Subgroups — C₅×#7890 #11930610

Order 1280.

830 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	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	—	2	1	no	—
6	C₂×C₅×#117	C₄	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#431	—	2	2	no	—
9	C₅×#431	C₄	1	4	yes	complemented
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₄×C₅×C₈	—	2	1	no	—
13	C₂×C₄×C₅×C₈	—	2	2	no	—
14	C₂×C₄×C₅×C₈	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
20	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
21	C₅×#376	—	2	1	no	—
22	C₄³×C₅	—	2	1	no	—
23	#7890	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	—	2	8	no	—
28	C₂²×C₅×C₈	—	2	4	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₅×#117	—	2	4	no	—
32	C₅×#117	C₈	1	8	yes	complemented
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₅	—	2	2	no	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	Q₈	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
48	C₅×C₂²⋊C₈	—	2	4	no	—
49	C₄×C₅×C₈	—	2	4	no	—
50	C₄×C₅×C₈	—	2	4	no	—
51	C₄×C₅×C₈	—	2	2	no	—
52	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
53	C₅×#95	—	2	2	no	—
54	C₅×#95	—	2	2	no	—
55	C₂×#431	C₂×C₅	1	1	yes	characteristic
56	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
57	C₄×#117	C₂×C₅	1	1	yes	characteristic
58	#1279	C₂×C₅	1	4	yes	complemented
59	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	C₄²	1	2	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₅	M₁₆	1	4	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	—	2	8	no	—
78	C₂×C₅×C₈	—	2	16	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	4	4	no	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	complemented
84	C₅×C₂²⋊C₄	—	2	8	no	—
85	C₄²×C₅	—	2	8	no	—
86	C₄²×C₅	—	4	4	no	—
87	C₄²×C₅	—	4	4	no	—
88	C₄²×C₅	C₄²	1	2	yes	—
89	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₅	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅	C₄⋊C₄	1	2	yes	—
93	C₄²×C₅	C₄⋊C₄	1	2	yes	—
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
98	C₂³×C₈	—	2	1	no	—
99	C₂×#117	C₄×C₅	1	2	yes	—
100	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
101	#431	—	2	2	no	—
102	#431	C₄×C₅	1	4	yes	complemented
103	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×C₂²⋊C₈	—	2	1	no	—
105	C₂×C₄×C₈	—	2	1	no	—
106	C₂×C₄×C₈	—	2	2	no	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
109	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
110	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
111	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
112	C₄×C₂²⋊C₄	—	2	1	no	—
113	C₄×C₂²⋊C₄	—	2	1	no	—
114	#376	—	2	1	no	—
115	C₄³	—	2	1	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	#95	1	2	yes	—
119	C₂³×C₅	C₂²⋊C₈	1	4	yes	—
120	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	8	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	#95	1	2	yes	—
131	C₂×C₄×C₅	#95	1	2	yes	—
132	C₂×C₄×C₅	#95	1	2	yes	—
133	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
134	C₂×C₄×C₅	#97	1	4	yes	—
135	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
136	C₂×C₄×C₅	#104	1	4	yes	—
137	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
138	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₅	#135	1	4	yes	—
147	C₅×C₈	—	2	8	no	—
148	C₅×C₈	—	4	4	no	—
149	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
151	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	#117	—	2	4	no	—
157	#117	C₅×C₈	1	8	yes	complemented
158	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
166	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
167	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
168	C₂×C₄²	C₅×D₄	1	2	yes	—
169	C₂×C₄²	C₅×Q₈	1	2	yes	—
170	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
173	C₂²⋊C₈	—	2	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂²×C₅	—	2	8	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	#336	1	4	yes	—
184	C₂²×C₅	C₂×#95	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₂²⋊C₈	1	2	yes	—
186	C₂²×C₅	#416	1	2	yes	—
187	C₂²×C₅	C₂×#104	1	2	yes	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	8	no	—
192	C₄×C₅	#336	1	4	yes	—
193	C₄×C₅	#339	1	4	yes	—
194	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
195	C₄×C₅	C₂×#97	1	1	yes	characteristic
196	C₄×C₅	#416	1	2	yes	—
197	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
198	C₄×C₅	#427	1	2	yes	—
199	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₄²×C₅	1	2	yes	—
204	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
206	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
210	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
211	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
212	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
213	C₂²×C₄	C₅×M₁₆	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	16	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	complemented
224	C₂²⋊C₄	—	2	8	no	—
225	C₄²	—	2	8	no	—
226	C₄²	—	4	4	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₄²×C₅	1	2	yes	—
229	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
230	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
231	C₄²	C₅×C₄⋊C₄	1	2	yes	—
232	C₄²	C₅×C₄⋊C₄	1	2	yes	—
233	C₄²	C₅×C₄⋊C₄	1	2	yes	—
234	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
237	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
238	C₂×C₅	—	2	4	no	—
239	C₂×C₅	#1476	1	4	yes	—
240	C₂×C₅	C₂×#336	1	1	yes	characteristic
241	C₂×C₅	C₂×#339	1	1	yes	characteristic
242	C₂×C₅	#1735	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	C₅×#95	1	2	yes	—
246	C₂³	C₅×C₂²⋊C₈	1	4	yes	—
247	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₅×#95	1	2	yes	—
258	C₂×C₄	C₅×#95	1	2	yes	—
259	C₂×C₄	C₅×#95	1	2	yes	—
260	C₂×C₄	C₄×C₅×C₈	1	4	yes	—
261	C₂×C₄	C₅×#97	1	4	yes	—
262	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
263	C₂×C₄	C₅×#104	1	4	yes	—
264	C₂×C₄	C₅×C₄⋊C₈	1	8	yes	—
265	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
272	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
273	C₂×C₄	C₅×#135	1	4	yes	—
274	C₈	—	2	8	no	—
275	C₈	—	4	4	no	—
276	C₅	#7890	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	C₅×#336	1	4	yes	—
281	C₂²	C₂×C₅×#95	1	1	yes	characteristic
282	C₂²	C₂×C₅×C₂²⋊C₈	1	2	yes	—
283	C₂²	C₅×#416	1	2	yes	—
284	C₂²	C₂×C₅×#104	1	2	yes	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	8	no	—
289	C₄	C₅×#336	1	4	yes	—
290	C₄	C₅×#339	1	4	yes	—
291	C₄	C₂×C₄×C₅×C₈	1	1	yes	characteristic
292	C₄	C₂×C₅×#97	1	1	yes	characteristic
293	C₄	C₅×#416	1	2	yes	—
294	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
295	C₄	C₅×#427	1	2	yes	—
296	C₂	—	2	4	no	—
297	C₂	C₅×#1476	1	4	yes	—
298	C₂	C₂×C₅×#336	1	1	yes	characteristic
299	C₂	C₂×C₅×#339	1	1	yes	characteristic
300	C₂	C₅×#1735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₅	D₄
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1279	C₂	semidirect
9. C₅×#431	C₄	semidirect
15. C₂×C₄×C₅×C₈	C₄	semidirect
23. #7890	C₅	direct (N × H)
32. C₅×#117	C₈	semidirect
58. #1279	C₂×C₅	semidirect
83. C₅×C₄⋊C₄	C₂×C₈	semidirect
102. #431	C₄×C₅	semidirect
108. C₂×C₄×C₈	C₄×C₅	semidirect
157. #117	C₅×C₈	semidirect
223. C₄⋊C₄	C₂×C₅×C₈	semidirect
276. C₅	#7890	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#431	2
C₂²×C₄×C₅×C₈	2
C₄×C₅×#117	2
C₅×#1279	2
#7890	5

Subgroups — C5×#7890 #11930610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7890 #11930610