Subgroups — C₃×#10602519 #14305888

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 340 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	C₂×C₃×#3967	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#3964	C₂	1	2	yes	maximal
4	C₃×#65224	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602519	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#124	—	2	2	no	—
10	C₃×C₅×#124	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
14	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂²×C₃×#168	C₂²	1	2	yes	—
16	C₃×#3967	—	2	4	no	—
17	C₃×#3964	—	2	4	no	—
18	C₃×#3935	—	2	2	no	—
19	C₃×#1959	—	5	1	no	maximal, Hall
20	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
21	C₂×#3967	C₂×C₃	1	2	yes	complemented
22	C₂×#3964	C₂×C₃	1	2	yes	—
23	#65224	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3935	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁴×C₃×C₅	—	2	1	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₃×Dic₅	—	2	2	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄⋊Dic₅	—	2	2	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄⋊Dic₅	—	4	4	no	—
44	C₂×C₃×#168	—	2	2	no	—
45	C₂×C₃×#168	—	2	2	no	—
46	C₂×C₃×#168	D₄	1	4	yes	—
47	C₂×C₃×#124	D₅	1	1	yes	characteristic
48	C₂×C₃×#103	—	5	2	no	—
49	C₂×C₃×#102	—	5	2	no	—
50	C₂×C₃×#97	—	5	1	no	—
51	C₃×#384	—	5	1	no	—
52	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₅×#124	—	2	2	no	—
55	C₅×#124	C₃×C₄	1	4	yes	complemented
56	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
57	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄⋊Dic₅	—	2	2	no	—
59	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
60	C₂²×#168	C₂²×C₃	1	2	yes	—
61	#3967	—	2	4	no	—
62	#3964	—	2	4	no	—
63	#3935	—	2	2	no	—
64	#1959	—	5	1	no	Sylow
65	C₂³×C₃×C₅	—	2	2	no	—
66	C₂³×C₃×C₅	—	4	1	no	—
67	C₂³×C₃×C₅	—	4	1	no	—
68	C₂³×C₃×C₅	—	4	2	no	—
69	C₂³×C₃×C₅	—	4	2	no	—
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	2	yes	—
85	C₂×C₃×Dic₅	—	4	2	no	—
86	C₂×C₃×Dic₅	—	4	4	no	—
87	C₃×#168	—	2	4	no	—
88	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
89	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
90	C₂²×C₃×C₈	—	5	2	no	—
91	C₃×#124	—	2	2	no	—
92	C₃×#124	Dic₅	1	4	yes	complemented
93	C₂×C₃×C₄⋊C₄	—	5	2	no	—
94	C₂×C₃×C₄⋊C₄	—	10	2	no	—
95	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₃×#103	—	10	4	no	—
98	C₃×#102	—	10	4	no	—
99	C₃×#97	—	10	2	no	—
100	C₂⁴×C₅	—	2	1	no	—
101	C₂×C₅×Q₈	—	2	1	no	—
102	C₂×C₅×Q₈	—	2	2	no	—
103	C₂×C₅×Q₈	—	2	2	no	—
104	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
109	C₂²×C₄×C₅	—	2	1	no	—
110	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
111	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₂²×Dic₅	—	2	2	no	—
113	C₅×C₂²⋊C₄	—	2	8	no	—
114	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
115	C₄²×C₅	C₃×D₄	1	2	yes	—
116	C₄⋊Dic₅	—	2	2	no	—
117	C₄⋊Dic₅	—	2	2	no	—
118	C₄⋊Dic₅	—	4	4	no	—
119	C₂×#168	—	2	2	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#168	C₃×D₄	1	4	yes	—
122	C₂×#124	C₃×D₅	1	1	yes	characteristic
123	C₂×#103	—	5	2	no	—
124	C₂×#102	—	5	2	no	—
125	C₂×#97	—	5	1	no	—
126	#384	—	5	1	no	—
127	C₂²×C₃×C₅	—	4	2	no	—
128	C₂²×C₃×C₅	—	4	4	no	—
129	C₂²×C₃×C₅	—	8	2	no	—
130	C₂²×C₃×C₅	—	8	1	no	—
131	C₂²×C₃×C₅	—	8	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#124	1	1	yes	characteristic
134	C₂²×C₃×C₅	#127	1	1	yes	characteristic
135	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₂²×C₃×C₅	#137	1	2	yes	—
137	C₃×C₄×C₅	—	2	4	no	—
138	C₃×C₄×C₅	—	4	2	no	—
139	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₃×C₄×C₅	#121	1	2	yes	—
141	C₃×Dic₅	—	8	2	no	—
142	C₂⁴×C₃	—	2	1	no	—
143	C₂×C₃×Q₈	—	2	1	no	—
144	C₂×C₃×Q₈	—	2	2	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	—	2	2	no	—
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
151	C₂²×C₃×C₄	—	2	1	no	—
152	C₂²×C₃×C₄	—	10	2	no	—
153	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	—	5	4	no	—
156	C₂×C₃×C₈	—	10	2	no	—
157	C₂×C₃×C₈	—	10	2	no	—
158	C₃×C₄⋊C₄	—	10	2	no	—
159	C₃×C₄⋊C₄	—	10	2	no	—
160	C₃×C₄⋊C₄	—	20	4	no	—
161	C₃×C₂²⋊C₄	—	2	8	no	—
162	C₃×C₄²	C₂×Dic₅	1	2	yes	—
163	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
164	C₂³×C₅	—	2	2	no	—
165	C₂³×C₅	—	4	1	no	—
166	C₂³×C₅	—	4	1	no	—
167	C₂³×C₅	—	4	2	no	—
168	C₂³×C₅	—	4	2	no	—
169	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₅×Q₈	—	2	4	no	—
171	C₅×Q₈	—	4	2	no	—
172	C₅×D₄	—	2	4	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	1	no	—
179	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
180	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₃×#41	1	2	yes	—
184	C₂×Dic₅	—	4	2	no	—
185	C₂×Dic₅	—	4	4	no	—
186	#168	—	2	4	no	—
187	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
188	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
189	C₂²×C₈	—	5	2	no	—
190	#124	—	2	2	no	—
191	#124	C₃×Dic₅	1	4	yes	complemented
192	C₂×C₄⋊C₄	—	5	2	no	—
193	C₂×C₄⋊C₄	—	10	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
196	#103	—	10	4	no	—
197	#102	—	10	4	no	—
198	#97	—	10	2	no	—
199	C₂×C₃×C₅	—	8	2	no	—
200	C₂×C₃×C₅	#390	1	1	yes	characteristic
201	C₂×C₃×C₅	#417	1	2	yes	—
202	C₂×C₃×C₅	#489	1	2	yes	—
203	C₂×C₃×C₅	#496	1	1	yes	characteristic
204	C₂×C₃×C₅	#497	1	1	yes	characteristic
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	—	4	2	no	—
207	C₂³×C₃	—	4	1	no	—
208	C₂³×C₃	—	4	1	no	—
209	C₂³×C₃	—	4	2	no	—
210	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	4	2	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	4	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	20	2	no	—
221	C₂×C₃×C₄	—	20	4	no	—
222	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
223	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
224	C₂×C₃×C₄	#711	1	2	yes	—
225	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₃×C₈	—	10	4	no	—
228	C₂²×C₅	—	4	2	no	—
229	C₂²×C₅	—	4	4	no	—
230	C₂²×C₅	—	8	2	no	—
231	C₂²×C₅	—	8	1	no	—
232	C₂²×C₅	—	8	1	no	—
233	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₂²×C₅	C₃×#124	1	1	yes	characteristic
235	C₂²×C₅	C₃×#127	1	1	yes	characteristic
236	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
237	C₂²×C₅	C₃×#137	1	2	yes	—
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	—	4	2	no	—
240	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
241	C₄×C₅	C₃×#121	1	2	yes	—
242	Dic₅	—	8	2	no	—
243	C₂⁴	—	2	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	10	2	no	—
254	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
255	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
256	C₂×C₈	—	5	4	no	—
257	C₂×C₈	—	10	2	no	—
258	C₂×C₈	—	10	2	no	—
259	C₄⋊C₄	—	10	2	no	—
260	C₄⋊C₄	—	10	2	no	—
261	C₄⋊C₄	—	20	4	no	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₄²	C₂×C₃×Dic₅	1	2	yes	—
264	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
265	C₃×C₅	#1959	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	—	4	4	no	—
268	C₂²×C₃	—	8	2	no	—
269	C₂²×C₃	—	8	1	no	—
270	C₂²×C₃	—	8	1	no	—
271	C₂²×C₃	#4082	1	1	yes	characteristic
272	C₂²×C₃	#4086	1	1	yes	characteristic
273	C₂²×C₃	#4095	1	2	yes	—
274	C₂²×C₃	#4097	1	2	yes	—
275	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	4	2	no	—
278	C₃×C₄	—	40	2	no	—
279	C₃×C₄	D₄×Dic₅	1	2	yes	—
280	C₃×C₄	#4084	1	2	yes	—
281	C₂×C₅	—	8	2	no	—
282	C₂×C₅	C₃×#390	1	1	yes	characteristic
283	C₂×C₅	C₃×#417	1	2	yes	—
284	C₂×C₅	C₃×#489	1	2	yes	—
285	C₂×C₅	C₃×#496	1	1	yes	characteristic
286	C₂×C₅	C₃×#497	1	1	yes	characteristic
287	C₂³	—	2	2	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
293	Q₈	—	2	4	no	—
294	Q₈	—	4	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	4	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	—	20	4	no	—
304	C₂×C₄	C₃×C₂²⋊Dic₅	1	4	yes	—
305	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
306	C₂×C₄	C₃×#711	1	2	yes	—
307	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
309	C₈	—	10	4	no	—
310	C₂×C₃	—	8	2	no	—
311	C₂×C₃	#65344	1	2	yes	—
312	C₂×C₃	#65347	1	1	yes	characteristic
313	C₂×C₃	#65349	1	1	yes	characteristic
314	C₂×C₃	#65510	1	1	yes	characteristic
315	C₂×C₃	#65521	1	2	yes	—
316	C₅	C₃×#1959	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	4	2	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₃×#4082	1	1	yes	characteristic
323	C₂²	C₃×#4086	1	1	yes	characteristic
324	C₂²	C₃×#4095	1	2	yes	—
325	C₂²	C₃×#4097	1	2	yes	—
326	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
327	C₄	—	2	4	no	—
328	C₄	—	4	2	no	—
329	C₄	—	40	2	no	—
330	C₄	C₃×D₄×Dic₅	1	2	yes	—
331	C₄	C₃×#4084	1	2	yes	—
332	C₃	#10602519	1	1	yes	characteristic, direct product factor, Sylow
333	C₂	—	8	2	no	—
334	C₂	C₃×#65344	1	2	yes	—
335	C₂	C₃×#65347	1	1	yes	characteristic
336	C₂	C₃×#65349	1	1	yes	characteristic
337	C₂	C₃×#65510	1	1	yes	characteristic
338	C₂	C₃×#65521	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₃×#1959
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3967	C₂	semidirect
4. C₃×#65224	C₂	semidirect
5. C₂×C₃×#3935	C₂	semidirect
6. #10602519	C₃	direct (N × H)
10. C₃×C₅×#124	C₄	semidirect
21. C₂×#3967	C₂×C₃	semidirect
23. #65224	C₂×C₃	semidirect
24. C₂×#3935	C₂×C₃	semidirect
55. C₅×#124	C₃×C₄	semidirect
92. C₃×#124	Dic₅	semidirect
191. #124	C₃×Dic₅	semidirect
265. C₃×C₅	#1959	semidirect
316. C₅	C₃×#1959	semidirect
332. C₃	#10602519	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
C₂×C₃×#3967	2
C₂×C₃×#3964	2
C₃×#65224	2
C₂×C₃×#3935	2
#10602519	3
C₃×#1959	5

Subgroups — C3×#10602519 #14305888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602519 #14305888