Subgroups — C₅×D₃×#491 #14437947

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×D₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5082	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5069	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×D₃×#103	C₂	1	4	yes	maximal
7	C₅×#4797	C₂	1	4	yes	maximal, complemented
8	C₅×D₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₅×#4703	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#491	—	3	1	no	maximal, Hall
11	C₂×C₅×D₃×Q₈	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#125	C₂²	1	1	yes	characteristic
14	C₅×#944	C₂²	1	2	yes	—
15	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
16	C₅×D₃×C₄⋊C₄	—	2	2	no	—
17	C₅×D₃×C₄⋊C₄	—	2	2	no	—
18	C₅×D₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₅×#890	C₂²	1	2	yes	—
20	C₅×#889	—	2	2	no	—
21	C₅×#888	—	2	2	no	—
22	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
23	C₅×#870	C₂²	1	1	yes	characteristic
24	C₅×#869	C₂²	1	1	yes	characteristic
25	C₃×C₅×#103	C₂²	1	4	yes	—
26	C₃×C₅×#97	C₂²	1	1	yes	characteristic
27	C₅×#835	C₂²	1	4	yes	—
28	C₅×#816	C₂²	1	4	yes	—
29	C₅×#815	C₂²	1	2	yes	—
30	C₅×#810	C₂²	1	4	yes	—
31	C₅×#803	C₂²	1	1	yes	characteristic
32	D₃×#491	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
34	C₂×C₅×#125	—	3	1	no	—
35	C₅×#491	—	3	4	no	—
36	C₅×#491	—	3	1	no	—
37	C₅×#491	—	3	1	no	—
38	C₅×#491	—	3	1	no	—
39	C₅×#491	D₃	1	1	yes	characteristic, direct product factor
40	C₂×C₅×#103	—	3	4	no	—
41	C₂×C₅×#97	—	3	1	no	—
42	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
43	C₅×D₃×Q₈	—	2	8	no	—
44	C₂×C₄×C₅×D₃	—	2	2	no	—
45	C₂×C₄×C₅×D₃	—	2	2	no	—
46	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
47	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₆	—	2	2	no	—
49	C₂×C₅×Dic₆	C₂³	1	2	yes	—
50	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
51	C₃×C₅×C₄⋊C₄	—	2	2	no	—
52	C₃×C₅×C₄⋊C₄	—	2	2	no	—
53	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
54	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×C₄⋊Dic₃	—	2	2	no	—
56	C₅×C₄⋊Dic₃	—	2	2	no	—
57	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
58	C₅×Dic₃⋊C₄	—	2	2	no	—
59	C₅×Dic₃⋊C₄	—	2	4	no	—
60	C₅×Dic₃⋊C₄	—	2	2	no	—
61	C₅×Dic₃⋊C₄	—	2	2	no	—
62	C₄×C₅×Dic₃	D₄	1	2	yes	—
63	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
65	C₅×D₃×C₈	—	2	4	no	—
66	D₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
67	D₃×#125	C₂×C₅	1	1	yes	characteristic
68	C₃×#491	C₂×C₅	1	1	yes	characteristic, complemented
69	#5082	C₂×C₅	1	1	yes	characteristic, complemented
70	#5069	C₂×C₅	1	1	yes	characteristic, complemented
71	D₃×#103	C₂×C₅	1	4	yes	—
72	#4797	C₂×C₅	1	4	yes	complemented
73	D₃×#97	C₂×C₅	1	1	yes	characteristic
74	#4703	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂²×C₅×Q₈	—	3	2	no	—
76	C₂²×C₅×C₈	—	3	2	no	—
77	C₅×C₄⋊Q₈	—	3	2	no	—
78	C₅×C₄⋊Q₈	—	3	1	no	—
79	C₅×C₄⋊Q₈	—	6	2	no	—
80	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
81	C₅×#125	—	3	2	no	—
82	C₅×#125	—	3	1	no	—
83	C₅×#125	—	6	2	no	—
84	C₅×#125	C₂×D₃	1	1	yes	characteristic
85	C₂×C₅×C₄⋊C₄	—	3	2	no	—
86	C₂×C₅×C₄⋊C₄	—	6	2	no	—
87	C₂×C₅×C₄⋊C₄	—	6	2	no	—
88	C₂×C₄²×C₅	—	3	1	no	—
89	C₅×#103	—	3	4	no	—
90	C₅×#103	—	3	4	no	—
91	C₅×#103	—	3	4	no	—
92	C₅×#103	C₂×D₃	1	4	yes	—
93	C₅×#97	—	3	2	no	—
94	C₅×#97	—	3	1	no	—
95	C₅×#97	C₂×D₃	1	1	yes	characteristic
96	C₂×#491	—	3	1	no	Sylow
97	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
98	C₃×C₅×Q₈	—	2	4	no	—
99	C₂×C₃×C₄×C₅	—	2	2	no	—
100	C₂×C₃×C₄×C₅	—	2	2	no	—
101	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×C₅×Dic₃	—	2	2	no	—
104	C₂×C₅×Dic₃	—	2	2	no	—
105	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
106	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
107	C₄×C₅×D₃	—	2	4	no	—
108	C₄×C₅×D₃	—	4	4	no	—
109	C₄×C₅×D₃	—	4	4	no	—
110	C₄×C₅×D₃	#41	1	4	yes	—
111	C₅×Dic₆	—	2	4	no	—
112	C₅×Dic₆	—	4	4	no	—
113	C₃×C₅×C₈	—	2	2	no	—
114	C₅×C₃⋊C₈	—	2	2	no	—
115	C₂×D₃×Q₈	C₂²×C₅	1	2	yes	—
116	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
117	C₃×#125	C₂²×C₅	1	1	yes	characteristic
118	#944	C₂²×C₅	1	2	yes	—
119	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
120	D₃×C₄⋊C₄	—	2	2	no	—
121	D₃×C₄⋊C₄	—	2	2	no	—
122	D₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
123	#890	C₂²×C₅	1	2	yes	—
124	#889	—	2	2	no	—
125	#888	—	2	2	no	—
126	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
127	#870	C₂²×C₅	1	1	yes	characteristic
128	#869	C₂²×C₅	1	1	yes	characteristic
129	C₃×#103	C₂²×C₅	1	4	yes	—
130	C₃×#97	C₂²×C₅	1	1	yes	characteristic
131	#835	C₂²×C₅	1	4	yes	—
132	#816	C₂²×C₅	1	4	yes	—
133	#815	C₂²×C₅	1	2	yes	—
134	#810	C₂²×C₅	1	4	yes	—
135	#803	C₂²×C₅	1	1	yes	characteristic
136	C₂×C₅×Q₈	—	3	2	no	—
137	C₂×C₅×Q₈	—	6	2	no	—
138	C₂×C₅×Q₈	—	6	8	no	—
139	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
140	C₂²×C₄×C₅	—	3	1	no	—
141	C₂²×C₄×C₅	—	3	2	no	—
142	C₂²×C₄×C₅	—	6	2	no	—
143	C₂²×C₄×C₅	—	6	2	no	—
144	C₂×C₅×C₈	—	3	2	no	—
145	C₂×C₅×C₈	—	6	4	no	—
146	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
147	C₅×C₄⋊C₄	—	2	2	no	—
148	C₅×C₄⋊C₄	—	2	2	no	—
149	C₅×C₄⋊C₄	—	3	2	no	—
150	C₅×C₄⋊C₄	—	6	2	no	—
151	C₅×C₄⋊C₄	—	6	2	no	—
152	C₅×C₄⋊C₄	—	6	2	no	—
153	C₅×C₄⋊C₄	—	6	4	no	—
154	C₅×C₄⋊C₄	—	6	2	no	—
155	C₅×C₄⋊C₄	—	6	2	no	—
156	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
157	C₄²×C₅	—	3	2	no	—
158	C₄²×C₅	—	3	1	no	—
159	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	—	3	1	no	—
161	C₂×#125	—	3	1	no	—
162	#491	—	3	1	no	—
163	#491	—	3	1	no	—
164	#491	—	3	4	no	—
165	#491	—	3	1	no	—
166	#491	C₅×D₃	1	1	yes	characteristic, direct product factor
167	C₂×#103	—	3	4	no	—
168	C₂×#97	—	3	1	no	—
169	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₅×D₃	#124	1	2	yes	—
171	C₂×C₅×D₃	#137	1	4	yes	—
172	C₃×C₄×C₅	—	2	2	no	—
173	C₃×C₄×C₅	—	4	2	no	—
174	C₃×C₄×C₅	—	4	2	no	—
175	C₃×C₄×C₅	C₂×#41	1	2	yes	—
176	C₅×Dic₃	—	2	2	no	—
177	C₅×Dic₃	—	4	2	no	—
178	C₅×Dic₃	—	4	2	no	—
179	C₅×Dic₃	#124	1	2	yes	—
180	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
181	D₃×Q₈	—	2	8	no	—
182	C₂×C₄×D₃	—	2	2	no	—
183	C₂×C₄×D₃	—	2	2	no	—
184	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
185	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
186	C₂×Dic₆	—	2	2	no	—
187	C₂×Dic₆	C₂³×C₅	1	2	yes	—
188	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
189	C₃×C₄⋊C₄	—	2	2	no	—
190	C₃×C₄⋊C₄	—	2	2	no	—
191	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
192	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
193	C₄⋊Dic₃	—	2	2	no	—
194	C₄⋊Dic₃	—	2	2	no	—
195	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
196	Dic₃⋊C₄	—	2	2	no	—
197	Dic₃⋊C₄	—	2	4	no	—
198	Dic₃⋊C₄	—	2	2	no	—
199	Dic₃⋊C₄	—	2	2	no	—
200	C₄×Dic₃	C₅×D₄	1	2	yes	—
201	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
202	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
203	D₃×C₈	—	2	4	no	—
204	C₂³×C₅	—	3	1	no	—
205	C₅×Q₈	—	2	4	no	—
206	C₅×Q₈	—	6	4	no	—
207	C₅×Q₈	—	12	4	no	—
208	C₂×C₄×C₅	—	2	2	no	—
209	C₂×C₄×C₅	—	2	2	no	—
210	C₂×C₄×C₅	—	3	1	no	—
211	C₂×C₄×C₅	—	3	4	no	—
212	C₂×C₄×C₅	—	3	2	no	—
213	C₂×C₄×C₅	—	6	4	no	—
214	C₂×C₄×C₅	—	6	2	no	—
215	C₂×C₄×C₅	—	6	2	no	—
216	C₂×C₄×C₅	—	12	4	no	—
217	C₂×C₄×C₅	—	12	4	no	—
218	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
219	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
220	C₅×C₈	—	2	2	no	—
221	C₅×C₈	—	6	2	no	—
222	C₂²×Q₈	—	3	2	no	—
223	C₂²×C₈	—	3	2	no	—
224	C₄⋊Q₈	—	3	2	no	—
225	C₄⋊Q₈	—	3	1	no	—
226	C₄⋊Q₈	—	6	2	no	—
227	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
228	#125	—	3	2	no	—
229	#125	—	3	1	no	—
230	#125	—	6	2	no	—
231	#125	C₂×C₅×D₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	—	3	2	no	—
233	C₂×C₄⋊C₄	—	6	2	no	—
234	C₂×C₄⋊C₄	—	6	2	no	—
235	C₂×C₄²	—	3	1	no	—
236	#103	—	3	4	no	—
237	#103	—	3	4	no	—
238	#103	—	3	4	no	—
239	#103	C₂×C₅×D₃	1	4	yes	—
240	#97	—	3	2	no	—
241	#97	—	3	1	no	—
242	#97	C₂×C₅×D₃	1	1	yes	characteristic
243	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
244	C₂×C₃×C₅	C₂×#137	1	2	yes	—
245	C₅×D₃	#491	1	4	yes	direct product factor
246	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
247	C₃×Q₈	—	2	4	no	—
248	C₂×C₃×C₄	—	2	2	no	—
249	C₂×C₃×C₄	—	2	2	no	—
250	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
251	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
252	C₂×Dic₃	—	2	2	no	—
253	C₂×Dic₃	—	2	2	no	—
254	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
255	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
256	C₄×D₃	—	2	4	no	—
257	C₄×D₃	—	4	4	no	—
258	C₄×D₃	—	4	4	no	—
259	C₄×D₃	C₅×#41	1	4	yes	—
260	Dic₆	—	2	4	no	—
261	Dic₆	—	4	4	no	—
262	C₃×C₈	—	2	2	no	—
263	C₃⋊C₈	—	2	2	no	—
264	C₂²×C₅	—	3	2	no	—
265	C₂²×C₅	—	3	4	no	—
266	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
267	C₄×C₅	—	2	2	no	—
268	C₄×C₅	—	3	2	no	—
269	C₄×C₅	—	4	2	no	—
270	C₄×C₅	—	4	2	no	—
271	C₄×C₅	—	6	2	no	—
272	C₄×C₅	—	12	2	no	—
273	C₄×C₅	—	12	2	no	—
274	C₄×C₅	D₃×#41	1	2	yes	—
275	C₂×Q₈	—	3	2	no	—
276	C₂×Q₈	—	6	2	no	—
277	C₂×Q₈	—	6	8	no	—
278	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
279	C₂²×C₄	—	3	1	no	—
280	C₂²×C₄	—	3	2	no	—
281	C₂²×C₄	—	6	2	no	—
282	C₂²×C₄	—	6	2	no	—
283	C₂×C₈	—	3	2	no	—
284	C₂×C₈	—	6	4	no	—
285	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	3	2	no	—
289	C₄⋊C₄	—	6	2	no	—
290	C₄⋊C₄	—	6	2	no	—
291	C₄⋊C₄	—	6	2	no	—
292	C₄⋊C₄	—	6	4	no	—
293	C₄⋊C₄	—	6	2	no	—
294	C₄⋊C₄	—	6	2	no	—
295	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
296	C₄²	—	3	2	no	—
297	C₄²	—	3	1	no	—
298	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
299	C₃×C₅	C₂×#491	1	1	yes	characteristic, complemented, Hall
300	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
301	C₂×D₃	C₅×#124	1	2	yes	—
302	C₂×D₃	C₅×#137	1	4	yes	—
303	C₃×C₄	—	2	2	no	—
304	C₃×C₄	—	4	2	no	—
305	C₃×C₄	—	4	2	no	—
306	C₃×C₄	C₂×C₅×#41	1	2	yes	—
307	Dic₃	—	2	2	no	—
308	Dic₃	—	4	2	no	—
309	Dic₃	—	4	2	no	—
310	Dic₃	C₅×#124	1	2	yes	—
311	C₂×C₅	—	3	4	no	—
312	C₂×C₅	D₃×#124	1	1	yes	characteristic
313	C₂×C₅	D₃×#137	1	2	yes	—
314	C₂³	—	3	1	no	—
315	Q₈	—	2	4	no	—
316	Q₈	—	6	4	no	—
317	Q₈	—	12	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	3	1	no	—
321	C₂×C₄	—	3	4	no	—
322	C₂×C₄	—	3	2	no	—
323	C₂×C₄	—	6	4	no	—
324	C₂×C₄	—	6	2	no	—
325	C₂×C₄	—	6	2	no	—
326	C₂×C₄	—	12	4	no	—
327	C₂×C₄	—	12	4	no	—
328	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
329	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	6	2	no	—
332	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
333	C₂×C₃	C₂×C₅×#137	1	2	yes	—
334	D₃	C₅×#491	1	4	yes	direct product factor
335	C₅	D₃×#491	1	1	yes	characteristic, direct product factor, Sylow
336	C₂²	—	3	2	no	—
337	C₂²	—	3	4	no	—
338	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	3	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	6	2	no	—
344	C₄	—	12	2	no	—
345	C₄	—	12	2	no	—
346	C₄	C₅×D₃×#41	1	2	yes	—
347	C₃	C₂×C₅×#491	1	1	yes	characteristic, complemented, Sylow
348	C₂	—	3	4	no	—
349	C₂	C₅×D₃×#124	1	1	yes	characteristic
350	C₂	C₅×D₃×#137	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#491	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₅×#491	D₃
hypocenter	C₃	C₂×C₅×#491
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#491

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#491	C₂	semidirect
4. C₅×#5082	C₂	semidirect
5. C₅×#5069	C₂	semidirect
7. C₅×#4797	C₂	semidirect
9. C₅×#4703	C₂	semidirect
32. D₃×#491	C₅	direct (N × H)
39. C₅×#491	D₃	direct (N × H)
68. C₃×#491	C₂×C₅	semidirect
69. #5082	C₂×C₅	semidirect
70. #5069	C₂×C₅	semidirect
72. #4797	C₂×C₅	semidirect
74. #4703	C₂×C₅	semidirect
166. #491	C₅×D₃	direct (N × H)
245. C₅×D₃	#491	direct (N × H)
299. C₃×C₅	C₂×#491	semidirect
334. D₃	C₅×#491	direct (N × H)
335. C₅	D₃×#491	direct (N × H)
347. C₃	C₂×C₅×#491	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₃×C₄⋊Q₈	2
C₅×D₃×#125	2
C₃×C₅×#491	2
C₅×#5082	2
C₅×#5069	2
C₅×D₃×#103	2
C₅×#4797	2
C₅×D₃×#97	2
C₅×#4703	2
C₂×C₅×#491	3
D₃×#491	5

Subgroups — C5×D3×#491 #14437947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×#491 #14437947