Subgroups — C₅×#20802 #11943522

Order 1280.

1070 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#428	C₂	1	2	yes	maximal
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
5	C₅×#2317	C₂	1	2	yes	maximal
6	C₅×#1847	C₂	1	4	yes	maximal, complemented
7	C₅×#1846	C₂	1	2	yes	maximal, complemented
8	C₅×#1844	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#117	C₂²	1	2	yes	—
12	C₂×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₅×#428	—	2	8	no	—
15	C₅×#427	—	2	4	no	—
16	C₂×C₅×#107	C₂²	1	4	yes	—
17	C₂×C₅×#107	C₂²	1	2	yes	—
18	C₂×C₅×#106	—	2	4	no	—
19	C₂×C₅×#106	C₂²	1	1	yes	characteristic
20	C₂×C₅×#106	C₂²	1	2	yes	—
21	C₂×C₅×#106	C₂²	1	4	yes	—
22	C₂×C₅×#106	C₂²	1	1	yes	characteristic
23	C₂×C₅×#106	C₂²	1	2	yes	—
24	C₅×#405	C₄	1	8	yes	complemented
25	C₂×C₅×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
27	C₅×#389	—	2	4	no	—
28	C₅×#385	—	2	4	no	—
29	C₅×#384	C₂²	1	4	yes	—
30	C₅×#382	C₂²	1	4	yes	—
31	#20802	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₄×C₅	—	2	2	no	—
33	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₅×M₁₆	D₄	1	4	yes	—
36	C₂×C₅×M₁₆	Q₈	1	4	yes	—
37	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₅×C₈	D₄	1	4	yes	—
39	C₂²×C₅×C₈	Q₈	1	4	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×#117	—	2	4	no	—
44	C₅×#117	—	2	4	no	—
45	C₅×#117	C₂×C₄	1	4	yes	—
46	C₅×#117	Q₈	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	—	2	4	no	—
48	C₂×C₅×C₄⋊C₄	—	2	4	no	—
49	C₂×C₅×C₄⋊C₄	—	2	2	no	—
50	C₂×C₅×C₄⋊C₄	—	2	8	no	—
51	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
54	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
56	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
57	C₂×C₄²×C₅	—	2	2	no	—
58	C₂×C₄²×C₅	C₂³	1	2	yes	—
59	C₅×#107	—	2	4	no	—
60	C₅×#107	—	2	8	no	—
61	C₅×#106	—	2	4	no	—
62	C₅×#106	—	2	4	no	—
63	C₅×#106	—	2	2	no	—
64	C₅×#106	—	2	2	no	—
65	C₅×#106	—	2	8	no	—
66	C₅×#97	C₂×C₄	1	8	yes	complemented
67	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
68	C₅×#95	—	2	8	no	—
69	C₂×#428	C₂×C₅	1	2	yes	—
70	C₂×#427	C₂×C₅	1	1	yes	characteristic
71	C₂²×#106	C₂×C₅	1	1	yes	characteristic
72	C₂×#405	C₂×C₅	1	1	yes	characteristic
73	#2317	C₂×C₅	1	2	yes	—
74	#1847	C₂×C₅	1	4	yes	complemented
75	#1846	C₂×C₅	1	2	yes	complemented
76	#1844	C₂×C₅	1	2	yes	complemented
77	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	4	4	no	—
82	C₂²×C₄×C₅	—	8	4	no	—
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×D₄	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	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₅×M₁₆	C₄⋊C₄	1	16	yes	complemented
93	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	complemented
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
100	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
101	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
102	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	4	4	no	—
110	C₅×C₄⋊C₄	—	8	8	no	—
111	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
112	C₅×C₂²⋊C₄	—	4	4	no	—
113	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
114	C₄²×C₅	—	4	4	no	—
115	C₄²×C₅	C₂²×C₄	1	4	yes	—
116	C₄²×C₅	C₂×Q₈	1	4	yes	—
117	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
119	C₂×#117	C₂²×C₅	1	2	yes	—
120	C₂×#117	C₂²×C₅	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
122	#428	—	2	8	no	—
123	#427	—	2	4	no	—
124	C₂×#107	C₂²×C₅	1	4	yes	—
125	C₂×#107	C₂²×C₅	1	2	yes	—
126	C₂×#106	—	2	4	no	—
127	C₂×#106	C₂²×C₅	1	1	yes	characteristic
128	C₂×#106	C₂²×C₅	1	2	yes	—
129	C₂×#106	C₂²×C₅	1	4	yes	—
130	C₂×#106	C₂²×C₅	1	1	yes	characteristic
131	C₂×#106	C₂²×C₅	1	2	yes	—
132	#405	C₄×C₅	1	8	yes	complemented
133	C₂×#97	C₂²×C₅	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
135	#389	—	2	4	no	—
136	#385	—	2	4	no	—
137	#384	C₂²×C₅	1	4	yes	—
138	#382	C₂²×C₅	1	4	yes	—
139	C₂³×C₅	—	2	4	no	—
140	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
141	C₂³×C₅	#127	1	2	yes	—
142	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
143	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	—	2	8	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	—	4	4	no	—
149	C₂×C₄×C₅	—	4	8	no	—
150	C₂×C₄×C₅	—	4	4	no	—
151	C₂×C₄×C₅	—	8	8	no	—
152	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
153	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₅	#127	1	2	yes	—
157	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
158	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
159	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
160	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
167	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
168	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
171	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
172	C₂×M₁₆	C₅×D₄	1	4	yes	—
173	C₂×M₁₆	C₅×Q₈	1	4	yes	—
174	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
175	C₂²×C₈	C₅×D₄	1	4	yes	—
176	C₂²×C₈	C₅×Q₈	1	4	yes	—
177	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	#117	—	2	4	no	—
181	#117	—	2	4	no	—
182	#117	C₂×C₄×C₅	1	4	yes	—
183	#117	C₅×Q₈	1	4	yes	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₅	1	2	yes	—
196	#107	—	2	4	no	—
197	#107	—	2	8	no	—
198	#106	—	2	4	no	—
199	#106	—	2	4	no	—
200	#106	—	2	2	no	—
201	#106	—	2	2	no	—
202	#106	—	2	8	no	—
203	#97	C₂×C₄×C₅	1	8	yes	complemented
204	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
205	#95	—	2	8	no	—
206	C₂²×C₅	—	2	4	no	—
207	C₂²×C₅	—	2	4	no	—
208	C₂²×C₅	—	2	4	no	—
209	C₂²×C₅	#384	1	4	yes	—
210	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
211	C₂²×C₅	C₂×#127	1	1	yes	characteristic
212	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
213	C₂²×C₅	#577	1	4	yes	—
214	C₄×C₅	—	2	8	no	—
215	C₄×C₅	—	8	2	no	—
216	C₄×C₅	—	8	2	no	—
217	C₄×C₅	—	8	4	no	—
218	C₄×C₅	#384	1	4	yes	—
219	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
222	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
223	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	8	4	no	—
229	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
230	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
233	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
234	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
235	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
236	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
238	M₁₆	C₅×C₄⋊C₄	1	16	yes	complemented
239	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	complemented
240	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
242	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
243	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
244	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
245	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
246	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
247	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
248	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	8	8	no	—
257	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
260	C₄²	—	4	4	no	—
261	C₄²	C₂²×C₄×C₅	1	4	yes	—
262	C₄²	C₂×C₅×Q₈	1	4	yes	—
263	C₂×C₅	—	2	4	no	—
264	C₂×C₅	C₂×#384	1	1	yes	characteristic
265	C₂×C₅	#2912	1	2	yes	—
266	C₂×C₅	#3154	1	2	yes	—
267	C₂×C₅	#3163	1	2	yes	—
268	C₂³	—	2	4	no	—
269	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
270	C₂³	C₅×#127	1	2	yes	—
271	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
272	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	8	8	no	—
281	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
282	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
284	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
285	C₂×C₄	C₅×#127	1	2	yes	—
286	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
287	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
288	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
289	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
295	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
296	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
297	C₈	C₂×C₅×C₄⋊C₄	1	8	yes	—
298	C₅	#20802	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₅×#384	1	4	yes	—
303	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
304	C₂²	C₂×C₅×#127	1	1	yes	characteristic
305	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
306	C₂²	C₅×#577	1	4	yes	—
307	C₄	—	2	8	no	—
308	C₄	—	8	4	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₅×#384	1	4	yes	—
312	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
313	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
314	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
315	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×C₅×#384	1	1	yes	characteristic
318	C₂	C₅×#2912	1	2	yes	—
319	C₂	C₅×#3154	1	2	yes	—
320	C₂	C₅×#3163	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
6. C₅×#1847	C₂	semidirect
7. C₅×#1846	C₂	semidirect
8. C₅×#1844	C₂	semidirect
24. C₅×#405	C₄	semidirect
31. #20802	C₅	direct (N × H)
66. C₅×#97	C₂×C₄	semidirect
67. C₄×C₅×C₈	C₂×C₄	semidirect
74. #1847	C₂×C₅	semidirect
75. #1846	C₂×C₅	semidirect
76. #1844	C₂×C₅	semidirect
92. C₅×M₁₆	C₄⋊C₄	semidirect
93. C₂×C₅×C₈	C₄⋊C₄	semidirect
132. #405	C₄×C₅	semidirect
203. #97	C₂×C₄×C₅	semidirect
204. C₄×C₈	C₂×C₄×C₅	semidirect
238. M₁₆	C₅×C₄⋊C₄	semidirect
239. C₂×C₈	C₅×C₄⋊C₄	semidirect
298. C₅	#20802	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×#428	2
C₂×C₅×#427	2
C₂²×C₅×#106	2
C₂×C₅×#405	2
C₅×#2317	2
C₅×#1847	2
C₅×#1846	2
C₅×#1844	2
#20802	5

Subgroups — C5×#20802 #11943522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20802 #11943522