Subgroups — C₅×#71808 #14285126

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 324 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×#4926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4925	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×#1553	—	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	C₅×#830	—	2	1	no	—
17	C₅×#830	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₅×#803	—	2	1	no	—
21	C₄×C₅×C₃⋊C₈	—	2	1	no	—
22	#71808	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#532	D₃	1	1	yes	characteristic
24	C₅×#432	—	3	1	no	—
25	C₅×#431	—	3	1	no	—
26	C₅×#332	—	3	2	no	—
27	C₅×#331	—	3	2	no	—
28	C₃×C₅×#41	—	4	4	no	—
29	C₂×C₃×C₅×Q₈	—	2	1	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₃×#532	C₂×C₅	1	1	yes	characteristic
49	#4926	C₂×C₅	1	1	yes	characteristic, complemented
50	#4925	C₂×C₅	1	1	yes	characteristic, complemented
51	#4547	C₂×C₅	1	2	yes	complemented
52	#4544	C₂×C₅	1	2	yes	complemented
53	C₂×C₅×#41	—	2	1	no	—
54	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
55	C₅×#127	C₂×D₃	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	Dic₃	1	2	yes	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	—	3	2	no	—
63	C₅×C₄⋊C₈	—	3	2	no	—
64	C₅×C₂²⋊C₈	—	6	1	no	—
65	C₅×C₂²⋊C₈	—	6	1	no	—
66	C₅×#97	—	6	1	no	—
67	C₄×C₅×C₈	—	6	1	no	—
68	#1553	—	3	1	no	Sylow
69	C₂³×C₃×C₅	—	2	2	no	—
70	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₃×C₅×D₄	—	4	4	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	4	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₅×C₃⋊C₈	—	4	2	no	—
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₂×C₃×#41	—	2	1	no	—
90	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₃×#127	C₂²×C₅	1	1	yes	characteristic
92	C₃×#124	—	2	1	no	—
93	C₃×#121	—	2	1	no	—
94	C₃×#121	C₄×C₅	1	2	yes	—
95	C₃×C₄×D₄	—	2	1	no	—
96	C₃×C₄×D₄	C₄×C₅	1	2	yes	—
97	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	#830	—	2	1	no	—
99	#830	—	2	1	no	—
100	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
101	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
102	#803	—	2	1	no	—
103	C₄×C₃⋊C₈	—	2	1	no	—
104	C₅×#41	—	4	4	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	1	no	—
108	C₂×C₅×D₄	—	2	1	no	—
109	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
110	C₂²×C₄×C₅	—	2	2	no	—
111	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	#67	1	1	yes	characteristic
113	C₂²×C₄×C₅	#67	1	1	yes	characteristic
114	C₂×C₅×C₈	—	6	2	no	—
115	C₂×C₅×C₈	—	6	2	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
118	C₅×C₂²⋊C₄	—	2	2	no	—
119	C₅×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
124	#532	C₅×D₃	1	1	yes	characteristic
125	#432	—	3	1	no	—
126	#431	—	3	1	no	—
127	#332	—	3	2	no	—
128	#331	—	3	2	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	4	2	no	—
133	C₂²×C₃×C₅	—	4	2	no	—
134	C₂²×C₃×C₅	—	4	2	no	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	#135	1	2	yes	—
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₃×#41	—	4	4	no	—
144	C₂×C₃×Q₈	—	2	1	no	—
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	1	no	—
147	C₂×C₃×D₄	—	2	1	no	—
148	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
149	C₂²×C₃×C₄	—	2	2	no	—
150	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
158	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
159	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃⋊C₈	—	2	2	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂³×C₅	—	2	2	no	—
164	C₂³×C₅	#217	1	1	yes	characteristic
165	C₅×Q₈	—	4	2	no	—
166	C₅×D₄	—	4	2	no	—
167	C₅×D₄	—	4	4	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	—	4	4	no	—
175	C₂×C₄×C₅	#217	1	1	yes	characteristic
176	C₂×C₄×C₅	#217	1	1	yes	characteristic
177	C₂×C₄×C₅	#217	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
181	C₅×C₈	—	12	2	no	—
182	C₅×C₈	—	12	2	no	—
183	C₂×#41	—	2	1	no	—
184	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
185	#127	C₂×C₅×D₃	1	1	yes	characteristic
186	#124	—	2	1	no	—
187	#121	—	2	1	no	—
188	#121	C₅×Dic₃	1	2	yes	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	C₅×Dic₃	1	2	yes	—
191	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
192	C₄⋊C₈	—	3	2	no	—
193	C₄⋊C₈	—	3	2	no	—
194	C₂²⋊C₈	—	6	1	no	—
195	C₂²⋊C₈	—	6	1	no	—
196	#97	—	6	1	no	—
197	C₄×C₈	—	6	1	no	—
198	C₂×C₃×C₅	—	4	1	no	—
199	C₂×C₃×C₅	—	8	2	no	—
200	C₂×C₃×C₅	#413	1	1	yes	characteristic
201	C₂×C₃×C₅	#416	1	1	yes	characteristic
202	C₂×C₃×C₅	#418	1	1	yes	characteristic
203	C₂³×C₃	—	2	2	no	—
204	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₃×Q₈	—	4	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	4	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	4	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	4	no	—
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₃⋊C₈	—	4	2	no	—
222	C₃⋊C₈	—	4	2	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	4	2	no	—
227	C₂²×C₅	—	4	2	no	—
228	C₂²×C₅	—	4	2	no	—
229	C₂²×C₅	C₂×#217	1	1	yes	characteristic
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	#932	1	2	yes	—
236	C₄×C₅	#950	1	2	yes	—
237	#41	—	4	4	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
245	C₂²×C₄	C₅×#67	1	1	yes	characteristic
246	C₂²×C₄	C₅×#67	1	1	yes	characteristic
247	C₂×C₈	—	6	2	no	—
248	C₂×C₈	—	6	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
254	C₄²	C₅×#67	1	1	yes	characteristic
255	C₄²	C₅×#67	1	1	yes	characteristic
256	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
257	C₃×C₅	#1553	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	4	2	no	—
270	C₃×C₄	C₅×#135	1	2	yes	—
271	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₅	—	4	1	no	—
273	C₂×C₅	—	8	2	no	—
274	C₂×C₅	#5215	1	1	yes	characteristic
275	C₂×C₅	#5233	1	1	yes	characteristic
276	C₂×C₅	#5237	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₅×#217	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₅×#217	1	1	yes	characteristic
290	C₂×C₄	C₅×#217	1	1	yes	characteristic
291	C₂×C₄	C₅×#217	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
294	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
295	C₈	—	12	2	no	—
296	C₈	—	12	2	no	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	—	8	2	no	—
299	C₂×C₃	C₅×#413	1	1	yes	characteristic
300	C₂×C₃	C₅×#416	1	1	yes	characteristic
301	C₂×C₃	C₅×#418	1	1	yes	characteristic
302	C₅	#71808	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂×C₅×#217	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	2	no	—
315	C₄	C₅×#932	1	2	yes	—
316	C₄	C₅×#950	1	2	yes	—
317	C₃	C₅×#1553	1	1	yes	characteristic, complemented, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₅×#5215	1	1	yes	characteristic
321	C₂	C₅×#5233	1	1	yes	characteristic
322	C₂	C₅×#5237	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×#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×#1553
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#532

Complements

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

N	H ≅ G/N	type
2. C₅×#4926	C₂	semidirect
3. C₅×#4925	C₂	semidirect
4. C₅×#4547	C₂	semidirect
5. C₅×#4544	C₂	semidirect
22. #71808	C₅	direct (N × H)
49. #4926	C₂×C₅	semidirect
50. #4925	C₂×C₅	semidirect
51. #4547	C₂×C₅	semidirect
52. #4544	C₂×C₅	semidirect
257. C₃×C₅	#1553	semidirect
302. C₅	#71808	direct (N × H)
317. C₃	C₅×#1553	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×#4926	2
C₅×#4925	2
C₅×#4547	2
C₅×#4544	2
C₅×#1553	3
#71808	5

Subgroups — C5×#71808 #14285126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71808 #14285126