Subgroups — C₅×#9640 #11932360

Order 1280.

582 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×D₄⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₅×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	—	2	2	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×#391	C₄	1	1	yes	characteristic
13	C₅×#390	C₄	1	1	yes	characteristic
14	C₅×#388	—	2	2	no	—
15	C₅×#382	C₄	1	1	yes	characteristic
16	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
17	C₄³×C₅	C₂²	1	1	yes	characteristic
18	C₅×#336	—	2	2	no	—
19	C₅×Q₈⋊C₈	—	2	4	no	—
20	C₅×D₄⋊C₈	—	2	4	no	—
21	#9640	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	—	2	1	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₅×#124	—	2	2	no	—
28	C₅×#124	C₈	1	4	yes	complemented
29	C₄×C₅×Q₈	—	2	2	no	—
30	C₄×C₅×Q₈	—	2	2	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	—	2	1	no	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	4	4	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	D₄	1	4	yes	—
50	C₅×#95	—	2	2	no	—
51	C₅×#95	—	2	2	no	—
52	C₄×#124	C₂×C₅	1	1	yes	characteristic
53	#1842	C₂×C₅	1	1	yes	characteristic, complemented
54	#1744	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×Q₈⋊C₈	C₂×C₅	1	2	yes	complemented
56	C₂×D₄⋊C₈	C₂×C₅	1	2	yes	—
57	C₂⁴×C₅	—	2	1	no	—
58	C₂×C₅×Q₈	—	2	1	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	1	no	—
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	4	4	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	4	no	—
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₅×C₂²⋊C₄	—	2	8	no	—
86	C₅×C₂²⋊C₄	—	4	4	no	—
87	C₅×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	4	2	no	—
92	C₄²×C₅	—	4	1	no	—
93	C₄²×C₅	—	4	1	no	—
94	C₄²×C₅	C₂×C₈	1	2	yes	—
95	C₄²×C₅	M₁₆	1	2	yes	—
96	C₄²×C₅	C₂×D₄	1	2	yes	—
97	C₄²×C₅	#41	1	2	yes	—
98	C₂×#124	C₄×C₅	1	1	yes	characteristic
99	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄⋊C₈	—	2	2	no	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
104	#391	C₄×C₅	1	1	yes	characteristic
105	#390	C₄×C₅	1	1	yes	characteristic
106	#388	—	2	2	no	—
107	#382	C₄×C₅	1	1	yes	characteristic
108	C₄×C₂²⋊C₄	—	2	2	no	—
109	C₄³	C₂²×C₅	1	1	yes	characteristic
110	#336	—	2	2	no	—
111	Q₈⋊C₈	—	2	4	no	—
112	D₄⋊C₈	—	2	4	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	4	1	no	—
115	C₂³×C₅	—	4	1	no	—
116	C₂³×C₅	—	4	2	no	—
117	C₂³×C₅	—	4	2	no	—
118	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₅×Q₈	—	2	4	no	—
120	C₅×Q₈	—	4	2	no	—
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	4	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	8	2	no	—
137	C₂×C₄×C₅	—	8	1	no	—
138	C₂×C₄×C₅	—	8	1	no	—
139	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
140	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₅	#121	1	2	yes	—
142	C₂×C₄×C₅	#124	1	1	yes	characteristic
143	C₂×C₄×C₅	#127	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅	#131	1	2	yes	—
147	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₅	#137	1	2	yes	—
149	C₅×C₈	—	4	4	no	—
150	C₅×C₈	—	8	2	no	—
151	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂³×C₄	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	#124	—	2	2	no	—
157	#124	C₅×C₈	1	4	yes	complemented
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	2	2	no	—
166	C₂×C₂²⋊C₄	—	2	2	no	—
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
171	C₂×C₄²	C₅×D₄	1	2	yes	—
172	C₂×C₄²	C₅×D₄	1	2	yes	—
173	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	4	4	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	C₅×D₄	1	4	yes	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	4	no	—
183	C₂²×C₅	—	8	1	no	—
184	C₂²×C₅	—	8	1	no	—
185	C₂²×C₅	—	8	2	no	—
186	C₂²×C₅	#390	1	1	yes	characteristic
187	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
188	C₂²×C₅	#408	1	1	yes	characteristic
189	C₂²×C₅	#417	1	2	yes	—
190	C₂²×C₅	#421	1	2	yes	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	4	no	—
193	C₄×C₅	—	4	2	no	—
194	C₄×C₅	—	4	2	no	—
195	C₄×C₅	—	8	1	no	—
196	C₄×C₅	—	8	1	no	—
197	C₄×C₅	C₈×D₄	1	2	yes	—
198	C₄×C₅	#436	1	2	yes	—
199	C₄×C₅	#489	1	2	yes	—
200	C₄×C₅	#496	1	1	yes	characteristic
201	C₄×C₅	#497	1	1	yes	characteristic
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
219	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	4	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×C₅×C₈	1	2	yes	—
240	C₄²	C₅×M₁₆	1	2	yes	—
241	C₄²	C₂×C₅×D₄	1	2	yes	—
242	C₄²	C₅×#41	1	2	yes	—
243	C₂×C₅	—	8	2	no	—
244	C₂×C₅	#1477	1	2	yes	—
245	C₂×C₅	#1495	1	2	yes	—
246	C₂×C₅	#1954	1	1	yes	characteristic
247	C₂×C₅	#1959	1	1	yes	characteristic
248	C₂×C₅	#1963	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
255	Q₈	—	2	4	no	—
256	Q₈	—	4	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	4	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
276	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
277	C₂×C₄	C₅×#121	1	2	yes	—
278	C₂×C₄	C₅×#124	1	1	yes	characteristic
279	C₂×C₄	C₅×#127	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
282	C₂×C₄	C₅×#131	1	2	yes	—
283	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
284	C₂×C₄	C₅×#137	1	2	yes	—
285	C₈	—	4	4	no	—
286	C₈	—	8	2	no	—
287	C₅	#9640	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	4	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	C₅×#390	1	1	yes	characteristic
294	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
295	C₂²	C₅×#408	1	1	yes	characteristic
296	C₂²	C₅×#417	1	2	yes	—
297	C₂²	C₅×#421	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	1	no	—
303	C₄	—	8	1	no	—
304	C₄	C₅×C₈×D₄	1	2	yes	—
305	C₄	C₅×#436	1	2	yes	—
306	C₄	C₅×#489	1	2	yes	—
307	C₄	C₅×#496	1	1	yes	characteristic
308	C₄	C₅×#497	1	1	yes	characteristic
309	C₂	—	8	2	no	—
310	C₂	C₅×#1477	1	2	yes	—
311	C₂	C₅×#1495	1	2	yes	—
312	C₂	C₅×#1954	1	1	yes	characteristic
313	C₂	C₅×#1959	1	1	yes	characteristic
314	C₂	C₅×#1963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1842	C₂	semidirect
3. C₅×#1744	C₂	semidirect
4. C₂×C₅×Q₈⋊C₈	C₂	semidirect
21. #9640	C₅	direct (N × H)
28. C₅×#124	C₈	semidirect
53. #1842	C₂×C₅	semidirect
54. #1744	C₂×C₅	semidirect
55. C₂×Q₈⋊C₈	C₂×C₅	semidirect
157. #124	C₅×C₈	semidirect
287. C₅	#9640	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#124	2
C₅×#1842	2
C₅×#1744	2
C₂×C₅×Q₈⋊C₈	2
C₂×C₅×D₄⋊C₈	2
#9640	5

Subgroups — C5×#9640 #11932360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9640 #11932360