Subgroups — C₃×#10602612 #14313412

Order 1920.

516 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₃×#65529	C₂	1	2	yes	maximal, complemented
3	C₃×#64765	C₂	1	2	yes	maximal, complemented
4	C₃×#64761	C₂	1	2	yes	maximal, complemented
5	#10602612	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	C₄	1	1	yes	characteristic, complemented
11	C₃×C₅×#124	C₂²	1	1	yes	characteristic
12	C₃×C₅×#121	—	2	2	no	—
13	C₃×C₄×C₅×D₄	—	2	2	no	—
14	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic, complemented
15	C₃×#4072	—	2	2	no	—
16	C₃×#3966	—	2	2	no	—
17	C₃×#3966	C₂²	1	4	yes	—
18	C₃×#2126	—	5	1	no	maximal, Hall
19	C₅×#532	C₂×C₃	1	1	yes	characteristic
20	#65529	C₂×C₃	1	2	yes	complemented
21	#64765	C₂×C₃	1	2	yes	complemented
22	#64761	C₂×C₃	1	2	yes	complemented
23	C₃×C₅×#41	—	4	1	no	—
24	C₃×C₅×#41	—	4	1	no	—
25	C₃×C₅×#41	—	4	1	no	—
26	C₃×C₅×#41	—	4	1	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	—	2	1	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
33	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	2	1	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₂²⋊Dic₅	—	2	4	no	—
45	C₃×C₄⋊Dic₅	—	4	2	no	—
46	C₃×C₄×Dic₅	—	4	2	no	—
47	C₃×#532	D₅	1	1	yes	characteristic
48	C₃×#410	—	5	2	no	—
49	C₃×#354	—	5	2	no	—
50	C₃×#353	—	5	2	no	—
51	C₂×C₅×#41	C₃×C₄	1	1	yes	characteristic, complemented
52	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic, complemented
54	C₅×#127	C₂²×C₃	1	1	yes	characteristic
55	C₅×#124	C₃×C₄	1	1	yes	characteristic, complemented
56	C₅×#124	C₂²×C₃	1	1	yes	characteristic
57	C₅×#121	—	2	2	no	—
58	C₄×C₅×D₄	—	2	2	no	—
59	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic, complemented
60	#4072	—	2	2	no	—
61	#3966	—	2	2	no	—
62	#3966	C₂²×C₃	1	4	yes	—
63	#2126	—	5	1	no	Sylow
64	C₂³×C₃×C₅	—	2	1	no	—
65	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
67	C₃×C₅×Q₈	—	4	1	no	—
68	C₃×C₅×Q₈	—	4	1	no	—
69	C₃×C₅×D₄	—	4	2	no	—
70	C₃×C₅×D₄	—	4	1	no	—
71	C₃×C₅×D₄	—	4	4	no	—
72	C₃×C₅×D₄	—	4	1	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	4	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	2	yes	—
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×Dic₅	—	4	4	no	—
88	C₂×C₃×#41	Dic₅	1	1	yes	characteristic, complemented
89	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
90	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
91	C₃×#127	C₂×D₅	1	1	yes	characteristic
92	C₃×#124	Dic₅	1	1	yes	characteristic, complemented
93	C₃×#124	C₂×D₅	1	1	yes	characteristic
94	C₃×#121	—	2	2	no	—
95	C₃×C₄×D₄	—	2	2	no	—
96	C₃×#117	—	10	2	no	—
97	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic, complemented
98	C₃×#99	—	5	4	no	—
99	C₃×#99	—	10	2	no	—
100	C₅×#41	—	4	1	no	—
101	C₅×#41	—	4	1	no	—
102	C₅×#41	—	4	1	no	—
103	C₅×#41	—	4	1	no	—
104	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
110	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₅×C₂²⋊C₄	—	2	2	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂²⋊Dic₅	—	2	4	no	—
122	C₄⋊Dic₅	—	4	2	no	—
123	C₄×Dic₅	—	4	2	no	—
124	#532	C₃×D₅	1	1	yes	characteristic
125	#410	—	5	2	no	—
126	#354	—	5	2	no	—
127	#353	—	5	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	4	1	no	—
131	C₂²×C₃×C₅	—	4	2	no	—
132	C₂²×C₃×C₅	—	4	2	no	—
133	C₂²×C₃×C₅	—	4	1	no	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	—	2	2	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	C₃×C₄×C₅	#99	1	1	yes	characteristic
141	C₃×C₄×C₅	#99	1	1	yes	characteristic
142	C₃×Dic₅	—	8	4	no	—
143	C₃×#41	—	4	1	no	—
144	C₃×#41	—	4	1	no	—
145	C₃×#41	—	4	1	no	—
146	C₃×#41	—	4	1	no	—
147	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
148	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
153	C₂×C₃×D₄	C₂²×D₅	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	—	2	1	no	—
156	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
158	C₃×C₄⋊C₄	—	2	2	no	—
159	C₃×C₄⋊C₄	—	20	2	no	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₂²⋊C₄	—	2	2	no	—
162	C₃×C₂²⋊C₄	—	10	4	no	—
163	C₃×C₄²	—	2	1	no	—
164	C₃×C₄²	—	20	2	no	—
165	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
166	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
167	C₂³×C₅	—	2	1	no	—
168	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
170	C₅×Q₈	—	4	1	no	—
171	C₅×Q₈	—	4	1	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	1	no	—
174	C₅×D₄	—	4	1	no	—
175	C₅×D₄	—	4	4	no	—
176	C₅×D₄	—	4	2	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	2	1	no	—
183	C₂×C₄×C₅	—	2	1	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
188	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×Dic₅	—	4	4	no	—
191	C₂×#41	C₃×Dic₅	1	1	yes	characteristic, complemented
192	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
193	C₄⋊Q₈	C₃×Dic₅	1	1	yes	characteristic, complemented
194	#127	C₂×C₃×D₅	1	1	yes	characteristic
195	#124	C₃×Dic₅	1	1	yes	characteristic, complemented
196	#124	C₂×C₃×D₅	1	1	yes	characteristic
197	#121	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	#117	—	10	2	no	—
200	C₂×C₄²	C₃×Dic₅	1	1	yes	characteristic, complemented
201	#99	—	5	4	no	—
202	#99	—	10	2	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	4	2	no	—
205	C₂×C₃×C₅	—	4	1	no	—
206	C₂×C₃×C₅	—	8	1	no	—
207	C₂×C₃×C₅	C₂×#99	1	1	yes	characteristic
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
210	C₂³×C₃	C₂×C₂²⋊D₅	1	2	yes	—
211	C₃×Q₈	—	4	1	no	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×D₄	—	4	2	no	—
214	C₃×D₄	—	4	1	no	—
215	C₃×D₄	—	4	1	no	—
216	C₃×D₄	—	4	4	no	—
217	C₃×D₄	—	4	2	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	4	2	no	—
226	C₂×C₃×C₄	—	4	1	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	—	20	4	no	—
229	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
230	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	4	1	no	—
235	C₂²×C₅	—	4	2	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	—	4	1	no	—
238	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	—	4	1	no	—
244	C₄×C₅	C₃×#99	1	1	yes	characteristic
245	C₄×C₅	C₃×#99	1	1	yes	characteristic
246	Dic₅	—	8	4	no	—
247	#41	—	4	1	no	—
248	#41	—	4	1	no	—
249	#41	—	4	1	no	—
250	#41	—	4	1	no	—
251	C₂×Q₈	C₂×C₃×Dic₅	1	1	yes	characteristic
252	C₂×Q₈	C₂×C₃×Dic₅	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	C₂×C₃×Dic₅	1	1	yes	characteristic
256	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
257	C₂×D₄	C₂²×C₃×D₅	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
261	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	20	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	10	4	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	20	2	no	—
269	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
270	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
271	C₃×C₅	#2126	1	1	yes	characteristic, complemented, Hall
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	—	4	2	no	—
276	C₂²×C₃	—	4	2	no	—
277	C₂²×C₃	—	4	1	no	—
278	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	4	1	no	—
283	C₃×C₄	—	4	1	no	—
284	C₃×C₄	—	40	4	no	—
285	C₃×C₄	#3966	1	1	yes	characteristic
286	C₃×C₄	#3966	1	1	yes	characteristic
287	C₂×C₅	—	2	1	no	—
288	C₂×C₅	—	4	2	no	—
289	C₂×C₅	—	4	1	no	—
290	C₂×C₅	—	8	1	no	—
291	C₂×C₅	C₂×C₃×#99	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
294	C₂³	C₂×C₃×C₂²⋊D₅	1	2	yes	—
295	Q₈	—	4	1	no	—
296	Q₈	—	4	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	20	4	no	—
313	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
314	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	4	2	no	—
318	C₂×C₃	—	4	1	no	—
319	C₂×C₃	—	8	1	no	—
320	C₂×C₃	C₂×#3966	1	1	yes	characteristic
321	C₅	C₃×#2126	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	40	4	no	—
335	C₄	C₃×#3966	1	1	yes	characteristic
336	C₄	C₃×#3966	1	1	yes	characteristic
337	C₃	#10602612	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	4	2	no	—
340	C₂	—	4	1	no	—
341	C₂	—	8	1	no	—
342	C₂	C₂×C₃×#3966	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#3966
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×D₄	C₂²×C₃×D₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂×C₃×C₅	C₂×#99
cosocle	C₂×C₅×D₄	C₂³×C₃
hypercenter	C₃×#532	D₅
hypocenter	C₅	C₃×#2126
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41 ◁ C₃×#532

Complements

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

N	H ≅ G/N	type
2. C₃×#65529	C₂	semidirect
3. C₃×#64765	C₂	semidirect
4. C₃×#64761	C₂	semidirect
5. #10602612	C₃	direct (N × H)
6. C₂×C₃×C₅×#41	C₄	semidirect
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
10. C₃×C₅×#124	C₄	semidirect
14. C₂×C₃×C₄²×C₅	C₄	semidirect
20. #65529	C₂×C₃	semidirect
21. #64765	C₂×C₃	semidirect
22. #64761	C₂×C₃	semidirect
51. C₂×C₅×#41	C₃×C₄	semidirect
53. C₅×C₄⋊Q₈	C₃×C₄	semidirect
55. C₅×#124	C₃×C₄	semidirect
59. C₂×C₄²×C₅	C₃×C₄	semidirect
88. C₂×C₃×#41	Dic₅	semidirect
90. C₃×C₄⋊Q₈	Dic₅	semidirect
92. C₃×#124	Dic₅	semidirect
97. C₂×C₃×C₄²	Dic₅	semidirect
191. C₂×#41	C₃×Dic₅	semidirect
193. C₄⋊Q₈	C₃×Dic₅	semidirect
195. #124	C₃×Dic₅	semidirect
200. C₂×C₄²	C₃×Dic₅	semidirect
271. C₃×C₅	#2126	semidirect
321. C₅	C₃×#2126	semidirect
337. C₃	#10602612	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×#532	2
C₃×#65529	2
C₃×#64765	2
C₃×#64761	2
#10602612	3
C₃×#2126	5

Subgroups — C3×#10602612 #14313412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602612 #14313412