Subgroups — C₅×#71800 #14281772

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₅×#1480	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×#41	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₃×C₅×#127	C₄	1	1	yes	characteristic
9	C₃×C₅×#124	—	2	1	no	—
10	C₃×C₅×#121	—	2	1	no	—
11	C₃×C₅×#121	C₄	1	2	yes	—
12	C₃×C₄×C₅×D₄	—	2	1	no	—
13	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
16	C₅×#830	—	2	1	no	—
17	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
21	#71800	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	D₃	1	1	yes	characteristic
23	C₅×#431	—	3	1	no	—
24	C₂×C₅×C₄⋊C₈	—	3	1	no	—
25	C₅×D₄⋊C₈	—	3	4	no	—
26	C₃×C₅×#41	—	4	2	no	—
27	C₃×C₅×#41	—	4	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	—	4	2	no	—
48	C₃×#532	C₂×C₅	1	1	yes	characteristic
49	#4925	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
51	D₄⋊(C₃⋊C₈)	C₂×C₅	1	4	yes	complemented
52	C₂×C₅×#41	—	2	1	no	—
53	C₂²×C₅×C₈	—	6	1	no	—
54	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
55	C₅×#127	Dic₃	1	1	yes	characteristic
56	C₅×#124	—	2	1	no	—
57	C₅×#121	—	2	1	no	—
58	C₅×#121	Dic₃	1	2	yes	—
59	C₄×C₅×D₄	—	2	1	no	—
60	C₄×C₅×D₄	C₂×D₃	1	2	yes	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	—	3	2	no	—
63	C₅×C₄⋊C₈	—	6	1	no	—
64	C₅×C₄⋊C₈	—	6	1	no	—
65	C₅×C₂²⋊C₈	—	6	1	no	—
66	C₄×C₅×C₈	—	3	2	no	—
67	#1480	—	3	1	no	Sylow
68	C₂³×C₃×C₅	—	2	2	no	—
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×D₄	—	2	4	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₃×C₅×D₄	—	4	4	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	—	4	2	no	—
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
84	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₅×C₃⋊C₈	—	4	2	no	—
90	C₂×C₃×#41	—	2	1	no	—
91	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
92	C₃×#127	C₄×C₅	1	1	yes	characteristic
93	C₃×#124	—	2	1	no	—
94	C₃×#121	—	2	1	no	—
95	C₃×#121	C₄×C₅	1	2	yes	—
96	C₃×C₄×D₄	—	2	1	no	—
97	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
98	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
99	C₂²×C₃⋊C₈	—	2	1	no	—
100	#830	—	2	1	no	—
101	C₄⋊(C₃⋊C₈)	—	2	1	no	—
102	C₄⋊(C₃⋊C₈)	—	2	1	no	—
103	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
104	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
105	C₅×#41	—	4	2	no	—
106	C₅×#41	—	4	2	no	—
107	C₂×C₅×Q₈	—	2	1	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	—	2	1	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
112	C₂²×C₄×C₅	—	2	2	no	—
113	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×C₄×C₅	#67	1	1	yes	characteristic
115	C₂²×C₄×C₅	#67	1	1	yes	characteristic
116	C₂×C₅×C₈	—	6	2	no	—
117	C₂×C₅×C₈	—	6	2	no	—
118	C₂×C₅×C₈	—	12	2	no	—
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
121	C₅×C₂²⋊C₄	—	2	2	no	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
124	C₄²×C₅	#67	1	1	yes	characteristic
125	C₄²×C₅	#67	1	1	yes	characteristic
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	#532	C₅×D₃	1	1	yes	characteristic
128	#431	—	3	1	no	—
129	C₂×C₄⋊C₈	—	3	1	no	—
130	D₄⋊C₈	—	3	4	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	—	4	2	no	—
134	C₂²×C₃×C₅	—	4	2	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	#102	1	2	yes	—
137	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	4	2	no	—
142	C₃×C₄×C₅	#102	1	2	yes	—
143	C₃×C₄×C₅	#131	1	2	yes	—
144	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
146	C₃×#41	—	4	2	no	—
147	C₃×#41	—	4	2	no	—
148	C₂×C₃×Q₈	—	2	1	no	—
149	C₂×C₃×D₄	—	2	2	no	—
150	C₂×C₃×D₄	—	2	1	no	—
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
153	C₂²×C₃×C₄	—	2	2	no	—
154	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
159	C₃×C₂²⋊C₄	—	2	2	no	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
162	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
164	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
165	C₂×C₃⋊C₈	—	2	2	no	—
166	C₂×C₃⋊C₈	—	2	2	no	—
167	C₂×C₃⋊C₈	—	4	2	no	—
168	C₂³×C₅	—	2	2	no	—
169	C₂³×C₅	#217	1	1	yes	characteristic
170	C₅×Q₈	—	4	2	no	—
171	C₅×D₄	—	2	4	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	4	no	—
174	C₅×D₄	—	4	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	#213	1	2	yes	—
181	C₂×C₄×C₅	#214	1	2	yes	—
182	C₂×C₄×C₅	#217	1	1	yes	characteristic
183	C₂×C₄×C₅	#217	1	1	yes	characteristic
184	C₂×C₄×C₅	#217	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
188	C₅×C₈	—	12	2	no	—
189	C₅×C₈	—	12	2	no	—
190	C₂×#41	—	2	1	no	—
191	C₂²×C₈	—	6	1	no	—
192	C₄⋊Q₈	C₅×Dic₃	1	1	yes	characteristic
193	#127	C₅×Dic₃	1	1	yes	characteristic
194	#124	—	2	1	no	—
195	#121	—	2	1	no	—
196	#121	C₅×Dic₃	1	2	yes	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	C₂×C₅×D₃	1	2	yes	—
199	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
200	C₄⋊C₈	—	3	2	no	—
201	C₄⋊C₈	—	6	1	no	—
202	C₄⋊C₈	—	6	1	no	—
203	C₂²⋊C₈	—	6	1	no	—
204	C₄×C₈	—	3	2	no	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	—	8	2	no	—
207	C₂×C₃×C₅	#408	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
209	C₂×C₃×C₅	#421	1	1	yes	characteristic
210	C₂³×C₃	—	2	2	no	—
211	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₃×Q₈	—	4	2	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	4	no	—
216	C₃×D₄	—	4	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
226	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
227	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₃⋊C₈	—	4	2	no	—
231	C₃⋊C₈	—	4	2	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	—	4	2	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	#832	1	2	yes	—
238	C₂²×C₅	C₂×#217	1	1	yes	characteristic
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	#832	1	2	yes	—
244	C₄×C₅	C₂×#213	1	1	yes	characteristic
245	C₄×C₅	C₂×#214	1	1	yes	characteristic
246	C₄×C₅	#948	1	2	yes	—
247	#41	—	4	2	no	—
248	#41	—	4	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
256	C₂²×C₄	C₅×#67	1	1	yes	characteristic
257	C₂²×C₄	C₅×#67	1	1	yes	characteristic
258	C₂×C₈	—	6	2	no	—
259	C₂×C₈	—	6	2	no	—
260	C₂×C₈	—	12	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
266	C₄²	C₅×#67	1	1	yes	characteristic
267	C₄²	C₅×#67	1	1	yes	characteristic
268	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
269	C₃×C₅	#1480	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	4	2	no	—
274	C₂²×C₃	—	4	2	no	—
275	C₂²×C₃	C₅×#102	1	2	yes	—
276	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	4	2	no	—
281	C₃×C₄	C₅×#102	1	2	yes	—
282	C₃×C₄	C₅×#131	1	2	yes	—
283	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
284	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	8	2	no	—
287	C₂×C₅	C₂×#832	1	1	yes	characteristic
288	C₂×C₅	#5234	1	1	yes	characteristic
289	C₂×C₅	#5236	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	C₅×#217	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	4	no	—
296	D₄	—	4	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₅×#213	1	2	yes	—
303	C₂×C₄	C₅×#214	1	2	yes	—
304	C₂×C₄	C₅×#217	1	1	yes	characteristic
305	C₂×C₄	C₅×#217	1	1	yes	characteristic
306	C₂×C₄	C₅×#217	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
309	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
310	C₈	—	12	2	no	—
311	C₈	—	12	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	8	2	no	—
314	C₂×C₃	C₅×#408	1	1	yes	characteristic
315	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
316	C₂×C₃	C₅×#421	1	1	yes	characteristic
317	C₅	#71800	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₅×#832	1	2	yes	—
324	C₂²	C₂×C₅×#217	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	2	no	—
329	C₄	C₅×#832	1	2	yes	—
330	C₄	C₂×C₅×#213	1	1	yes	characteristic
331	C₄	C₂×C₅×#214	1	1	yes	characteristic
332	C₄	C₅×#948	1	2	yes	—
333	C₃	C₅×#1480	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	8	2	no	—
336	C₂	C₂×C₅×#832	1	1	yes	characteristic
337	C₂	C₅×#5234	1	1	yes	characteristic
338	C₂	C₅×#5236	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, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#532	D₃
hypocenter	C₃	C₅×#1480
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#532

Complements

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

N	H ≅ G/N	type
2. C₅×#4925	C₂	semidirect
3. C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₅×D₄⋊(C₃⋊C₈)	C₂	semidirect
21. #71800	C₅	direct (N × H)
49. #4925	C₂×C₅	semidirect
50. C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
51. D₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
269. C₃×C₅	#1480	semidirect
317. C₅	#71800	direct (N × H)
333. C₃	C₅×#1480	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₅×#4925	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₅×D₄⋊(C₃⋊C₈)	2
C₅×#1480	3
#71800	5

Subgroups — C5×#71800 #14281772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71800 #14281772