Subgroups — C₅×#20820 #11943540

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₅×#2314	C₂	1	2	yes	maximal
3	C₅×#1846	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1836	C₂	1	2	yes	maximal, complemented
6	C₅×#1809	C₂	1	2	yes	maximal
7	C₅×#1807	C₂	1	2	yes	maximal
8	C₅×#1776	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×#106	C₂	1	2	yes	maximal, complemented
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₅×#107	C₂²	1	2	yes	—
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₅×#106	C₂²	1	2	yes	—
17	C₅×#405	C₄	1	8	yes	complemented
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#388	—	2	4	no	—
23	C₅×#387	—	2	4	no	—
24	C₅×#384	C₂²	1	2	yes	—
25	C₅×#382	C₂²	1	2	yes	—
26	C₅×#380	C₂²	1	4	yes	—
27	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
28	C₅×#340	C₂²	1	4	yes	—
29	C₅×#340	C₂²	1	4	yes	—
30	#20820	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	—	2	2	no	—
37	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×#117	C₂×C₄	1	4	yes	—
43	C₅×#117	Q₈	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	—	2	4	no	—
45	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
50	C₂×C₄²×C₅	—	2	4	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×#107	—	2	4	no	—
54	C₅×#106	—	2	4	no	—
55	C₅×#106	—	2	4	no	—
56	C₅×#106	—	2	4	no	—
57	C₅×#97	C₂×C₄	1	4	yes	—
58	C₅×#97	C₂×C₄	1	4	yes	—
59	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
61	C₅×#95	—	2	4	no	—
62	C₅×#95	—	2	4	no	—
63	C₅×#95	—	2	4	no	—
64	C₂×#405	C₂×C₅	1	1	yes	characteristic
65	#2314	C₂×C₅	1	2	yes	—
66	#1846	C₂×C₅	1	1	yes	characteristic, complemented
67	#1844	C₂×C₅	1	1	yes	characteristic, complemented
68	#1836	C₂×C₅	1	2	yes	complemented
69	#1809	C₂×C₅	1	2	yes	—
70	#1807	C₂×C₅	1	2	yes	—
71	#1776	C₂×C₅	1	2	yes	complemented
72	C₄×#106	C₂×C₅	1	2	yes	complemented
73	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	2	1	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	—	2	4	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₅×M₁₆	—	2	8	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	8	no	—
92	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₅×C₈	#41	1	2	yes	—
97	C₂×C₅×C₈	#41	1	2	yes	—
98	C₂×C₅×C₈	#41	1	2	yes	—
99	C₂×C₅×C₈	#41	1	2	yes	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	8	4	no	—
105	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
106	C₅×C₂²⋊C₄	—	8	4	no	—
107	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
108	C₄²×C₅	—	8	4	no	—
109	C₄²×C₅	C₂²×C₄	1	2	yes	—
110	C₄²×C₅	C₂²×C₄	1	2	yes	—
111	C₄²×C₅	C₂×Q₈	1	2	yes	—
112	C₄²×C₅	C₂×Q₈	1	2	yes	—
113	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×#117	C₂²×C₅	1	1	yes	characteristic
116	C₂×#107	C₂²×C₅	1	2	yes	—
117	C₂×#106	C₂²×C₅	1	2	yes	—
118	C₂×#106	C₂²×C₅	1	2	yes	—
119	C₂×#106	C₂²×C₅	1	2	yes	—
120	#405	C₄×C₅	1	8	yes	complemented
121	C₂×#97	C₂²×C₅	1	1	yes	characteristic
122	C₂×#97	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
125	#388	—	2	4	no	—
126	#387	—	2	4	no	—
127	#384	C₂²×C₅	1	2	yes	—
128	#382	C₂²×C₅	1	2	yes	—
129	#380	C₂²×C₅	1	4	yes	—
130	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
131	#340	C₂²×C₅	1	4	yes	—
132	#340	C₂²×C₅	1	4	yes	—
133	C₂³×C₅	—	2	1	no	—
134	C₂³×C₅	—	2	1	no	—
135	C₂³×C₅	—	2	4	no	—
136	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
137	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	—	4	4	no	—
149	C₂×C₄×C₅	—	4	4	no	—
150	C₂×C₄×C₅	—	4	4	no	—
151	C₂×C₄×C₅	—	4	4	no	—
152	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
153	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	2	2	no	—
163	C₅×C₈	#117	1	4	yes	—
164	C₅×C₈	#117	1	4	yes	—
165	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
176	#117	C₂×C₄×C₅	1	4	yes	—
177	#117	C₅×Q₈	1	4	yes	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
181	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#106	—	2	4	no	—
189	#106	—	2	4	no	—
190	#106	—	2	4	no	—
191	#97	C₂×C₄×C₅	1	4	yes	—
192	#97	C₂×C₄×C₅	1	4	yes	—
193	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
194	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
195	#95	—	2	4	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	—	4	2	no	—
202	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₂²×C₅	#532	1	1	yes	characteristic
204	C₂²×C₅	#532	1	1	yes	characteristic
205	C₂²×C₅	#577	1	4	yes	—
206	C₄×C₅	—	2	4	no	—
207	C₄×C₅	—	2	4	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	8	4	no	—
210	C₄×C₅	—	8	4	no	—
211	C₄×C₅	C₂×#117	1	1	yes	characteristic
212	C₄×C₅	C₂×#117	1	1	yes	characteristic
213	C₄×C₅	#533	1	1	yes	characteristic
214	C₄×C₅	#533	1	1	yes	characteristic
215	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
223	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
225	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
226	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
228	M₁₆	—	2	8	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
237	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
238	C₂×C₈	C₅×#41	1	2	yes	—
239	C₂×C₈	C₅×#41	1	2	yes	—
240	C₂×C₈	C₅×#41	1	2	yes	—
241	C₂×C₈	C₅×#41	1	2	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	8	4	no	—
247	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
248	C₂²⋊C₄	—	8	4	no	—
249	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
250	C₄²	—	8	4	no	—
251	C₄²	C₂²×C₄×C₅	1	2	yes	—
252	C₄²	C₂²×C₄×C₅	1	2	yes	—
253	C₄²	C₂×C₅×Q₈	1	2	yes	—
254	C₄²	C₂×C₅×Q₈	1	2	yes	—
255	C₂×C₅	—	4	2	no	—
256	C₂×C₅	#2285	1	1	yes	characteristic
257	C₂×C₅	#2912	1	2	yes	—
258	C₂×C₅	#3173	1	2	yes	—
259	C₂×C₅	#3174	1	2	yes	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	4	no	—
263	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
264	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
280	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
281	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
282	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	C₅×#117	1	4	yes	—
291	C₈	C₅×#117	1	4	yes	—
292	C₅	#20820	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
298	C₂²	C₅×#532	1	1	yes	characteristic
299	C₂²	C₅×#532	1	1	yes	characteristic
300	C₂²	C₅×#577	1	4	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	8	4	no	—
305	C₄	—	8	4	no	—
306	C₄	C₂×C₅×#117	1	1	yes	characteristic
307	C₄	C₂×C₅×#117	1	1	yes	characteristic
308	C₄	C₅×#533	1	1	yes	characteristic
309	C₄	C₅×#533	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	C₅×#2285	1	1	yes	characteristic
312	C₂	C₅×#2912	1	2	yes	—
313	C₂	C₅×#3173	1	2	yes	—
314	C₂	C₅×#3174	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1846	C₂	semidirect
4. C₅×#1844	C₂	semidirect
5. C₅×#1836	C₂	semidirect
8. C₅×#1776	C₂	semidirect
9. C₄×C₅×#106	C₂	semidirect
17. C₅×#405	C₄	semidirect
30. #20820	C₅	direct (N × H)
66. #1846	C₂×C₅	semidirect
67. #1844	C₂×C₅	semidirect
68. #1836	C₂×C₅	semidirect
71. #1776	C₂×C₅	semidirect
72. C₄×#106	C₂×C₅	semidirect
120. #405	C₄×C₅	semidirect
292. C₅	#20820	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×#405	2
C₅×#2314	2
C₅×#1846	2
C₅×#1844	2
C₅×#1836	2
C₅×#1809	2
C₅×#1807	2
C₅×#1776	2
C₄×C₅×#106	2
#20820	5

Subgroups — C5×#20820 #11943540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20820 #11943540