Subgroups — C₅×#21440 #11944160

Order 1280.

1708 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#435	C₂	1	3	yes	maximal, complemented
3	C₄×C₅×#120	C₂	1	1	yes	maximal, characteristic
4	C₅×#1869	C₂	1	3	yes	maximal, complemented
5	C₅×#1850	C₂	1	3	yes	maximal, complemented
6	C₅×#1779	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1752	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₅×C₈	—	2	12	no	—
11	C₂³×C₅×C₈	C₂²	1	3	yes	—
12	C₂³×C₅×C₈	C₂²	1	3	yes	—
13	C₂×C₅×#120	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×D₄	C₂²	1	3	yes	—
15	C₅×#435	—	2	12	no	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	3	yes	complemented
17	C₂×C₅×C₂²⋊C₈	C₂²	1	3	yes	—
18	C₂×C₅×C₂²⋊C₈	C₂²	1	3	yes	—
19	C₂×C₅×C₂²⋊C₈	C₂²	1	3	yes	complemented
20	C₂×C₅×#97	C₂²	1	3	yes	complemented
21	C₅×#386	C₄	1	3	yes	—
22	C₅×#385	C₄	1	3	yes	—
23	C₅×#379	C₄	1	1	yes	characteristic
24	C₄×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
25	C₅×#336	C₂²	1	6	yes	complemented
26	#21440	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×D₄	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₅	—	2	6	no	—
30	C₂³×C₄×C₅	—	2	6	no	—
31	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₅	C₂³	1	3	yes	—
34	C₂×C₅×M₁₆	—	2	6	no	—
35	C₂²×C₅×C₈	—	2	32	no	—
36	C₂²×C₅×C₈	—	2	6	no	—
37	C₂²×C₅×C₈	—	2	12	no	—
38	C₂²×C₅×C₈	—	2	12	no	—
39	C₂²×C₅×C₈	D₄	1	12	yes	complemented
40	C₂²×C₅×C₈	C₂³	1	3	yes	—
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	3	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#120	—	2	4	no	—
45	C₄×C₅×D₄	—	2	6	no	—
46	C₄×C₅×D₄	—	2	6	no	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
48	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	3	yes	—
49	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	3	yes	—
50	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	3	yes	—
51	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	3	yes	—
52	C₂×C₄²×C₅	C₂³	1	3	yes	—
53	C₅×C₄⋊C₈	—	2	6	no	—
54	C₅×C₂²⋊C₈	—	2	6	no	—
55	C₅×C₂²⋊C₈	—	2	6	no	—
56	C₅×C₂²⋊C₈	—	2	6	no	—
57	C₅×#97	—	2	6	no	—
58	C₅×#95	C₂×C₄	1	6	yes	—
59	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×#435	C₂×C₅	1	3	yes	complemented
61	C₄×#120	C₂×C₅	1	1	yes	characteristic
62	#1869	C₂×C₅	1	3	yes	complemented
63	#1850	C₂×C₅	1	3	yes	complemented
64	#1779	C₂×C₅	1	1	yes	characteristic, complemented
65	#1752	C₂×C₅	1	3	yes	complemented
66	C₂⁴×C₅	—	2	6	no	—
67	C₂⁴×C₅	—	2	6	no	—
68	C₂⁴×C₅	M₁₆	1	4	yes	complemented
69	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂⁴×C₅	C₂²×C₄	1	3	yes	—
71	C₂×C₅×D₄	—	2	6	no	—
72	C₂×C₅×D₄	—	2	6	no	—
73	C₂×C₅×D₄	—	2	6	no	—
74	C₂²×C₄×C₅	—	2	8	no	—
75	C₂²×C₄×C₅	—	2	24	no	—
76	C₂²×C₄×C₅	—	2	6	no	—
77	C₂²×C₄×C₅	—	2	12	no	—
78	C₂²×C₄×C₅	—	2	6	no	—
79	C₂²×C₄×C₅	—	2	6	no	—
80	C₂²×C₄×C₅	—	2	3	no	—
81	C₂²×C₄×C₅	—	2	3	no	—
82	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₅	#41	1	6	yes	—
88	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₅×M₁₆	—	4	4	no	—
90	C₂×C₅×C₈	—	2	24	no	—
91	C₂×C₅×C₈	—	2	6	no	—
92	C₂×C₅×C₈	—	2	24	no	—
93	C₂×C₅×C₈	—	2	3	no	—
94	C₂×C₅×C₈	—	2	3	no	—
95	C₂×C₅×C₈	—	2	6	no	—
96	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
98	C₅×C₄⋊C₄	—	2	6	no	—
99	C₅×C₂²⋊C₄	—	2	6	no	—
100	C₅×C₂²⋊C₄	—	2	6	no	—
101	C₅×C₂²⋊C₄	—	2	6	no	—
102	C₅×C₂²⋊C₄	—	2	6	no	—
103	C₄²×C₅	—	2	3	no	—
104	C₄²×C₅	—	2	3	no	—
105	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
106	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
107	C₂³×C₈	—	2	12	no	—
108	C₂³×C₈	C₂²×C₅	1	3	yes	—
109	C₂³×C₈	C₂²×C₅	1	3	yes	—
110	C₂×#120	C₄×C₅	1	1	yes	characteristic
111	C₂×C₄×D₄	C₂²×C₅	1	3	yes	—
112	#435	—	2	12	no	—
113	C₂×C₄⋊C₈	C₂²×C₅	1	3	yes	complemented
114	C₂×C₂²⋊C₈	C₂²×C₅	1	3	yes	—
115	C₂×C₂²⋊C₈	C₂²×C₅	1	3	yes	—
116	C₂×C₂²⋊C₈	C₂²×C₅	1	3	yes	complemented
117	C₂×#97	C₂²×C₅	1	3	yes	complemented
118	#386	C₄×C₅	1	3	yes	—
119	#385	C₄×C₅	1	3	yes	—
120	#379	C₄×C₅	1	1	yes	characteristic
121	C₄×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
122	#336	C₂²×C₅	1	6	yes	complemented
123	C₂³×C₅	—	2	8	no	—
124	C₂³×C₅	—	2	24	no	—
125	C₂³×C₅	—	2	6	no	—
126	C₂³×C₅	—	2	6	no	—
127	C₂³×C₅	—	2	24	no	—
128	C₂³×C₅	—	2	3	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	—	2	3	no	—
131	C₂³×C₅	C₄×D₄	1	6	yes	—
132	C₂³×C₅	C₂×M₁₆	1	6	yes	—
133	C₂³×C₅	#131	1	6	yes	—
134	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₅×D₄	—	4	12	no	—
136	C₂×C₄×C₅	—	2	24	no	—
137	C₂×C₄×C₅	—	2	6	no	—
138	C₂×C₄×C₅	—	2	6	no	—
139	C₂×C₄×C₅	—	2	6	no	—
140	C₂×C₄×C₅	—	2	12	no	—
141	C₂×C₄×C₅	—	2	3	no	—
142	C₂×C₄×C₅	—	2	3	no	—
143	C₂×C₄×C₅	—	2	3	no	—
144	C₂×C₄×C₅	—	2	3	no	—
145	C₂×C₄×C₅	—	2	6	no	—
146	C₂×C₄×C₅	—	2	3	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	3	no	—
149	C₂×C₄×C₅	C₄×D₄	1	6	yes	—
150	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
151	C₂×C₄×C₅	C₂×#41	1	3	yes	—
152	C₅×C₈	—	2	6	no	—
153	C₅×C₈	—	4	1	no	—
154	C₅×C₈	—	4	3	no	—
155	C₅×C₈	#120	1	4	yes	complemented
156	C₂⁵	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂²×D₄	C₂×C₄×C₅	1	3	yes	—
158	C₂³×C₄	—	2	6	no	—
159	C₂³×C₄	—	2	6	no	—
160	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
161	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₅	1	3	yes	—
163	C₂×M₁₆	—	2	6	no	—
164	C₂²×C₈	—	2	32	no	—
165	C₂²×C₈	—	2	6	no	—
166	C₂²×C₈	—	2	12	no	—
167	C₂²×C₈	—	2	12	no	—
168	C₂²×C₈	C₅×D₄	1	12	yes	complemented
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅	1	3	yes	—
171	C₂²×C₈	C₂³×C₅	1	3	yes	—
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	#120	—	2	4	no	—
174	C₄×D₄	—	2	6	no	—
175	C₄×D₄	—	2	6	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₅	1	3	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	3	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	3	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	3	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	3	yes	—
181	C₂×C₄²	C₂³×C₅	1	3	yes	—
182	C₄⋊C₈	—	2	6	no	—
183	C₂²⋊C₈	—	2	6	no	—
184	C₂²⋊C₈	—	2	6	no	—
185	C₂²⋊C₈	—	2	6	no	—
186	#97	—	2	6	no	—
187	#95	C₂×C₄×C₅	1	6	yes	—
188	C₂²×C₅	—	2	32	no	—
189	C₂²×C₅	—	2	12	no	—
190	C₂²×C₅	—	2	6	no	—
191	C₂²×C₅	—	2	12	no	—
192	C₂²×C₅	—	2	6	no	—
193	C₂²×C₅	—	4	3	no	—
194	C₂²×C₅	—	4	3	no	—
195	C₂²×C₅	—	4	1	no	—
196	C₂²×C₅	#435	1	12	yes	complemented
197	C₂²×C₅	C₂×C₄×D₄	1	3	yes	—
198	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
199	C₂²×C₅	C₂×#131	1	3	yes	—
200	C₄×C₅	—	2	6	no	—
201	C₄×C₅	—	4	3	no	—
202	C₄×C₅	—	4	3	no	—
203	C₄×C₅	—	4	1	no	—
204	C₄×C₅	C₂×#120	1	1	yes	characteristic
205	C₄×C₅	#525	1	3	yes	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	2	6	no	—
208	C₂⁴	C₅×M₁₆	1	4	yes	complemented
209	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂⁴	C₂²×C₄×C₅	1	3	yes	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	—	2	6	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	24	no	—
216	C₂²×C₄	—	2	6	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	2	6	no	—
219	C₂²×C₄	—	2	6	no	—
220	C₂²×C₄	—	2	3	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
223	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
224	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
226	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
227	C₂²×C₄	C₅×#41	1	6	yes	—
228	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
229	M₁₆	—	4	4	no	—
230	C₂×C₈	—	2	24	no	—
231	C₂×C₈	—	2	6	no	—
232	C₂×C₈	—	2	24	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	—	2	6	no	—
236	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
237	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
238	C₄⋊C₄	—	2	6	no	—
239	C₂²⋊C₄	—	2	6	no	—
240	C₂²⋊C₄	—	2	6	no	—
241	C₂²⋊C₄	—	2	6	no	—
242	C₂²⋊C₄	—	2	6	no	—
243	C₄²	—	2	3	no	—
244	C₄²	—	2	3	no	—
245	C₂×C₅	—	2	12	no	—
246	C₂×C₅	—	8	1	no	—
247	C₂×C₅	C₄×#120	1	1	yes	characteristic
248	C₂×C₅	C₂×#435	1	3	yes	—
249	C₂×C₅	#2929	1	3	yes	—
250	C₂³	—	2	8	no	—
251	C₂³	—	2	24	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	6	no	—
254	C₂³	—	2	6	no	—
255	C₂³	—	2	3	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	3	no	—
258	C₂³	C₄×C₅×D₄	1	6	yes	—
259	C₂³	C₂×C₅×M₁₆	1	6	yes	—
260	C₂³	C₅×#131	1	6	yes	—
261	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
262	D₄	—	4	12	no	—
263	C₂×C₄	—	2	24	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	12	no	—
268	C₂×C₄	—	2	3	no	—
269	C₂×C₄	—	2	3	no	—
270	C₂×C₄	—	2	3	no	—
271	C₂×C₄	—	2	3	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	2	3	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	3	no	—
276	C₂×C₄	C₄×C₅×D₄	1	6	yes	—
277	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
278	C₂×C₄	C₂×C₅×#41	1	3	yes	—
279	C₈	—	2	6	no	—
280	C₈	—	4	1	no	—
281	C₈	—	4	3	no	—
282	C₈	C₅×#120	1	4	yes	complemented
283	C₅	#21440	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	32	no	—
285	C₂²	—	2	6	no	—
286	C₂²	—	2	12	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	12	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	3	no	—
291	C₂²	—	4	3	no	—
292	C₂²	C₅×#435	1	12	yes	complemented
293	C₂²	C₂×C₄×C₅×D₄	1	3	yes	—
294	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
295	C₂²	C₂×C₅×#131	1	3	yes	—
296	C₄	—	2	6	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	3	no	—
299	C₄	—	4	3	no	—
300	C₄	C₂×C₅×#120	1	1	yes	characteristic
301	C₄	C₅×#525	1	3	yes	—
302	C₂	—	2	12	no	—
303	C₂	—	8	1	no	—
304	C₂	C₄×C₅×#120	1	1	yes	characteristic
305	C₂	C₂×C₅×#435	1	3	yes	—
306	C₂	C₅×#2929	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂×C₅×#120	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×C₈	C₂	semidirect
2. C₂×C₅×#435	C₂	semidirect
4. C₅×#1869	C₂	semidirect
5. C₅×#1850	C₂	semidirect
6. C₅×#1779	C₂	semidirect
7. C₅×#1752	C₂	semidirect
9. C₂²×C₅×M₁₆	C₂²	semidirect
16. C₂×C₅×C₄⋊C₈	C₂²	semidirect
19. C₂×C₅×C₂²⋊C₈	C₂²	semidirect
20. C₂×C₅×#97	C₂²	semidirect
25. C₅×#336	C₂²	semidirect
26. #21440	C₅	direct (N × H)
39. C₂²×C₅×C₈	D₄	semidirect
59. C₂⁴×C₈	C₂×C₅	semidirect
60. C₂×#435	C₂×C₅	semidirect
62. #1869	C₂×C₅	semidirect
63. #1850	C₂×C₅	semidirect
64. #1779	C₂×C₅	semidirect
65. #1752	C₂×C₅	semidirect
68. C₂⁴×C₅	M₁₆	semidirect
106. C₂²×M₁₆	C₂²×C₅	semidirect
113. C₂×C₄⋊C₈	C₂²×C₅	semidirect
116. C₂×C₂²⋊C₈	C₂²×C₅	semidirect
117. C₂×#97	C₂²×C₅	semidirect
122. #336	C₂²×C₅	semidirect
155. C₅×C₈	#120	semidirect
168. C₂²×C₈	C₅×D₄	semidirect
196. C₂²×C₅	#435	semidirect
208. C₂⁴	C₅×M₁₆	semidirect
282. C₈	C₅×#120	semidirect
283. C₅	#21440	direct (N × H)
292. C₂²	C₅×#435	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₈	2
C₂×C₅×#435	2
C₄×C₅×#120	2
C₅×#1869	2
C₅×#1850	2
C₅×#1779	2
C₅×#1752	2
#21440	5

Subgroups — C5×#21440 #11944160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21440 #11944160