Subgroups — C₅×#22231 #11944951

Order 1280.

726 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#491	C₂	1	1	yes	maximal, characteristic
2	C₅×#2330	C₂	1	1	yes	maximal, characteristic
3	C₅×#2327	C₂	1	1	yes	maximal, characteristic
4	C₅×#1991	C₂	1	1	yes	maximal, characteristic
5	C₅×#1956	C₂	1	1	yes	maximal, characteristic
6	C₅×#1928	C₂	1	2	yes	maximal
7	C₅×#1927	C₂	1	2	yes	maximal
8	C₅×#1865	C₂	1	2	yes	maximal
9	C₅×#1863	C₂	1	2	yes	maximal
10	C₅×#1847	C₂	1	1	yes	maximal, characteristic
11	C₅×#1746	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#125	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
16	C₅×#491	C₄	1	8	yes	complemented
17	C₂×C₅×#107	C₂²	1	2	yes	—
18	C₂×C₅×#106	C₂²	1	2	yes	—
19	C₂×C₅×#103	C₂²	1	2	yes	—
20	C₂×C₅×#103	C₂²	1	2	yes	—
21	C₂×C₅×#103	C₂²	1	2	yes	—
22	C₂×C₅×#103	C₂²	1	2	yes	—
23	C₂×C₅×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#391	C₂²	1	1	yes	characteristic
26	C₅×#391	C₂²	1	1	yes	characteristic
27	C₅×#389	—	2	2	no	—
28	C₅×#389	C₂²	1	2	yes	—
29	C₅×#387	—	2	2	no	—
30	C₅×#387	—	2	2	no	—
31	C₅×#387	—	2	2	no	—
32	C₅×#384	C₂²	1	1	yes	characteristic
33	C₅×#382	C₂²	1	1	yes	characteristic
34	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
35	C₅×#376	C₂²	1	1	yes	characteristic
36	C₅×#340	C₂²	1	2	yes	—
37	C₅×#340	C₂²	1	2	yes	—
38	C₅×#336	C₂²	1	2	yes	—
39	C₅×#336	C₂²	1	2	yes	—
40	#22231	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
47	C₅×#125	—	2	2	no	—
48	C₅×#125	C₂×C₄	1	4	yes	—
49	C₄×C₅×Q₈	—	4	2	no	—
50	C₄×C₅×Q₈	—	4	2	no	—
51	C₂×C₅×C₄⋊C₄	—	2	2	no	—
52	C₂×C₅×C₄⋊C₄	—	2	2	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	2	no	—
56	C₂×C₅×C₄⋊C₄	—	2	2	no	—
57	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
61	C₂×C₄²×C₅	—	2	1	no	—
62	C₂×C₄²×C₅	—	2	1	no	—
63	C₂×C₄²×C₅	—	2	2	no	—
64	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₅×#107	—	4	2	no	—
68	C₅×#106	—	4	2	no	—
69	C₅×#103	—	4	2	no	—
70	C₅×#103	—	4	2	no	—
71	C₅×#103	C₂×C₄	1	8	yes	—
72	C₅×#103	C₂×C₄	1	8	yes	—
73	C₅×#97	C₂×C₄	1	4	yes	—
74	C₄×C₅×C₈	—	2	2	no	—
75	C₅×#95	—	2	2	no	—
76	C₅×#95	—	2	2	no	—
77	C₅×#95	—	2	2	no	—
78	C₅×#95	—	2	2	no	—
79	C₅×#95	—	2	2	no	—
80	C₅×#95	D₄	1	2	yes	—
81	C₅×#95	D₄	1	2	yes	—
82	C₂×#491	C₂×C₅	1	1	yes	characteristic
83	#2330	C₂×C₅	1	1	yes	characteristic
84	#2327	C₂×C₅	1	1	yes	characteristic
85	#1991	C₂×C₅	1	1	yes	characteristic
86	#1956	C₂×C₅	1	1	yes	characteristic
87	#1928	C₂×C₅	1	2	yes	—
88	#1927	C₂×C₅	1	2	yes	—
89	#1865	C₂×C₅	1	2	yes	—
90	#1863	C₂×C₅	1	2	yes	—
91	#1847	C₂×C₅	1	1	yes	characteristic
92	#1746	C₂×C₅	1	1	yes	characteristic
93	C₂×C₅×Q₈	—	2	1	no	—
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	2	2	no	—
98	C₂×C₅×Q₈	—	2	2	no	—
99	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	—	2	1	no	—
103	C₂²×C₄×C₅	—	2	1	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	2	1	no	—
107	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
118	C₅×C₄⋊C₄	—	2	8	no	—
119	C₅×C₄⋊C₄	—	2	8	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	—	4	2	no	—
126	C₅×C₄⋊C₄	—	4	2	no	—
127	C₅×C₄⋊C₄	—	8	2	no	—
128	C₅×C₄⋊C₄	—	8	2	no	—
129	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
130	C₄²×C₅	—	2	2	no	—
131	C₄²×C₅	—	2	2	no	—
132	C₄²×C₅	—	4	2	no	—
133	C₄²×C₅	—	4	2	no	—
134	C₄²×C₅	—	8	2	no	—
135	C₄²×C₅	C₂²×C₄	1	2	yes	—
136	C₄²×C₅	#41	1	2	yes	—
137	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
138	C₂×#125	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
140	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
141	#491	C₄×C₅	1	8	yes	complemented
142	C₂×#107	C₂²×C₅	1	2	yes	—
143	C₂×#106	C₂²×C₅	1	2	yes	—
144	C₂×#103	C₂²×C₅	1	2	yes	—
145	C₂×#103	C₂²×C₅	1	2	yes	—
146	C₂×#103	C₂²×C₅	1	2	yes	—
147	C₂×#103	C₂²×C₅	1	2	yes	—
148	C₂×#97	C₂²×C₅	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
150	#391	C₂²×C₅	1	1	yes	characteristic
151	#391	C₂²×C₅	1	1	yes	characteristic
152	#389	—	2	2	no	—
153	#389	C₂²×C₅	1	2	yes	—
154	#387	—	2	2	no	—
155	#387	—	2	2	no	—
156	#387	—	2	2	no	—
157	#384	C₂²×C₅	1	1	yes	characteristic
158	#382	C₂²×C₅	1	1	yes	characteristic
159	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
160	#376	C₂²×C₅	1	1	yes	characteristic
161	#340	C₂²×C₅	1	2	yes	—
162	#340	C₂²×C₅	1	2	yes	—
163	#336	C₂²×C₅	1	2	yes	—
164	#336	C₂²×C₅	1	2	yes	—
165	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
166	C₅×Q₈	—	2	4	no	—
167	C₅×Q₈	—	2	4	no	—
168	C₅×Q₈	—	4	2	no	—
169	C₅×Q₈	—	4	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	4	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	1	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	4	2	no	—
187	C₂×C₄×C₅	—	4	2	no	—
188	C₂×C₄×C₅	—	4	1	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	4	2	no	—
192	C₂×C₄×C₅	—	4	2	no	—
193	C₂×C₄×C₅	—	4	2	no	—
194	C₂×C₄×C₅	—	4	2	no	—
195	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
196	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
199	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
200	C₅×C₈	—	2	4	no	—
201	C₅×C₈	—	4	2	no	—
202	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
203	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₅	1	2	yes	—
205	C₂²×C₈	C₂³×C₅	1	2	yes	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
208	#125	—	2	2	no	—
209	#125	C₂×C₄×C₅	1	4	yes	—
210	C₄×Q₈	—	4	2	no	—
211	C₄×Q₈	—	4	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
228	#107	—	4	2	no	—
229	#106	—	4	2	no	—
230	#103	—	4	2	no	—
231	#103	—	4	2	no	—
232	#103	C₂×C₄×C₅	1	8	yes	—
233	#103	C₂×C₄×C₅	1	8	yes	—
234	#97	C₂×C₄×C₅	1	4	yes	—
235	C₄×C₈	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	C₅×D₄	1	2	yes	—
242	#95	C₅×D₄	1	2	yes	—
243	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²×C₅	#535	1	1	yes	characteristic
245	C₂²×C₅	#536	1	1	yes	characteristic
246	C₂²×C₅	#577	1	2	yes	—
247	C₂²×C₅	#578	1	2	yes	—
248	C₄×C₅	—	2	4	no	—
249	C₄×C₅	—	4	2	no	—
250	C₄×C₅	—	4	2	no	—
251	C₄×C₅	—	4	4	no	—
252	C₄×C₅	—	4	1	no	—
253	C₄×C₅	—	4	1	no	—
254	C₄×C₅	—	4	2	no	—
255	C₄×C₅	—	8	1	no	—
256	C₄×C₅	—	8	2	no	—
257	C₄×C₅	—	8	1	no	—
258	C₄×C₅	#520	1	2	yes	—
259	C₄×C₅	#542	1	2	yes	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
267	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
275	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
278	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	8	2	no	—
295	C₄⋊C₄	—	8	2	no	—
296	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	—	8	2	no	—
302	C₄²	C₂²×C₄×C₅	1	2	yes	—
303	C₄²	C₅×#41	1	2	yes	—
304	C₂×C₅	#2326	1	1	yes	characteristic
305	C₂×C₅	#2947	1	2	yes	—
306	C₂×C₅	#3221	1	1	yes	characteristic
307	C₂×C₅	#3223	1	1	yes	characteristic
308	C₂×C₅	#3242	1	2	yes	—
309	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
340	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
341	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
342	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
343	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
344	C₈	—	2	4	no	—
345	C₈	—	4	2	no	—
346	C₅	#22231	1	1	yes	characteristic, direct product factor, Sylow
347	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
348	C₂²	C₅×#535	1	1	yes	characteristic
349	C₂²	C₅×#536	1	1	yes	characteristic
350	C₂²	C₅×#577	1	2	yes	—
351	C₂²	C₅×#578	1	2	yes	—
352	C₄	—	2	4	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	4	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	1	no	—
362	C₄	C₅×#520	1	2	yes	—
363	C₄	C₅×#542	1	2	yes	—
364	C₂	C₅×#2326	1	1	yes	characteristic
365	C₂	C₅×#2947	1	2	yes	—
366	C₂	C₅×#3221	1	1	yes	characteristic
367	C₂	C₅×#3223	1	1	yes	characteristic
368	C₂	C₅×#3242	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
16. C₅×#491	C₄	semidirect
40. #22231	C₅	direct (N × H)
141. #491	C₄×C₅	semidirect
346. C₅	#22231	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×#491	2
C₅×#2330	2
C₅×#2327	2
C₅×#1991	2
C₅×#1956	2
C₅×#1928	2
C₅×#1927	2
C₅×#1865	2
C₅×#1863	2
C₅×#1847	2
C₅×#1746	2
#22231	5

Subgroups — C5×#22231 #11944951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22231 #11944951