Subgroups — C₅×#35336 #11958056

Order 1280.

1538 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2851	C₂	1	2	yes	maximal
4	C₅×#2647	C₂	1	2	yes	maximal, complemented
5	C₅×#2629	C₂	1	2	yes	maximal
6	C₅×#2590	C₂	1	8	yes	maximal, complemented
7	C₅×#2587	C₂	1	8	yes	maximal
8	C₅×#2330	C₂	1	2	yes	maximal, complemented
9	C₅×#2314	C₂	1	2	yes	maximal
10	C₄×C₅×#125	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×#125	C₂²	1	2	yes	—
13	C₂×C₅×#125	C₂²	1	2	yes	—
14	C₂×C₅×#117	C₂²	1	1	yes	characteristic
15	C₂×C₅×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₅×#400	C₂²	1	8	yes	—
18	C₅×#398	C₂²	1	8	yes	—
19	C₅×#398	C₂²	1	8	yes	—
20	C₅×#396	C₂²	1	8	yes	—
21	C₅×#389	C₂²	1	4	yes	—
22	C₅×#389	C₂²	1	8	yes	—
23	C₅×#389	C₂²	1	4	yes	—
24	C₅×#389	C₂²	1	8	yes	—
25	C₅×#389	C₂²	1	8	yes	—
26	C₅×#388	C₂²	1	8	yes	—
27	C₅×#387	C₂²	1	8	yes	—
28	C₅×#387	C₂²	1	8	yes	—
29	C₅×#387	C₂²	1	8	yes	—
30	C₅×#387	C₂²	1	8	yes	—
31	C₅×#385	C₂²	1	8	yes	—
32	C₅×#384	C₂²	1	2	yes	—
33	C₅×#384	C₂²	1	2	yes	—
34	C₅×#382	C₂²	1	2	yes	—
35	C₅×#382	C₂²	1	2	yes	—
36	C₅×#380	C₂²	1	4	yes	—
37	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
38	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
39	C₄×C₅×C₄⋊C₄	C₂²	1	8	yes	—
40	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
41	C₅×#376	C₂²	1	1	yes	characteristic
42	C₅×#376	C₂²	1	1	yes	characteristic
43	C₄³×C₅	C₂²	1	1	yes	characteristic
44	C₄³×C₅	C₂²	1	1	yes	characteristic
45	#35336	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅	C₂³	1	2	yes	—
48	C₅×#125	—	2	8	no	—
49	C₅×#125	—	2	8	no	—
50	C₅×#117	—	2	4	no	—
51	C₅×#117	Q₈	1	8	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	4	yes	—
55	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
56	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
57	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
58	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
59	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
60	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
61	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
62	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
63	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
64	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
65	C₂×C₄²×C₅	—	2	4	no	—
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅	C₂³	1	2	yes	—
69	C₂×C₄²×C₅	C₂³	1	4	yes	—
70	C₂×C₄²×C₅	C₂³	1	4	yes	—
71	C₂×C₄²×C₅	C₂³	1	8	yes	—
72	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
76	C₅×#95	C₂³	1	4	yes	—
77	C₅×#95	C₂³	1	4	yes	—
78	C₅×#95	C₂³	1	8	yes	—
79	C₅×#95	C₂³	1	8	yes	—
80	C₅×#95	C₂³	1	8	yes	—
81	C₅×#95	C₂³	1	8	yes	—
82	C₅×#95	C₂³	1	8	yes	—
83	C₅×#95	C₂³	1	8	yes	—
84	#2859	C₂×C₅	1	1	yes	characteristic, complemented
85	#2859	C₂×C₅	1	1	yes	characteristic, complemented
86	#2851	C₂×C₅	1	2	yes	—
87	#2647	C₂×C₅	1	2	yes	complemented
88	#2629	C₂×C₅	1	2	yes	—
89	#2590	C₂×C₅	1	8	yes	complemented
90	#2587	C₂×C₅	1	8	yes	—
91	#2330	C₂×C₅	1	2	yes	complemented
92	#2314	C₂×C₅	1	2	yes	—
93	C₄×#125	C₂×C₅	1	2	yes	complemented
94	C₄×#117	C₂×C₅	1	1	yes	characteristic
95	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	2	2	no	—
98	C₂²×C₄×C₅	—	2	4	no	—
99	C₂²×C₄×C₅	—	4	4	no	—
100	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
103	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
104	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
105	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
106	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
107	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	2	16	no	—
113	C₅×C₄⋊C₄	—	2	16	no	—
114	C₅×C₄⋊C₄	—	2	8	no	—
115	C₅×C₄⋊C₄	—	2	8	no	—
116	C₅×C₄⋊C₄	—	4	8	no	—
117	C₅×C₄⋊C₄	—	4	8	no	—
118	C₅×C₄⋊C₄	—	4	8	no	—
119	C₅×C₄⋊C₄	—	4	8	no	—
120	C₅×C₄⋊C₄	C₂×Q₈	1	8	yes	—
121	C₅×C₂²⋊C₄	—	2	4	no	—
122	C₅×C₂²⋊C₄	—	4	8	no	—
123	C₅×C₂²⋊C₄	C₂×Q₈	1	8	yes	—
124	C₄²×C₅	—	2	4	no	—
125	C₄²×C₅	—	2	2	no	—
126	C₄²×C₅	—	2	2	no	—
127	C₄²×C₅	—	4	4	no	—
128	C₄²×C₅	—	4	8	no	—
129	C₄²×C₅	—	4	4	no	—
130	C₄²×C₅	C₂×Q₈	1	4	yes	—
131	C₄²×C₅	C₂×Q₈	1	4	yes	—
132	C₄²×C₅	#41	1	4	yes	—
133	C₄²×C₅	#41	1	4	yes	—
134	C₂×#125	C₂²×C₅	1	2	yes	—
135	C₂×#125	C₂²×C₅	1	2	yes	—
136	C₂×#117	C₂²×C₅	1	1	yes	characteristic
137	C₂×#117	C₂²×C₅	1	1	yes	characteristic
138	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
139	#400	C₂²×C₅	1	8	yes	—
140	#398	C₂²×C₅	1	8	yes	—
141	#398	C₂²×C₅	1	8	yes	—
142	#396	C₂²×C₅	1	8	yes	—
143	#389	C₂²×C₅	1	4	yes	—
144	#389	C₂²×C₅	1	8	yes	—
145	#389	C₂²×C₅	1	4	yes	—
146	#389	C₂²×C₅	1	8	yes	—
147	#389	C₂²×C₅	1	8	yes	—
148	#388	C₂²×C₅	1	8	yes	—
149	#387	C₂²×C₅	1	8	yes	—
150	#387	C₂²×C₅	1	8	yes	—
151	#387	C₂²×C₅	1	8	yes	—
152	#387	C₂²×C₅	1	8	yes	—
153	#385	C₂²×C₅	1	8	yes	—
154	#384	C₂²×C₅	1	2	yes	—
155	#384	C₂²×C₅	1	2	yes	—
156	#382	C₂²×C₅	1	2	yes	—
157	#382	C₂²×C₅	1	2	yes	—
158	#380	C₂²×C₅	1	4	yes	—
159	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
160	C₄×C₄⋊C₄	C₂²×C₅	1	8	yes	—
161	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
162	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
163	#376	C₂²×C₅	1	1	yes	characteristic
164	#376	C₂²×C₅	1	1	yes	characteristic
165	C₄³	C₂²×C₅	1	1	yes	characteristic
166	C₄³	C₂²×C₅	1	1	yes	characteristic
167	C₂³×C₅	—	2	1	no	—
168	C₂³×C₅	—	2	1	no	—
169	C₂³×C₅	—	2	4	no	—
170	C₂³×C₅	C₂²×Q₈	1	2	yes	—
171	C₂³×C₅	C₂⁵	1	1	yes	characteristic
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	4	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	—	2	4	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	2	8	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	8	no	—
182	C₂×C₄×C₅	—	2	8	no	—
183	C₂×C₄×C₅	—	2	8	no	—
184	C₂×C₄×C₅	—	2	4	no	—
185	C₂×C₄×C₅	—	2	8	no	—
186	C₂×C₄×C₅	—	2	8	no	—
187	C₂×C₄×C₅	—	2	8	no	—
188	C₂×C₄×C₅	—	2	4	no	—
189	C₂×C₄×C₅	—	4	8	no	—
190	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
191	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
192	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
193	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
194	C₂×C₄×C₅	C₂×#41	1	4	yes	—
195	C₂×C₄×C₅	C₂×#41	1	4	yes	—
196	C₂×C₄×C₅	C₂×#41	1	2	yes	—
197	C₂×C₄×C₅	C₂×#41	1	2	yes	—
198	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₅	1	2	yes	—
200	#125	—	2	8	no	—
201	#125	—	2	8	no	—
202	#117	—	2	4	no	—
203	#117	C₅×Q₈	1	8	yes	—
204	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
206	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
207	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
208	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
209	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
210	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
211	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
212	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
213	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
214	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
215	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
216	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₅	1	2	yes	—
221	C₂×C₄²	C₂³×C₅	1	4	yes	—
222	C₂×C₄²	C₂³×C₅	1	4	yes	—
223	C₂×C₄²	C₂³×C₅	1	8	yes	—
224	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
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	#95	C₂³×C₅	1	4	yes	—
229	#95	C₂³×C₅	1	4	yes	—
230	#95	C₂³×C₅	1	8	yes	—
231	#95	C₂³×C₅	1	8	yes	—
232	#95	C₂³×C₅	1	8	yes	—
233	#95	C₂³×C₅	1	8	yes	—
234	#95	C₂³×C₅	1	8	yes	—
235	#95	C₂³×C₅	1	8	yes	—
236	C₂²×C₅	—	2	2	no	—
237	C₂²×C₅	—	2	4	no	—
238	C₂²×C₅	—	4	2	no	—
239	C₂²×C₅	—	4	2	no	—
240	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
241	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
242	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
243	C₂²×C₅	#585	1	4	yes	—
244	C₄×C₅	—	2	2	no	—
245	C₄×C₅	—	2	4	no	—
246	C₄×C₅	—	2	4	no	—
247	C₄×C₅	—	2	4	no	—
248	C₄×C₅	—	2	4	no	—
249	C₄×C₅	—	4	8	no	—
250	C₄×C₅	—	4	2	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	4	8	no	—
253	C₄×C₅	#533	1	4	yes	—
254	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	4	4	no	—
259	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
260	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
262	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
263	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
264	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
265	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
267	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
268	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
269	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	16	no	—
272	C₄⋊C₄	—	2	16	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	4	8	no	—
276	C₄⋊C₄	—	4	8	no	—
277	C₄⋊C₄	—	4	8	no	—
278	C₄⋊C₄	—	4	8	no	—
279	C₄⋊C₄	C₂×C₅×Q₈	1	8	yes	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	4	8	no	—
282	C₂²⋊C₄	C₂×C₅×Q₈	1	8	yes	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	4	no	—
287	C₄²	—	4	8	no	—
288	C₄²	—	4	4	no	—
289	C₄²	C₂×C₅×Q₈	1	4	yes	—
290	C₄²	C₂×C₅×Q₈	1	4	yes	—
291	C₄²	C₅×#41	1	4	yes	—
292	C₄²	C₅×#41	1	4	yes	—
293	C₂×C₅	—	4	2	no	—
294	C₂×C₅	C₂×#533	1	1	yes	characteristic
295	C₂×C₅	#3455	1	2	yes	—
296	C₂×C₅	#3504	1	2	yes	—
297	C₂×C₅	#3512	1	2	yes	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	4	no	—
301	C₂³	C₂²×C₅×Q₈	1	2	yes	—
302	C₂³	C₂⁵×C₅	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	2	8	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	8	no	—
321	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
322	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
323	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
324	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
325	C₂×C₄	C₂×C₅×#41	1	4	yes	—
326	C₂×C₄	C₂×C₅×#41	1	4	yes	—
327	C₂×C₄	C₂×C₅×#41	1	2	yes	—
328	C₂×C₄	C₂×C₅×#41	1	2	yes	—
329	C₅	#35336	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	2	2	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₂³×C₅×Q₈	1	1	yes	characteristic
335	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
336	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
337	C₂²	C₅×#585	1	4	yes	—
338	C₄	—	2	2	no	—
339	C₄	—	2	4	no	—
340	C₄	—	2	4	no	—
341	C₄	—	2	4	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	8	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	8	no	—
347	C₄	C₅×#533	1	4	yes	—
348	C₂	—	4	2	no	—
349	C₂	C₂×C₅×#533	1	1	yes	characteristic
350	C₂	C₅×#3455	1	2	yes	—
351	C₂	C₅×#3504	1	2	yes	—
352	C₂	C₅×#3512	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂⁴×C₅	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2859	C₂	semidirect
2. C₅×#2859	C₂	semidirect
4. C₅×#2647	C₂	semidirect
6. C₅×#2590	C₂	semidirect
8. C₅×#2330	C₂	semidirect
10. C₄×C₅×#125	C₂	semidirect
45. #35336	C₅	direct (N × H)
84. #2859	C₂×C₅	semidirect
85. #2859	C₂×C₅	semidirect
87. #2647	C₂×C₅	semidirect
89. #2590	C₂×C₅	semidirect
91. #2330	C₂×C₅	semidirect
93. C₄×#125	C₂×C₅	semidirect
329. C₅	#35336	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2859	2
C₅×#2851	2
C₅×#2647	2
C₅×#2629	2
C₅×#2590	2
C₅×#2587	2
C₅×#2330	2
C₅×#2314	2
C₄×C₅×#125	2
C₄×C₅×#117	2
#35336	5

Subgroups — C5×#35336 #11958056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35336 #11958056