Subgroups — C₄×C₅×#360 #11935363

Order 1280.

466 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×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#360	C₂	1	2	yes	maximal
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	1	yes	characteristic
7	C₄×C₅×#41	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
10	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₄×C₅×SD₁₆	—	2	1	no	—
12	C₄×C₅×SD₁₆	—	2	1	no	—
13	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
14	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
15	C₅×#427	C₂²	1	1	yes	characteristic
16	C₂×C₅×#106	C₂²	1	2	yes	—
17	C₅×#416	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₄×C₅×C₄⋊C₄	—	2	1	no	—
20	C₅×#360	C₄	1	8	yes	direct product factor
21	C₅×#346	—	2	1	no	—
22	C₄×C₅×C₁₆	—	2	1	no	—
23	C₅×#340	—	2	1	no	—
24	C₄×#360	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	—	2	1	no	—
26	C₅×#135	—	2	2	no	—
27	C₅×#135	C₂×C₄	1	4	yes	—
28	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×SD₁₆	—	2	1	no	—
30	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₅×Q₈	—	2	1	no	—
34	C₄×C₅×Q₈	—	2	1	no	—
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	4	1	no	—
38	C₄×C₅×D₄	—	4	1	no	—
39	C₅×#117	—	2	1	no	—
40	C₅×#117	—	4	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₅	—	4	1	no	—
43	C₂×C₄²×C₅	—	4	1	no	—
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₅×M₃₂	—	2	2	no	—
46	C₅×M₃₂	C₂×C₄	1	4	yes	complemented
47	C₂×C₅×C₁₆	—	2	2	no	—
48	C₅×#107	C₂×C₄	1	1	yes	characteristic
49	C₅×#107	C₂×C₄	1	1	yes	characteristic
50	C₅×#106	—	2	1	no	—
51	C₅×#106	—	2	2	no	—
52	C₅×#106	C₂×C₄	1	4	yes	complemented
53	C₅×#103	—	2	1	no	—
54	C₅×#103	—	2	1	no	—
55	C₅×#102	—	2	1	no	—
56	C₅×#102	—	2	1	no	—
57	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	D₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₅×#95	—	4	1	no	—
62	C₄×#135	C₂×C₅	1	1	yes	characteristic
63	C₂×#360	C₂×C₅	1	2	yes	—
64	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
65	C₄×#106	C₂×C₅	1	1	yes	characteristic, complemented
66	#1386	C₂×C₅	1	2	yes	complemented
67	C₅×#41	—	2	4	no	—
68	C₅×#41	—	4	2	no	—
69	C₂×C₅×Q₈	—	2	1	no	—
70	C₂×C₅×D₄	—	2	1	no	—
71	C₂×C₅×D₄	—	4	1	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×Q₁₆	—	2	1	no	—
80	C₅×Q₁₆	C₄²	1	2	yes	complemented
81	C₅×SD₁₆	—	2	4	no	—
82	C₅×D₈	—	2	1	no	—
83	C₅×D₈	C₄²	1	2	yes	complemented
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	#41	1	2	yes	—
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	1	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₂²⋊C₄	—	4	1	no	—
98	C₅×C₂²⋊C₄	—	4	1	no	—
99	C₅×C₂²⋊C₄	—	4	1	no	—
100	C₄²×C₅	—	4	1	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₄²×C₅	—	4	1	no	—
103	C₄²×C₅	—	8	2	no	—
104	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	SD₁₆	1	1	yes	characteristic
107	C₄²×C₅	SD₁₆	1	1	yes	characteristic
108	C₅×C₁₆	—	2	4	no	—
109	C₂×#135	C₂²×C₅	1	1	yes	characteristic
110	C₄×#41	—	2	1	no	—
111	C₂×M₃₂	C₂²×C₅	1	2	yes	—
112	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
113	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
114	C₄×SD₁₆	—	2	1	no	—
115	C₄×SD₁₆	—	2	1	no	—
116	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
117	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
118	#427	C₂²×C₅	1	1	yes	characteristic
119	C₂×#106	C₂²×C₅	1	2	yes	—
120	#416	—	2	1	no	—
121	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	2	1	no	—
123	#360	C₄×C₅	1	8	yes	direct product factor
124	#346	—	2	1	no	—
125	C₄×C₁₆	—	2	1	no	—
126	#340	—	2	1	no	—
127	C₂³×C₅	—	4	1	no	—
128	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
129	C₅×Q₈	—	2	2	no	—
130	C₅×Q₈	—	4	1	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	4	4	no	—
133	C₅×D₄	—	4	1	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	8	2	no	—
142	C₂×C₄×C₅	—	8	2	no	—
143	C₂×C₄×C₅	—	8	2	no	—
144	C₂×C₄×C₅	—	8	2	no	—
145	C₂×C₄×C₅	#102	1	1	yes	characteristic
146	C₂×C₄×C₅	#102	1	1	yes	characteristic
147	C₂×C₄×C₅	#102	1	1	yes	characteristic
148	C₂×C₄×C₅	#102	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
152	C₂×C₄×C₅	#135	1	2	yes	—
153	C₅×C₈	—	2	1	no	—
154	C₅×C₈	—	2	1	no	—
155	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
156	C₅×C₈	#117	1	1	yes	characteristic
157	C₅×C₈	C₄×D₄	1	2	yes	—
158	C₂×#41	—	2	1	no	—
159	#135	—	2	2	no	—
160	#135	C₂×C₄×C₅	1	4	yes	—
161	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂×SD₁₆	—	2	1	no	—
163	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
166	C₄×Q₈	—	2	1	no	—
167	C₄×Q₈	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	#117	—	2	1	no	—
173	#117	—	4	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	4	1	no	—
176	C₂×C₄²	—	4	1	no	—
177	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
178	M₃₂	—	2	2	no	—
179	M₃₂	C₂×C₄×C₅	1	4	yes	complemented
180	C₂×C₁₆	—	2	2	no	—
181	#107	C₂×C₄×C₅	1	1	yes	characteristic
182	#107	C₂×C₄×C₅	1	1	yes	characteristic
183	#106	—	2	1	no	—
184	#106	—	2	2	no	—
185	#106	C₂×C₄×C₅	1	4	yes	complemented
186	#103	—	2	1	no	—
187	#103	—	2	1	no	—
188	#102	—	2	1	no	—
189	#102	—	2	1	no	—
190	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
191	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
192	C₄×C₈	C₅×D₄	1	1	yes	characteristic
193	C₄×C₈	C₅×D₄	1	1	yes	characteristic
194	#95	—	4	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	4	2	no	—
198	C₂²×C₅	—	8	1	no	—
199	C₂²×C₅	—	8	1	no	—
200	C₂²×C₅	C₂×#102	1	1	yes	characteristic
201	C₂²×C₅	C₄×D₈	1	2	yes	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	4	2	no	—
206	C₄×C₅	—	8	4	no	—
207	C₄×C₅	—	8	1	no	—
208	C₄×C₅	—	8	1	no	—
209	C₄×C₅	#360	1	2	yes	direct product factor
210	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
211	C₄×C₅	#416	1	1	yes	characteristic
212	C₄×C₅	C₄×SD₁₆	1	2	yes	—
213	#41	—	2	4	no	—
214	#41	—	4	2	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
223	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	Q₁₆	—	2	1	no	—
226	Q₁₆	C₄²×C₅	1	2	yes	complemented
227	SD₁₆	—	2	4	no	—
228	D₈	—	2	1	no	—
229	D₈	C₄²×C₅	1	2	yes	complemented
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
235	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₈	C₅×#41	1	2	yes	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	8	2	no	—
250	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₄²	C₅×SD₁₆	1	1	yes	characteristic
253	C₄²	C₅×SD₁₆	1	1	yes	characteristic
254	C₁₆	—	2	4	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	8	2	no	—
257	C₂×C₅	C₄×#102	1	1	yes	characteristic
258	C₂×C₅	C₂×#360	1	1	yes	characteristic
259	C₂×C₅	#2145	1	1	yes	characteristic
260	C₂³	—	4	1	no	—
261	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	4	1	no	—
264	D₄	—	2	2	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	C₅×#102	1	1	yes	characteristic
279	C₂×C₄	C₅×#102	1	1	yes	characteristic
280	C₂×C₄	C₅×#102	1	1	yes	characteristic
281	C₂×C₄	C₅×#102	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
284	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
285	C₂×C₄	C₅×#135	1	2	yes	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
289	C₈	C₅×#117	1	1	yes	characteristic
290	C₈	C₄×C₅×D₄	1	2	yes	—
291	C₅	C₄×#360	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	C₂×C₅×#102	1	1	yes	characteristic
298	C₂²	C₄×C₅×D₈	1	2	yes	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	4	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	C₅×#360	1	2	yes	direct product factor
307	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
308	C₄	C₅×#416	1	1	yes	characteristic
309	C₄	C₄×C₅×SD₁₆	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	8	2	no	—
312	C₂	C₄×C₅×#102	1	1	yes	characteristic
313	C₂	C₂×C₅×#360	1	1	yes	characteristic
314	C₂	C₅×#2145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×M₃₂	C₂	semidirect
4. C₄×C₅×#106	C₂	semidirect
5. C₅×#1386	C₂	semidirect
9. C₄×C₅×Q₁₆	C₄	semidirect
10. C₄×C₅×Q₁₆	C₄	semidirect
13. C₄×C₅×D₈	C₄	semidirect
14. C₄×C₅×D₈	C₄	semidirect
20. C₅×#360	C₄	direct (N × H)
24. C₄×#360	C₅	direct (N × H)
46. C₅×M₃₂	C₂×C₄	semidirect
52. C₅×#106	C₂×C₄	semidirect
64. C₄×M₃₂	C₂×C₅	semidirect
65. C₄×#106	C₂×C₅	semidirect
66. #1386	C₂×C₅	semidirect
80. C₅×Q₁₆	C₄²	semidirect
83. C₅×D₈	C₄²	semidirect
112. C₄×Q₁₆	C₄×C₅	semidirect
113. C₄×Q₁₆	C₄×C₅	semidirect
116. C₄×D₈	C₄×C₅	semidirect
117. C₄×D₈	C₄×C₅	semidirect
123. #360	C₄×C₅	direct (N × H)
179. M₃₂	C₂×C₄×C₅	semidirect
185. #106	C₂×C₄×C₅	semidirect
209. C₄×C₅	#360	direct (N × H)
226. Q₁₆	C₄²×C₅	semidirect
229. D₈	C₄²×C₅	semidirect
291. C₅	C₄×#360	direct (N × H)
306. C₄	C₅×#360	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×#135	2
C₂×C₅×#360	2
C₄×C₅×M₃₂	2
C₄×C₅×#106	2
C₅×#1386	2
C₄×#360	5

Subgroups — C4×C5×#360 #11935363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#360 #11935363