Subgroups — C₅×#68866 #14294552

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#892	C₂	1	2	yes	maximal
3	C₃×C₅×#340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4551	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4529	C₂	1	2	yes	maximal, complemented
6	C₅×#1735	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	—	2	2	no	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₃×C₈	—	2	4	no	—
15	C₂×C₅×D₃×C₈	C₄	1	4	yes	complemented
16	C₅×#892	—	2	4	no	—
17	C₅×#892	C₄	1	8	yes	complemented
18	#68866	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₅×C₈	—	3	1	no	—
20	C₂×C₅×#117	—	3	2	no	—
21	C₅×#340	—	3	1	no	—
22	C₅×#340	—	3	2	no	—
23	C₅×#340	D₃	1	1	yes	characteristic, complemented
24	C₂³×C₅×D₃	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₅×Dic₃	—	2	2	no	—
28	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
29	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
31	C₂×C₄×C₅×D₃	D₄	1	4	yes	—
32	C₂×C₄×C₅×D₃	Q₈	1	2	yes	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
38	C₅×C₄⋊Dic₃	—	2	2	no	—
39	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
40	C₄×C₅×Dic₃	—	2	8	no	—
41	C₂×C₅×C₃⋊C₈	—	2	2	no	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₅×D₃×C₈	—	2	8	no	—
44	C₂²×D₃×C₈	C₂×C₅	1	1	yes	characteristic
45	C₂×#892	C₂×C₅	1	2	yes	—
46	C₃×#340	C₂×C₅	1	1	yes	characteristic, complemented
47	#4551	C₂×C₅	1	1	yes	characteristic, complemented
48	#4529	C₂×C₅	1	2	yes	complemented
49	C₂³×C₄×C₅	—	3	1	no	—
50	C₂²×C₅×C₈	—	3	1	no	—
51	C₂²×C₅×C₈	—	3	4	no	—
52	C₂²×C₅×C₈	—	6	4	no	—
53	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	C₅×#117	—	3	8	no	—
55	C₅×#117	—	6	4	no	—
56	C₂×C₅×C₄⋊C₄	—	3	2	no	—
57	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
58	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
59	C₂×C₄²×C₅	—	6	2	no	—
60	#1735	—	3	1	no	Sylow
61	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₅×D₃	—	2	1	no	—
63	C₂²×C₅×D₃	—	2	2	no	—
64	C₂²×C₅×D₃	—	2	1	no	—
65	C₂²×C₅×D₃	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₅×D₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	4	2	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₅×Dic₃	—	2	4	no	—
75	C₂×C₅×Dic₃	—	4	2	no	—
76	C₂×C₅×Dic₃	—	4	2	no	—
77	C₂×C₅×Dic₃	C₂²⋊C₄	1	4	yes	—
78	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
79	C₄×C₅×D₃	C₄²	1	4	yes	—
80	C₄×C₅×D₃	C₂²⋊C₄	1	4	yes	—
81	C₄×C₅×D₃	C₄⋊C₄	1	8	yes	—
82	C₃×C₅×C₈	—	2	2	no	—
83	C₅×C₃⋊C₈	—	2	2	no	—
84	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
87	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
88	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
89	C₂×C₄×Dic₃	—	2	2	no	—
90	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×D₃×C₈	—	2	4	no	—
92	C₂×D₃×C₈	C₄×C₅	1	4	yes	complemented
93	#892	—	2	4	no	—
94	#892	C₄×C₅	1	8	yes	complemented
95	C₂⁴×C₅	—	3	1	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂²×C₄×C₅	—	3	4	no	—
99	C₂²×C₄×C₅	—	3	2	no	—
100	C₂²×C₄×C₅	—	3	4	no	—
101	C₂²×C₄×C₅	—	3	2	no	—
102	C₂²×C₄×C₅	—	6	2	no	—
103	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	3	2	no	—
106	C₂×C₅×C₈	—	6	8	no	—
107	C₂×C₅×C₈	—	6	2	no	—
108	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	3	4	no	—
111	C₅×C₄⋊C₄	—	6	2	no	—
112	C₅×C₄⋊C₄	C₄×D₃	1	4	yes	—
113	C₅×C₂²⋊C₄	—	6	8	no	—
114	C₄²×C₅	—	6	8	no	—
115	C₂³×C₈	—	3	1	no	—
116	C₂×#117	—	3	2	no	—
117	#340	—	3	1	no	—
118	#340	—	3	2	no	—
119	#340	C₅×D₃	1	1	yes	characteristic, complemented
120	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#135	1	1	yes	characteristic
123	C₂²×C₃×C₅	#135	1	2	yes	—
124	C₂²×C₃×C₅	#135	1	1	yes	characteristic
125	C₂×C₅×D₃	—	2	4	no	—
126	C₂×C₅×D₃	—	2	4	no	—
127	C₂×C₅×D₃	—	2	2	no	—
128	C₂×C₅×D₃	—	2	2	no	—
129	C₂×C₅×D₃	#95	1	4	yes	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
134	C₅×Dic₃	—	4	4	no	—
135	C₅×Dic₃	#95	1	4	yes	—
136	C₂³×D₃	C₅×D₄	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	2	2	no	—
138	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂²×Dic₃	—	2	2	no	—
140	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
141	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
142	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
143	C₂×C₄×D₃	C₅×D₄	1	4	yes	—
144	C₂×C₄×D₃	C₅×Q₈	1	2	yes	—
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
149	(C₂×D₃)⋊C₄	—	2	8	no	—
150	C₄⋊Dic₃	—	2	2	no	—
151	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
152	C₄×Dic₃	—	2	8	no	—
153	C₂×C₃⋊C₈	—	2	2	no	—
154	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
155	D₃×C₈	—	2	8	no	—
156	C₂³×C₅	—	3	4	no	—
157	C₂³×C₅	—	3	2	no	—
158	C₂³×C₅	—	6	1	no	—
159	C₂³×C₅	—	6	2	no	—
160	C₂³×C₅	—	6	1	no	—
161	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	3	4	no	—
164	C₂×C₄×C₅	—	3	2	no	—
165	C₂×C₄×C₅	—	3	4	no	—
166	C₂×C₄×C₅	—	3	4	no	—
167	C₂×C₄×C₅	—	3	8	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	6	4	no	—
171	C₂×C₄×C₅	—	12	2	no	—
172	C₂×C₄×C₅	—	12	2	no	—
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
175	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
177	C₅×C₈	—	2	2	no	—
178	C₅×C₈	—	6	2	no	—
179	C₂³×C₄	—	3	1	no	—
180	C₂²×C₈	—	3	1	no	—
181	C₂²×C₈	—	3	4	no	—
182	C₂²×C₈	—	6	4	no	—
183	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
184	#117	—	3	8	no	—
185	#117	—	6	4	no	—
186	C₂×C₄⋊C₄	—	3	2	no	—
187	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
188	C₂×C₂²⋊C₄	—	6	2	no	—
189	C₂×C₄²	—	6	2	no	—
190	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
191	C₂×C₃×C₅	#416	1	2	yes	—
192	C₂×C₃×C₅	#416	1	2	yes	—
193	C₂×C₃×C₅	#427	1	1	yes	characteristic
194	C₂×C₃×C₅	#427	1	1	yes	characteristic
195	C₅×D₃	—	2	4	no	—
196	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×D₃	—	2	1	no	—
198	C₂²×D₃	—	2	2	no	—
199	C₂²×D₃	—	2	1	no	—
200	C₂²×D₃	C₅×C₂²⋊C₄	1	4	yes	—
201	C₂²×D₃	C₅×C₄⋊C₄	1	2	yes	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
209	C₂×Dic₃	—	2	4	no	—
210	C₂×Dic₃	—	4	2	no	—
211	C₂×Dic₃	—	4	2	no	—
212	C₂×Dic₃	C₅×C₂²⋊C₄	1	4	yes	—
213	C₂×Dic₃	C₅×C₄⋊C₄	1	2	yes	—
214	C₄×D₃	C₄²×C₅	1	4	yes	—
215	C₄×D₃	C₅×C₂²⋊C₄	1	4	yes	—
216	C₄×D₃	C₅×C₄⋊C₄	1	8	yes	—
217	C₃×C₈	—	2	2	no	—
218	C₃⋊C₈	—	2	2	no	—
219	C₂²×C₅	—	3	4	no	—
220	C₂²×C₅	—	6	2	no	—
221	C₂²×C₅	—	6	4	no	—
222	C₂²×C₅	—	6	4	no	—
223	C₂²×C₅	—	6	2	no	—
224	C₂²×C₅	D₃×C₂²⋊C₄	1	2	yes	—
225	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
226	C₂²×C₅	#912	1	1	yes	characteristic
227	C₂²×C₅	#916	1	2	yes	—
228	C₂²×C₅	#919	1	1	yes	characteristic
229	C₄×C₅	—	3	4	no	—
230	C₄×C₅	—	4	4	no	—
231	C₄×C₅	—	12	4	no	—
232	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
233	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
235	C₂⁴	—	3	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	3	1	no	—
238	C₂²×C₄	—	3	4	no	—
239	C₂²×C₄	—	3	2	no	—
240	C₂²×C₄	—	3	4	no	—
241	C₂²×C₄	—	3	2	no	—
242	C₂²×C₄	—	6	2	no	—
243	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	3	2	no	—
246	C₂×C₈	—	6	8	no	—
247	C₂×C₈	—	6	2	no	—
248	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	3	4	no	—
251	C₄⋊C₄	—	6	2	no	—
252	C₄⋊C₄	C₄×C₅×D₃	1	4	yes	—
253	C₂²⋊C₄	—	6	8	no	—
254	C₄²	—	6	8	no	—
255	C₃×C₅	#1735	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
257	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
258	C₂²×C₃	C₅×#135	1	1	yes	characteristic
259	C₂²×C₃	C₅×#135	1	2	yes	—
260	C₂²×C₃	C₅×#135	1	1	yes	characteristic
261	C₂×D₃	—	2	4	no	—
262	C₂×D₃	—	2	4	no	—
263	C₂×D₃	—	2	2	no	—
264	C₂×D₃	—	2	2	no	—
265	C₂×D₃	C₅×#95	1	4	yes	—
266	C₃×C₄	—	4	4	no	—
267	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
268	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
270	Dic₃	—	4	4	no	—
271	Dic₃	C₅×#95	1	4	yes	—
272	C₂×C₅	—	6	4	no	—
273	C₂×C₅	D₃×#95	1	1	yes	characteristic
274	C₂×C₅	#4772	1	2	yes	—
275	C₂×C₅	#4804	1	2	yes	—
276	C₂×C₅	#4860	1	1	yes	characteristic
277	C₂×C₅	#4880	1	1	yes	characteristic
278	C₂³	—	3	4	no	—
279	C₂³	—	3	2	no	—
280	C₂³	—	6	1	no	—
281	C₂³	—	6	2	no	—
282	C₂³	—	6	1	no	—
283	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	3	4	no	—
286	C₂×C₄	—	3	2	no	—
287	C₂×C₄	—	3	4	no	—
288	C₂×C₄	—	3	4	no	—
289	C₂×C₄	—	3	8	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	6	4	no	—
293	C₂×C₄	—	12	2	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
297	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
298	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	6	2	no	—
301	C₂×C₃	C₂×C₅×#95	1	1	yes	characteristic
302	C₂×C₃	C₅×#416	1	2	yes	—
303	C₂×C₃	C₅×#416	1	2	yes	—
304	C₂×C₃	C₅×#427	1	1	yes	characteristic
305	C₂×C₃	C₅×#427	1	1	yes	characteristic
306	D₃	—	2	4	no	—
307	C₅	#68866	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	3	4	no	—
309	C₂²	—	6	4	no	—
310	C₂²	—	6	4	no	—
311	C₂²	—	6	2	no	—
312	C₂²	—	6	2	no	—
313	C₂²	C₅×D₃×C₂²⋊C₄	1	2	yes	—
314	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
315	C₂²	C₅×#912	1	1	yes	characteristic
316	C₂²	C₅×#916	1	2	yes	—
317	C₂²	C₅×#919	1	1	yes	characteristic
318	C₄	—	3	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	12	4	no	—
321	C₄	C₄²×C₅×D₃	1	1	yes	characteristic
322	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
323	C₄	C₅×D₃×C₄⋊C₄	1	2	yes	—
324	C₃	C₅×#1735	1	1	yes	characteristic, complemented, Sylow
325	C₂	—	6	4	no	—
326	C₂	C₅×D₃×#95	1	1	yes	characteristic
327	C₂	C₅×#4772	1	2	yes	—
328	C₂	C₅×#4804	1	2	yes	—
329	C₂	C₅×#4860	1	1	yes	characteristic
330	C₂	C₅×#4880	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#340	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×C₅×D₃	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#340	D₃
hypocenter	C₃	C₅×#1735
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#340

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#340	C₂	semidirect
4. C₅×#4551	C₂	semidirect
5. C₅×#4529	C₂	semidirect
15. C₂×C₅×D₃×C₈	C₄	semidirect
17. C₅×#892	C₄	semidirect
18. #68866	C₅	direct (N × H)
23. C₅×#340	D₃	semidirect
46. C₃×#340	C₂×C₅	semidirect
47. #4551	C₂×C₅	semidirect
48. #4529	C₂×C₅	semidirect
92. C₂×D₃×C₈	C₄×C₅	semidirect
94. #892	C₄×C₅	semidirect
119. #340	C₅×D₃	semidirect
255. C₃×C₅	#1735	semidirect
307. C₅	#68866	direct (N × H)
324. C₃	C₅×#1735	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₃×C₈	2
C₂×C₅×#892	2
C₃×C₅×#340	2
C₅×#4551	2
C₅×#4529	2
C₅×#1735	3
#68866	5

Subgroups — C5×#68866 #14294552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68866 #14294552