂²⋊D₅	—	4	2	no	—
201	C₂×Dic₅	—	2	1	no	—
202	C₂×Dic₅	—	2	1	no	—
203	C₂×Dic₅	—	2	1	no	—
204	D₂₀	—	4	1	no	—
205	C₄×D₅	—	4	2	no	—
206	Dic₁₀	—	4	1	no	—
207	C₅×C₈	—	2	4	no	—
208	#168	—	4	2	no	—
209	C₂×M₁₆	—	2	1	no	—
210	#126	—	10	1	no	—
211	#125	—	5	1	no	—
212	#124	—	5	1	no	—
213	#123	—	10	1	no	—
214	C₂²⋊Q₈	—	5	1	no	—
215	#121	—	5	1	no	—
216	C₄×Q₈	—	5	1	no	—
217	C₄×D₄	—	5	1	no	—
218	C₄×D₄	—	10	1	no	—
219	#117	—	10	1	no	—
220	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
221	C₄⋊C₈	—	10	1	no	—
222	C₂²⋊C₈	—	10	1	no	—
223	#97	—	5	2	no	—
224	#97	C₂×C₃×D₅	1	2	yes	—
225	C₄×C₈	—	2	1	no	—
226	C₂×C₃×C₅	—	4	1	no	—
227	C₂×C₃×C₅	#413	1	1	yes	characteristic
228	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
229	C₂×C₃×C₅	#421	1	1	yes	characteristic
230	C₃×D₅	—	8	1	no	—
231	C₂³×C₃	—	10	1	no	—
232	C₂³×C₃	#685	1	1	yes	characteristic
233	C₃×Q₈	—	20	1	no	—
234	C₃×D₄	—	20	1	no	—
235	C₃×D₄	—	20	2	no	—
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	4	1	no	—
238	C₂×C₃×C₄	—	4	1	no	—
239	C₂×C₃×C₄	—	10	1	no	—
240	C₂×C₃×C₄	—	10	1	no	—
241	C₂×C₃×C₄	—	10	1	no	—
242	C₂×C₃×C₄	—	20	2	no	—
243	C₂×C₃×C₄	#685	1	1	yes	characteristic
244	C₂×C₃×C₄	#685	1	1	yes	characteristic
245	C₂×C₃×C₄	#685	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
248	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
249	C₃×C₈	—	2	4	no	—
250	C₃×C₈	—	20	2	no	—
251	C₂²×C₅	—	2	1	no	—
252	C₂²×C₅	—	2	1	no	—
253	C₂²×C₅	—	2	1	no	—
254	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×D₅	—	4	1	no	—
256	C₂×D₅	—	4	1	no	—
257	C₂×D₅	—	4	1	no	—
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	2	1	no	—
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	C₃×#104	1	2	yes	—
264	Dic₅	—	4	1	no	—
265	Dic₅	—	4	1	no	—
266	Dic₅	—	4	1	no	—
267	C₂×Q₈	—	5	1	no	—
268	C₂×D₄	—	5	1	no	—
269	C₂×D₄	—	10	1	no	—
270	C₂²×C₄	—	10	1	no	—
271	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
272	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
273	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
274	M₁₆	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	10	2	no	—
277	C₄⋊C₄	—	5	1	no	—
278	C₄⋊C₄	—	5	1	no	—
279	C₄⋊C₄	—	10	1	no	—
280	C₄⋊C₄	—	10	1	no	—
281	C₄⋊C₄	—	10	1	no	—
282	C₄⋊C₄	—	10	1	no	—
283	C₂²⋊C₄	—	10	1	no	—
284	C₂²⋊C₄	—	10	1	no	—
285	C₂²⋊C₄	—	10	1	no	—
286	C₂²⋊C₄	—	10	1	no	—
287	C₂²⋊C₄	—	10	1	no	—
288	C₄²	—	10	1	no	—
289	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
290	C₄²	C₃×D₂₀	1	1	yes	characteristic
291	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
292	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
293	C₃×C₅	#1529	1	1	yes	characteristic, complemented, Hall
294	C₂²×C₃	—	2	1	no	—
295	C₂²×C₃	—	2	1	no	—
296	C₂²×C₃	—	2	1	no	—
297	C₂²×C₃	—	20	1	no	—
298	C₂²×C₃	—	20	1	no	—
299	C₂²×C₃	—	20	1	no	—
300	C₂²×C₃	C₂×#685	1	1	yes	characteristic
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	20	1	no	—
307	C₃×C₄	—	20	1	no	—
308	C₃×C₄	—	20	1	no	—
309	C₃×C₄	#3937	1	2	yes	—
310	C₂×C₅	—	4	1	no	—
311	C₂×C₅	C₃×#413	1	1	yes	characteristic
312	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
313	C₂×C₅	C₃×#421	1	1	yes	characteristic
314	D₅	—	8	1	no	—
315	C₂³	—	10	1	no	—
316	C₂³	C₃×#685	1	1	yes	characteristic
317	Q₈	—	20	1	no	—
318	D₄	—	20	1	no	—
319	D₄	—	20	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	10	1	no	—
324	C₂×C₄	—	10	1	no	—
325	C₂×C₄	—	10	1	no	—
326	C₂×C₄	—	20	2	no	—
327	C₂×C₄	C₃×#685	1	1	yes	characteristic
328	C₂×C₄	C₃×#685	1	1	yes	characteristic
329	C₂×C₄	C₃×#685	1	1	yes	characteristic
330	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
331	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
332	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
333	C₈	—	2	4	no	—
334	C₈	—	20	2	no	—
335	C₂×C₃	—	4	1	no	—
336	C₂×C₃	—	40	1	no	—
337	C₂×C₃	C₂×#3937	1	1	yes	characteristic
338	C₂×C₃	#65421	1	1	yes	characteristic
339	C₂×C₃	#65428	1	1	yes	characteristic
340	C₅	C₃×#1529	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	20	1	no	—
345	C₂²	—	20	1	no	—
346	C₂²	—	20	1	no	—
347	C₂²	C₂×C₃×#685	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	20	1	no	—
354	C₄	—	20	1	no	—
355	C₄	—	20	1	no	—
356	C₄	C₃×#3937	1	2	yes	—
357	C₃	#10600654	1	1	yes	characteristic, direct product factor, Sylow
358	C₂	—	4	1	no	—
359	C₂	—	40	1	no	—
360	C₂	C₂×C₃×#3937	1	1	yes	characteristic
361	C₂	C₃×#65421	1	1	yes	characteristic
362	C₂	C₃×#65428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#685
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#4076	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×#1529
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. C₃×#64685	C₂	semidirect
5. C₃×#64684	C₂	semidirect
6. #10600654	C₃	direct (N × H)
26. C₄×C₅×M₁₆	C₂×C₃	semidirect
27. #65215	C₂×C₃	semidirect
28. #64685	C₂×C₃	semidirect
29. #64684	C₂×C₃	semidirect
58. C₃×C₄×M₁₆	D₅	semidirect
144. C₄×M₁₆	C₃×D₅	semidirect
293. C₃×C₅	#1529	semidirect
340. C₅	C₃×#1529	semidirect
357. C₃	#10600654	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×#65813	2
C₃×C₄×C₅×M₁₆	2
C₃×#65215	2
C₃×#64685	2
C₃×#64684	2
#10600654	3
C₃×#1529	5

Subgroups — C3×#10600654 #14284121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600654 #14284121