Subgroups — C₅×#69511 #14297870

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic
3	C₅×#5212	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#808	C₂	1	2	yes	maximal, complemented
5	C₅×#4529	C₂	1	2	yes	maximal, complemented
6	C₅×#1807	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
10	C₂×C₅×#217	—	2	1	no	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₅×C₃⋊C₈	—	2	4	no	—
13	C₂²×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#95	—	2	1	no	—
17	C₅×C₃⋊#95	—	2	1	no	—
18	C₅×#808	—	2	4	no	—
19	#69511	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂³×C₅×C₈	—	3	1	no	—
21	C₂×C₅×#106	—	3	2	no	—
22	C₅×#380	—	3	1	no	—
23	C₅×#380	D₃	1	1	yes	characteristic
24	C₅×#340	—	3	2	no	—
25	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×Dic₃	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
37	C₅×#217	—	4	2	no	—
38	C₅×C₄⋊Dic₃	—	2	2	no	—
39	C₅×C₄⋊Dic₃	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×C₅×C₃⋊C₈	—	2	8	no	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
45	C₂³×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
46	C₃×#380	C₂×C₅	1	1	yes	characteristic
47	#5212	C₂×C₅	1	1	yes	characteristic
48	C₂×#808	C₂×C₅	1	2	yes	complemented
49	#4529	C₂×C₅	1	2	yes	complemented
50	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₅×C₈	—	3	1	no	—
52	C₂²×C₅×C₈	—	3	4	no	—
53	C₂²×C₅×C₈	—	3	1	no	—
54	C₂²×C₅×C₈	—	6	4	no	—
55	C₂×C₅×C₄⋊C₄	—	3	2	no	—
56	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
57	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
58	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
59	C₅×#106	—	6	4	no	—
60	C₅×#95	—	2	1	no	—
61	C₅×#95	—	6	1	no	—
62	#1807	—	3	1	no	Sylow
63	C₂³×C₃×C₅	—	2	2	no	—
64	C₂³×C₃×C₅	—	2	2	no	—
65	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
66	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
67	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
68	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	4	2	no	—
76	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₅×Dic₃	—	4	2	no	—
82	C₂×C₅×Dic₃	—	4	2	no	—
83	C₂×C₅×Dic₃	—	4	2	no	—
84	C₅×C₃⋊C₈	—	2	2	no	—
85	C₅×C₃⋊C₈	C₂²⋊C₄	1	4	yes	complemented
86	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
88	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
89	C₂×#217	—	2	1	no	—
90	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
91	C₂²×C₃⋊C₈	—	2	4	no	—
92	C₂²×C₃⋊C₈	C₄×C₅	1	4	yes	complemented
93	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₃×#95	—	2	1	no	—
96	C₃⋊#95	—	2	1	no	—
97	#808	—	2	4	no	—
98	C₂⁴×C₅	#67	1	1	yes	characteristic
99	C₂²×C₄×C₅	—	2	2	no	—
100	C₂²×C₄×C₅	—	2	2	no	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	—	6	2	no	—
103	C₂²×C₄×C₅	Dic₆	1	2	yes	—
104	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₅×C₈	—	3	2	no	—
108	C₂×C₅×C₈	—	3	4	no	—
109	C₂×C₅×C₈	—	3	2	no	—
110	C₂×C₅×C₈	—	6	8	no	—
111	C₂×C₅×C₈	—	6	2	no	—
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	6	2	no	—
115	C₅×C₄⋊C₄	—	6	2	no	—
116	C₅×C₂²⋊C₄	—	4	2	no	—
117	C₅×C₂²⋊C₄	—	12	2	no	—
118	C₂³×C₈	—	3	1	no	—
119	C₂×#106	—	3	2	no	—
120	#380	—	3	1	no	—
121	#380	C₅×D₃	1	1	yes	characteristic
122	#340	—	3	2	no	—
123	C₂²×C₃×C₅	—	2	2	no	—
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	—	2	4	no	—
126	C₂²×C₃×C₅	—	2	4	no	—
127	C₂²×C₃×C₅	#106	1	4	yes	complemented
128	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₃×C₅	#121	1	2	yes	—
130	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
132	C₂²×C₃×C₅	#135	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	—	8	2	no	—
135	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₃×C₄×C₅	#117	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
138	C₅×Dic₃	—	8	2	no	—
139	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
140	C₂²×C₃×C₄	—	2	2	no	—
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
144	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
146	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₂²⋊C₄	—	4	2	no	—
151	#217	—	4	2	no	—
152	C₄⋊Dic₃	—	2	2	no	—
153	C₄⋊Dic₃	—	2	2	no	—
154	C₂×C₃⋊C₈	—	2	2	no	—
155	C₂×C₃⋊C₈	—	2	8	no	—
156	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
157	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
159	C₂³×C₅	—	2	2	no	—
160	C₂³×C₅	—	2	2	no	—
161	C₂³×C₅	Dic₃⋊C₄	1	2	yes	—
162	C₂³×C₅	#214	1	2	yes	—
163	C₂³×C₅	#215	1	2	yes	—
164	C₂³×C₅	C₂×#67	1	1	yes	characteristic
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	4	2	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	12	2	no	—
173	C₂×C₄×C₅	—	12	2	no	—
174	C₂×C₄×C₅	—	12	2	no	—
175	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
176	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
179	C₂×C₄×C₅	#237	1	2	yes	—
180	C₅×C₈	—	3	4	no	—
181	C₅×C₈	—	6	2	no	—
182	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
183	C₂²×C₈	—	3	1	no	—
184	C₂²×C₈	—	3	4	no	—
185	C₂²×C₈	—	3	1	no	—
186	C₂²×C₈	—	6	4	no	—
187	C₂×C₄⋊C₄	—	3	2	no	—
188	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₂²⋊C₄	—	6	1	no	—
191	#106	—	6	4	no	—
192	#95	—	2	1	no	—
193	#95	—	6	1	no	—
194	C₂×C₃×C₅	—	2	4	no	—
195	C₂×C₃×C₅	#380	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
197	C₂×C₃×C₅	#427	1	1	yes	characteristic
198	C₂×C₃×C₅	#465	1	2	yes	—
199	C₂×C₃×C₅	#465	1	2	yes	—
200	C₂³×C₃	—	2	2	no	—
201	C₂³×C₃	—	2	2	no	—
202	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
203	C₂³×C₃	C₅×SD₁₆	1	2	yes	—
204	C₂³×C₃	C₅×SD₁₆	1	2	yes	—
205	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	4	2	no	—
211	C₂×C₃×C₄	—	4	2	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
214	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×#41	1	2	yes	—
218	C₂×Dic₃	—	4	2	no	—
219	C₂×Dic₃	—	4	2	no	—
220	C₂×Dic₃	—	4	2	no	—
221	C₃⋊C₈	—	2	2	no	—
222	C₃⋊C₈	C₅×C₂²⋊C₄	1	4	yes	complemented
223	C₂²×C₅	—	2	2	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	2	4	no	—
226	C₂²×C₅	—	2	4	no	—
227	C₂²×C₅	#808	1	4	yes	complemented
228	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
229	C₂²×C₅	C₂×#214	1	1	yes	characteristic
230	C₂²×C₅	#939	1	2	yes	—
231	C₂²×C₅	C₂×#215	1	1	yes	characteristic
232	C₂²×C₅	#950	1	2	yes	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	8	2	no	—
235	C₄×C₅	—	24	2	no	—
236	C₄×C₅	#877	1	1	yes	characteristic
237	C₄×C₅	#878	1	2	yes	—
238	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
239	C₂⁴	C₅×#67	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	6	2	no	—
244	C₂²×C₄	C₅×Dic₆	1	2	yes	—
245	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
246	C₂²×C₄	C₅×#67	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
248	C₂×C₈	—	3	2	no	—
249	C₂×C₈	—	3	4	no	—
250	C₂×C₈	—	3	2	no	—
251	C₂×C₈	—	6	8	no	—
252	C₂×C₈	—	6	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	6	2	no	—
256	C₄⋊C₄	—	6	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	12	2	no	—
259	C₃×C₅	#1807	1	1	yes	characteristic, complemented, Hall
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	4	no	—
263	C₂²×C₃	—	2	4	no	—
264	C₂²×C₃	C₅×#106	1	4	yes	complemented
265	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
266	C₂²×C₃	C₅×#121	1	2	yes	—
267	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
268	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₂²×C₃	C₅×#135	1	2	yes	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	8	2	no	—
272	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
273	C₃×C₄	C₅×#117	1	1	yes	characteristic
274	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
275	Dic₃	—	8	2	no	—
276	C₂×C₅	—	2	4	no	—
277	C₂×C₅	#4942	1	1	yes	characteristic
278	C₂×C₅	C₂×#808	1	1	yes	characteristic
279	C₂×C₅	#4998	1	1	yes	characteristic
280	C₂×C₅	#5041	1	2	yes	—
281	C₂×C₅	#5048	1	2	yes	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	C₅×Dic₃⋊C₄	1	2	yes	—
285	C₂³	C₅×#214	1	2	yes	—
286	C₂³	C₅×#215	1	2	yes	—
287	C₂³	C₂×C₅×#67	1	1	yes	characteristic
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	12	2	no	—
296	C₂×C₄	—	12	2	no	—
297	C₂×C₄	—	12	2	no	—
298	C₂×C₄	C₅×Dic₃⋊C₄	1	2	yes	—
299	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
301	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
302	C₂×C₄	C₅×#237	1	2	yes	—
303	C₈	—	3	4	no	—
304	C₈	—	6	2	no	—
305	C₂×C₃	—	2	4	no	—
306	C₂×C₃	C₅×#380	1	1	yes	characteristic
307	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
308	C₂×C₃	C₅×#427	1	1	yes	characteristic
309	C₂×C₃	C₅×#465	1	2	yes	—
310	C₂×C₃	C₅×#465	1	2	yes	—
311	C₅	#69511	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	C₅×#808	1	4	yes	complemented
317	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
318	C₂²	C₂×C₅×#214	1	1	yes	characteristic
319	C₂²	C₅×#939	1	2	yes	—
320	C₂²	C₂×C₅×#215	1	1	yes	characteristic
321	C₂²	C₅×#950	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	24	2	no	—
325	C₄	C₅×#877	1	1	yes	characteristic
326	C₄	C₅×#878	1	2	yes	—
327	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
328	C₃	C₅×#1807	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	4	no	—
330	C₂	C₅×#4942	1	1	yes	characteristic
331	C₂	C₂×C₅×#808	1	1	yes	characteristic
332	C₂	C₅×#4998	1	1	yes	characteristic
333	C₂	C₅×#5041	1	2	yes	—
334	C₂	C₅×#5048	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#380	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#380	D₃
hypocenter	C₃	C₅×#1807
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₅×#380

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#808	C₂	semidirect
5. C₅×#4529	C₂	semidirect
13. C₂²×C₅×C₃⋊C₈	C₄	semidirect
19. #69511	C₅	direct (N × H)
48. C₂×#808	C₂×C₅	semidirect
49. #4529	C₂×C₅	semidirect
85. C₅×C₃⋊C₈	C₂²⋊C₄	semidirect
92. C₂²×C₃⋊C₈	C₄×C₅	semidirect
127. C₂²×C₃×C₅	#106	semidirect
222. C₃⋊C₈	C₅×C₂²⋊C₄	semidirect
227. C₂²×C₅	#808	semidirect
259. C₃×C₅	#1807	semidirect
264. C₂²×C₃	C₅×#106	semidirect
311. C₅	#69511	direct (N × H)
316. C₂²	C₅×#808	semidirect
328. C₃	C₅×#1807	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₃⋊C₈	2
C₃×C₅×#380	2
C₅×#5212	2
C₂×C₅×#808	2
C₅×#4529	2
C₅×#1807	3
#69511	5

Subgroups — C5×#69511 #14297870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69511 #14297870