Subgroups — C₅×#70880 #14286976

Order 1920.

364 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	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#5012	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4457	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1586	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×#208	—	2	1	no	—
13	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
14	C₃×C₅×#107	C₄	1	1	yes	characteristic
15	C₃×C₅×#102	C₄	1	1	yes	characteristic
16	C₃×C₅×#102	C₄	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#830	—	2	1	no	—
19	C₅×#830	—	2	1	no	—
20	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#70880	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₄×C₅×D₈	D₃	1	1	yes	characteristic
26	C₅×#436	—	3	1	no	—
27	C₅×C₈×D₄	—	3	1	no	—
28	C₅×#334	—	3	1	no	—
29	C₅×D₄⋊C₈	—	3	1	no	—
30	C₅×D₄⋊C₈	—	3	1	no	—
31	C₅×#322	—	3	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₃×C₅×D₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#208	—	4	2	no	—
45	C₂×C₅×C₃⋊C₈	—	2	1	no	—
46	C₂×C₅×C₃⋊C₈	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	4	2	no	—
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
52	#5012	C₂×C₅	1	1	yes	characteristic, complemented
53	D₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	D₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
55	D₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
56	#4457	C₂×C₅	1	1	yes	characteristic, complemented
57	#4454	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×D₈	Dic₃	1	1	yes	characteristic
59	C₂×C₅×M₁₆	—	6	1	no	—
60	C₂²×C₅×C₈	—	6	1	no	—
61	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
63	C₅×#107	Dic₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	—	3	1	no	—
66	C₅×#102	Dic₃	1	1	yes	characteristic
67	C₅×#102	Dic₃	1	1	yes	characteristic
68	C₅×C₂²⋊C₈	—	6	1	no	—
69	C₅×C₂²⋊C₈	—	6	1	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	C₄×C₅×C₈	—	3	1	no	—
72	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
73	#1586	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂³×C₃×C₅	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₃×C₅×D₄	—	2	1	no	—
79	C₃×C₅×D₄	M₁₆	1	2	yes	complemented
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	—	4	1	no	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₅×C₃⋊C₈	—	2	2	no	—
92	C₅×C₃⋊C₈	—	2	1	no	—
93	C₅×C₃⋊C₈	—	2	1	no	—
94	C₅×C₃⋊C₈	—	4	1	no	—
95	C₅×C₃⋊C₈	—	4	1	no	—
96	C₂×C₃×D₈	C₄×C₅	1	1	yes	characteristic
97	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
99	C₂×#208	—	2	1	no	—
100	C₂²×C₃⋊C₈	—	2	1	no	—
101	C₃×#107	C₄×C₅	1	1	yes	characteristic
102	C₃×#102	C₄×C₅	1	1	yes	characteristic
103	C₃×#102	C₄×C₅	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	#830	—	2	1	no	—
106	#830	—	2	1	no	—
107	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
108	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
109	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
110	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
113	C₂²×C₄×C₅	—	2	1	no	—
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₅×D₈	—	2	2	no	—
116	C₅×M₁₆	—	12	2	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	3	1	no	—
119	C₂×C₅×C₈	—	6	1	no	—
120	C₂×C₅×C₈	—	6	1	no	—
121	C₂×C₅×C₈	—	6	1	no	—
122	C₂×C₅×C₈	—	12	2	no	—
123	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
125	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
126	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
127	C₅×C₂²⋊C₄	—	2	1	no	—
128	C₅×C₂²⋊C₄	—	2	1	no	—
129	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
130	C₄×D₈	C₅×D₃	1	1	yes	characteristic
131	#436	—	3	1	no	—
132	C₈×D₄	—	3	1	no	—
133	#334	—	3	1	no	—
134	D₄⋊C₈	—	3	1	no	—
135	D₄⋊C₈	—	3	1	no	—
136	#322	—	3	1	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	—	4	1	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	1	no	—
141	C₂²×C₃×C₅	—	4	1	no	—
142	C₂²×C₃×C₅	—	4	1	no	—
143	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	4	1	no	—
146	C₃×C₄×C₅	—	4	1	no	—
147	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
148	C₃×C₄×C₅	#131	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₃×D₈	—	2	2	no	—
156	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₂²⋊C₄	—	2	1	no	—
162	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
163	#208	—	4	2	no	—
164	C₂×C₃⋊C₈	—	2	1	no	—
165	C₂×C₃⋊C₈	—	2	1	no	—
166	C₂×C₃⋊C₈	—	2	1	no	—
167	C₂×C₃⋊C₈	—	4	2	no	—
168	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
169	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	—	2	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	—	2	1	no	—
174	C₅×D₄	—	2	1	no	—
175	C₅×D₄	#208	1	2	yes	complemented
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	4	1	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	—	4	1	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	#237	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	6	2	no	—
188	C₅×C₈	—	6	1	no	—
189	C₅×C₈	—	6	1	no	—
190	C₅×C₈	—	12	1	no	—
191	C₅×C₈	—	12	1	no	—
192	C₂×D₈	C₅×Dic₃	1	1	yes	characteristic
193	C₂×M₁₆	—	6	1	no	—
194	C₂²×C₈	—	6	1	no	—
195	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
196	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
197	#107	C₅×Dic₃	1	1	yes	characteristic
198	C₄⋊C₈	—	3	1	no	—
199	C₄⋊C₈	—	3	1	no	—
200	#102	C₅×Dic₃	1	1	yes	characteristic
201	#102	C₅×Dic₃	1	1	yes	characteristic
202	C₂²⋊C₈	—	6	1	no	—
203	C₂²⋊C₈	—	6	1	no	—
204	C₄×C₈	—	3	1	no	—
205	C₄×C₈	—	3	1	no	—
206	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
207	C₂×C₃×C₅	—	4	2	no	—
208	C₂×C₃×C₅	—	8	1	no	—
209	C₂×C₃×C₅	#435	1	1	yes	characteristic
210	C₂×C₃×C₅	#442	1	1	yes	characteristic
211	C₂×C₃×C₅	#443	1	1	yes	characteristic
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	—	2	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₃×D₄	—	2	1	no	—
217	C₃×D₄	C₅×M₁₆	1	2	yes	complemented
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	4	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃⋊C₈	—	2	2	no	—
230	C₃⋊C₈	—	2	1	no	—
231	C₃⋊C₈	—	2	1	no	—
232	C₃⋊C₈	—	4	1	no	—
233	C₃⋊C₈	—	4	1	no	—
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	—	4	1	no	—
238	C₂²×C₅	—	4	1	no	—
239	C₂²×C₅	—	4	1	no	—
240	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	—	4	1	no	—
244	C₄×C₅	#911	1	1	yes	characteristic
245	C₄×C₅	#911	1	1	yes	characteristic
246	C₄×C₅	C₂×#208	1	1	yes	characteristic
247	C₄×C₅	#948	1	1	yes	characteristic
248	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
249	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	D₈	—	2	2	no	—
253	M₁₆	—	12	2	no	—
254	C₂×C₈	—	3	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	—	6	1	no	—
259	C₂×C₈	—	12	2	no	—
260	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
262	C₄⋊C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₃×C₅	#1586	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	1	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
279	C₃×C₄	C₅×#131	1	1	yes	characteristic
280	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂×C₅	—	4	2	no	—
283	C₂×C₅	—	8	1	no	—
284	C₂×C₅	#5011	1	1	yes	characteristic
285	C₂×C₅	#5152	1	1	yes	characteristic
286	C₂×C₅	#5196	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	C₅×#208	1	2	yes	complemented
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
300	C₂×C₄	C₅×#237	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	6	2	no	—
305	C₈	—	6	1	no	—
306	C₈	—	6	1	no	—
307	C₈	—	12	1	no	—
308	C₈	—	12	1	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	—	8	1	no	—
311	C₂×C₃	C₅×#435	1	1	yes	characteristic
312	C₂×C₃	C₅×#442	1	1	yes	characteristic
313	C₂×C₃	C₅×#443	1	1	yes	characteristic
314	C₅	#70880	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₅×D₄×Dic₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	C₅×#911	1	1	yes	characteristic
326	C₄	C₅×#911	1	1	yes	characteristic
327	C₄	C₂×C₅×#208	1	1	yes	characteristic
328	C₄	C₅×#948	1	1	yes	characteristic
329	C₃	C₅×#1586	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	C₅×#5011	1	1	yes	characteristic
333	C₂	C₅×#5152	1	1	yes	characteristic
334	C₂	C₅×#5196	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×D₈	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₄×C₅×D₈	D₃
hypocenter	C₃	C₅×#1586
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×D₈

Complements

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

N	H ≅ G/N	type
2. C₅×#5012	C₂	semidirect
3. C₅×D₄×C₃⋊C₈	C₂	semidirect
4. C₅×D₄⋊(C₃⋊C₈)	C₂	semidirect
5. C₅×D₄⋊(C₃⋊C₈)	C₂	semidirect
6. C₅×#4457	C₂	semidirect
7. C₅×#4454	C₂	semidirect
24. #70880	C₅	direct (N × H)
52. #5012	C₂×C₅	semidirect
53. D₄×C₃⋊C₈	C₂×C₅	semidirect
54. D₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
55. D₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
56. #4457	C₂×C₅	semidirect
57. #4454	C₂×C₅	semidirect
79. C₃×C₅×D₄	M₁₆	semidirect
175. C₅×D₄	#208	semidirect
217. C₃×D₄	C₅×M₁₆	semidirect
267. C₃×C₅	#1586	semidirect
292. D₄	C₅×#208	semidirect
314. C₅	#70880	direct (N × H)
329. C₃	C₅×#1586	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₈	2
C₅×#5012	2
C₅×D₄×C₃⋊C₈	2
C₅×D₄⋊(C₃⋊C₈)	2
C₅×#4457	2
C₅×#4454	2
C₅×#1586	3
#70880	5

Subgroups — C5×#70880 #14286976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70880 #14286976