Subgroups — C₃×#10602513 #14281240

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64745	C₂	1	4	yes	maximal, complemented
5	#10602513	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×D₄	C₄	1	1	yes	characteristic
8	C₃×C₅×#124	—	2	1	no	—
9	C₃×C₅×#124	—	2	1	no	—
10	C₃×C₅×#124	—	2	1	no	—
11	C₃×C₅×#124	—	2	1	no	—
12	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊#168	C₂²	1	2	yes	—
16	C₃×C₂²⋊#168	C₂²	1	2	yes	—
17	C₃×C₄⋊#168	—	2	1	no	—
18	C₃×C₄⋊#168	—	2	1	no	—
19	C₃×#3935	—	2	1	no	—
20	C₃×C₄×#168	—	2	1	no	—
21	C₃×#1469	—	5	1	no	maximal, Hall
22	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
23	#65215	C₂×C₃	1	1	yes	characteristic, complemented
24	#65214	C₂×C₃	1	1	yes	characteristic, complemented
25	#64745	C₂×C₃	1	4	yes	complemented
26	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	—	4	2	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	4	2	no	—
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₃×#168	—	2	2	no	—
46	C₂×C₃×#168	—	2	2	no	—
47	C₂×C₃×#124	D₅	1	1	yes	characteristic
48	C₃×#432	—	5	1	no	—
49	C₃×#431	—	5	1	no	—
50	C₃×#323	—	5	4	no	—
51	C₂²×C₅×Q₈	C₃×C₄	1	1	yes	characteristic
52	C₂²×C₅×D₄	C₃×C₄	1	1	yes	characteristic
53	C₅×#124	—	2	1	no	—
54	C₅×#124	—	2	1	no	—
55	C₅×#124	—	2	1	no	—
56	C₅×#124	—	2	1	no	—
57	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
58	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂²⋊#168	C₂²×C₃	1	2	yes	—
61	C₂²⋊#168	C₂²×C₃	1	2	yes	—
62	C₄⋊#168	—	2	1	no	—
63	C₄⋊#168	—	2	1	no	—
64	#3935	—	2	1	no	—
65	C₄×#168	—	2	1	no	—
66	#1469	—	5	1	no	Sylow
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	2	4	no	—
69	C₂³×C₃×C₅	—	4	2	no	—
70	C₂³×C₃×C₅	—	4	2	no	—
71	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×Q₈	—	4	2	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	2	no	—
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₃×#168	—	4	2	no	—
88	C₃×#168	—	4	2	no	—
89	C₂²×C₃×Q₈	Dic₅	1	1	yes	characteristic
90	C₂²×C₃×D₄	Dic₅	1	1	yes	characteristic
91	C₃×#124	—	2	1	no	—
92	C₃×#124	—	2	1	no	—
93	C₃×#124	—	2	1	no	—
94	C₃×#124	—	2	1	no	—
95	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
96	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	—	10	1	no	—
99	C₃×C₄⋊C₈	—	10	1	no	—
100	C₃×C₂²⋊C₈	—	5	2	no	—
101	C₃×C₂²⋊C₈	—	5	2	no	—
102	C₃×#97	—	10	1	no	—
103	C₃×C₄×C₈	—	10	1	no	—
104	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	2	1	no	—
107	C₂×C₅×Q₈	—	4	2	no	—
108	C₂×C₅×D₄	—	2	1	no	—
109	C₂×C₅×D₄	—	2	1	no	—
110	C₂×C₅×D₄	—	4	2	no	—
111	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₅×C₂²⋊C₄	—	2	2	no	—
118	C₅×C₂²⋊C₄	—	2	2	no	—
119	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
123	C₂×#168	—	2	2	no	—
124	C₂×#168	—	2	2	no	—
125	C₂×#124	C₃×D₅	1	1	yes	characteristic
126	#432	—	5	1	no	—
127	#431	—	5	1	no	—
128	#323	—	5	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	4	2	no	—
132	C₂²×C₃×C₅	—	4	2	no	—
133	C₂²×C₃×C₅	—	4	2	no	—
134	C₂²×C₃×C₅	—	8	4	no	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₃×C₅	#131	1	2	yes	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	4	2	no	—
143	C₃×C₄×C₅	#100	1	2	yes	—
144	C₂⁴×C₃	C₂×Dic₅	1	2	yes	—
145	C₂×C₃×Q₈	—	2	1	no	—
146	C₂×C₃×Q₈	—	2	1	no	—
147	C₂×C₃×Q₈	—	4	2	no	—
148	C₂×C₃×D₄	—	2	1	no	—
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	—	4	2	no	—
151	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	—	10	2	no	—
156	C₂×C₃×C₈	—	10	2	no	—
157	C₃×C₂²⋊C₄	—	2	2	no	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₂²⋊C₄	—	2	2	no	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
165	C₂³×C₅	—	2	2	no	—
166	C₂³×C₅	—	2	4	no	—
167	C₂³×C₅	—	4	2	no	—
168	C₂³×C₅	—	4	2	no	—
169	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₅×Q₈	—	4	2	no	—
171	C₅×Q₈	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
185	#168	—	4	2	no	—
186	#168	—	4	2	no	—
187	C₂²×Q₈	C₃×Dic₅	1	1	yes	characteristic
188	C₂²×D₄	C₃×Dic₅	1	1	yes	characteristic
189	#124	—	2	1	no	—
190	#124	—	2	1	no	—
191	#124	—	2	1	no	—
192	#124	—	2	1	no	—
193	C₂×C₂²⋊C₄	C₃×Dic₅	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
195	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
196	C₄⋊C₈	—	10	1	no	—
197	C₄⋊C₈	—	10	1	no	—
198	C₂²⋊C₈	—	5	2	no	—
199	C₂²⋊C₈	—	5	2	no	—
200	#97	—	10	1	no	—
201	C₄×C₈	—	10	1	no	—
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	—	8	2	no	—
204	C₂×C₃×C₅	#408	1	1	yes	characteristic
205	C₂×C₃×C₅	#410	1	1	yes	characteristic
206	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
207	C₂³×C₃	—	2	2	no	—
208	C₂³×C₃	—	2	4	no	—
209	C₂³×C₃	—	4	2	no	—
210	C₂³×C₃	—	4	2	no	—
211	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
212	C₃×Q₈	—	4	2	no	—
213	C₃×Q₈	—	4	2	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₃×C₈	—	20	2	no	—
228	C₃×C₈	—	20	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	4	2	no	—
232	C₂²×C₅	—	4	2	no	—
233	C₂²×C₅	—	4	2	no	—
234	C₂²×C₅	—	8	4	no	—
235	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₅	C₃×#131	1	2	yes	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	C₃×#100	1	2	yes	—
244	C₂⁴	C₂×C₃×Dic₅	1	2	yes	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
254	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
255	C₂×C₈	—	10	2	no	—
256	C₂×C₈	—	10	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
262	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
263	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
264	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
265	C₃×C₅	#1469	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	4	2	no	—
269	C₂²×C₃	—	4	2	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	8	4	no	—
272	C₂²×C₃	#4094	1	2	yes	—
273	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	4	2	no	—
280	C₃×C₄	#3965	1	2	yes	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	—	8	2	no	—
283	C₂×C₅	C₃×#408	1	1	yes	characteristic
284	C₂×C₅	C₃×#410	1	1	yes	characteristic
285	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	—	2	4	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
301	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
302	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
303	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
304	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
305	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
306	C₈	—	20	2	no	—
307	C₈	—	20	2	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	—	8	2	no	—
310	C₂×C₃	C₂×#3965	1	1	yes	characteristic
311	C₂×C₃	#65523	1	1	yes	characteristic
312	C₂×C₃	#65529	1	1	yes	characteristic
313	C₅	C₃×#1469	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	4	no	—
320	C₂²	C₃×#4094	1	2	yes	—
321	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	2	no	—
328	C₄	C₃×#3965	1	2	yes	—
329	C₃	#10602513	1	1	yes	characteristic, direct product factor, Sylow
330	C₂	—	2	2	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂×C₃×#3965	1	1	yes	characteristic
333	C₂	C₃×#65523	1	1	yes	characteristic
334	C₂	C₃×#65529	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊Dic₅
commutator	C₂³×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×C₅×D₄	C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₃×#1469
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
2. C₃×#65215	C₂	semidirect
3. C₃×#65214	C₂	semidirect
4. C₃×#64745	C₂	semidirect
5. #10602513	C₃	direct (N × H)
23. #65215	C₂×C₃	semidirect
24. #65214	C₂×C₃	semidirect
25. #64745	C₂×C₃	semidirect
265. C₃×C₅	#1469	semidirect
313. C₅	C₃×#1469	semidirect
329. C₃	#10602513	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#124	2
C₃×#65215	2
C₃×#65214	2
C₃×#64745	2
#10602513	3
C₃×#1469	5

Subgroups — C3×#10602513 #14281240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602513 #14281240