Subgroups — C₅×#71804 #14284580

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 348 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×#4547	C₂	1	2	yes	maximal, complemented
5	C₅×#4544	C₂	1	2	yes	maximal, complemented
6	C₅×#1540	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	—	2	1	no	—
11	C₃×C₅×#121	—	2	1	no	—
12	C₃×C₅×#121	C₄	1	2	yes	—
13	C₃×C₄×C₅×D₄	—	2	1	no	—
14	C₃×C₄×C₅×D₄	C₄	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
17	C₅×#830	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
21	C₄×C₅×C₃⋊C₈	—	2	1	no	—
22	#71804	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#532	D₃	1	1	yes	characteristic
24	C₅×#431	—	3	1	no	—
25	C₂×C₅×C₄⋊C₈	—	3	1	no	—
26	C₅×#332	—	3	2	no	—
27	C₅×#331	—	3	2	no	—
28	C₃×C₅×#41	—	4	2	no	—
29	C₃×C₅×#41	—	4	2	no	—
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×C₅×C₃⋊C₈	—	2	2	no	—
49	C₂×C₅×C₃⋊C₈	—	4	2	no	—
50	C₃×#532	C₂×C₅	1	1	yes	characteristic
51	#4925	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
53	#4547	C₂×C₅	1	2	yes	complemented
54	#4544	C₂×C₅	1	2	yes	complemented
55	C₂×C₅×#41	—	2	1	no	—
56	C₂²×C₅×C₈	—	6	1	no	—
57	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
58	C₅×#127	C₂×D₃	1	1	yes	characteristic
59	C₅×#124	—	2	1	no	—
60	C₅×#121	—	2	1	no	—
61	C₅×#121	Dic₃	1	2	yes	—
62	C₄×C₅×D₄	—	2	1	no	—
63	C₄×C₅×D₄	Dic₃	1	2	yes	—
64	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	—	3	2	no	—
66	C₅×C₄⋊C₈	—	3	2	no	—
67	C₅×C₄⋊C₈	—	6	1	no	—
68	C₅×C₂²⋊C₈	—	6	1	no	—
69	C₄×C₅×C₈	—	6	1	no	—
70	#1540	—	3	1	no	Sylow
71	C₂³×C₃×C₅	—	2	2	no	—
72	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	4	1	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₃×C₅×D₄	—	4	4	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₃×C₅×D₄	—	4	1	no	—
79	C₃×C₅×D₄	—	4	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	4	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
88	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×C₃⋊C₈	—	4	2	no	—
93	C₅×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃×#41	—	2	1	no	—
95	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×#127	C₂²×C₅	1	1	yes	characteristic
97	C₃×#124	—	2	1	no	—
98	C₃×#121	—	2	1	no	—
99	C₃×#121	C₄×C₅	1	2	yes	—
100	C₃×C₄×D₄	—	2	1	no	—
101	C₃×C₄×D₄	C₄×C₅	1	2	yes	—
102	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
103	C₂²×C₃⋊C₈	—	2	1	no	—
104	#830	—	2	1	no	—
105	C₄⋊(C₃⋊C₈)	—	2	1	no	—
106	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
107	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
108	C₄×C₃⋊C₈	—	2	1	no	—
109	C₅×#41	—	4	2	no	—
110	C₅×#41	—	4	2	no	—
111	C₂×C₅×Q₈	—	2	1	no	—
112	C₂×C₅×D₄	—	2	2	no	—
113	C₂×C₅×D₄	—	2	1	no	—
114	C₂×C₅×D₄	—	2	1	no	—
115	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
116	C₂²×C₄×C₅	—	2	2	no	—
117	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
118	C₂²×C₄×C₅	#67	1	1	yes	characteristic
119	C₂²×C₄×C₅	#67	1	1	yes	characteristic
120	C₂×C₅×C₈	—	6	2	no	—
121	C₂×C₅×C₈	—	6	2	no	—
122	C₂×C₅×C₈	—	12	2	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
125	C₅×C₂²⋊C₄	—	2	2	no	—
126	C₅×C₂²⋊C₄	—	2	2	no	—
127	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
128	C₄²×C₅	#67	1	1	yes	characteristic
129	C₄²×C₅	#67	1	1	yes	characteristic
130	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
131	#532	C₅×D₃	1	1	yes	characteristic
132	#431	—	3	1	no	—
133	C₂×C₄⋊C₈	—	3	1	no	—
134	#332	—	3	2	no	—
135	#331	—	3	2	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	—	2	1	no	—
138	C₂²×C₃×C₅	—	4	2	no	—
139	C₂²×C₃×C₅	—	4	2	no	—
140	C₂²×C₃×C₅	—	4	2	no	—
141	C₂²×C₃×C₅	#102	1	2	yes	—
142	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	4	2	no	—
147	C₃×C₄×C₅	#102	1	2	yes	—
148	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₃×C₄×C₅	#135	1	2	yes	—
151	C₃×#41	—	4	2	no	—
152	C₃×#41	—	4	2	no	—
153	C₂×C₃×Q₈	—	2	1	no	—
154	C₂×C₃×D₄	—	2	2	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	2	1	no	—
157	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
161	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	2	1	no	—
163	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
164	C₃×C₂²⋊C₄	—	2	2	no	—
165	C₃×C₂²⋊C₄	—	2	2	no	—
166	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
167	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
168	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
169	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₂×C₃⋊C₈	—	2	2	no	—
171	C₂×C₃⋊C₈	—	2	2	no	—
172	C₂×C₃⋊C₈	—	4	2	no	—
173	C₂³×C₅	—	2	2	no	—
174	C₂³×C₅	#217	1	1	yes	characteristic
175	C₅×Q₈	—	4	1	no	—
176	C₅×Q₈	—	4	1	no	—
177	C₅×D₄	—	4	2	no	—
178	C₅×D₄	—	4	4	no	—
179	C₅×D₄	—	4	1	no	—
180	C₅×D₄	—	4	1	no	—
181	C₅×D₄	—	4	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	4	no	—
186	C₂×C₄×C₅	#213	1	2	yes	—
187	C₂×C₄×C₅	#214	1	2	yes	—
188	C₂×C₄×C₅	#217	1	1	yes	characteristic
189	C₂×C₄×C₅	#217	1	1	yes	characteristic
190	C₂×C₄×C₅	#217	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
194	C₅×C₈	—	12	2	no	—
195	C₅×C₈	—	12	2	no	—
196	C₂×#41	—	2	1	no	—
197	C₂²×C₈	—	6	1	no	—
198	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
199	#127	C₂×C₅×D₃	1	1	yes	characteristic
200	#124	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	C₅×Dic₃	1	2	yes	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	C₅×Dic₃	1	2	yes	—
205	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
206	C₄⋊C₈	—	3	2	no	—
207	C₄⋊C₈	—	3	2	no	—
208	C₄⋊C₈	—	6	1	no	—
209	C₂²⋊C₈	—	6	1	no	—
210	C₄×C₈	—	6	1	no	—
211	C₂×C₃×C₅	—	2	2	no	—
212	C₂×C₃×C₅	—	8	2	no	—
213	C₂×C₃×C₅	#413	1	1	yes	characteristic
214	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
215	C₂×C₃×C₅	#416	1	1	yes	characteristic
216	C₂³×C₃	—	2	2	no	—
217	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₃×Q₈	—	4	1	no	—
219	C₃×Q₈	—	4	1	no	—
220	C₃×D₄	—	4	2	no	—
221	C₃×D₄	—	4	4	no	—
222	C₃×D₄	—	4	1	no	—
223	C₃×D₄	—	4	1	no	—
224	C₃×D₄	—	4	2	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	—	4	4	no	—
229	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
233	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
234	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
237	C₃⋊C₈	—	4	2	no	—
238	C₃⋊C₈	—	4	2	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	4	2	no	—
243	C₂²×C₅	—	4	2	no	—
244	C₂²×C₅	#832	1	2	yes	—
245	C₂²×C₅	C₂×#217	1	1	yes	characteristic
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	4	2	no	—
250	C₄×C₅	#832	1	2	yes	—
251	C₄×C₅	C₂×#213	1	1	yes	characteristic
252	C₄×C₅	C₂×#214	1	1	yes	characteristic
253	C₄×C₅	#950	1	2	yes	—
254	#41	—	4	2	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
263	C₂²×C₄	C₅×#67	1	1	yes	characteristic
264	C₂²×C₄	C₅×#67	1	1	yes	characteristic
265	C₂×C₈	—	6	2	no	—
266	C₂×C₈	—	6	2	no	—
267	C₂×C₈	—	12	2	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
273	C₄²	C₅×#67	1	1	yes	characteristic
274	C₄²	C₅×#67	1	1	yes	characteristic
275	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
276	C₃×C₅	#1540	1	1	yes	characteristic, complemented, Hall
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	4	2	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	C₅×#102	1	2	yes	—
283	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	C₅×#102	1	2	yes	—
289	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
290	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
291	C₃×C₄	C₅×#135	1	2	yes	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	8	2	no	—
294	C₂×C₅	C₂×#832	1	1	yes	characteristic
295	C₂×C₅	#5233	1	1	yes	characteristic
296	C₂×C₅	#5237	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	C₅×#217	1	1	yes	characteristic
299	Q₈	—	4	1	no	—
300	Q₈	—	4	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	C₅×#213	1	2	yes	—
311	C₂×C₄	C₅×#214	1	2	yes	—
312	C₂×C₄	C₅×#217	1	1	yes	characteristic
313	C₂×C₄	C₅×#217	1	1	yes	characteristic
314	C₂×C₄	C₅×#217	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
317	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
318	C₈	—	12	2	no	—
319	C₈	—	12	2	no	—
320	C₂×C₃	—	2	2	no	—
321	C₂×C₃	—	8	2	no	—
322	C₂×C₃	C₅×#413	1	1	yes	characteristic
323	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
324	C₂×C₃	C₅×#416	1	1	yes	characteristic
325	C₅	#71804	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₅×#832	1	2	yes	—
332	C₂²	C₂×C₅×#217	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	2	no	—
337	C₄	C₅×#832	1	2	yes	—
338	C₄	C₂×C₅×#213	1	1	yes	characteristic
339	C₄	C₂×C₅×#214	1	1	yes	characteristic
340	C₄	C₅×#950	1	2	yes	—
341	C₃	C₅×#1540	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×C₅×#832	1	1	yes	characteristic
345	C₂	C₅×#5233	1	1	yes	characteristic
346	C₂	C₅×#5237	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#217
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×#1540
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×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×#4547	C₂	semidirect
5. C₅×#4544	C₂	semidirect
22. #71804	C₅	direct (N × H)
51. #4925	C₂×C₅	semidirect
52. C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
53. #4547	C₂×C₅	semidirect
54. #4544	C₂×C₅	semidirect
276. C₃×C₅	#1540	semidirect
325. C₅	#71804	direct (N × H)
341. C₃	C₅×#1540	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×#4547	2
C₅×#4544	2
C₅×#1540	3
#71804	5

Subgroups — C5×#71804 #14284580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71804 #14284580