Subgroups — C₅×#20541 #11943261

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 346 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	C₅×#2296	C₂	1	1	yes	maximal, characteristic
3	C₅×#1771	C₂	1	2	yes	maximal, complemented
4	C₅×#1771	C₂	1	2	yes	maximal, complemented
5	C₅×#1770	C₂	1	2	yes	maximal, complemented
6	C₅×C₈×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₅×#1750	C₂	1	2	yes	maximal
8	C₅×#1749	C₂	1	1	yes	maximal, characteristic
9	C₄×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
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	C₄	1	16	yes	complemented
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₅×#97	C₂²	1	1	yes	characteristic
21	C₂×C₅×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
26	C₅×#389	C₄	1	2	yes	—
27	C₅×#389	C₄	1	2	yes	—
28	C₅×#385	C₄	1	4	yes	—
29	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
30	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	2	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	2	yes	—
35	#20541	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₅×#117	—	4	1	no	—
43	C₅×#117	—	4	1	no	—
44	C₅×#117	C₂×C₄	1	8	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅	—	4	1	no	—
53	C₂×C₄²×C₅	—	4	1	no	—
54	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅	C₂³	1	2	yes	—
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅	C₂³	1	2	yes	—
59	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
60	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
61	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
62	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
63	C₅×#97	—	4	1	no	—
64	C₅×#97	—	4	1	no	—
65	C₅×#97	—	4	2	no	—
66	C₅×#97	C₂×C₄	1	8	yes	complemented
67	C₄×C₅×C₈	—	4	1	no	—
68	C₄×C₅×C₈	—	4	1	no	—
69	C₄×C₅×C₈	—	4	2	no	—
70	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
71	C₅×#95	C₂×C₄	1	4	yes	—
72	C₅×#95	C₂×C₄	1	4	yes	—
73	C₂×#433	C₂×C₅	1	2	yes	—
74	#2296	C₂×C₅	1	1	yes	characteristic
75	#1771	C₂×C₅	1	2	yes	complemented
76	#1771	C₂×C₅	1	2	yes	complemented
77	#1770	C₂×C₅	1	2	yes	complemented
78	C₈×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
79	#1750	C₂×C₅	1	2	yes	—
80	#1749	C₂×C₅	1	1	yes	characteristic
81	C₄×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
82	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	2	1	no	—
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	4	no	—
87	C₂²×C₄×C₅	—	4	2	no	—
88	C₂²×C₄×C₅	—	4	2	no	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
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	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
109	C₅×C₄⋊C₄	—	2	8	no	—
110	C₅×C₄⋊C₄	—	4	4	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	C₄²	1	4	yes	—
113	C₅×C₄⋊C₄	C₄²	1	4	yes	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	C₄²	1	8	yes	—
116	C₄²×C₅	—	2	1	no	—
117	C₄²×C₅	—	2	1	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	—	2	2	no	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	—	4	2	no	—
123	C₄²×C₅	C₂²×C₄	1	4	yes	—
124	C₄²×C₅	C₂²×C₄	1	4	yes	—
125	C₂×#117	C₂²×C₅	1	1	yes	characteristic
126	C₂×#117	C₂²×C₅	1	1	yes	characteristic
127	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
128	#433	C₄×C₅	1	16	yes	complemented
129	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
131	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
132	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
133	C₂×#97	C₂²×C₅	1	1	yes	characteristic
134	C₂×#97	C₂²×C₅	1	1	yes	characteristic
135	C₂×#97	C₂²×C₅	1	2	yes	—
136	C₂×#97	C₂²×C₅	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
141	#389	C₄×C₅	1	2	yes	—
142	#389	C₄×C₅	1	2	yes	—
143	#385	C₄×C₅	1	4	yes	—
144	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
145	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
146	#336	C₂²×C₅	1	2	yes	—
147	#336	C₂²×C₅	1	2	yes	—
148	#336	C₂²×C₅	1	2	yes	—
149	#336	C₂²×C₅	1	2	yes	—
150	C₂³×C₅	—	2	1	no	—
151	C₂³×C₅	—	2	1	no	—
152	C₂³×C₅	—	2	2	no	—
153	C₂³×C₅	—	2	2	no	—
154	C₂³×C₅	C₂×C₄²	1	2	yes	—
155	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	4	8	no	—
171	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
172	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
173	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
174	C₂×C₄×C₅	#131	1	4	yes	—
175	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
176	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
179	C₅×C₈	—	2	4	no	—
180	C₅×C₈	—	2	4	no	—
181	C₅×C₈	—	2	4	no	—
182	C₅×C₈	—	2	4	no	—
183	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
184	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₅	1	2	yes	—
186	C₂²×C₈	C₂³×C₅	1	2	yes	—
187	C₂²×C₈	C₂³×C₅	1	2	yes	—
188	C₂²×C₈	C₂³×C₅	1	2	yes	—
189	#117	—	4	1	no	—
190	#117	—	4	1	no	—
191	#117	C₂×C₄×C₅	1	8	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
198	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
199	C₂×C₄²	—	4	1	no	—
200	C₂×C₄²	—	4	1	no	—
201	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
202	C₂×C₄²	C₂³×C₅	1	2	yes	—
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₅	1	2	yes	—
206	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	complemented
207	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	complemented
208	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
209	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
210	#97	—	4	1	no	—
211	#97	—	4	1	no	—
212	#97	—	4	2	no	—
213	#97	C₂×C₄×C₅	1	8	yes	complemented
214	C₄×C₈	—	4	1	no	—
215	C₄×C₈	—	4	1	no	—
216	C₄×C₈	—	4	2	no	—
217	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
218	#95	C₂×C₄×C₅	1	4	yes	—
219	#95	C₂×C₄×C₅	1	4	yes	—
220	C₂²×C₅	—	2	2	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	4	2	no	—
224	C₂²×C₅	—	4	2	no	—
225	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
226	C₂²×C₅	#518	1	1	yes	characteristic
227	C₂²×C₅	#519	1	1	yes	characteristic
228	C₂²×C₅	C₂×#131	1	2	yes	—
229	C₂²×C₅	#568	1	2	yes	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	—	2	4	no	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	4	4	no	—
235	C₄×C₅	—	4	2	no	—
236	C₄×C₅	#520	1	2	yes	—
237	C₄×C₅	#520	1	2	yes	—
238	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
251	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
262	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
263	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
264	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	C₄²×C₅	1	4	yes	—
269	C₄⋊C₄	C₄²×C₅	1	4	yes	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	C₄²×C₅	1	8	yes	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	C₂²×C₄×C₅	1	4	yes	—
280	C₄²	C₂²×C₄×C₅	1	4	yes	—
281	C₂×C₅	—	4	2	no	—
282	C₂×C₅	#2274	1	1	yes	characteristic
283	C₂×C₅	C₄×#131	1	1	yes	characteristic
284	C₂×C₅	#2875	1	1	yes	characteristic
285	C₂×C₅	#2976	1	2	yes	—
286	C₂×C₅	#2985	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×C₄²×C₅	1	2	yes	—
292	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	8	no	—
308	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
309	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
310	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
311	C₂×C₄	C₅×#131	1	4	yes	—
312	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
313	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
314	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
315	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
316	C₈	—	2	4	no	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₅	#20541	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
327	C₂²	C₅×#518	1	1	yes	characteristic
328	C₂²	C₅×#519	1	1	yes	characteristic
329	C₂²	C₂×C₅×#131	1	2	yes	—
330	C₂²	C₅×#568	1	2	yes	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	4	no	—
337	C₄	C₅×#520	1	2	yes	—
338	C₄	C₅×#520	1	2	yes	—
339	C₂	—	4	2	no	—
340	C₂	C₅×#2274	1	1	yes	characteristic
341	C₂	C₄×C₅×#131	1	1	yes	characteristic
342	C₂	C₅×#2875	1	1	yes	characteristic
343	C₂	C₅×#2976	1	2	yes	—
344	C₂	C₅×#2985	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×#1771	C₂	semidirect
4. C₅×#1771	C₂	semidirect
5. C₅×#1770	C₂	semidirect
6. C₅×C₈×C₄⋊C₄	C₂	semidirect
13. C₅×#433	C₄	semidirect
35. #20541	C₅	direct (N × H)
59. C₅×C₄⋊C₈	C₂×C₄	semidirect
60. C₅×C₄⋊C₈	C₂×C₄	semidirect
66. C₅×#97	C₂×C₄	semidirect
70. C₄×C₅×C₈	C₂×C₄	semidirect
75. #1771	C₂×C₅	semidirect
76. #1771	C₂×C₅	semidirect
77. #1770	C₂×C₅	semidirect
78. C₈×C₄⋊C₄	C₂×C₅	semidirect
128. #433	C₄×C₅	semidirect
206. C₄⋊C₈	C₂×C₄×C₅	semidirect
207. C₄⋊C₈	C₂×C₄×C₅	semidirect
213. #97	C₂×C₄×C₅	semidirect
217. C₄×C₈	C₂×C₄×C₅	semidirect
320. C₅	#20541	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
C₅×#2296	2
C₅×#1771	2
C₅×#1770	2
C₅×C₈×C₄⋊C₄	2
C₅×#1750	2
C₅×#1749	2
C₄×C₅×C₂²⋊C₈	2
#20541	5

Subgroups — C5×#20541 #11943261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20541 #11943261