Subgroups — C₅×#8257 #11930977

Order 1280.

474 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1540	C₂	1	1	yes	maximal, characteristic
4	C₅×#1502	C₂	1	1	yes	maximal, characteristic
5	C₅×#1452	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	—	2	1	no	—
9	C₅×#532	C₂²	1	1	yes	characteristic
10	C₅×#436	—	2	1	no	—
11	C₅×#436	C₄	1	2	yes	complemented
12	C₅×C₈×D₄	—	2	1	no	—
13	C₅×C₈×D₄	C₄	1	2	yes	complemented
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₅×#431	C₂²	1	1	yes	characteristic
16	C₅×#423	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₅×#408	—	2	1	no	—
19	C₂×C₄×C₅×C₈	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅×#336	—	2	1	no	—
23	C₅×#332	C₄	1	2	yes	complemented
24	C₅×#331	C₄	1	2	yes	complemented
25	C₅×#327	C₄	1	4	yes	—
26	C₅×#322	—	2	1	no	—
27	C₅×#322	—	2	1	no	—
28	#8257	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅×#41	—	2	1	no	—
30	C₅×#131	—	4	2	no	—
31	C₅×#131	—	4	2	no	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	1	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₅×#127	C₂×C₄	1	1	yes	characteristic
40	C₅×#124	—	2	1	no	—
41	C₅×#121	—	2	1	no	—
42	C₅×#121	C₂×C₄	1	2	yes	—
43	C₄×C₅×D₄	—	2	1	no	—
44	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄²×C₅	—	2	1	no	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₅×C₂²⋊C₈	—	2	1	no	—
55	C₅×C₂²⋊C₈	—	2	2	no	—
56	C₅×C₂²⋊C₈	—	2	2	no	—
57	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₄×C₅×C₈	—	4	2	no	—
61	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	D₄	1	1	yes	characteristic
65	C₅×#95	C₂×C₄	1	2	yes	—
66	#2931	C₂×C₅	1	1	yes	characteristic
67	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
68	#1540	C₂×C₅	1	1	yes	characteristic
69	#1502	C₂×C₅	1	1	yes	characteristic
70	#1452	C₂×C₅	1	1	yes	characteristic, complemented
71	#1303	C₂×C₅	1	1	yes	characteristic, complemented
72	#1301	C₂×C₅	1	1	yes	characteristic, complemented
73	C₅×#41	—	4	1	no	—
74	C₅×#41	—	4	1	no	—
75	C₅×#41	—	4	1	no	—
76	C₅×#41	—	4	1	no	—
77	C₂×C₅×Q₈	—	2	1	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	2	1	no	—
80	C₂×C₅×D₄	—	2	1	no	—
81	C₂×C₅×D₄	C₄²	1	2	yes	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	#41	1	1	yes	characteristic
87	C₅×M₁₆	—	4	4	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₅×M₁₆	—	4	1	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
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	—	4	4	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	C₄²	1	2	yes	—
108	C₅×C₂²⋊C₄	—	2	2	no	—
109	C₅×C₂²⋊C₄	—	2	2	no	—
110	C₄²×C₅	—	4	1	no	—
111	C₄²×C₅	—	4	1	no	—
112	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅	#41	1	1	yes	characteristic
116	C₂×#131	—	2	1	no	—
117	#532	C₂²×C₅	1	1	yes	characteristic
118	#436	—	2	1	no	—
119	#436	C₄×C₅	1	2	yes	complemented
120	C₈×D₄	—	2	1	no	—
121	C₈×D₄	C₄×C₅	1	2	yes	complemented
122	#431	C₂²×C₅	1	1	yes	characteristic
123	#431	C₂²×C₅	1	1	yes	characteristic
124	#423	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
126	#408	—	2	1	no	—
127	C₂×C₄×C₈	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
130	#336	—	2	1	no	—
131	#332	C₄×C₅	1	2	yes	complemented
132	#331	C₄×C₅	1	2	yes	complemented
133	#327	C₄×C₅	1	4	yes	—
134	#322	—	2	1	no	—
135	#322	—	2	1	no	—
136	C₂³×C₅	—	2	2	no	—
137	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
138	C₅×Q₈	—	4	1	no	—
139	C₅×Q₈	—	4	1	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	1	no	—
144	C₅×D₄	—	4	1	no	—
145	C₅×D₄	—	4	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	#117	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₅	#137	1	1	yes	characteristic
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	4	2	no	—
169	C₅×C₈	—	4	2	no	—
170	C₅×C₈	—	4	2	no	—
171	C₅×C₈	—	4	2	no	—
172	C₂×#41	—	2	1	no	—
173	#131	—	4	2	no	—
174	#131	—	4	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
180	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
182	#127	C₂×C₄×C₅	1	1	yes	characteristic
183	#124	—	2	1	no	—
184	#121	—	2	1	no	—
185	#121	C₂×C₄×C₅	1	2	yes	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
195	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
196	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	4	2	no	—
204	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
205	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
206	C₄×C₈	C₅×D₄	1	1	yes	characteristic
207	C₄×C₈	C₅×D₄	1	1	yes	characteristic
208	#95	C₂×C₄×C₅	1	2	yes	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	4	2	no	—
212	C₂²×C₅	—	4	2	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₅	#440	1	1	yes	characteristic
216	C₂²×C₅	#442	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	4	4	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	#417	1	1	yes	characteristic
223	C₄×C₅	#418	1	1	yes	characteristic
224	C₄×C₅	#440	1	1	yes	characteristic
225	C₄×C₅	#442	1	1	yes	characteristic
226	C₄×C₅	#443	1	2	yes	—
227	#41	—	4	1	no	—
228	#41	—	4	1	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₄²×C₅	1	2	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₅×#41	1	1	yes	characteristic
241	M₁₆	—	4	4	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	C₄²×C₅	1	2	yes	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
267	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
268	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
269	C₄²	C₅×#41	1	1	yes	characteristic
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	8	2	no	—
273	C₂×C₅	#1757	1	1	yes	characteristic
274	C₂×C₅	#1762	1	1	yes	characteristic
275	C₂×C₅	#1764	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	Q₈	—	4	1	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₅×#117	1	1	yes	characteristic
297	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
299	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂×C₄	C₅×#137	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₅	#8257	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
319	C₂²	C₅×#440	1	1	yes	characteristic
320	C₂²	C₅×#442	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	2	no	—
326	C₄	C₅×#417	1	1	yes	characteristic
327	C₄	C₅×#418	1	1	yes	characteristic
328	C₄	C₅×#440	1	1	yes	characteristic
329	C₄	C₅×#442	1	1	yes	characteristic
330	C₄	C₅×#443	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₅×#1757	1	1	yes	characteristic
335	C₂	C₅×#1762	1	1	yes	characteristic
336	C₂	C₅×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×C₄⋊C₄	C₂	semidirect
5. C₅×#1452	C₂	semidirect
6. C₅×#1303	C₂	semidirect
7. C₅×#1301	C₂	semidirect
11. C₅×#436	C₄	semidirect
13. C₅×C₈×D₄	C₄	semidirect
23. C₅×#332	C₄	semidirect
24. C₅×#331	C₄	semidirect
28. #8257	C₅	direct (N × H)
67. C₈×C₄⋊C₄	C₂×C₅	semidirect
70. #1452	C₂×C₅	semidirect
71. #1303	C₂×C₅	semidirect
72. #1301	C₂×C₅	semidirect
119. #436	C₄×C₅	semidirect
121. C₈×D₄	C₄×C₅	semidirect
131. #332	C₄×C₅	semidirect
132. #331	C₄×C₅	semidirect
312. C₅	#8257	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×#2931	2
C₅×C₈×C₄⋊C₄	2
C₅×#1540	2
C₅×#1502	2
C₅×#1452	2
C₅×#1303	2
C₅×#1301	2
#8257	5

Subgroups — C5×#8257 #11930977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8257 #11930977