Subgroups — C₅×#71807 #14285024

Order 1920.

504 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×#4926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4546	C₂	1	2	yes	maximal, complemented
5	C₅×#4544	C₂	1	2	yes	maximal, complemented
6	C₅×#1551	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₄	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	C₄	1	2	yes	—
11	C₃×C₅×#121	—	2	1	no	—
12	C₃×C₅×#121	—	2	1	no	—
13	C₃×C₄×C₅×D₄	—	2	1	no	—
14	C₃×C₄×C₅×D₄	—	2	1	no	—
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	C₅×#803	—	2	1	no	—
22	#71807	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	C₂×C₅×C₄⋊C₈	—	3	1	no	—
26	C₅×#333	—	3	2	no	—
27	C₅×#331	—	3	2	no	—
28	C₃×C₅×#41	—	4	4	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
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	C₂×C₅×C₃⋊C₈	—	4	2	no	—
49	C₃×#532	C₂×C₅	1	1	yes	characteristic
50	#4926	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic, complemented
52	#4546	C₂×C₅	1	2	yes	complemented
53	#4544	C₂×C₅	1	2	yes	complemented
54	C₂×C₅×#41	Dic₃	1	2	yes	—
55	C₂²×C₅×C₈	—	6	1	no	—
56	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
57	C₅×#127	C₂×D₃	1	1	yes	characteristic
58	C₅×#124	Dic₃	1	2	yes	—
59	C₅×#121	—	2	1	no	—
60	C₅×#121	—	2	1	no	—
61	C₄×C₅×D₄	—	2	1	no	—
62	C₄×C₅×D₄	—	2	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	3	2	no	—
65	C₅×C₄⋊C₈	—	3	2	no	—
66	C₅×C₄⋊C₈	—	6	1	no	—
67	C₅×C₂²⋊C₈	—	6	1	no	—
68	C₅×#97	—	6	1	no	—
69	#1551	—	3	1	no	Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	4	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₃×C₅×D₄	—	4	1	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	2	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	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×C₃⋊C₈	—	4	2	no	—
91	C₅×C₃⋊C₈	—	4	2	no	—
92	C₂×C₃×#41	C₄×C₅	1	2	yes	—
93	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
94	C₃×#127	C₂²×C₅	1	1	yes	characteristic
95	C₃×#124	C₄×C₅	1	2	yes	—
96	C₃×#121	—	2	1	no	—
97	C₃×#121	—	2	1	no	—
98	C₃×C₄×D₄	—	2	1	no	—
99	C₃×C₄×D₄	—	2	1	no	—
100	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
101	C₂²×C₃⋊C₈	—	2	1	no	—
102	#830	—	2	1	no	—
103	C₄⋊(C₃⋊C₈)	—	2	1	no	—
104	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
105	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
106	#803	—	2	1	no	—
107	C₅×#41	—	4	4	no	—
108	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	—	2	1	no	—
112	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
113	C₂²×C₄×C₅	—	2	2	no	—
114	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
115	C₂²×C₄×C₅	#67	1	1	yes	characteristic
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₂×C₅×C₈	—	6	2	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₂×C₅×C₈	—	12	2	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₅×C₂²⋊C₄	—	2	2	no	—
124	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
125	C₄²×C₅	#67	1	1	yes	characteristic
126	C₄²×C₅	#67	1	1	yes	characteristic
127	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
128	#532	C₅×D₃	1	1	yes	characteristic
129	#432	—	3	1	no	—
130	C₂×C₄⋊C₈	—	3	1	no	—
131	#333	—	3	2	no	—
132	#331	—	3	2	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	—	4	2	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	#100	1	2	yes	—
139	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	4	2	no	—
145	C₃×C₄×C₅	#135	1	2	yes	—
146	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₃×C₄×C₅	#137	1	1	yes	characteristic
148	C₃×#41	—	4	4	no	—
149	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	—	2	1	no	—
153	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₂²⋊C₄	—	2	2	no	—
162	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
163	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
164	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
165	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₂×C₃⋊C₈	—	2	2	no	—
167	C₂×C₃⋊C₈	—	2	2	no	—
168	C₂×C₃⋊C₈	—	4	2	no	—
169	C₂³×C₅	—	2	2	no	—
170	C₂³×C₅	#217	1	1	yes	characteristic
171	C₅×Q₈	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	4	no	—
174	C₅×D₄	—	4	1	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	4	1	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	4	no	—
183	C₂×C₄×C₅	#217	1	1	yes	characteristic
184	C₂×C₄×C₅	#217	1	1	yes	characteristic
185	C₂×C₄×C₅	#217	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
189	C₅×C₈	—	12	2	no	—
190	C₅×C₈	—	12	2	no	—
191	C₂×#41	C₅×Dic₃	1	2	yes	—
192	C₂²×C₈	—	6	1	no	—
193	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
194	#127	C₂×C₅×D₃	1	1	yes	characteristic
195	#124	C₅×Dic₃	1	2	yes	—
196	#121	—	2	1	no	—
197	#121	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
201	C₄⋊C₈	—	3	2	no	—
202	C₄⋊C₈	—	3	2	no	—
203	C₄⋊C₈	—	6	1	no	—
204	C₂²⋊C₈	—	6	1	no	—
205	#97	—	6	1	no	—
206	C₂×C₃×C₅	—	2	2	no	—
207	C₂×C₃×C₅	—	8	2	no	—
208	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
209	C₂×C₃×C₅	#416	1	1	yes	characteristic
210	C₂×C₃×C₅	#417	1	1	yes	characteristic
211	C₂³×C₃	—	2	2	no	—
212	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₃×Q₈	—	4	2	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	4	no	—
216	C₃×D₄	—	4	2	no	—
217	C₃×D₄	—	4	1	no	—
218	C₃×D₄	—	4	1	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	4	4	no	—
225	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₃⋊C₈	—	4	2	no	—
232	C₃⋊C₈	—	4	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	4	2	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	—	4	2	no	—
238	C₂²×C₅	#833	1	2	yes	—
239	C₂²×C₅	C₂×#217	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	4	2	no	—
245	C₄×C₅	#949	1	1	yes	characteristic
246	C₄×C₅	#950	1	2	yes	—
247	C₄×C₅	#951	1	1	yes	characteristic
248	#41	—	4	4	no	—
249	C₂×Q₈	C₂×C₅×Dic₃	1	2	yes	—
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	—	2	1	no	—
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	#1551	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×#100	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	—	2	1	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	C₅×#135	1	2	yes	—
283	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₃×C₄	C₅×#137	1	1	yes	characteristic
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	8	2	no	—
287	C₂×C₅	C₂×#833	1	1	yes	characteristic
288	C₂×C₅	#5232	1	1	yes	characteristic
289	C₂×C₅	#5233	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	C₅×#217	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	4	no	—
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	C₂×C₅×#100	1	1	yes	characteristic
315	C₂×C₃	C₅×#416	1	1	yes	characteristic
316	C₂×C₃	C₅×#417	1	1	yes	characteristic
317	C₅	#71807	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×#833	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	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	C₅×#949	1	1	yes	characteristic
331	C₄	C₅×#950	1	2	yes	—
332	C₄	C₅×#951	1	1	yes	characteristic
333	C₃	C₅×#1551	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	8	2	no	—
336	C₂	C₂×C₅×#833	1	1	yes	characteristic
337	C₂	C₅×#5232	1	1	yes	characteristic
338	C₂	C₅×#5233	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×#1551
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. C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₅×#4546	C₂	semidirect
5. C₅×#4544	C₂	semidirect
22. #71807	C₅	direct (N × H)
50. #4926	C₂×C₅	semidirect
51. C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
52. #4546	C₂×C₅	semidirect
53. #4544	C₂×C₅	semidirect
269. C₃×C₅	#1551	semidirect
317. C₅	#71807	direct (N × H)
333. C₃	C₅×#1551	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
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₅×#4546	2
C₅×#4544	2
C₅×#1551	3
#71807	5

Subgroups — C5×#71807 #14285024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71807 #14285024