Subgroups — C₅×#8026 #11930746

Order 1280.

742 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#382	C₂	1	1	yes	maximal, characteristic
4	C₅×#1294	C₂	1	4	yes	maximal
5	C₂×C₅×#117	—	2	1	no	—
6	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#431	—	2	2	no	—
9	C₅×#431	C₄	1	4	yes	complemented
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	—	2	1	no	—
13	C₅×#384	C₄	1	2	yes	complemented
14	C₅×#382	—	2	2	no	—
15	C₅×#382	—	2	1	no	—
16	C₅×#382	C₄	1	4	yes	complemented
17	C₅×#380	—	2	1	no	—
18	C₅×#380	C₂²	1	2	yes	—
19	C₄×C₅×C₄⋊C₄	—	2	1	no	—
20	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	complemented
21	C₂×C₅×#95	—	2	1	no	—
22	C₂×C₅×#95	C₂²	1	2	yes	—
23	#8026	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₅×#117	—	4	2	no	—
30	C₅×#117	—	4	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	4	2	no	—
35	C₂×C₅×C₄⋊C₄	—	4	2	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₅	—	2	2	no	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	2	yes	—
46	C₂×C₄²×C₅	Q₈	1	2	yes	—
47	C₅×C₄⋊C₈	—	2	4	no	—
48	C₅×C₂²⋊C₈	—	2	2	no	—
49	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
50	C₄×C₅×C₈	—	2	4	no	—
51	C₅×#95	—	2	4	no	—
52	C₅×#95	—	2	2	no	—
53	C₅×#95	—	2	2	no	—
54	C₅×#95	—	2	2	no	—
55	C₅×#95	C₂×C₄	1	4	yes	—
56	C₂×#431	C₂×C₅	1	1	yes	characteristic
57	#2285	C₂×C₅	1	1	yes	characteristic
58	C₂×#382	C₂×C₅	1	1	yes	characteristic
59	#1294	C₂×C₅	1	4	yes	—
60	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	4	no	—
66	C₂²×C₄×C₅	—	4	4	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₅	D₈	1	2	yes	—
78	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	4	4	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₄⋊C₄	—	4	4	no	—
91	C₅×C₄⋊C₄	—	4	8	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₅×C₂²⋊C₄	—	4	2	no	—
95	C₄²×C₅	—	4	4	no	—
96	C₄²×C₅	C₄²	1	2	yes	—
97	C₄²×C₅	C₄²	1	2	yes	—
98	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₅	C₄⋊C₄	1	2	yes	—
101	C₄²×C₅	C₄⋊C₄	1	2	yes	—
102	C₄²×C₅	C₄⋊C₄	1	2	yes	—
103	C₄²×C₅	C₄⋊C₄	1	2	yes	—
104	C₂×#117	—	2	1	no	—
105	C₂²×C₄⋊C₄	—	2	1	no	—
106	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
107	#431	—	2	2	no	—
108	#431	C₄×C₅	1	4	yes	complemented
109	C₂×C₄⋊C₈	—	2	1	no	—
110	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
111	C₂×C₄×C₈	—	2	1	no	—
112	#384	C₄×C₅	1	2	yes	complemented
113	#382	—	2	2	no	—
114	#382	—	2	1	no	—
115	#382	C₄×C₅	1	4	yes	complemented
116	#380	—	2	1	no	—
117	#380	C₂²×C₅	1	2	yes	—
118	C₄×C₄⋊C₄	—	2	1	no	—
119	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	complemented
120	C₂×#95	—	2	1	no	—
121	C₂×#95	C₂²×C₅	1	2	yes	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
128	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	8	4	no	—
148	C₂×C₄×C₅	—	8	4	no	—
149	C₂×C₄×C₅	#95	1	2	yes	—
150	C₂×C₄×C₅	#95	1	2	yes	—
151	C₂×C₄×C₅	#95	1	2	yes	—
152	C₂×C₄×C₅	#95	1	2	yes	—
153	C₂×C₄×C₅	#102	1	2	yes	—
154	C₂×C₄×C₅	#102	1	4	yes	—
155	C₂×C₄×C₅	#102	1	2	yes	—
156	C₂×C₄×C₅	#103	1	4	yes	—
157	C₂×C₄×C₅	#103	1	2	yes	—
158	C₂×C₄×C₅	#103	1	2	yes	—
159	C₂×C₄×C₅	#106	1	4	yes	—
160	C₂×C₄×C₅	#107	1	4	yes	—
161	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₅×C₈	—	4	4	no	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
168	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
169	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	#117	—	4	2	no	—
172	#117	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	4	2	no	—
177	C₂×C₄⋊C₄	—	4	2	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
182	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
186	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
187	C₂×C₄²	C₅×D₄	1	2	yes	—
188	C₂×C₄²	C₅×Q₈	1	2	yes	—
189	C₄⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
192	C₄×C₈	—	2	4	no	—
193	#95	—	2	4	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	C₂×C₄×C₅	1	4	yes	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	C₂×#95	1	1	yes	characteristic
207	C₂²×C₅	#410	1	2	yes	—
208	C₂²×C₅	C₂×#102	1	1	yes	characteristic
209	C₂²×C₅	C₂×#102	1	1	yes	characteristic
210	C₂²×C₅	C₂×#103	1	1	yes	characteristic
211	C₂²×C₅	C₂×#103	1	1	yes	characteristic
212	C₂²×C₅	#421	1	2	yes	—
213	C₂²×C₅	C₂×#106	1	1	yes	characteristic
214	C₂²×C₅	C₂×#107	1	1	yes	characteristic
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	8	4	no	—
222	C₄×C₅	—	8	4	no	—
223	C₄×C₅	#340	1	4	yes	—
224	C₄×C₅	#340	1	4	yes	—
225	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
237	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
239	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
241	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
242	C₂²×C₄	C₅×D₈	1	2	yes	—
243	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
244	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
245	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	8	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	4	no	—
261	C₄²	C₄²×C₅	1	2	yes	—
262	C₄²	C₄²×C₅	1	2	yes	—
263	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
264	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
265	C₄²	C₅×C₄⋊C₄	1	2	yes	—
266	C₄²	C₅×C₄⋊C₄	1	2	yes	—
267	C₄²	C₅×C₄⋊C₄	1	2	yes	—
268	C₄²	C₅×C₄⋊C₄	1	2	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	2	1	no	—
273	C₂×C₅	#1517	1	2	yes	—
274	C₂×C₅	#1518	1	2	yes	—
275	C₂×C₅	#1733	1	1	yes	characteristic
276	C₂×C₅	C₂×#340	1	1	yes	characteristic
277	C₂×C₅	#1740	1	1	yes	characteristic
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
283	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
284	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
285	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
286	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
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	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	—	8	4	no	—
305	C₂×C₄	C₅×#95	1	2	yes	—
306	C₂×C₄	C₅×#95	1	2	yes	—
307	C₂×C₄	C₅×#95	1	2	yes	—
308	C₂×C₄	C₅×#95	1	2	yes	—
309	C₂×C₄	C₅×#102	1	2	yes	—
310	C₂×C₄	C₅×#102	1	4	yes	—
311	C₂×C₄	C₅×#102	1	2	yes	—
312	C₂×C₄	C₅×#103	1	4	yes	—
313	C₂×C₄	C₅×#103	1	2	yes	—
314	C₂×C₄	C₅×#103	1	2	yes	—
315	C₂×C₄	C₅×#106	1	4	yes	—
316	C₂×C₄	C₅×#107	1	4	yes	—
317	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
318	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
321	C₈	—	4	4	no	—
322	C₅	#8026	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	C₂×C₅×#95	1	1	yes	characteristic
332	C₂²	C₅×#410	1	2	yes	—
333	C₂²	C₂×C₅×#102	1	1	yes	characteristic
334	C₂²	C₂×C₅×#102	1	1	yes	characteristic
335	C₂²	C₂×C₅×#103	1	1	yes	characteristic
336	C₂²	C₂×C₅×#103	1	1	yes	characteristic
337	C₂²	C₅×#421	1	2	yes	—
338	C₂²	C₂×C₅×#106	1	1	yes	characteristic
339	C₂²	C₂×C₅×#107	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	4	no	—
347	C₄	—	8	4	no	—
348	C₄	C₅×#340	1	4	yes	—
349	C₄	C₅×#340	1	4	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	C₅×#1517	1	2	yes	—
355	C₂	C₅×#1518	1	2	yes	—
356	C₂	C₅×#1733	1	1	yes	characteristic
357	C₂	C₂×C₅×#340	1	1	yes	characteristic
358	C₂	C₅×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅×#431	C₄	semidirect
13. C₅×#384	C₄	semidirect
16. C₅×#382	C₄	semidirect
20. C₄×C₅×C₄⋊C₄	C₄	semidirect
23. #8026	C₅	direct (N × H)
108. #431	C₄×C₅	semidirect
112. #384	C₄×C₅	semidirect
115. #382	C₄×C₅	semidirect
119. C₄×C₄⋊C₄	C₄×C₅	semidirect
322. C₅	#8026	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×#431	2
C₅×#2285	2
C₂×C₅×#382	2
C₅×#1294	2
#8026	5

Subgroups — C5×#8026 #11930746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8026 #11930746