Subgroups — C₅×#20813 #11943533

Order 1280.

854 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#433	C₂	1	2	yes	maximal
2	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
3	C₅×#1843	C₂	1	2	yes	maximal, complemented
4	C₅×#1831	C₂	1	2	yes	maximal, complemented
5	C₅×#1798	C₂	1	2	yes	maximal
6	C₅×#1797	C₂	1	2	yes	maximal
7	C₅×#1768	C₂	1	2	yes	maximal, complemented
8	C₅×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂×C₅×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#433	—	2	8	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
18	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
22	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
23	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
24	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
25	C₅×#376	C₂²	1	1	yes	characteristic
26	C₅×#376	C₂²	1	1	yes	characteristic
27	C₄³×C₅	C₂²	1	1	yes	characteristic
28	C₄³×C₅	C₂²	1	1	yes	characteristic
29	C₅×#336	C₂²	1	2	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	4	yes	—
32	#20813	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	4	yes	—
36	C₂²×C₅×C₈	C₂³	1	4	yes	—
37	C₅×#117	—	2	4	no	—
38	C₅×#117	D₄	1	4	yes	—
39	C₅×#117	Q₈	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅	—	2	4	no	—
47	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₅×C₄⋊C₈	—	2	8	no	—
59	C₅×C₄⋊C₈	—	2	8	no	—
60	C₅×C₄⋊C₈	—	2	4	no	—
61	C₅×C₄⋊C₈	—	2	4	no	—
62	C₅×C₂²⋊C₈	—	2	8	no	—
63	C₅×#97	—	2	4	no	—
64	C₄×C₅×C₈	—	2	4	no	—
65	C₅×#95	C₂×C₄	1	2	yes	—
66	C₅×#95	C₂×C₄	1	2	yes	—
67	C₅×#95	C₂×C₄	1	2	yes	—
68	C₅×#95	C₂×C₄	1	2	yes	—
69	C₂×#433	C₂×C₅	1	2	yes	—
70	C₄×#117	C₂×C₅	1	1	yes	characteristic
71	#1843	C₂×C₅	1	2	yes	complemented
72	#1831	C₂×C₅	1	2	yes	complemented
73	#1798	C₂×C₅	1	2	yes	—
74	#1797	C₂×C₅	1	2	yes	—
75	#1768	C₂×C₅	1	2	yes	complemented
76	#1767	C₂×C₅	1	1	yes	characteristic, complemented
77	C₄×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	4	2	no	—
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	—	2	4	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
105	C₅×C₂²⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	2	2	no	—
107	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
108	C₄²×C₅	—	2	4	no	—
109	C₄²×C₅	—	2	4	no	—
110	C₄²×C₅	—	2	4	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	4	2	no	—
114	C₄²×C₅	—	4	2	no	—
115	C₄²×C₅	C₂×D₄	1	2	yes	—
116	C₄²×C₅	C₂×D₄	1	2	yes	—
117	C₄²×C₅	C₂×Q₈	1	2	yes	—
118	C₄²×C₅	C₂×Q₈	1	2	yes	—
119	C₄²×C₅	#41	1	4	yes	—
120	C₄²×C₅	#41	1	4	yes	—
121	C₂×#117	C₂²×C₅	1	1	yes	characteristic
122	C₂×#117	C₂²×C₅	1	1	yes	characteristic
123	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
124	#433	—	2	8	no	—
125	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
126	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
129	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
130	C₂×#97	C₂²×C₅	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
132	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
133	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
134	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
135	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
136	#376	C₂²×C₅	1	1	yes	characteristic
137	#376	C₂²×C₅	1	1	yes	characteristic
138	C₄³	C₂²×C₅	1	1	yes	characteristic
139	C₄³	C₂²×C₅	1	1	yes	characteristic
140	#336	C₂²×C₅	1	2	yes	—
141	#336	C₂²×C₅	1	2	yes	—
142	#336	C₂²×C₅	1	4	yes	—
143	C₂³×C₅	—	2	1	no	—
144	C₂³×C₅	—	2	1	no	—
145	C₂³×C₅	—	2	2	no	—
146	C₂³×C₅	—	2	2	no	—
147	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
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	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	8	no	—
168	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
169	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
170	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
171	C₂×C₄×C₅	#117	1	4	yes	—
172	C₂×C₄×C₅	#117	1	2	yes	—
173	C₂×C₄×C₅	#117	1	2	yes	—
174	C₂×C₄×C₅	#131	1	4	yes	—
175	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×#41	1	2	yes	—
178	C₂×C₄×C₅	C₂×#41	1	2	yes	—
179	C₅×C₈	—	4	4	no	—
180	C₅×C₈	—	4	4	no	—
181	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
182	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₅	1	4	yes	—
184	C₂²×C₈	C₂³×C₅	1	4	yes	—
185	#117	—	2	4	no	—
186	#117	C₅×D₄	1	4	yes	—
187	#117	C₅×Q₈	1	4	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
200	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
206	C₄⋊C₈	—	2	8	no	—
207	C₄⋊C₈	—	2	8	no	—
208	C₄⋊C₈	—	2	4	no	—
209	C₄⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	8	no	—
211	#97	—	2	4	no	—
212	C₄×C₈	—	2	4	no	—
213	#95	C₂×C₄×C₅	1	2	yes	—
214	#95	C₂×C₄×C₅	1	2	yes	—
215	#95	C₂×C₄×C₅	1	2	yes	—
216	#95	C₂×C₄×C₅	1	2	yes	—
217	C₂²×C₅	—	2	4	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	4	1	no	—
220	C₂²×C₅	—	4	1	no	—
221	C₂²×C₅	—	4	2	no	—
222	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₂²×C₅	C₂×#117	1	1	yes	characteristic
224	C₂²×C₅	C₂×#117	1	1	yes	characteristic
225	C₂²×C₅	C₂×#131	1	2	yes	—
226	C₂²×C₅	#568	1	2	yes	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	2	4	no	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	—	2	2	no	—
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	—	2	4	no	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	4	1	no	—
235	C₄×C₅	—	4	1	no	—
236	C₄×C₅	#532	1	2	yes	—
237	C₄×C₅	#533	1	2	yes	—
238	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
253	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
254	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×C₅×D₄	1	2	yes	—
276	C₄²	C₂×C₅×D₄	1	2	yes	—
277	C₄²	C₂×C₅×Q₈	1	2	yes	—
278	C₄²	C₂×C₅×Q₈	1	2	yes	—
279	C₄²	C₅×#41	1	4	yes	—
280	C₄²	C₅×#41	1	4	yes	—
281	C₂×C₅	—	4	2	no	—
282	C₂×C₅	#2285	1	1	yes	characteristic
283	C₂×C₅	#2906	1	2	yes	—
284	C₂×C₅	#2922	1	1	yes	characteristic
285	C₂×C₅	#2923	1	1	yes	characteristic
286	C₂×C₅	#2924	1	2	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
292	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	8	no	—
312	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
313	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
314	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
315	C₂×C₄	C₅×#117	1	4	yes	—
316	C₂×C₄	C₅×#117	1	2	yes	—
317	C₂×C₄	C₅×#117	1	2	yes	—
318	C₂×C₄	C₅×#131	1	4	yes	—
319	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
321	C₂×C₄	C₂×C₅×#41	1	2	yes	—
322	C₂×C₄	C₂×C₅×#41	1	2	yes	—
323	C₈	—	4	4	no	—
324	C₈	—	4	4	no	—
325	C₅	#20813	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	2	2	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
332	C₂²	C₂×C₅×#117	1	1	yes	characteristic
333	C₂²	C₂×C₅×#117	1	1	yes	characteristic
334	C₂²	C₂×C₅×#131	1	2	yes	—
335	C₂²	C₅×#568	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	4	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	C₅×#532	1	2	yes	—
346	C₄	C₅×#533	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	C₅×#2285	1	1	yes	characteristic
349	C₂	C₅×#2906	1	2	yes	—
350	C₂	C₅×#2922	1	1	yes	characteristic
351	C₂	C₅×#2923	1	1	yes	characteristic
352	C₂	C₅×#2924	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1843	C₂	semidirect
4. C₅×#1831	C₂	semidirect
7. C₅×#1768	C₂	semidirect
8. C₅×#1767	C₂	semidirect
9. C₄×C₅×C₄⋊C₈	C₂	semidirect
32. #20813	C₅	direct (N × H)
71. #1843	C₂×C₅	semidirect
72. #1831	C₂×C₅	semidirect
75. #1768	C₂×C₅	semidirect
76. #1767	C₂×C₅	semidirect
77. C₄×C₄⋊C₈	C₂×C₅	semidirect
325. C₅	#20813	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#433	2
C₄×C₅×#117	2
C₅×#1843	2
C₅×#1831	2
C₅×#1798	2
C₅×#1797	2
C₅×#1768	2
C₅×#1767	2
C₄×C₅×C₄⋊C₈	2
#20813	5

Subgroups — C5×#20813 #11943533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20813 #11943533