Subgroups — C₅×D₃×#500 #14439801

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 320 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	2	yes	maximal
2	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5078	C₂	1	2	yes	maximal, complemented
4	C₅×D₃×#107	C₂	1	4	yes	maximal
5	C₅×#4874	C₂	1	4	yes	maximal, complemented
6	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₅×#4682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#500	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×Q₈	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₅×#944	—	2	2	no	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₅×D₃×C₄⋊C₄	—	2	2	no	—
14	C₅×D₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₅×#888	—	2	2	no	—
16	C₅×#888	C₂²	1	4	yes	—
17	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₅×#869	C₂²	1	2	yes	—
19	C₃×C₅×#107	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#818	C₂²	1	4	yes	—
22	C₅×C₈×Dic₃	C₂²	1	2	yes	—
23	C₅×#807	C₂²	1	8	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	D₃×#500	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂×C₅×C₄⋊Q₈	—	3	2	no	—
27	C₅×#500	—	3	4	no	—
28	C₅×#500	—	3	1	no	—
29	C₅×#500	—	3	2	no	—
30	C₅×#500	D₃	1	1	yes	characteristic, direct product factor
31	C₂×C₅×#107	—	3	4	no	—
32	C₂×C₄×C₅×C₈	—	3	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₅×D₃×Q₈	—	4	8	no	—
35	C₂×C₄×C₅×D₃	—	2	4	no	—
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
39	C₂×C₅×Dic₆	—	2	2	no	—
40	C₂×C₅×Dic₆	—	2	4	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊Dic₃	—	2	2	no	—
46	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
47	C₅×Dic₃⋊C₄	—	2	4	no	—
48	C₅×Dic₃⋊C₄	—	2	4	no	—
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
52	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
53	C₅×D₃×C₈	Q₈	1	8	yes	complemented
54	D₃×C₄⋊Q₈	C₂×C₅	1	2	yes	—
55	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
56	#5078	C₂×C₅	1	2	yes	complemented
57	D₃×#107	C₂×C₅	1	4	yes	—
58	#4874	C₂×C₅	1	4	yes	complemented
59	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
60	#4682	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂²×C₅×Q₈	—	6	2	no	—
62	C₂²×C₅×C₈	—	3	2	no	—
63	C₅×C₄⋊Q₈	—	3	4	no	—
64	C₅×C₄⋊Q₈	—	3	2	no	—
65	C₅×C₄⋊Q₈	—	6	2	no	—
66	C₅×C₄⋊Q₈	—	6	2	no	—
67	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
68	C₂×C₅×C₄⋊C₄	—	3	4	no	—
69	C₂×C₅×C₄⋊C₄	—	6	2	no	—
70	C₂×C₄²×C₅	—	3	1	no	—
71	C₅×#107	—	3	4	no	—
72	C₅×#107	—	3	8	no	—
73	C₅×#107	C₂×D₃	1	4	yes	—
74	C₄×C₅×C₈	—	3	2	no	—
75	C₄×C₅×C₈	—	3	1	no	—
76	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
77	C₂×#500	—	3	1	no	Sylow
78	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
79	C₃×C₅×Q₈	—	4	4	no	—
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	2	4	no	—
85	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
88	C₄×C₅×D₃	—	4	8	no	—
89	C₄×C₅×D₃	D₈	1	4	yes	—
90	C₄×C₅×D₃	Q₁₆	1	4	yes	—
91	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
92	C₅×Dic₆	—	4	4	no	—
93	C₅×Dic₆	—	4	8	no	—
94	C₃×C₅×C₈	C₂×Q₈	1	4	yes	complemented
95	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	complemented
96	C₂×D₃×Q₈	—	2	2	no	—
97	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
98	#944	—	2	2	no	—
99	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
100	D₃×C₄⋊C₄	—	2	2	no	—
101	D₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
102	#888	—	2	2	no	—
103	#888	C₂²×C₅	1	4	yes	—
104	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
105	#869	C₂²×C₅	1	2	yes	—
106	C₃×#107	C₂²×C₅	1	4	yes	—
107	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
108	#818	C₂²×C₅	1	4	yes	—
109	C₈×Dic₃	C₂²×C₅	1	2	yes	—
110	#807	C₂²×C₅	1	8	yes	—
111	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₅×Q₈	—	2	2	no	—
113	C₂×C₅×Q₈	—	6	4	no	—
114	C₂×C₅×Q₈	—	6	2	no	—
115	C₂×C₅×Q₈	—	12	8	no	—
116	C₂²×C₄×C₅	—	3	1	no	—
117	C₂²×C₄×C₅	—	3	1	no	—
118	C₂²×C₄×C₅	—	3	1	no	—
119	C₂²×C₄×C₅	—	6	4	no	—
120	C₂×C₅×C₈	—	3	2	no	—
121	C₂×C₅×C₈	—	3	8	no	—
122	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	3	4	no	—
125	C₅×C₄⋊C₄	—	6	4	no	—
126	C₅×C₄⋊C₄	—	6	4	no	—
127	C₅×C₄⋊C₄	—	6	2	no	—
128	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
129	C₄²×C₅	—	3	1	no	—
130	C₄²×C₅	—	3	1	no	—
131	C₄²×C₅	—	3	1	no	—
132	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	—	3	2	no	—
134	#500	—	3	2	no	—
135	#500	—	3	4	no	—
136	#500	—	3	1	no	—
137	#500	C₅×D₃	1	1	yes	characteristic, direct product factor
138	C₂×#107	—	3	4	no	—
139	C₂×C₄×C₈	—	3	1	no	—
140	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₅×D₃	C₄⋊Q₈	1	2	yes	—
142	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
143	C₂×C₅×D₃	C₂×Q₁₆	1	2	yes	—
144	C₃×C₄×C₅	—	4	4	no	—
145	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
146	C₃×C₄×C₅	C₂×Q₁₆	1	2	yes	—
147	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
148	C₅×Dic₃	—	4	4	no	—
149	C₅×Dic₃	C₄⋊Q₈	1	2	yes	—
150	C₅×Dic₃	C₂×D₈	1	2	yes	—
151	C₅×Dic₃	C₂×Q₁₆	1	2	yes	—
152	C₂×C₃×Q₈	—	2	2	no	—
153	D₃×Q₈	—	4	8	no	—
154	C₂×C₄×D₃	—	2	4	no	—
155	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
156	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
157	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
158	C₂×Dic₆	—	2	2	no	—
159	C₂×Dic₆	—	2	4	no	—
160	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
161	C₃×C₄⋊C₄	—	2	2	no	—
162	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
163	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
164	C₄⋊Dic₃	—	2	2	no	—
165	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
166	Dic₃⋊C₄	—	2	4	no	—
167	Dic₃⋊C₄	—	2	4	no	—
168	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
169	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
170	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
171	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
172	D₃×C₈	C₅×Q₈	1	8	yes	complemented
173	C₂³×C₅	—	3	1	no	—
174	C₅×Q₈	—	4	4	no	—
175	C₅×Q₈	—	12	8	no	—
176	C₅×Q₈	—	12	4	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	3	1	no	—
179	C₂×C₄×C₅	—	3	1	no	—
180	C₂×C₄×C₅	—	3	1	no	—
181	C₂×C₄×C₅	—	3	4	no	—
182	C₂×C₄×C₅	—	3	4	no	—
183	C₂×C₄×C₅	—	3	4	no	—
184	C₂×C₄×C₅	—	6	4	no	—
185	C₂×C₄×C₅	—	12	8	no	—
186	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
189	C₅×C₈	—	3	4	no	—
190	C₅×C₈	D₃×Q₈	1	4	yes	complemented
191	C₂²×Q₈	—	6	2	no	—
192	C₂²×C₈	—	3	2	no	—
193	C₄⋊Q₈	—	3	4	no	—
194	C₄⋊Q₈	—	3	2	no	—
195	C₄⋊Q₈	—	6	2	no	—
196	C₄⋊Q₈	—	6	2	no	—
197	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
198	C₂×C₄⋊C₄	—	3	4	no	—
199	C₂×C₄⋊C₄	—	6	2	no	—
200	C₂×C₄²	—	3	1	no	—
201	#107	—	3	4	no	—
202	#107	—	3	8	no	—
203	#107	C₂×C₅×D₃	1	4	yes	—
204	C₄×C₈	—	3	2	no	—
205	C₄×C₈	—	3	1	no	—
206	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
209	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
210	C₅×D₃	#500	1	4	yes	direct product factor
211	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
212	C₃×Q₈	—	4	4	no	—
213	C₂×C₃×C₄	—	2	4	no	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
217	C₂×Dic₃	—	2	4	no	—
218	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
221	C₄×D₃	—	4	8	no	—
222	C₄×D₃	C₅×D₈	1	4	yes	—
223	C₄×D₃	C₅×Q₁₆	1	4	yes	—
224	C₄×D₃	C₂×C₅×Q₈	1	4	yes	—
225	Dic₆	—	4	4	no	—
226	Dic₆	—	4	8	no	—
227	C₃×C₈	C₂×C₅×Q₈	1	4	yes	complemented
228	C₃⋊C₈	C₂×C₅×Q₈	1	4	yes	complemented
229	C₂²×C₅	—	3	2	no	—
230	C₂²×C₅	—	3	2	no	—
231	C₂²×C₅	—	3	2	no	—
232	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
233	C₄×C₅	—	3	2	no	—
234	C₄×C₅	—	3	2	no	—
235	C₄×C₅	—	3	2	no	—
236	C₄×C₅	—	4	4	no	—
237	C₄×C₅	—	12	4	no	—
238	C₄×C₅	D₃×D₈	1	2	yes	—
239	C₄×C₅	D₃×Q₁₆	1	2	yes	—
240	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	6	4	no	—
243	C₂×Q₈	—	6	2	no	—
244	C₂×Q₈	—	12	8	no	—
245	C₂²×C₄	—	3	1	no	—
246	C₂²×C₄	—	3	1	no	—
247	C₂²×C₄	—	3	1	no	—
248	C₂²×C₄	—	6	4	no	—
249	C₂×C₈	—	3	2	no	—
250	C₂×C₈	—	3	8	no	—
251	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	3	4	no	—
254	C₄⋊C₄	—	6	4	no	—
255	C₄⋊C₄	—	6	4	no	—
256	C₄⋊C₄	—	6	2	no	—
257	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
258	C₄²	—	3	1	no	—
259	C₄²	—	3	1	no	—
260	C₄²	—	3	1	no	—
261	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
262	C₃×C₅	C₂×#500	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
264	C₂×D₃	C₅×C₄⋊Q₈	1	2	yes	—
265	C₂×D₃	C₂×C₅×D₈	1	2	yes	—
266	C₂×D₃	C₂×C₅×Q₁₆	1	2	yes	—
267	C₃×C₄	—	4	4	no	—
268	C₃×C₄	C₂×C₅×D₈	1	2	yes	—
269	C₃×C₄	C₂×C₅×Q₁₆	1	2	yes	—
270	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
271	Dic₃	—	4	4	no	—
272	Dic₃	C₅×C₄⋊Q₈	1	2	yes	—
273	Dic₃	C₂×C₅×D₈	1	2	yes	—
274	Dic₃	C₂×C₅×Q₁₆	1	2	yes	—
275	C₂×C₅	—	3	4	no	—
276	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
277	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
278	C₂×C₅	C₂×D₃×Q₁₆	1	1	yes	characteristic
279	C₂³	—	3	1	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	12	8	no	—
282	Q₈	—	12	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	3	1	no	—
285	C₂×C₄	—	3	1	no	—
286	C₂×C₄	—	3	1	no	—
287	C₂×C₄	—	3	4	no	—
288	C₂×C₄	—	3	4	no	—
289	C₂×C₄	—	3	4	no	—
290	C₂×C₄	—	6	4	no	—
291	C₂×C₄	—	12	8	no	—
292	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
293	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
295	C₈	—	3	4	no	—
296	C₈	C₅×D₃×Q₈	1	4	yes	complemented
297	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
298	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
299	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
300	D₃	C₅×#500	1	4	yes	direct product factor
301	C₅	D₃×#500	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	3	2	no	—
303	C₂²	—	3	2	no	—
304	C₂²	—	3	2	no	—
305	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
306	C₄	—	3	2	no	—
307	C₄	—	3	2	no	—
308	C₄	—	3	2	no	—
309	C₄	—	4	4	no	—
310	C₄	—	12	4	no	—
311	C₄	C₅×D₃×D₈	1	2	yes	—
312	C₄	C₅×D₃×Q₁₆	1	2	yes	—
313	C₄	C₂×C₅×D₃×Q₈	1	2	yes	—
314	C₃	C₂×C₅×#500	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	3	4	no	—
316	C₂	C₅×D₃×C₄⋊Q₈	1	1	yes	characteristic
317	C₂	C₂×C₅×D₃×D₈	1	1	yes	characteristic
318	C₂	C₂×C₅×D₃×Q₁₆	1	1	yes	characteristic

	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×#500	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₅×#500	D₃
hypocenter	C₃	C₂×C₅×#500
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#500

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#500	C₂	semidirect
3. C₅×#5078	C₂	semidirect
5. C₅×#4874	C₂	semidirect
7. C₅×#4682	C₂	semidirect
25. D₃×#500	C₅	direct (N × H)
30. C₅×#500	D₃	direct (N × H)
53. C₅×D₃×C₈	Q₈	semidirect
55. C₃×#500	C₂×C₅	semidirect
56. #5078	C₂×C₅	semidirect
58. #4874	C₂×C₅	semidirect
60. #4682	C₂×C₅	semidirect
94. C₃×C₅×C₈	C₂×Q₈	semidirect
95. C₅×C₃⋊C₈	C₂×Q₈	semidirect
137. #500	C₅×D₃	direct (N × H)
172. D₃×C₈	C₅×Q₈	semidirect
190. C₅×C₈	D₃×Q₈	semidirect
210. C₅×D₃	#500	direct (N × H)
227. C₃×C₈	C₂×C₅×Q₈	semidirect
228. C₃⋊C₈	C₂×C₅×Q₈	semidirect
262. C₃×C₅	C₂×#500	semidirect
296. C₈	C₅×D₃×Q₈	semidirect
300. D₃	C₅×#500	direct (N × H)
301. C₅	D₃×#500	direct (N × H)
314. C₃	C₂×C₅×#500	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
C₃×C₅×#500	2
C₅×#5078	2
C₅×D₃×#107	2
C₅×#4874	2
C₄×C₅×D₃×C₈	2
C₅×#4682	2
C₂×C₅×#500	3
D₃×#500	5

Subgroups — C5×D3×#500 #14439801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×#500 #14439801