Subgroups — C₅×#70866 #14287311

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#214	C₂	1	1	yes	maximal, characteristic
3	C₅×#4679	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₅×#4484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1590	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×M₁₆	—	2	1	no	—
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×#214	C₄	1	1	yes	characteristic
12	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#832	C₄	1	1	yes	characteristic
17	C₅×#815	—	2	1	no	—
18	C₅×Dic₃⋊C₈	—	2	1	no	—
19	C₅×#810	C₄	1	1	yes	characteristic
20	C₅×#808	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	#70866	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#446	—	3	1	no	—
26	C₄×C₅×SD₁₆	—	3	1	no	—
27	C₅×#436	D₃	1	1	yes	characteristic
28	C₅×#334	—	3	1	no	—
29	C₅×Q₈⋊C₈	—	3	1	no	—
30	C₅×D₄⋊C₈	—	3	1	no	—
31	C₅×C₈²	—	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×Dic₆	C₂×C₄	1	1	yes	characteristic
35	C₃×C₅×M₁₆	—	4	2	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#214	—	2	2	no	—
43	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₅×Dic₃⋊C₄	—	2	1	no	—
45	C₄×C₅×Dic₃	—	2	1	no	—
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	C₃×#436	C₂×C₅	1	1	yes	characteristic
51	C₄×#214	C₂×C₅	1	1	yes	characteristic
52	#4679	C₂×C₅	1	1	yes	characteristic, complemented
53	D₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
54	#4484	C₂×C₅	1	1	yes	characteristic, complemented
55	#4480	C₂×C₅	1	1	yes	characteristic, complemented
56	C₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×SD₁₆	—	3	1	no	—
58	C₂×C₅×M₁₆	—	2	1	no	—
59	C₄×C₅×Q₈	—	3	1	no	—
60	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
61	C₅×#106	—	3	1	no	—
62	C₅×C₄⋊C₈	—	3	1	no	—
63	C₅×C₄⋊C₈	—	6	1	no	—
64	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₅×#103	—	3	1	no	—
66	C₅×#102	—	3	1	no	—
67	C₅×C₂²⋊C₈	—	2	1	no	—
68	C₅×#97	—	6	1	no	—
69	C₄×C₅×C₈	—	3	1	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#1590	—	3	1	no	Sylow
73	C₂³×C₃×C₅	—	2	1	no	—
74	C₃×C₅×D₄	—	2	1	no	—
75	C₃×C₅×D₄	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
82	C₂×C₅×Dic₃	—	2	1	no	—
83	C₂×C₅×Dic₃	—	2	1	no	—
84	C₅×Dic₆	—	2	1	no	—
85	C₅×Dic₆	—	2	1	no	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
89	C₅×C₃⋊C₈	—	2	2	no	—
90	C₅×C₃⋊C₈	—	2	1	no	—
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	M₁₆	1	2	yes	complemented
93	C₂×C₃×M₁₆	—	2	1	no	—
94	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
95	C₂×#214	C₄×C₅	1	1	yes	characteristic
96	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₂²⋊C₈	—	2	1	no	—
99	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#832	C₄×C₅	1	1	yes	characteristic
101	#815	—	2	1	no	—
102	Dic₃⋊C₈	—	2	1	no	—
103	#810	C₄×C₅	1	1	yes	characteristic
104	#808	C₄×C₅	1	1	yes	characteristic
105	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
106	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
107	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₅×Q₈	—	3	1	no	—
109	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	—	2	1	no	—
111	C₅×SD₁₆	—	6	2	no	—
112	C₅×M₁₆	—	4	2	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	6	1	no	—
117	C₂×C₅×C₈	—	6	1	no	—
118	C₂×C₅×C₈	#67	1	1	yes	characteristic
119	C₂×C₅×C₈	#67	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	3	1	no	—
121	C₅×C₄⋊C₄	—	6	1	no	—
122	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
123	C₅×C₂²⋊C₄	—	2	1	no	—
124	C₄²×C₅	—	6	1	no	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	#446	—	3	1	no	—
127	C₄×SD₁₆	—	3	1	no	—
128	#436	C₅×D₃	1	1	yes	characteristic
129	#334	—	3	1	no	—
130	Q₈⋊C₈	—	3	1	no	—
131	D₄⋊C₈	—	3	1	no	—
132	C₈²	—	3	1	no	—
133	C₂²×C₃×C₅	—	4	1	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
140	C₃×C₄×C₅	#131	1	1	yes	characteristic
141	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	#135	1	1	yes	characteristic
143	C₅×Dic₃	—	4	1	no	—
144	C₅×Dic₃	—	4	1	no	—
145	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂²×C₃×C₄	—	2	1	no	—
147	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
148	C₃×M₁₆	—	4	2	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
151	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
152	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
153	C₃×C₂²⋊C₄	—	2	1	no	—
154	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
155	#214	—	2	2	no	—
156	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
157	Dic₃⋊C₄	—	2	1	no	—
158	C₄×Dic₃	—	2	1	no	—
159	C₂×C₃⋊C₈	—	2	1	no	—
160	C₂×C₃⋊C₈	—	2	1	no	—
161	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂³×C₅	—	2	1	no	—
164	C₅×Q₈	—	6	1	no	—
165	C₅×Q₈	—	6	1	no	—
166	C₅×D₄	—	2	1	no	—
167	C₅×D₄	—	2	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	6	1	no	—
172	C₂×C₄×C₅	—	6	1	no	—
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄×C₅	#235	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	3	2	no	—
178	C₅×C₈	—	4	1	no	—
179	C₅×C₈	—	6	2	no	—
180	C₅×C₈	—	6	1	no	—
181	C₅×C₈	—	12	1	no	—
182	C₅×C₈	#214	1	2	yes	complemented
183	C₂×SD₁₆	—	3	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₄×Q₈	—	3	1	no	—
186	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
187	#106	—	3	1	no	—
188	C₄⋊C₈	—	3	1	no	—
189	C₄⋊C₈	—	6	1	no	—
190	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
191	#103	—	3	1	no	—
192	#102	—	3	1	no	—
193	C₂²⋊C₈	—	2	1	no	—
194	#97	—	6	1	no	—
195	C₄×C₈	—	3	1	no	—
196	C₄×C₈	—	3	1	no	—
197	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
198	C₂×C₃×C₅	—	8	1	no	—
199	C₂×C₃×C₅	#436	1	1	yes	characteristic
200	C₂×C₃×C₅	C₄×SD₁₆	1	1	yes	characteristic
201	C₂×C₃×C₅	#443	1	1	yes	characteristic
202	C₂³×C₃	—	2	1	no	—
203	C₃×D₄	—	2	1	no	—
204	C₃×D₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
211	C₂×Dic₃	—	2	1	no	—
212	C₂×Dic₃	—	2	1	no	—
213	Dic₆	—	2	1	no	—
214	Dic₆	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
218	C₃⋊C₈	—	2	2	no	—
219	C₃⋊C₈	—	2	1	no	—
220	C₃⋊C₈	—	4	1	no	—
221	C₃⋊C₈	C₅×M₁₆	1	2	yes	complemented
222	C₂²×C₅	—	4	1	no	—
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	C₄×#67	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	1	no	—
228	C₄×C₅	—	12	1	no	—
229	C₄×C₅	—	12	1	no	—
230	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
231	C₄×C₅	#907	1	1	yes	characteristic
232	C₄×C₅	C₂×#214	1	1	yes	characteristic
233	C₄×C₅	#950	1	1	yes	characteristic
234	C₂×Q₈	—	3	1	no	—
235	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	SD₁₆	—	6	2	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	—	6	1	no	—
243	C₂×C₈	—	6	1	no	—
244	C₂×C₈	C₅×#67	1	1	yes	characteristic
245	C₂×C₈	C₅×#67	1	1	yes	characteristic
246	C₄⋊C₄	—	3	1	no	—
247	C₄⋊C₄	—	6	1	no	—
248	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₄²	—	6	1	no	—
251	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₃×C₅	#1590	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	4	1	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
260	C₃×C₄	C₅×#131	1	1	yes	characteristic
261	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
262	C₃×C₄	C₅×#135	1	1	yes	characteristic
263	Dic₃	—	4	1	no	—
264	Dic₃	—	4	1	no	—
265	C₂×C₅	—	8	1	no	—
266	C₂×C₅	C₄×#214	1	1	yes	characteristic
267	C₂×C₅	#5128	1	1	yes	characteristic
268	C₂×C₅	#5144	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	Q₈	—	6	1	no	—
271	Q₈	—	6	1	no	—
272	D₄	—	2	1	no	—
273	D₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	6	1	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
280	C₂×C₄	C₅×#235	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	3	2	no	—
284	C₈	—	4	1	no	—
285	C₈	—	6	2	no	—
286	C₈	—	6	1	no	—
287	C₈	—	12	1	no	—
288	C₈	C₅×#214	1	2	yes	complemented
289	C₂×C₃	—	8	1	no	—
290	C₂×C₃	C₅×#436	1	1	yes	characteristic
291	C₂×C₃	C₄×C₅×SD₁₆	1	1	yes	characteristic
292	C₂×C₃	C₅×#443	1	1	yes	characteristic
293	C₅	#70866	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₄×C₅×#67	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	12	1	no	—
301	C₄	—	12	1	no	—
302	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
303	C₄	C₅×#907	1	1	yes	characteristic
304	C₄	C₂×C₅×#214	1	1	yes	characteristic
305	C₄	C₅×#950	1	1	yes	characteristic
306	C₃	C₅×#1590	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	8	1	no	—
308	C₂	C₄×C₅×#214	1	1	yes	characteristic
309	C₂	C₅×#5128	1	1	yes	characteristic
310	C₂	C₅×#5144	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#436	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×D₄	C₂×C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#436	D₃
hypocenter	C₃	C₅×#1590
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#436

Complements

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

N	H ≅ G/N	type
3. C₅×#4679	C₂	semidirect
5. C₅×#4484	C₂	semidirect
6. C₅×#4480	C₂	semidirect
7. C₅×C₈×C₃⋊C₈	C₂	semidirect
24. #70866	C₅	direct (N × H)
52. #4679	C₂×C₅	semidirect
54. #4484	C₂×C₅	semidirect
55. #4480	C₂×C₅	semidirect
56. C₈×C₃⋊C₈	C₂×C₅	semidirect
88. C₃×C₅×C₈	SD₁₆	semidirect
92. C₅×C₃⋊C₈	M₁₆	semidirect
182. C₅×C₈	#214	semidirect
217. C₃×C₈	C₅×SD₁₆	semidirect
221. C₃⋊C₈	C₅×M₁₆	semidirect
252. C₃×C₅	#1590	semidirect
288. C₈	C₅×#214	semidirect
293. C₅	#70866	direct (N × H)
306. C₃	C₅×#1590	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#436	2
C₄×C₅×#214	2
C₅×#4679	2
C₅×D₄⋊(C₃⋊C₈)	2
C₅×#4484	2
C₅×#4480	2
C₅×C₈×C₃⋊C₈	2
C₅×#1590	3
#70866	5

Subgroups — C5×#70866 #14287311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70866 #14287311