Subgroups — C₅×#20810 #11943530

Order 1280.

1110 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×#432	C₂	1	2	yes	maximal
2	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#1842	C₂	1	2	yes	maximal, complemented
4	C₅×#1831	C₂	1	2	yes	maximal, complemented
5	C₅×#1797	C₂	1	4	yes	maximal
6	C₅×#1767	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄²×C₅	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#432	—	2	8	no	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₅×#97	C₂²	1	4	yes	—
17	C₄³×C₅	C₄	1	1	yes	characteristic
18	C₄³×C₅	C₄	1	2	yes	—
19	C₄³×C₅	C₄	1	1	yes	characteristic
20	C₄³×C₅	C₂²	1	2	yes	—
21	C₄³×C₅	C₂²	1	1	yes	characteristic
22	C₄³×C₅	C₂²	1	1	yes	characteristic
23	C₅×#336	C₂²	1	8	yes	—
24	#20810	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	8	yes	—
31	C₂×C₄²×C₅	—	2	8	no	—
32	C₂×C₄²×C₅	—	2	4	no	—
33	C₂×C₄²×C₅	—	2	4	no	—
34	C₂×C₄²×C₅	—	2	8	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	D₄	1	4	yes	—
50	C₂×C₄²×C₅	Q₈	1	4	yes	—
51	C₂×C₄²×C₅	C₂³	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	2	yes	—
55	C₅×C₄⋊C₈	—	2	8	no	—
56	C₅×C₄⋊C₈	—	2	16	no	—
57	C₅×C₂²⋊C₈	—	2	8	no	—
58	C₅×#97	—	2	8	no	—
59	C₂×#432	C₂×C₅	1	2	yes	—
60	C₂×C₄³	C₂×C₅	1	1	yes	characteristic
61	#1842	C₂×C₅	1	2	yes	complemented
62	#1831	C₂×C₅	1	2	yes	complemented
63	#1797	C₂×C₅	1	4	yes	—
64	#1767	C₂×C₅	1	4	yes	complemented
65	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₅	M₁₆	1	8	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₅×C₈	—	2	8	no	—
91	C₂×C₅×C₈	—	2	16	no	—
92	C₄²×C₅	—	2	4	no	—
93	C₄²×C₅	—	2	4	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	2	4	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	4	no	—
100	C₄²×C₅	—	2	4	no	—
101	C₄²×C₅	—	2	4	no	—
102	C₄²×C₅	—	2	4	no	—
103	C₄²×C₅	C₄⋊C₄	1	4	yes	—
104	C₄²×C₅	C₄⋊C₄	1	4	yes	—
105	C₄²×C₅	M₁₆	1	4	yes	—
106	C₄²×C₅	M₁₆	1	4	yes	—
107	C₄²×C₅	C₂×D₄	1	2	yes	—
108	C₄²×C₅	C₂×D₄	1	2	yes	—
109	C₄²×C₅	C₂×Q₈	1	2	yes	—
110	C₄²×C₅	C₂×Q₈	1	2	yes	—
111	C₄²×C₅	#41	1	8	yes	—
112	C₂²×C₄²	C₄×C₅	1	2	yes	—
113	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
114	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
115	C₂²×C₄²	C₂²×C₅	1	2	yes	—
116	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
117	#432	—	2	8	no	—
118	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
119	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
120	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
121	C₂×#97	C₂²×C₅	1	4	yes	—
122	C₄³	C₄×C₅	1	1	yes	characteristic
123	C₄³	C₄×C₅	1	2	yes	—
124	C₄³	C₄×C₅	1	1	yes	characteristic
125	C₄³	C₂²×C₅	1	1	yes	characteristic
126	C₄³	C₂²×C₅	1	1	yes	characteristic
127	C₄³	C₂²×C₅	1	2	yes	—
128	#336	C₂²×C₅	1	8	yes	—
129	C₂³×C₅	—	2	2	no	—
130	C₂³×C₅	—	2	1	no	—
131	C₂³×C₅	—	2	1	no	—
132	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂³×C₅	C₂×M₁₆	1	4	yes	—
134	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	—	2	16	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	1	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	—	2	2	no	—
147	C₂×C₄×C₅	—	2	8	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	2	8	no	—
151	C₂×C₄×C₅	—	2	16	no	—
152	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
157	C₂×C₄×C₅	#117	1	4	yes	—
158	C₂×C₄×C₅	#117	1	4	yes	—
159	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
160	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
161	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
162	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
163	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
164	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
165	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×#41	1	4	yes	—
168	C₅×C₈	—	4	8	no	—
169	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
170	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
172	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
173	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₅	1	8	yes	—
175	C₂×C₄²	—	2	8	no	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	—	2	4	no	—
178	C₂×C₄²	—	2	8	no	—
179	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
182	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
185	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₅×D₄	1	4	yes	—
194	C₂×C₄²	C₅×Q₈	1	4	yes	—
195	C₂×C₄²	C₂³×C₅	1	2	yes	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₅	1	2	yes	—
199	C₄⋊C₈	—	2	8	no	—
200	C₄⋊C₈	—	2	16	no	—
201	C₂²⋊C₈	—	2	8	no	—
202	#97	—	2	8	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	4	no	—
205	C₂²×C₅	—	2	4	no	—
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	#423	1	4	yes	—
208	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₅	C₂×#117	1	2	yes	—
210	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
211	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
212	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	2	4	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	#423	1	4	yes	—
221	C₄×C₅	#432	1	8	yes	complemented
222	C₄×C₅	#532	1	2	yes	—
223	C₄×C₅	#533	1	2	yes	—
224	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
235	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
236	C₂²×C₄	C₅×M₁₆	1	8	yes	—
237	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
246	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
247	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
248	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
249	C₂×C₈	—	2	8	no	—
250	C₂×C₈	—	2	16	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	C₅×C₄⋊C₄	1	4	yes	—
263	C₄²	C₅×C₄⋊C₄	1	4	yes	—
264	C₄²	C₅×M₁₆	1	4	yes	—
265	C₄²	C₅×M₁₆	1	4	yes	—
266	C₄²	C₂×C₅×D₄	1	2	yes	—
267	C₄²	C₂×C₅×D₄	1	2	yes	—
268	C₄²	C₂×C₅×Q₈	1	2	yes	—
269	C₄²	C₂×C₅×Q₈	1	2	yes	—
270	C₄²	C₅×#41	1	8	yes	—
271	C₂×C₅	—	2	4	no	—
272	C₂×C₅	#2285	1	1	yes	characteristic
273	C₂×C₅	C₂×#423	1	1	yes	characteristic
274	C₂×C₅	C₂×#423	1	1	yes	characteristic
275	C₂×C₅	C₂×#432	1	2	yes	—
276	C₂×C₅	#2920	1	2	yes	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
281	C₂³	C₂×C₅×M₁₆	1	4	yes	—
282	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	16	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
301	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
302	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
303	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
304	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
305	C₂×C₄	C₅×#117	1	4	yes	—
306	C₂×C₄	C₅×#117	1	4	yes	—
307	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
308	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
309	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
310	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
311	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
312	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
313	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#41	1	4	yes	—
316	C₈	—	4	8	no	—
317	C₅	#20810	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	2	4	no	—
322	C₂²	C₅×#423	1	4	yes	—
323	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
324	C₂²	C₂×C₅×#117	1	2	yes	—
325	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
326	C₂²	C₂²×C₅×M₁₆	1	2	yes	—
327	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	C₅×#423	1	4	yes	—
336	C₄	C₅×#432	1	8	yes	complemented
337	C₄	C₅×#532	1	2	yes	—
338	C₄	C₅×#533	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₅×#2285	1	1	yes	characteristic
341	C₂	C₂×C₅×#423	1	1	yes	characteristic
342	C₂	C₂×C₅×#423	1	1	yes	characteristic
343	C₂	C₂×C₅×#432	1	2	yes	—
344	C₂	C₅×#2920	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×#1842	C₂	semidirect
4. C₅×#1831	C₂	semidirect
6. C₅×#1767	C₂	semidirect
24. #20810	C₅	direct (N × H)
61. #1842	C₂×C₅	semidirect
62. #1831	C₂×C₅	semidirect
64. #1767	C₂×C₅	semidirect
221. C₄×C₅	#432	semidirect
317. C₅	#20810	direct (N × H)
336. C₄	C₅×#432	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#432	2
C₂×C₄³×C₅	2
C₅×#1842	2
C₅×#1831	2
C₅×#1797	2
C₅×#1767	2
#20810	5

Subgroups — C5×#20810 #11943530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20810 #11943530