Subgroups — C₅×#8025 #11930745

Order 1280.

654 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	C₅×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#383	C₂	1	1	yes	maximal, characteristic
4	C₅×#1294	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#431	—	2	2	no	—
7	C₅×#431	C₄	1	4	yes	complemented
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	—	2	1	no	—
11	C₅×#383	—	2	2	no	—
12	C₅×#383	—	2	1	no	—
13	C₅×#383	C₄	1	4	yes	complemented
14	C₅×#382	C₄	1	2	yes	complemented
15	C₅×#381	—	2	1	no	—
16	C₅×#380	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	complemented
18	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
19	C₅×#376	—	2	1	no	—
20	C₂×C₅×#95	—	2	1	no	—
21	C₂×C₅×#95	—	2	1	no	—
22	C₂×C₅×#95	C₂²	1	2	yes	—
23	#8025	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
31	C₂×C₄²×C₅	—	2	2	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	2	yes	—
39	C₂×C₄²×C₅	Q₈	1	2	yes	—
40	C₅×C₄⋊C₈	—	2	4	no	—
41	C₅×C₂²⋊C₈	—	2	2	no	—
42	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
43	C₄×C₅×C₈	—	2	4	no	—
44	C₅×#95	—	2	4	no	—
45	C₅×#95	—	2	4	no	—
46	C₅×#95	—	2	2	no	—
47	C₅×#95	—	2	2	no	—
48	C₅×#95	—	2	2	no	—
49	C₅×#95	—	2	2	no	—
50	C₅×#95	—	2	2	no	—
51	C₅×#95	C₂×C₄	1	4	yes	—
52	C₂×#431	C₂×C₅	1	1	yes	characteristic
53	#2283	C₂×C₅	1	1	yes	characteristic
54	C₂×#383	C₂×C₅	1	1	yes	characteristic
55	#1294	C₂×C₅	1	4	yes	—
56	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	—	4	4	no	—
63	C₂²×C₄×C₅	—	4	4	no	—
64	C₂²×C₄×C₅	—	4	4	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	4	no	—
82	C₅×C₂²⋊C₄	—	4	4	no	—
83	C₄²×C₅	—	4	4	no	—
84	C₄²×C₅	C₄²	1	2	yes	—
85	C₄²×C₅	C₄²	1	2	yes	—
86	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅	C₄⋊C₄	1	2	yes	—
92	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
93	#431	—	2	2	no	—
94	#431	C₄×C₅	1	4	yes	complemented
95	C₂×C₄⋊C₈	—	2	1	no	—
96	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
97	C₂×C₄×C₈	—	2	1	no	—
98	#383	—	2	2	no	—
99	#383	—	2	1	no	—
100	#383	C₄×C₅	1	4	yes	complemented
101	#382	C₄×C₅	1	2	yes	complemented
102	#381	—	2	1	no	—
103	#380	C₂²×C₅	1	2	yes	—
104	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	complemented
105	C₄×C₂²⋊C₄	—	2	1	no	—
106	#376	—	2	1	no	—
107	C₂×#95	—	2	1	no	—
108	C₂×#95	—	2	1	no	—
109	C₂×#95	C₂²×C₅	1	2	yes	—
110	C₂³×C₅	—	2	2	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₅	#135	1	4	yes	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
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	—	4	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	—	8	8	no	—
137	C₂×C₄×C₅	#95	1	2	yes	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₅×C₈	—	4	4	no	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
148	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
152	C₂×C₂²⋊C₄	—	2	2	no	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₂×C₄²	C₅×D₄	1	2	yes	—
161	C₂×C₄²	C₅×Q₈	1	2	yes	—
162	C₄⋊C₈	—	2	4	no	—
163	C₂²⋊C₈	—	2	2	no	—
164	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
165	C₄×C₈	—	2	4	no	—
166	#95	—	2	4	no	—
167	#95	—	2	4	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	#95	C₂×C₄×C₅	1	4	yes	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	8	no	—
179	C₂²×C₅	C₂×#95	1	1	yes	characteristic
180	C₂²×C₅	#410	1	2	yes	—
181	C₂²×C₅	#416	1	2	yes	—
182	C₂²×C₅	#416	1	2	yes	—
183	C₂²×C₅	#421	1	2	yes	—
184	C₂²×C₅	#427	1	2	yes	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	8	4	no	—
194	C₄×C₅	—	8	4	no	—
195	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
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×C₂²⋊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	C₂²×C₄×C₅	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
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	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₄²×C₅	1	2	yes	—
224	C₄²	C₄²×C₅	1	2	yes	—
225	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
226	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
227	C₄²	C₅×C₄⋊C₄	1	2	yes	—
228	C₄²	C₅×C₄⋊C₄	1	2	yes	—
229	C₄²	C₅×C₄⋊C₄	1	2	yes	—
230	C₄²	C₅×C₄⋊C₄	1	2	yes	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	#1521	1	4	yes	—
234	C₂×C₅	#1733	1	1	yes	characteristic
235	C₂×C₅	#1735	1	1	yes	characteristic
236	C₂×C₅	#1740	1	1	yes	characteristic
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
241	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₂³	C₅×#135	1	4	yes	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	C₅×#95	1	2	yes	—
265	C₂×C₄	C₅×#95	1	2	yes	—
266	C₂×C₄	C₅×#95	1	2	yes	—
267	C₂×C₄	C₅×#95	1	2	yes	—
268	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
272	C₈	—	4	4	no	—
273	C₅	#8025	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	8	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂×C₅×#95	1	1	yes	characteristic
280	C₂²	C₅×#410	1	2	yes	—
281	C₂²	C₅×#416	1	2	yes	—
282	C₂²	C₅×#416	1	2	yes	—
283	C₂²	C₅×#421	1	2	yes	—
284	C₂²	C₅×#427	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	4	no	—
294	C₄	—	8	4	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	C₅×#1521	1	4	yes	—
298	C₂	C₅×#1733	1	1	yes	characteristic
299	C₂	C₅×#1735	1	1	yes	characteristic
300	C₂	C₅×#1740	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₅×#431	C₄	semidirect
13. C₅×#383	C₄	semidirect
14. C₅×#382	C₄	semidirect
17. C₄×C₅×C₄⋊C₄	C₄	semidirect
23. #8025	C₅	direct (N × H)
94. #431	C₄×C₅	semidirect
100. #383	C₄×C₅	semidirect
101. #382	C₄×C₅	semidirect
104. C₄×C₄⋊C₄	C₄×C₅	semidirect
273. C₅	#8025	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
C₅×#2283	2
C₂×C₅×#383	2
C₅×#1294	2
#8025	5

Subgroups — C5×#8025 #11930745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8025 #11930745