Subgroups — C₅×#68663 #14297470

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₅×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4524	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1798	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
11	C₂×C₅×Dic₃⋊C₄	C₄	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₅×#872	—	2	4	no	—
15	C₅×(C₂×D₃)⋊C₈	—	2	4	no	—
16	C₅×Dic₃⋊C₈	—	2	4	no	—
17	#68663	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#117	—	3	1	no	—
19	C₂×C₅×C₄⋊C₈	—	3	2	no	—
20	C₂×C₅×C₂²⋊C₈	—	3	2	no	—
21	C₅×#336	—	3	1	no	—
22	C₅×#336	D₃	1	1	yes	characteristic, complemented
23	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
27	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅×D₃	—	2	2	no	—
29	C₂×C₄×C₅×D₃	—	2	2	no	—
30	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
31	C₂×C₄×C₅×D₃	Q₈	1	2	yes	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₄²×C₅	—	2	2	no	—
36	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
37	C₅×Dic₃⋊C₄	—	2	4	no	—
38	C₄×C₅×Dic₃	D₄	1	4	yes	—
39	C₂×C₅×C₃⋊C₈	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×C₅×C₃⋊C₈	—	2	2	no	—
42	C₂×#872	C₂×C₅	1	1	yes	characteristic
43	C₂×(C₂×D₃)⋊C₈	C₂×C₅	1	2	yes	—
44	C₂×Dic₃⋊C₈	C₂×C₅	1	2	yes	complemented
45	C₃×#336	C₂×C₅	1	1	yes	characteristic, complemented
46	#4524	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂³×C₄×C₅	—	3	1	no	—
48	C₂²×C₅×C₈	—	3	2	no	—
49	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
50	C₅×#117	—	6	4	no	—
51	C₂×C₅×C₄⋊C₄	—	3	2	no	—
52	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
53	C₂×C₄²×C₅	—	3	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₅×C₄⋊C₈	—	6	4	no	—
56	C₅×C₂²⋊C₈	—	6	4	no	—
57	#1798	—	3	1	no	Sylow
58	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₅×D₃	—	2	1	no	—
60	C₂²×C₅×D₃	—	2	1	no	—
61	C₂²×C₅×D₃	—	2	2	no	—
62	C₂²×C₅×D₃	—	2	2	no	—
63	C₂²×C₅×D₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₂×C₅×Dic₃	—	2	1	no	—
72	C₂×C₅×Dic₃	—	2	1	no	—
73	C₂×C₅×Dic₃	—	2	2	no	—
74	C₂×C₅×Dic₃	—	2	2	no	—
75	C₂×C₅×Dic₃	C₂²⋊C₄	1	4	yes	—
76	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
77	C₄×C₅×D₃	—	2	4	no	—
78	C₄×C₅×D₃	—	2	4	no	—
79	C₃×C₅×C₈	—	4	2	no	—
80	C₅×C₃⋊C₈	—	4	2	no	—
81	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
82	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
84	C₂×(C₂×D₃)⋊C₄	C₄×C₅	1	2	yes	—
85	C₂×Dic₃⋊C₄	C₄×C₅	1	2	yes	—
86	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
88	#872	—	2	4	no	—
89	(C₂×D₃)⋊C₈	—	2	4	no	—
90	Dic₃⋊C₈	—	2	4	no	—
91	C₂⁴×C₅	—	3	1	no	—
92	C₂²×C₄×C₅	—	3	2	no	—
93	C₂²×C₄×C₅	—	3	1	no	—
94	C₂²×C₄×C₅	—	3	2	no	—
95	C₂²×C₄×C₅	—	3	2	no	—
96	C₂²×C₄×C₅	—	6	2	no	—
97	C₂²×C₄×C₅	—	6	2	no	—
98	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
99	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	6	2	no	—
104	C₂×C₅×C₈	—	6	2	no	—
105	C₂×C₅×C₈	—	6	2	no	—
106	C₅×C₄⋊C₄	—	6	4	no	—
107	C₅×C₂²⋊C₄	—	6	4	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	3	4	no	—
110	C₂×#117	—	3	1	no	—
111	C₂×C₄⋊C₈	—	3	2	no	—
112	C₂×C₂²⋊C₈	—	3	2	no	—
113	#336	—	3	1	no	—
114	#336	C₅×D₃	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#117	1	1	yes	characteristic
118	C₂²×C₃×C₅	#131	1	2	yes	—
119	C₂²×C₃×C₅	#131	1	2	yes	—
120	C₂×C₅×D₃	—	2	2	no	—
121	C₂×C₅×D₃	—	2	4	no	—
122	C₂×C₅×D₃	—	4	2	no	—
123	C₂×C₅×D₃	—	4	1	no	—
124	C₂×C₅×D₃	—	4	1	no	—
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	#121	1	2	yes	—
127	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
128	C₅×Dic₃	—	2	2	no	—
129	C₅×Dic₃	—	2	4	no	—
130	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
132	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
134	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂×C₄×D₃	—	2	2	no	—
136	C₂×C₄×D₃	—	2	2	no	—
137	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
138	C₂×C₄×D₃	C₅×Q₈	1	2	yes	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₃×C₄²	—	2	2	no	—
143	(C₂×D₃)⋊C₄	—	2	4	no	—
144	Dic₃⋊C₄	—	2	4	no	—
145	C₄×Dic₃	C₅×D₄	1	4	yes	—
146	C₂×C₃⋊C₈	—	2	2	no	—
147	C₂×C₃⋊C₈	—	2	2	no	—
148	C₂×C₃⋊C₈	—	2	2	no	—
149	C₂³×C₅	—	3	2	no	—
150	C₂³×C₅	—	6	1	no	—
151	C₂³×C₅	—	6	1	no	—
152	C₂³×C₅	—	6	2	no	—
153	C₂³×C₅	—	6	2	no	—
154	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	3	4	no	—
159	C₂×C₄×C₅	—	3	2	no	—
160	C₂×C₄×C₅	—	6	1	no	—
161	C₂×C₄×C₅	—	6	1	no	—
162	C₂×C₄×C₅	—	6	2	no	—
163	C₂×C₄×C₅	—	6	2	no	—
164	C₂×C₄×C₅	—	6	4	no	—
165	C₂×C₄×C₅	—	6	4	no	—
166	C₂×C₄×C₅	#235	1	2	yes	—
167	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
169	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
170	C₅×C₈	—	4	2	no	—
171	C₅×C₈	—	12	2	no	—
172	C₂³×C₄	—	3	1	no	—
173	C₂²×C₈	—	3	2	no	—
174	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
175	#117	—	6	4	no	—
176	C₂×C₄⋊C₄	—	3	2	no	—
177	C₂×C₂²⋊C₄	—	3	2	no	—
178	C₂×C₄²	—	3	1	no	—
179	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
180	C₄⋊C₈	—	6	4	no	—
181	C₂²⋊C₈	—	6	4	no	—
182	C₂×C₃×C₅	#380	1	1	yes	characteristic
183	C₂×C₃×C₅	#408	1	2	yes	—
184	C₂×C₃×C₅	#424	1	2	yes	—
185	C₂×C₃×C₅	#433	1	1	yes	characteristic
186	C₂×C₃×C₅	#433	1	1	yes	characteristic
187	C₅×D₃	—	4	2	no	—
188	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×D₃	—	2	1	no	—
190	C₂²×D₃	—	2	1	no	—
191	C₂²×D₃	—	2	2	no	—
192	C₂²×D₃	—	2	2	no	—
193	C₂²×D₃	C₅×C₄⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₅×#41	1	2	yes	—
201	C₂×Dic₃	—	2	1	no	—
202	C₂×Dic₃	—	2	1	no	—
203	C₂×Dic₃	—	2	2	no	—
204	C₂×Dic₃	—	2	2	no	—
205	C₂×Dic₃	C₅×C₂²⋊C₄	1	4	yes	—
206	C₂×Dic₃	C₅×C₄⋊C₄	1	2	yes	—
207	C₄×D₃	—	2	4	no	—
208	C₄×D₃	—	2	4	no	—
209	C₃×C₈	—	4	2	no	—
210	C₃⋊C₈	—	4	2	no	—
211	C₂²×C₅	—	6	2	no	—
212	C₂²×C₅	—	6	4	no	—
213	C₂²×C₅	—	12	1	no	—
214	C₂²×C₅	—	12	1	no	—
215	C₂²×C₅	—	12	2	no	—
216	C₂²×C₅	#872	1	1	yes	characteristic
217	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
219	C₂²×C₅	#901	1	2	yes	—
220	C₂²×C₅	#907	1	2	yes	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	—	6	2	no	—
223	C₄×C₅	—	6	4	no	—
224	C₄×C₅	#884	1	2	yes	—
225	C₄×C₅	#896	1	2	yes	—
226	C₂⁴	—	3	1	no	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	—	3	1	no	—
229	C₂²×C₄	—	3	2	no	—
230	C₂²×C₄	—	3	2	no	—
231	C₂²×C₄	—	6	2	no	—
232	C₂²×C₄	—	6	2	no	—
233	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
234	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	6	2	no	—
239	C₂×C₈	—	6	2	no	—
240	C₂×C₈	—	6	2	no	—
241	C₄⋊C₄	—	6	4	no	—
242	C₂²⋊C₄	—	6	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	3	4	no	—
245	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
246	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
248	C₂²×C₃	C₅×#117	1	1	yes	characteristic
249	C₂²×C₃	C₅×#131	1	2	yes	—
250	C₂²×C₃	C₅×#131	1	2	yes	—
251	C₂×D₃	—	2	2	no	—
252	C₂×D₃	—	2	4	no	—
253	C₂×D₃	—	4	2	no	—
254	C₂×D₃	—	4	1	no	—
255	C₂×D₃	—	4	1	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	C₅×#121	1	2	yes	—
258	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
259	Dic₃	—	2	2	no	—
260	Dic₃	—	2	4	no	—
261	C₂×C₅	—	12	2	no	—
262	C₂×C₅	#4667	1	1	yes	characteristic
263	C₂×C₅	#4690	1	1	yes	characteristic
264	C₂×C₅	#4709	1	1	yes	characteristic
265	C₂×C₅	#4727	1	2	yes	—
266	C₂×C₅	#4839	1	2	yes	—
267	C₂³	—	3	2	no	—
268	C₂³	—	6	1	no	—
269	C₂³	—	6	1	no	—
270	C₂³	—	6	2	no	—
271	C₂³	—	6	2	no	—
272	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	3	4	no	—
277	C₂×C₄	—	3	2	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	6	2	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	—	6	4	no	—
284	C₂×C₄	C₅×#235	1	2	yes	—
285	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
286	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
287	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
288	C₈	—	4	2	no	—
289	C₈	—	12	2	no	—
290	C₂×C₃	C₅×#380	1	1	yes	characteristic
291	C₂×C₃	C₅×#408	1	2	yes	—
292	C₂×C₃	C₅×#424	1	2	yes	—
293	C₂×C₃	C₅×#433	1	1	yes	characteristic
294	C₂×C₃	C₅×#433	1	1	yes	characteristic
295	D₃	—	4	2	no	—
296	C₅	#68663	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	6	2	no	—
298	C₂²	—	6	4	no	—
299	C₂²	—	12	1	no	—
300	C₂²	—	12	1	no	—
301	C₂²	—	12	2	no	—
302	C₂²	C₅×#872	1	1	yes	characteristic
303	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
304	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
305	C₂²	C₅×#901	1	2	yes	—
306	C₂²	C₅×#907	1	2	yes	—
307	C₄	—	4	2	no	—
308	C₄	—	6	2	no	—
309	C₄	—	6	4	no	—
310	C₄	C₅×#884	1	2	yes	—
311	C₄	C₅×#896	1	2	yes	—
312	C₃	C₅×#1798	1	1	yes	characteristic, complemented, Sylow
313	C₂	—	12	2	no	—
314	C₂	C₅×#4667	1	1	yes	characteristic
315	C₂	C₅×#4690	1	1	yes	characteristic
316	C₂	C₅×#4709	1	1	yes	characteristic
317	C₂	C₅×#4727	1	2	yes	—
318	C₂	C₅×#4839	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#336	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₅×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#336	D₃
hypocenter	C₃	C₅×#1798
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₅×#336

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×Dic₃⋊C₈	C₂	semidirect
4. C₃×C₅×#336	C₂	semidirect
5. C₅×#4524	C₂	semidirect
17. #68663	C₅	direct (N × H)
22. C₅×#336	D₃	semidirect
44. C₂×Dic₃⋊C₈	C₂×C₅	semidirect
45. C₃×#336	C₂×C₅	semidirect
46. #4524	C₂×C₅	semidirect
114. #336	C₅×D₃	semidirect
245. C₃×C₅	#1798	semidirect
296. C₅	#68663	direct (N × H)
312. C₃	C₅×#1798	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#872	2
C₂×C₅×(C₂×D₃)⋊C₈	2
C₂×C₅×Dic₃⋊C₈	2
C₃×C₅×#336	2
C₅×#4524	2
C₅×#1798	3
#68663	5

Subgroups — C5×#68663 #14297470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68663 #14297470