Subgroups — C₅×#10862 #11933582

Order 1280.

982 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#107	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#340	C₂	1	2	yes	maximal
4	C₅×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×#107	—	2	4	no	—
10	C₂×C₅×#107	—	2	4	no	—
11	C₂×C₅×#107	C₄	1	8	yes	complemented
12	C₂×C₄×C₅×C₈	—	2	4	no	—
13	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
14	C₂×C₅×#95	—	2	4	no	—
15	C₅×#340	—	2	4	no	—
16	#10862	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	—	2	4	no	—
18	C₂³×C₄×C₅	D₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	4	yes	—
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	Q₈	1	2	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	—	2	4	no	—
34	C₂×C₅×C₄⋊C₄	—	2	4	no	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	Q₈	1	2	yes	—
40	C₅×#107	—	2	16	no	—
41	C₄×C₅×C₈	—	2	8	no	—
42	C₅×#95	—	4	8	no	—
43	C₂²×#107	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
45	C₂×#340	C₂×C₅	1	2	yes	—
46	C₄⋊#95	C₂×C₅	1	2	yes	—
47	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	—	2	8	no	—
51	C₂²×C₄×C₅	—	4	4	no	—
52	C₂²×C₄×C₅	—	4	4	no	—
53	C₂²×C₄×C₅	—	4	8	no	—
54	C₂²×C₄×C₅	—	4	8	no	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅	D₈	1	4	yes	—
57	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	#41	1	2	yes	—
67	C₂²×C₄×C₅	#41	1	2	yes	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	C₄²	1	4	yes	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
75	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
76	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
77	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	2	8	no	—
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	4	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	—	2	4	no	—
85	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
88	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
89	C₂×#107	—	2	4	no	—
90	C₂×#107	—	2	4	no	—
91	C₂×#107	C₄×C₅	1	8	yes	complemented
92	C₂×C₄×C₈	—	2	4	no	—
93	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
94	C₂×#95	—	2	4	no	—
95	#340	—	2	4	no	—
96	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₅	C₄×D₄	1	2	yes	—
98	C₂³×C₅	#121	1	2	yes	—
99	C₂³×C₅	#127	1	1	yes	characteristic
100	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
101	C₂³×C₅	C₂×D₈	1	2	yes	—
102	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
103	C₂³×C₅	#135	1	4	yes	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	8	no	—
107	C₂×C₄×C₅	—	2	8	no	—
108	C₂×C₄×C₅	—	4	8	no	—
109	C₂×C₄×C₅	—	4	16	no	—
110	C₂×C₄×C₅	—	8	8	no	—
111	C₂×C₄×C₅	—	8	4	no	—
112	C₂×C₄×C₅	—	8	4	no	—
113	C₂×C₄×C₅	#107	1	8	yes	complemented
114	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#117	1	1	yes	characteristic
121	C₂×C₄×C₅	#117	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₅	#121	1	2	yes	—
126	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₅	#124	1	2	yes	—
129	C₂×C₄×C₅	#125	1	2	yes	—
130	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
132	C₅×C₈	—	2	4	no	—
133	C₅×C₈	#95	1	8	yes	complemented
134	C₂³×C₄	—	2	4	no	—
135	C₂³×C₄	C₅×D₄	1	2	yes	—
136	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
141	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
142	C₂²×C₈	C₅×D₄	1	2	yes	—
143	C₂²×C₈	C₅×D₄	1	4	yes	—
144	C₂²×C₈	C₅×D₄	1	2	yes	—
145	C₂²×C₈	C₅×Q₈	1	2	yes	—
146	C₂²×C₈	C₅×Q₈	1	2	yes	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	4	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₄²	C₅×Q₈	1	2	yes	—
157	#107	—	2	16	no	—
158	C₄×C₈	—	2	8	no	—
159	#95	—	4	8	no	—
160	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₅	#380	1	1	yes	characteristic
162	C₂²×C₅	#384	1	1	yes	characteristic
163	C₂²×C₅	#389	1	2	yes	—
164	C₂²×C₅	#390	1	2	yes	—
165	C₂²×C₅	C₂×#107	1	2	yes	—
166	C₂²×C₅	#427	1	2	yes	—
167	C₂²×C₅	C₄×D₈	1	4	yes	—
168	C₂²×C₅	C₄×Q₁₆	1	4	yes	—
169	C₂²×C₅	#466	1	4	yes	—
170	C₂²×C₅	#467	1	4	yes	—
171	C₂²×C₅	#493	1	1	yes	characteristic
172	C₂²×C₅	#494	1	1	yes	characteristic
173	C₂²×C₅	#495	1	2	yes	—
174	C₂²×C₅	#499	1	2	yes	—
175	C₂²×C₅	#500	1	2	yes	—
176	C₄×C₅	—	2	8	no	—
177	C₄×C₅	—	8	8	no	—
178	C₄×C₅	C₂×#95	1	1	yes	characteristic
179	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
180	C₄×C₅	#380	1	1	yes	characteristic
181	C₄×C₅	#382	1	1	yes	characteristic
182	C₄×C₅	#389	1	2	yes	—
183	C₄×C₅	#391	1	2	yes	—
184	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	—	4	8	no	—
192	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
193	C₂²×C₄	C₅×D₈	1	4	yes	—
194	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
197	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
203	C₂²×C₄	C₅×#41	1	2	yes	—
204	C₂²×C₄	C₅×#41	1	2	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₄²×C₅	1	4	yes	—
210	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
211	C₂×C₈	C₅×C₂²⋊C₄	1	8	yes	—
212	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
213	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
214	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	4	no	—
222	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
223	C₂×C₅	C₄×#107	1	2	yes	—
224	C₂×C₅	#1808	1	2	yes	—
225	C₂×C₅	#1845	1	1	yes	characteristic
226	C₂×C₅	#1846	1	1	yes	characteristic
227	C₂×C₅	#1942	1	4	yes	—
228	C₂×C₅	#1969	1	2	yes	—
229	C₂×C₅	#1970	1	2	yes	—
230	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂³	C₄×C₅×D₄	1	2	yes	—
232	C₂³	C₅×#121	1	2	yes	—
233	C₂³	C₅×#127	1	1	yes	characteristic
234	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
235	C₂³	C₂×C₅×D₈	1	2	yes	—
236	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
237	C₂³	C₅×#135	1	4	yes	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	16	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₅×#107	1	8	yes	complemented
248	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
249	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
253	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₅×#117	1	1	yes	characteristic
255	C₂×C₄	C₅×#117	1	1	yes	characteristic
256	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
257	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
258	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
259	C₂×C₄	C₅×#121	1	2	yes	—
260	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
261	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
262	C₂×C₄	C₅×#124	1	2	yes	—
263	C₂×C₄	C₅×#125	1	2	yes	—
264	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	C₅×#95	1	8	yes	complemented
268	C₅	#10862	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
270	C₂²	C₅×#380	1	1	yes	characteristic
271	C₂²	C₅×#384	1	1	yes	characteristic
272	C₂²	C₅×#389	1	2	yes	—
273	C₂²	C₅×#390	1	2	yes	—
274	C₂²	C₂×C₅×#107	1	2	yes	—
275	C₂²	C₅×#427	1	2	yes	—
276	C₂²	C₄×C₅×D₈	1	4	yes	—
277	C₂²	C₄×C₅×Q₁₆	1	4	yes	—
278	C₂²	C₅×#466	1	4	yes	—
279	C₂²	C₅×#467	1	4	yes	—
280	C₂²	C₅×#493	1	1	yes	characteristic
281	C₂²	C₅×#494	1	1	yes	characteristic
282	C₂²	C₅×#495	1	2	yes	—
283	C₂²	C₅×#499	1	2	yes	—
284	C₂²	C₅×#500	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	8	8	no	—
287	C₄	C₂×C₅×#95	1	1	yes	characteristic
288	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
289	C₄	C₅×#380	1	1	yes	characteristic
290	C₄	C₅×#382	1	1	yes	characteristic
291	C₄	C₅×#389	1	2	yes	—
292	C₄	C₅×#391	1	2	yes	—
293	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
294	C₂	C₄×C₅×#107	1	2	yes	—
295	C₂	C₅×#1808	1	2	yes	—
296	C₂	C₅×#1845	1	1	yes	characteristic
297	C₂	C₅×#1846	1	1	yes	characteristic
298	C₂	C₅×#1942	1	4	yes	—
299	C₂	C₅×#1969	1	2	yes	—
300	C₂	C₅×#1970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂×C₅×#107	C₄	semidirect
13. C₂×C₄×C₅×C₈	C₄	semidirect
16. #10862	C₅	direct (N × H)
91. C₂×#107	C₄×C₅	semidirect
93. C₂×C₄×C₈	C₄×C₅	semidirect
113. C₂×C₄×C₅	#107	semidirect
133. C₅×C₈	#95	semidirect
247. C₂×C₄	C₅×#107	semidirect
267. C₈	C₅×#95	semidirect
268. C₅	#10862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#107	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#340	2
C₅×C₄⋊#95	2
#10862	5

Subgroups — C5×#10862 #11933582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10862 #11933582