Subgroups — C₅×#2485 #10616080

Order 640.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#397	C₂	1	2	yes	maximal, complemented
4	C₅×#394	C₂	1	2	yes	maximal
5	C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#388	C₂	1	2	yes	maximal, complemented
8	C₅×#387	C₂	1	2	yes	maximal, complemented
9	C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₄	—	2	4	no	—
16	C₅×#117	—	2	4	no	—
17	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄²×C₅	C₂²	1	2	yes	—
31	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
32	C₅×#95	C₂²	1	2	yes	—
33	C₅×#95	C₂²	1	2	yes	—
34	C₅×#95	C₂²	1	2	yes	—
35	C₅×#95	C₂²	1	2	yes	—
36	C₅×#95	C₂²	1	2	yes	—
37	#2485	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₅	C₂³	1	1	yes	characteristic
39	C₂⁴×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×D₄	—	2	2	no	—
41	C₂×C₅×D₄	—	2	2	no	—
42	C₂×C₅×D₄	—	2	2	no	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	—	2	2	no	—
45	C₂²×C₄×C₅	—	2	2	no	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	D₄	1	4	yes	—
49	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂³	1	2	yes	—
51	C₂²×C₄×C₅	C₂³	1	2	yes	—
52	C₂²×C₄×C₅	C₂³	1	2	yes	—
53	C₂²×C₄×C₅	C₂³	1	2	yes	—
54	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	2	2	no	—
59	C₅×C₄⋊C₄	—	2	2	no	—
60	C₅×C₄⋊C₄	—	2	2	no	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	2	2	no	—
63	C₅×C₄⋊C₄	—	2	2	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₂²⋊C₄	—	2	4	no	—
66	C₅×C₂²⋊C₄	—	2	2	no	—
67	C₅×C₂²⋊C₄	—	2	4	no	—
68	C₅×C₂²⋊C₄	—	2	2	no	—
69	C₅×C₂²⋊C₄	—	4	2	no	—
70	C₅×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₅	—	2	4	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₂×C₄×D₄	C₂×C₅	1	1	yes	characteristic
74	C₂×#117	C₂×C₅	1	1	yes	characteristic, complemented
75	#397	C₂×C₅	1	2	yes	complemented
76	#394	C₂×C₅	1	2	yes	—
77	#389	C₂×C₅	1	1	yes	characteristic, complemented
78	#389	C₂×C₅	1	1	yes	characteristic, complemented
79	#388	C₂×C₅	1	2	yes	complemented
80	#387	C₂×C₅	1	2	yes	complemented
81	#380	C₂×C₅	1	1	yes	characteristic, complemented
82	#380	C₂×C₅	1	1	yes	characteristic, complemented
83	#380	C₂×C₅	1	1	yes	characteristic, complemented
84	C₂³×C₅	—	2	1	no	—
85	C₂³×C₅	—	2	1	no	—
86	C₂³×C₅	—	2	1	no	—
87	C₂³×C₅	—	2	1	no	—
88	C₂³×C₅	—	2	1	no	—
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	—	2	1	no	—
91	C₂³×C₅	—	2	1	no	—
92	C₂³×C₅	—	2	1	no	—
93	C₂³×C₅	—	2	1	no	—
94	C₂³×C₅	—	2	1	no	—
95	C₂³×C₅	—	2	1	no	—
96	C₂³×C₅	C₂×D₄	1	2	yes	—
97	C₂³×C₅	#41	1	2	yes	—
98	C₂³×C₅	C₂⁴	1	1	yes	characteristic
99	C₅×D₄	—	2	8	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₅	#41	1	2	yes	—
127	C₂×C₄×C₅	#41	1	2	yes	—
128	C₂×C₄×C₅	#41	1	2	yes	—
129	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
131	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
132	C₄×D₄	—	2	4	no	—
133	#117	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
147	C₂×C₄²	C₂²×C₅	1	2	yes	—
148	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
149	#95	C₂²×C₅	1	2	yes	—
150	#95	C₂²×C₅	1	2	yes	—
151	#95	C₂²×C₅	1	2	yes	—
152	#95	C₂²×C₅	1	2	yes	—
153	#95	C₂²×C₅	1	2	yes	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	4	1	no	—
161	C₂²×C₅	—	4	1	no	—
162	C₂²×C₅	—	4	1	no	—
163	C₂²×C₅	—	4	1	no	—
164	C₂²×C₅	—	4	1	no	—
165	C₂²×C₅	—	4	1	no	—
166	C₂²×C₅	—	4	1	no	—
167	C₂²×C₅	—	4	1	no	—
168	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂²×C₅	C₂×#41	1	1	yes	characteristic
170	C₂²×C₅	C₂×#41	1	1	yes	characteristic
171	C₂²×C₅	C₂×#41	1	1	yes	characteristic
172	C₂²×C₅	C₂×#41	1	1	yes	characteristic
173	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	4	1	no	—
178	C₄×C₅	—	4	2	no	—
179	C₄×C₅	—	4	1	no	—
180	C₄×C₅	—	4	2	no	—
181	C₄×C₅	—	4	2	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	#123	1	4	yes	—
184	C₂⁴	C₂³×C₅	1	1	yes	characteristic
185	C₂⁴	C₂³×C₅	1	1	yes	characteristic
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₅×D₄	1	4	yes	—
195	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₅	1	2	yes	—
197	C₂²×C₄	C₂³×C₅	1	2	yes	—
198	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₅	1	2	yes	—
200	C₂²×C₄	C₂³×C₅	1	2	yes	—
201	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₂×C₅	—	4	2	no	—
220	C₂×C₅	—	4	2	no	—
221	C₂×C₅	C₂×#123	1	1	yes	characteristic
222	C₂×C₅	#525	1	1	yes	characteristic
223	C₂×C₅	#537	1	1	yes	characteristic
224	C₂×C₅	#552	1	1	yes	characteristic
225	C₂×C₅	#553	1	1	yes	characteristic
226	C₂×C₅	#555	1	1	yes	characteristic
227	C₂×C₅	#556	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	C₂×C₅×D₄	1	2	yes	—
241	C₂³	C₅×#41	1	2	yes	—
242	C₂³	C₂⁴×C₅	1	1	yes	characteristic
243	D₄	—	2	8	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
269	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
270	C₂×C₄	C₅×#41	1	2	yes	—
271	C₂×C₄	C₅×#41	1	2	yes	—
272	C₂×C₄	C₅×#41	1	2	yes	—
273	C₅	#2485	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
289	C₂²	C₂×C₅×#41	1	1	yes	characteristic
290	C₂²	C₂×C₅×#41	1	1	yes	characteristic
291	C₂²	C₂×C₅×#41	1	1	yes	characteristic
292	C₂²	C₂×C₅×#41	1	1	yes	characteristic
293	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	C₅×#123	1	4	yes	—
304	C₂	—	4	2	no	—
305	C₂	—	4	2	no	—
306	C₂	C₂×C₅×#123	1	1	yes	characteristic
307	C₂	C₅×#525	1	1	yes	characteristic
308	C₂	C₅×#537	1	1	yes	characteristic
309	C₂	C₅×#552	1	1	yes	characteristic
310	C₂	C₅×#553	1	1	yes	characteristic
311	C₂	C₅×#555	1	1	yes	characteristic
312	C₂	C₅×#556	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#117	C₂	semidirect
3. C₅×#397	C₂	semidirect
5. C₅×#389	C₂	semidirect
6. C₅×#389	C₂	semidirect
7. C₅×#388	C₂	semidirect
8. C₅×#387	C₂	semidirect
9. C₅×#380	C₂	semidirect
10. C₅×#380	C₂	semidirect
11. C₅×#380	C₂	semidirect
37. #2485	C₅	direct (N × H)
74. C₂×#117	C₂×C₅	semidirect
75. #397	C₂×C₅	semidirect
77. #389	C₂×C₅	semidirect
78. #389	C₂×C₅	semidirect
79. #388	C₂×C₅	semidirect
80. #387	C₂×C₅	semidirect
81. #380	C₂×C₅	semidirect
82. #380	C₂×C₅	semidirect
83. #380	C₂×C₅	semidirect
273. C₅	#2485	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×C₄×C₅×D₄	2
C₂×C₅×#117	2
C₅×#397	2
C₅×#394	2
C₅×#389	2
C₅×#388	2
C₅×#387	2
C₅×#380	2
#2485	5

Subgroups — C5×#2485 #10616080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2485 #10616080