Subgroups — C₅×#22620 #11945340

Order 1280.

766 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₅×#2347	C₂	1	2	yes	maximal
3	C₅×#1943	C₂	1	2	yes	maximal, complemented
4	C₅×#1920	C₂	1	4	yes	maximal, complemented
5	C₅×#1825	C₂	1	2	yes	maximal
6	C₅×#1807	C₂	1	1	yes	maximal, characteristic
7	C₅×#1776	C₂	1	2	yes	maximal, complemented
8	C₅×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	2	yes	—
11	C₅×#427	C₄	1	8	yes	complemented
12	C₂×C₅×#107	C₂²	1	1	yes	characteristic
13	C₂×C₅×#107	C₂²	1	1	yes	characteristic
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₅×#389	C₂²	1	4	yes	—
19	C₅×#388	—	2	4	no	—
20	C₅×#383	—	2	2	no	—
21	C₅×#380	C₂²	1	2	yes	—
22	C₄×C₅×C₄⋊C₄	—	2	2	no	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
24	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×#340	C₂²	1	4	yes	—
26	C₅×#340	C₂²	1	2	yes	—
27	C₅×#340	C₂²	1	4	yes	—
28	C₅×#336	C₂²	1	2	yes	—
29	#22620	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	4	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#117	—	2	4	no	—
37	C₅×#117	C₂×C₄	1	8	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	4	no	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
45	C₂×C₄²×C₅	—	2	2	no	—
46	C₂×C₄²×C₅	—	2	4	no	—
47	C₂×C₄²×C₅	C₂³	1	2	yes	—
48	C₅×#107	C₂×C₄	1	4	yes	—
49	C₅×#107	C₂×C₄	1	4	yes	—
50	C₅×#106	—	2	4	no	—
51	C₅×#106	C₂×C₄	1	8	yes	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×#97	—	2	4	no	—
54	C₅×#95	—	2	2	no	—
55	C₅×#95	—	2	4	no	—
56	C₅×#95	—	2	4	no	—
57	C₅×#95	D₄	1	2	yes	—
58	C₂×#427	C₂×C₅	1	1	yes	characteristic
59	#2347	C₂×C₅	1	2	yes	—
60	#1943	C₂×C₅	1	2	yes	complemented
61	#1920	C₂×C₅	1	4	yes	complemented
62	#1825	C₂×C₅	1	2	yes	—
63	#1807	C₂×C₅	1	1	yes	characteristic
64	#1776	C₂×C₅	1	2	yes	complemented
65	#1752	C₂×C₅	1	1	yes	characteristic
66	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	8	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	#41	1	4	yes	—
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	4	8	no	—
90	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
92	C₅×C₂²⋊C₄	—	2	8	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₅×C₂²⋊C₄	—	8	2	no	—
95	C₄²×C₅	—	2	8	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	—	8	4	no	—
99	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#117	C₂²×C₅	1	2	yes	—
101	#427	C₄×C₅	1	8	yes	complemented
102	C₂×#107	C₂²×C₅	1	1	yes	characteristic
103	C₂×#107	C₂²×C₅	1	1	yes	characteristic
104	C₂×#106	C₂²×C₅	1	2	yes	—
105	C₂×#106	C₂²×C₅	1	2	yes	—
106	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
107	C₂×#97	C₂²×C₅	1	2	yes	—
108	#389	C₂²×C₅	1	4	yes	—
109	#388	—	2	4	no	—
110	#383	—	2	2	no	—
111	#380	C₂²×C₅	1	2	yes	—
112	C₄×C₄⋊C₄	—	2	2	no	—
113	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
114	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
115	#340	C₂²×C₅	1	2	yes	—
116	#340	C₂²×C₅	1	4	yes	—
117	#340	C₂²×C₅	1	4	yes	—
118	#336	C₂²×C₅	1	2	yes	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	C₄×D₄	1	2	yes	—
123	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	4	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	—	4	4	no	—
139	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×#41	1	2	yes	—
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	—	4	2	no	—
146	C₅×C₈	#117	1	4	yes	—
147	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
150	C₂²×C₈	C₂³×C₅	1	2	yes	—
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
153	#117	—	2	4	no	—
154	#117	C₂×C₄×C₅	1	8	yes	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
157	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
158	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₂²⋊C₄	—	2	2	no	—
161	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	C₂³×C₅	1	2	yes	—
165	#107	C₂×C₄×C₅	1	4	yes	—
166	#107	C₂×C₄×C₅	1	4	yes	—
167	#106	—	2	4	no	—
168	#106	C₂×C₄×C₅	1	8	yes	—
169	C₂²⋊C₈	—	2	2	no	—
170	#97	—	2	4	no	—
171	#95	—	2	2	no	—
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	#95	C₅×D₄	1	2	yes	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	4	4	no	—
179	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂²×C₅	#547	1	2	yes	—
181	C₂²×C₅	#575	1	2	yes	—
182	C₂²×C₅	#577	1	2	yes	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	4	4	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	—	8	4	no	—
188	C₄×C₅	C₂×#117	1	1	yes	characteristic
189	C₄×C₅	#519	1	1	yes	characteristic
190	C₄×C₅	#552	1	2	yes	—
191	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
198	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₅×#41	1	2	yes	—
201	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
209	C₂×C₈	C₅×#41	1	4	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	8	no	—
215	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
216	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	8	2	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	8	4	no	—
224	C₂×C₅	—	4	2	no	—
225	C₂×C₅	#2344	1	1	yes	characteristic
226	C₂×C₅	#2943	1	1	yes	characteristic
227	C₂×C₅	#2952	1	1	yes	characteristic
228	C₂×C₅	#3350	1	4	yes	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₄×C₅×D₄	1	2	yes	—
233	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
250	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
251	C₂×C₄	C₂×C₅×#41	1	2	yes	—
252	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
253	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₈	C₅×#117	1	4	yes	—
257	C₅	#22620	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
263	C₂²	C₅×#547	1	2	yes	—
264	C₂²	C₅×#575	1	2	yes	—
265	C₂²	C₅×#577	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₂×C₅×#117	1	1	yes	characteristic
272	C₄	C₅×#519	1	1	yes	characteristic
273	C₄	C₅×#552	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	C₅×#2344	1	1	yes	characteristic
276	C₂	C₅×#2943	1	1	yes	characteristic
277	C₂	C₅×#2952	1	1	yes	characteristic
278	C₂	C₅×#3350	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅×#1943	C₂	semidirect
4. C₅×#1920	C₂	semidirect
7. C₅×#1776	C₂	semidirect
11. C₅×#427	C₄	semidirect
29. #22620	C₅	direct (N × H)
60. #1943	C₂×C₅	semidirect
61. #1920	C₂×C₅	semidirect
64. #1776	C₂×C₅	semidirect
101. #427	C₄×C₅	semidirect
257. C₅	#22620	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×#427	2
C₅×#2347	2
C₅×#1943	2
C₅×#1920	2
C₅×#1825	2
C₅×#1807	2
C₅×#1776	2
C₅×#1752	2
#22620	5

Subgroups — C5×#22620 #11945340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22620 #11945340