Subgroups — C₅×#22229 #11944949

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#487	C₂	1	1	yes	maximal, characteristic
2	C₅×#2331	C₂	1	1	yes	maximal, characteristic
3	C₅×#2327	C₂	1	1	yes	maximal, characteristic
4	C₅×#1993	C₂	1	1	yes	maximal, characteristic
5	C₅×#1960	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×#1846	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×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
15	C₅×#487	C₄	1	8	yes	complemented
16	C₂×C₅×#107	C₂²	1	2	yes	—
17	C₂×C₅×#106	C₂²	1	2	yes	—
18	C₂×C₅×#103	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×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#391	—	2	2	no	—
25	C₅×#391	C₂²	1	2	yes	—
26	C₅×#389	C₂²	1	2	yes	—
27	C₅×#387	—	2	2	no	—
28	C₅×#387	—	2	2	no	—
29	C₅×#387	—	2	2	no	—
30	C₅×#382	C₂²	1	1	yes	characteristic
31	C₅×#382	C₂²	1	1	yes	characteristic
32	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
33	C₅×#376	C₂²	1	1	yes	characteristic
34	C₅×#340	C₂²	1	2	yes	—
35	C₅×#340	C₂²	1	2	yes	—
36	C₅×#336	C₂²	1	1	yes	characteristic
37	C₅×#336	C₂²	1	1	yes	characteristic
38	C₅×#336	C₂²	1	1	yes	characteristic
39	C₅×#336	C₂²	1	1	yes	characteristic
40	#22229	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₅×Q₈	—	2	1	no	—
42	C₂²×C₅×Q₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	—	2	2	no	—
48	C₅×C₄⋊Q₈	—	2	2	no	—
49	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
50	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
51	C₄×C₅×Q₈	—	4	2	no	—
52	C₄×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₄⋊C₄	—	2	2	no	—
57	C₂×C₅×C₄⋊C₄	—	2	2	no	—
58	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
59	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
60	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
61	C₂×C₄²×C₅	—	2	2	no	—
62	C₂×C₄²×C₅	—	2	2	no	—
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
66	C₅×#107	—	4	2	no	—
67	C₅×#106	—	4	2	no	—
68	C₅×#103	—	4	2	no	—
69	C₅×#103	—	4	2	no	—
70	C₅×#103	C₂×C₄	1	8	yes	—
71	C₅×#103	C₂×C₄	1	8	yes	—
72	C₅×#97	—	2	2	no	—
73	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
74	C₅×#95	—	2	2	no	—
75	C₅×#95	—	2	1	no	—
76	C₅×#95	—	2	1	no	—
77	C₅×#95	—	2	1	no	—
78	C₅×#95	—	2	1	no	—
79	C₅×#95	—	2	4	no	—
80	C₅×#95	—	2	2	no	—
81	C₅×#95	D₄	1	2	yes	—
82	C₅×#95	D₄	1	2	yes	—
83	C₂×#487	C₂×C₅	1	1	yes	characteristic
84	#2331	C₂×C₅	1	1	yes	characteristic
85	#2327	C₂×C₅	1	1	yes	characteristic
86	#1993	C₂×C₅	1	1	yes	characteristic
87	#1960	C₂×C₅	1	1	yes	characteristic
88	#1928	C₂×C₅	1	2	yes	—
89	#1927	C₂×C₅	1	2	yes	—
90	#1865	C₂×C₅	1	2	yes	—
91	#1863	C₂×C₅	1	2	yes	—
92	#1846	C₂×C₅	1	1	yes	characteristic
93	#1746	C₂×C₅	1	1	yes	characteristic
94	C₂×C₅×Q₈	—	2	4	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	—	2	2	no	—
100	C₂×C₅×Q₈	—	8	2	no	—
101	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	—	2	2	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	—	2	2	no	—
106	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
123	C₅×C₄⋊C₄	—	2	4	no	—
124	C₅×C₄⋊C₄	—	2	4	no	—
125	C₅×C₄⋊C₄	—	2	4	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	—	2	2	no	—
129	C₅×C₄⋊C₄	—	2	2	no	—
130	C₅×C₄⋊C₄	—	2	2	no	—
131	C₅×C₄⋊C₄	—	4	4	no	—
132	C₅×C₄⋊C₄	—	8	2	no	—
133	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
134	C₄²×C₅	—	2	1	no	—
135	C₄²×C₅	—	2	1	no	—
136	C₄²×C₅	—	2	1	no	—
137	C₄²×C₅	—	2	1	no	—
138	C₄²×C₅	—	4	4	no	—
139	C₄²×C₅	—	8	2	no	—
140	C₄²×C₅	C₂²×C₄	1	2	yes	—
141	C₄²×C₅	#41	1	2	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
145	#487	C₄×C₅	1	8	yes	complemented
146	C₂×#107	C₂²×C₅	1	2	yes	—
147	C₂×#106	C₂²×C₅	1	2	yes	—
148	C₂×#103	C₂²×C₅	1	2	yes	—
149	C₂×#103	C₂²×C₅	1	2	yes	—
150	C₂×#103	C₂²×C₅	1	2	yes	—
151	C₂×#103	C₂²×C₅	1	2	yes	—
152	C₂×#97	C₂²×C₅	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
154	#391	—	2	2	no	—
155	#391	C₂²×C₅	1	2	yes	—
156	#389	C₂²×C₅	1	2	yes	—
157	#387	—	2	2	no	—
158	#387	—	2	2	no	—
159	#387	—	2	2	no	—
160	#382	C₂²×C₅	1	1	yes	characteristic
161	#382	C₂²×C₅	1	1	yes	characteristic
162	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
163	#376	C₂²×C₅	1	1	yes	characteristic
164	#340	C₂²×C₅	1	2	yes	—
165	#340	C₂²×C₅	1	2	yes	—
166	#336	C₂²×C₅	1	1	yes	characteristic
167	#336	C₂²×C₅	1	1	yes	characteristic
168	#336	C₂²×C₅	1	1	yes	characteristic
169	#336	C₂²×C₅	1	1	yes	characteristic
170	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₅×Q₈	—	2	4	no	—
172	C₅×Q₈	—	2	4	no	—
173	C₅×Q₈	—	4	2	no	—
174	C₅×Q₈	—	4	2	no	—
175	C₅×Q₈	—	8	4	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	—	2	4	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	4	2	no	—
187	C₂×C₄×C₅	—	4	2	no	—
188	C₂×C₄×C₅	—	4	2	no	—
189	C₂×C₄×C₅	—	4	2	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	—	4	2	no	—
196	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
198	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
203	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
204	C₅×C₈	—	2	2	no	—
205	C₅×C₈	—	2	2	no	—
206	C₅×C₈	—	4	1	no	—
207	C₅×C₈	—	4	1	no	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×Q₈	C₂³×C₅	1	2	yes	—
210	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
217	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
218	C₄×Q₈	—	4	2	no	—
219	C₄×Q₈	—	4	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
233	#107	—	4	2	no	—
234	#106	—	4	2	no	—
235	#103	—	4	2	no	—
236	#103	—	4	2	no	—
237	#103	C₂×C₄×C₅	1	8	yes	—
238	#103	C₂×C₄×C₅	1	8	yes	—
239	#97	—	2	2	no	—
240	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
241	#95	—	2	2	no	—
242	#95	—	2	1	no	—
243	#95	—	2	1	no	—
244	#95	—	2	1	no	—
245	#95	—	2	1	no	—
246	#95	—	2	4	no	—
247	#95	—	2	2	no	—
248	#95	C₅×D₄	1	2	yes	—
249	#95	C₅×D₄	1	2	yes	—
250	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂²×C₅	#535	1	1	yes	characteristic
252	C₂²×C₅	#536	1	1	yes	characteristic
253	C₂²×C₅	#577	1	2	yes	—
254	C₂²×C₅	#578	1	2	yes	—
255	C₄×C₅	—	2	2	no	—
256	C₄×C₅	—	2	2	no	—
257	C₄×C₅	—	4	4	no	—
258	C₄×C₅	—	4	4	no	—
259	C₄×C₅	—	4	2	no	—
260	C₄×C₅	—	4	2	no	—
261	C₄×C₅	—	8	2	no	—
262	C₄×C₅	—	8	2	no	—
263	C₄×C₅	#520	1	1	yes	characteristic
264	C₄×C₅	#520	1	1	yes	characteristic
265	C₄×C₅	#542	1	1	yes	characteristic
266	C₄×C₅	#542	1	1	yes	characteristic
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	8	2	no	—
274	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
280	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
281	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
283	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
295	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	4	no	—
305	C₄⋊C₄	—	8	2	no	—
306	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
307	C₄²	—	2	1	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	4	4	no	—
312	C₄²	—	8	2	no	—
313	C₄²	C₂²×C₄×C₅	1	2	yes	—
314	C₄²	C₅×#41	1	2	yes	—
315	C₂×C₅	#2326	1	1	yes	characteristic
316	C₂×C₅	#2949	1	1	yes	characteristic
317	C₂×C₅	#2950	1	1	yes	characteristic
318	C₂×C₅	#3225	1	2	yes	—
319	C₂×C₅	#3242	1	2	yes	—
320	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
321	Q₈	—	2	4	no	—
322	Q₈	—	2	4	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	8	4	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
347	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
348	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
349	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
350	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
351	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
352	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
353	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
354	C₈	—	2	2	no	—
355	C₈	—	2	2	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	1	no	—
358	C₅	#22229	1	1	yes	characteristic, direct product factor, Sylow
359	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
360	C₂²	C₅×#535	1	1	yes	characteristic
361	C₂²	C₅×#536	1	1	yes	characteristic
362	C₂²	C₅×#577	1	2	yes	—
363	C₂²	C₅×#578	1	2	yes	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	4	no	—
368	C₄	—	4	4	no	—
369	C₄	—	4	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	8	2	no	—
372	C₄	C₅×#520	1	1	yes	characteristic
373	C₄	C₅×#520	1	1	yes	characteristic
374	C₄	C₅×#542	1	1	yes	characteristic
375	C₄	C₅×#542	1	1	yes	characteristic
376	C₂	C₅×#2326	1	1	yes	characteristic
377	C₂	C₅×#2949	1	1	yes	characteristic
378	C₂	C₅×#2950	1	1	yes	characteristic
379	C₂	C₅×#3225	1	2	yes	—
380	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
15. C₅×#487	C₄	semidirect
40. #22229	C₅	direct (N × H)
145. #487	C₄×C₅	semidirect
358. C₅	#22229	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×#487	2
C₅×#2331	2
C₅×#2327	2
C₅×#1993	2
C₅×#1960	2
C₅×#1928	2
C₅×#1927	2
C₅×#1865	2
C₅×#1863	2
C₅×#1846	2
C₅×#1746	2
#22229	5

Subgroups — C5×#22229 #11944949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22229 #11944949