Subgroups — C₅×#22240 #11944960

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#1984	C₂	1	1	yes	maximal, characteristic
5	C₅×#1984	C₂	1	1	yes	maximal, characteristic
6	C₅×#1917	C₂	1	4	yes	maximal
7	C₅×#1916	C₂	1	4	yes	maximal
8	C₅×#1831	C₂	1	1	yes	maximal, characteristic
9	C₅×#1744	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₅×#446	—	2	4	no	—
14	C₅×C₈×Q₈	—	2	4	no	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×#391	C₄	1	1	yes	characteristic
24	C₅×#391	C₄	1	1	yes	characteristic
25	C₅×#389	C₄	1	4	yes	—
26	C₅×#384	C₄	1	1	yes	characteristic
27	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
28	C₄³×C₅	C₂²	1	1	yes	characteristic
29	C₅×#336	C₂²	1	4	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	4	yes	—
32	C₅×#336	C₂²	1	4	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	#22240	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Q₈	—	2	4	no	—
43	C₄×C₅×Q₈	—	2	2	no	—
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₄×C₅×Q₈	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅	C₂³	1	2	yes	—
57	C₂×C₄²×C₅	C₂³	1	2	yes	—
58	C₂×C₄²×C₅	C₂³	1	2	yes	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₅×C₄⋊C₈	—	2	4	no	—
61	C₅×C₄⋊C₈	—	2	4	no	—
62	C₅×C₄⋊C₈	—	2	2	no	—
63	C₅×C₄⋊C₈	—	2	2	no	—
64	C₅×#97	—	2	4	no	—
65	C₄×C₅×C₈	—	2	4	no	—
66	C₄×C₅×C₈	—	2	2	no	—
67	C₄×C₅×C₈	D₄	1	4	yes	—
68	C₅×#95	C₂×C₄	1	4	yes	—
69	C₅×#95	C₂×C₄	1	4	yes	—
70	C₂×#446	C₂×C₅	1	1	yes	characteristic
71	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
72	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
73	#1984	C₂×C₅	1	1	yes	characteristic
74	#1984	C₂×C₅	1	1	yes	characteristic
75	#1917	C₂×C₅	1	4	yes	—
76	#1916	C₂×C₅	1	4	yes	—
77	#1831	C₂×C₅	1	1	yes	characteristic
78	#1744	C₂×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	M₁₆	1	4	yes	—
85	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
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	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	4	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	2	8	no	—
110	C₅×C₄⋊C₄	—	2	8	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	4	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	2	2	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	—	2	2	no	—
119	C₄²×C₅	—	2	2	no	—
120	C₄²×C₅	C₂×D₄	1	2	yes	—
121	C₄²×C₅	#41	1	2	yes	—
122	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
125	#446	—	2	4	no	—
126	C₈×Q₈	—	2	4	no	—
127	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
129	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
131	C₂×#97	C₂²×C₅	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
135	#391	C₄×C₅	1	1	yes	characteristic
136	#391	C₄×C₅	1	1	yes	characteristic
137	#389	C₄×C₅	1	4	yes	—
138	#384	C₄×C₅	1	1	yes	characteristic
139	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
140	C₄³	C₂²×C₅	1	1	yes	characteristic
141	#336	C₂²×C₅	1	4	yes	—
142	#336	C₂²×C₅	1	2	yes	—
143	#336	C₂²×C₅	1	4	yes	—
144	#336	C₂²×C₅	1	4	yes	—
145	#336	C₂²×C₅	1	2	yes	—
146	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₅×Q₈	—	2	8	no	—
148	C₅×Q₈	—	4	4	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	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	2	no	—
164	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
167	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
168	C₂×C₄×C₅	#131	1	4	yes	—
169	C₂×C₄×C₅	#131	1	2	yes	—
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
173	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
174	C₅×C₈	—	2	2	no	—
175	C₅×C₈	—	2	2	no	—
176	C₅×C₈	—	4	2	no	—
177	C₅×C₈	—	4	2	no	—
178	C₅×C₈	—	4	2	no	—
179	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₅	1	2	yes	—
182	C₂²×C₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₈	C₂³×C₅	1	2	yes	—
184	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	4	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
197	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₅	1	2	yes	—
201	C₂×C₄²	C₂³×C₅	1	2	yes	—
202	C₂×C₄²	C₂³×C₅	1	2	yes	—
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₄⋊C₈	—	2	4	no	—
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	#97	—	2	4	no	—
209	C₄×C₈	—	2	4	no	—
210	C₄×C₈	—	2	2	no	—
211	C₄×C₈	C₅×D₄	1	4	yes	—
212	#95	C₂×C₄×C₅	1	4	yes	—
213	#95	C₂×C₄×C₅	1	4	yes	—
214	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₅	#519	1	2	yes	—
216	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
217	C₂²×C₅	C₂×#131	1	1	yes	characteristic
218	C₂²×C₅	C₂×#131	1	2	yes	—
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	4	no	—
222	C₄×C₅	—	4	2	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	#435	1	4	yes	—
227	C₄×C₅	#536	1	1	yes	characteristic
228	C₄×C₅	#536	1	1	yes	characteristic
229	C₄×C₅	#541	1	2	yes	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	C₅×M₁₆	1	4	yes	—
236	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
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	1	yes	characteristic
244	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
255	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂×C₅×D₄	1	2	yes	—
272	C₄²	C₅×#41	1	2	yes	—
273	C₂×C₅	#2327	1	1	yes	characteristic
274	C₂×C₅	C₂×#435	1	1	yes	characteristic
275	C₂×C₅	#2931	1	1	yes	characteristic
276	C₂×C₅	#2984	1	2	yes	—
277	C₂×C₅	#2986	1	2	yes	—
278	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
279	Q₈	—	2	8	no	—
280	Q₈	—	4	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
297	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
298	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
299	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
300	C₂×C₄	C₅×#131	1	4	yes	—
301	C₂×C₄	C₅×#131	1	2	yes	—
302	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
304	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
305	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₅	#22240	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
313	C₂²	C₅×#519	1	2	yes	—
314	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
315	C₂²	C₂×C₅×#131	1	2	yes	—
316	C₂²	C₂×C₅×#131	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	C₅×#435	1	4	yes	—
325	C₄	C₅×#536	1	1	yes	characteristic
326	C₄	C₅×#536	1	1	yes	characteristic
327	C₄	C₅×#541	1	2	yes	—
328	C₂	C₅×#2327	1	1	yes	characteristic
329	C₂	C₂×C₅×#435	1	1	yes	characteristic
330	C₂	C₅×#2931	1	1	yes	characteristic
331	C₂	C₅×#2984	1	2	yes	—
332	C₂	C₅×#2986	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
34. #22240	C₅	direct (N × H)
311. C₅	#22240	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×#446	2
C₂×C₅×C₈×Q₈	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1984	2
C₅×#1917	2
C₅×#1916	2
C₅×#1831	2
C₅×#1744	2
#22240	5

Subgroups — C5×#22240 #11944960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22240 #11944960