Subgroups — C₅×#70840 #14286433

Order 1920.

384 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×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#213	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1580	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	—	2	1	no	—
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×#213	C₄	1	1	yes	characteristic
12	C₂×C₅×D₃×C₈	—	2	1	no	—
13	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#832	C₄	1	1	yes	characteristic
18	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
19	C₅×#809	C₄	1	1	yes	characteristic
20	C₅×#807	C₄	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	#70840	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₄×C₅×D₈	—	3	1	no	—
26	C₅×C₈×D₄	—	3	1	no	—
27	C₅×C₈×D₄	D₃	1	1	yes	characteristic, complemented
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	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	4	2	no	—
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	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×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#213	C₈	1	4	yes	complemented
44	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₅×D₃×C₈	—	4	2	no	—
51	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic, complemented
52	C₄×#213	C₂×C₅	1	1	yes	characteristic
53	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
54	D₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
55	#4483	C₂×C₅	1	1	yes	characteristic, complemented
56	#4480	C₂×C₅	1	1	yes	characteristic, complemented
57	#4454	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×D₈	—	3	1	no	—
59	C₂²×C₅×C₈	—	2	1	no	—
60	C₂²×C₅×C₈	—	6	1	no	—
61	C₄×C₅×D₄	—	3	1	no	—
62	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
63	C₅×#107	—	3	1	no	—
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×#102	—	3	1	no	—
67	C₅×#102	—	3	1	no	—
68	C₅×C₂²⋊C₈	—	2	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	#1580	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂²×C₅×D₃	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	2	yes	complemented
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	2	1	no	—
85	C₅×D₁₂	—	2	1	no	—
86	C₅×D₁₂	C₂×C₈	1	2	yes	complemented
87	C₄×C₅×D₃	—	4	1	no	—
88	C₄×C₅×D₃	—	4	1	no	—
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃×C₅×C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	—	2	1	no	—
93	C₅×C₃⋊C₈	—	4	1	no	—
94	C₅×C₃⋊C₈	—	4	1	no	—
95	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	complemented
96	C₂²×C₃×C₈	—	2	1	no	—
97	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
98	C₂×#213	C₄×C₅	1	1	yes	characteristic
99	C₂×D₃×C₈	—	2	1	no	—
100	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₂²⋊C₈	—	2	1	no	—
103	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	#832	C₄×C₅	1	1	yes	characteristic
105	(C₂×D₃)⋊C₈	—	2	1	no	—
106	#809	C₄×C₅	1	1	yes	characteristic
107	#807	C₄×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	—	3	1	no	—
112	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₄×C₅	—	2	1	no	—
114	C₂²×C₄×C₅	—	6	1	no	—
115	C₅×D₈	—	3	4	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	3	1	no	—
119	C₂×C₅×C₈	—	4	2	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	#67	1	1	yes	characteristic
124	C₂×C₅×C₈	#67	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	3	1	no	—
126	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
127	C₅×C₂²⋊C₄	—	2	1	no	—
128	C₅×C₂²⋊C₄	—	6	1	no	—
129	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
130	C₄×D₈	—	3	1	no	—
131	C₈×D₄	—	3	1	no	—
132	C₈×D₄	C₅×D₃	1	1	yes	characteristic, complemented
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	C₄×D₄	1	1	yes	characteristic
141	C₂×C₅×D₃	—	2	2	no	—
142	C₂×C₅×D₃	—	4	1	no	—
143	C₂×C₅×D₃	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	4	1	no	—
146	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
147	C₃×C₄×C₅	#131	1	1	yes	characteristic
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₅×Dic₃	—	4	1	no	—
151	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂×C₄×D₃	—	2	1	no	—
155	C₂×C₃×C₈	—	2	1	no	—
156	C₂×C₃×C₈	—	4	2	no	—
157	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
162	#213	C₅×C₈	1	4	yes	complemented
163	(C₂×D₃)⋊C₄	—	2	1	no	—
164	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×C₃⋊C₈	—	2	1	no	—
166	C₂×C₃⋊C₈	—	2	1	no	—
167	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
168	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
169	D₃×C₈	—	4	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	—	6	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	—	3	2	no	—
174	C₅×D₄	—	6	1	no	—
175	C₅×D₄	D₃×C₈	1	2	yes	complemented
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	—	4	1	no	—
179	C₂×C₄×C₅	—	6	1	no	—
180	C₂×C₄×C₅	—	12	1	no	—
181	C₂×C₄×C₅	—	12	1	no	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	#235	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	3	2	no	—
188	C₅×C₈	—	4	1	no	—
189	C₅×C₈	—	6	1	no	—
190	C₅×C₈	—	12	1	no	—
191	C₅×C₈	—	12	1	no	—
192	C₂×D₈	—	3	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	6	1	no	—
195	C₄×D₄	—	3	1	no	—
196	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
197	#107	—	3	1	no	—
198	C₄⋊C₈	—	3	1	no	—
199	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
200	#102	—	3	1	no	—
201	#102	—	3	1	no	—
202	C₂²⋊C₈	—	2	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	C₈×D₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#442	1	1	yes	characteristic
210	C₂×C₃×C₅	#444	1	1	yes	characteristic
211	C₅×D₃	—	4	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂²×D₃	—	2	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	C₂×C₅×C₈	1	2	yes	complemented
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
222	C₂×Dic₃	—	2	1	no	—
223	D₁₂	—	2	1	no	—
224	D₁₂	C₂×C₅×C₈	1	2	yes	complemented
225	C₄×D₃	—	4	1	no	—
226	C₄×D₃	—	4	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	4	1	no	—
230	C₃⋊C₈	—	2	1	no	—
231	C₃⋊C₈	—	4	1	no	—
232	C₃⋊C₈	—	4	1	no	—
233	C₃⋊C₈	C₂×C₅×C₈	1	2	yes	complemented
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	—	6	2	no	—
238	C₂²×C₅	—	12	1	no	—
239	C₂²×C₅	—	12	1	no	—
240	C₂²×C₅	C₄×#67	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	—	12	1	no	—
244	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
245	C₄×C₅	#901	1	1	yes	characteristic
246	C₄×C₅	#932	1	1	yes	characteristic
247	C₄×C₅	#949	1	1	yes	characteristic
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	6	1	no	—
252	D₈	—	3	4	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	3	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	—	6	1	no	—
259	C₂×C₈	—	12	2	no	—
260	C₂×C₈	C₅×#67	1	1	yes	characteristic
261	C₂×C₈	C₅×#67	1	1	yes	characteristic
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	6	1	no	—
266	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₃×C₅	#1580	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	C₄×C₅×D₄	1	1	yes	characteristic
272	C₂×D₃	—	2	2	no	—
273	C₂×D₃	—	4	1	no	—
274	C₂×D₃	—	4	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
278	C₃×C₄	C₅×#131	1	1	yes	characteristic
279	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
281	Dic₃	—	4	1	no	—
282	C₂×C₅	—	4	2	no	—
283	C₂×C₅	—	12	2	no	—
284	C₂×C₅	#5014	1	1	yes	characteristic
285	C₂×C₅	C₈×#67	1	1	yes	characteristic
286	C₂×C₅	#5145	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	6	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	3	2	no	—
291	D₄	—	6	1	no	—
292	D₄	C₅×D₃×C₈	1	2	yes	complemented
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
300	C₂×C₄	C₅×#235	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	3	2	no	—
305	C₈	—	4	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	C₅×C₈×D₄	1	1	yes	characteristic
311	C₂×C₃	C₅×#442	1	1	yes	characteristic
312	C₂×C₃	C₅×#444	1	1	yes	characteristic
313	D₃	—	4	2	no	—
314	C₅	#70840	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	—	6	2	no	—
319	C₂²	—	12	1	no	—
320	C₂²	—	12	1	no	—
321	C₂²	C₄×C₅×#67	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	12	1	no	—
325	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
326	C₄	C₅×#901	1	1	yes	characteristic
327	C₄	C₅×#932	1	1	yes	characteristic
328	C₄	C₅×#949	1	1	yes	characteristic
329	C₃	C₅×#1580	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	4	2	no	—
331	C₂	—	12	2	no	—
332	C₂	C₅×#5014	1	1	yes	characteristic
333	C₂	C₅×C₈×#67	1	1	yes	characteristic
334	C₂	C₅×#5145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×#213	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×C₈×D₄	D₃
hypocenter	C₃	C₅×#1580
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₈×D₄	C₂	semidirect
3. C₅×C₈×D₁₂	C₂	semidirect
4. C₅×D₄⋊(C₃⋊C₈)	C₂	semidirect
5. C₅×#4483	C₂	semidirect
6. C₅×#4480	C₂	semidirect
7. C₅×#4454	C₂	semidirect
24. #70840	C₅	direct (N × H)
27. C₅×C₈×D₄	D₃	semidirect
43. C₅×#213	C₈	semidirect
51. C₃×C₈×D₄	C₂×C₅	semidirect
53. C₈×D₁₂	C₂×C₅	semidirect
54. D₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
55. #4483	C₂×C₅	semidirect
56. #4480	C₂×C₅	semidirect
57. #4454	C₂×C₅	semidirect
77. C₃×C₅×D₄	C₂×C₈	semidirect
86. C₅×D₁₂	C₂×C₈	semidirect
95. C₅×C₃⋊C₈	C₂×C₈	semidirect
132. C₈×D₄	C₅×D₃	semidirect
162. #213	C₅×C₈	semidirect
175. C₅×D₄	D₃×C₈	semidirect
215. C₃×D₄	C₂×C₅×C₈	semidirect
224. D₁₂	C₂×C₅×C₈	semidirect
233. C₃⋊C₈	C₂×C₅×C₈	semidirect
267. C₃×C₅	#1580	semidirect
292. D₄	C₅×D₃×C₈	semidirect
314. C₅	#70840	direct (N × H)
329. C₃	C₅×#1580	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×D₄	2
C₄×C₅×#213	2
C₅×C₈×D₁₂	2
C₅×D₄⋊(C₃⋊C₈)	2
C₅×#4483	2
C₅×#4480	2
C₅×#4454	2
C₅×#1580	3
#70840	5

Subgroups — C5×#70840 #14286433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70840 #14286433