Subgroups — C₅×#2661 #10616256

Order 640.

598 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#394	C₂	1	2	yes	maximal, complemented
6	C₅×#394	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#392	C₂	1	2	yes	maximal, complemented
8	C₅×#392	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#390	C₂	1	2	yes	maximal, complemented
10	C₅×#388	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×D₄	C₂²	1	2	yes	—
13	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₅×#123	—	2	4	no	—
18	C₅×#121	—	2	4	no	—
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×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄²×C₅	C₂²	1	2	yes	—
33	C₅×#95	C₂²	1	1	yes	characteristic
34	C₅×#95	C₂²	1	1	yes	characteristic
35	C₅×#95	C₂²	1	2	yes	—
36	C₅×#95	C₂²	1	2	yes	—
37	#2661	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₅	C₂³	1	1	yes	characteristic
39	C₂⁴×C₅	C₂³	1	2	yes	—
40	C₂⁴×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₅×D₄	—	2	4	no	—
42	C₂×C₅×D₄	—	2	2	no	—
43	C₂×C₅×D₄	—	2	2	no	—
44	C₂×C₅×D₄	—	2	4	no	—
45	C₂×C₅×D₄	—	4	2	no	—
46	C₂×C₅×D₄	—	4	2	no	—
47	C₂×C₅×D₄	—	4	2	no	—
48	C₂×C₅×D₄	—	4	2	no	—
49	C₂×C₅×D₄	—	4	2	no	—
50	C₂×C₅×D₄	—	4	2	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	D₄	1	4	yes	complemented
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	2	yes	—
57	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂³	1	2	yes	—
61	C₅×C₄⋊C₄	—	2	2	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	2	no	—
66	C₅×C₂²⋊C₄	—	2	4	no	—
67	C₅×C₂²⋊C₄	—	2	2	no	—
68	C₅×C₂²⋊C₄	—	2	2	no	—
69	C₅×C₂²⋊C₄	—	2	4	no	—
70	C₅×C₂²⋊C₄	—	2	4	no	—
71	C₅×C₂²⋊C₄	—	4	2	no	—
72	C₅×C₂²⋊C₄	—	4	2	no	—
73	C₅×C₂²⋊C₄	—	4	2	no	—
74	C₄²×C₅	—	4	2	no	—
75	C₂×#123	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂×#121	C₂×C₅	1	1	yes	characteristic, complemented
77	#398	C₂×C₅	1	1	yes	characteristic, complemented
78	#397	C₂×C₅	1	1	yes	characteristic, complemented
79	#394	C₂×C₅	1	1	yes	characteristic, complemented
80	#394	C₂×C₅	1	2	yes	complemented
81	#392	C₂×C₅	1	1	yes	characteristic, complemented
82	#392	C₂×C₅	1	2	yes	complemented
83	#390	C₂×C₅	1	2	yes	complemented
84	#388	C₂×C₅	1	2	yes	complemented
85	C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
86	C₂³×C₅	—	2	2	no	—
87	C₂³×C₅	—	2	2	no	—
88	C₂³×C₅	—	2	1	no	—
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	—	2	1	no	—
91	C₂³×C₅	—	2	2	no	—
92	C₂³×C₅	—	2	2	no	—
93	C₂³×C₅	—	2	2	no	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	2	1	no	—
96	C₂³×C₅	—	2	2	no	—
97	C₂³×C₅	—	2	2	no	—
98	C₂³×C₅	—	2	2	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	1	no	—
102	C₂³×C₅	C₂×D₄	1	2	yes	—
103	C₂³×C₅	C₂⁴	1	1	yes	characteristic
104	C₅×D₄	—	4	4	no	—
105	C₅×D₄	—	4	8	no	—
106	C₅×D₄	—	4	4	no	—
107	C₅×D₄	—	4	8	no	—
108	C₂×C₄×C₅	—	2	2	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	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	1	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	1	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₅	#41	1	4	yes	—
129	C₂²×D₄	C₂²×C₅	1	2	yes	—
130	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
132	C₂²×D₄	C₂²×C₅	1	2	yes	—
133	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
134	#123	—	2	4	no	—
135	#121	—	2	4	no	—
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×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
149	C₂×C₄²	C₂²×C₅	1	2	yes	—
150	#95	C₂²×C₅	1	1	yes	characteristic
151	#95	C₂²×C₅	1	1	yes	characteristic
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	—	4	2	no	—
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	—	4	1	no	—
160	C₂²×C₅	—	4	2	no	—
161	C₂²×C₅	—	4	2	no	—
162	C₂²×C₅	—	4	1	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	—	4	1	no	—
165	C₂²×C₅	—	4	2	no	—
166	C₂²×C₅	—	4	2	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	2	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂²×C₅	C₂×#41	1	2	yes	—
173	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
174	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	4	no	—
179	C₄×C₅	—	4	1	no	—
180	C₄×C₅	—	4	1	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	—	4	2	no	—
184	C₂⁴	C₂³×C₅	1	1	yes	characteristic
185	C₂⁴	C₂³×C₅	1	2	yes	—
186	C₂⁴	C₂³×C₅	1	1	yes	characteristic
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₅×D₄	1	4	yes	complemented
199	C₂²×C₄	C₂³×C₅	1	2	yes	—
200	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₅	1	2	yes	—
202	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₅	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	—	4	2	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	—	8	2	no	—
223	C₂×C₅	—	8	1	no	—
224	C₂×C₅	#532	1	1	yes	characteristic
225	C₂×C₅	#534	1	1	yes	characteristic
226	C₂×C₅	#535	1	1	yes	characteristic
227	C₂×C₅	#538	1	2	yes	—
228	C₂×C₅	#540	1	2	yes	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	C₂×C₅×D₄	1	2	yes	—
246	C₂³	C₂⁴×C₅	1	1	yes	characteristic
247	D₄	—	4	4	no	—
248	D₄	—	4	8	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
270	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₅×#41	1	4	yes	—
272	C₅	#2661	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
291	C₂²	C₂×C₅×#41	1	2	yes	—
292	C₂²	C₅×H(5,𝔽₂)	1	2	yes	—
293	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	8	1	no	—
306	C₂	C₅×#532	1	1	yes	characteristic
307	C₂	C₅×#534	1	1	yes	characteristic
308	C₂	C₅×#535	1	1	yes	characteristic
309	C₂	C₅×#538	1	2	yes	—
310	C₂	C₅×#540	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
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
1. C₂×C₅×#123	C₂	semidirect
2. C₂×C₅×#121	C₂	semidirect
3. C₅×#398	C₂	semidirect
4. C₅×#397	C₂	semidirect
5. C₅×#394	C₂	semidirect
6. C₅×#394	C₂	semidirect
7. C₅×#392	C₂	semidirect
8. C₅×#392	C₂	semidirect
9. C₅×#390	C₂	semidirect
10. C₅×#388	C₂	semidirect
11. C₄×C₅×C₂²⋊C₄	C₂	semidirect
37. #2661	C₅	direct (N × H)
52. C₂²×C₄×C₅	D₄	semidirect
75. C₂×#123	C₂×C₅	semidirect
76. C₂×#121	C₂×C₅	semidirect
77. #398	C₂×C₅	semidirect
78. #397	C₂×C₅	semidirect
79. #394	C₂×C₅	semidirect
80. #394	C₂×C₅	semidirect
81. #392	C₂×C₅	semidirect
82. #392	C₂×C₅	semidirect
83. #390	C₂×C₅	semidirect
84. #388	C₂×C₅	semidirect
85. C₄×C₂²⋊C₄	C₂×C₅	semidirect
198. C₂²×C₄	C₅×D₄	semidirect
272. C₅	#2661	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×#123	2
C₂×C₅×#121	2
C₅×#398	2
C₅×#397	2
C₅×#394	2
C₅×#392	2
C₅×#390	2
C₅×#388	2
C₄×C₅×C₂²⋊C₄	2
#2661	5

Subgroups — C5×#2661 #10616256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2661 #10616256