Subgroups — C₅×#66166 #13106985

Order 1600.

784 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×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4093	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4086	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4085	C₂	1	4	yes	maximal, complemented
7	C₅×#4082	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅²×#41	—	2	4	no	—
11	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₅²×D₄	C₂²	1	2	yes	—
14	C₂²×C₄×C₅²	C₂²	1	2	yes	—
15	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	—
17	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
19	C₅×#709	—	2	4	no	—
20	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
22	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
24	C₅×#685	C₂²	1	4	yes	—
25	C₅×#683	C₂²	1	4	yes	—
26	C₄×C₅×Dic₅	C₂²	1	2	yes	—
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	C₅×#532	—	5	1	no	maximal
30	#66166	C₅	1	1	yes	maximal, characteristic, direct product factor
31	C₂³×C₅²	C₂³	1	2	yes	—
32	C₂³×C₅²	C₂³	1	1	yes	characteristic
33	C₂²×C₅×D₅	C₂³	1	2	yes	—
34	C₅²×Q₈	—	2	2	no	—
35	C₅²×D₄	—	2	4	no	—
36	C₅²×D₄	—	2	2	no	—
37	C₂×C₄×C₅²	—	2	4	no	—
38	C₂×C₄×C₅²	D₄	1	4	yes	complemented
39	C₂×C₄×C₅²	C₂³	1	2	yes	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₅×C₂²⋊D₅	—	2	4	no	—
43	C₅×C₂²⋊D₅	—	2	8	no	—
44	C₂×C₅×Dic₅	—	2	4	no	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₅×D₂₀	—	2	2	no	—
49	C₄×C₅×D₅	—	2	4	no	—
50	C₅×Dic₁₀	—	2	2	no	—
51	C₂×C₅×#41	—	2	2	no	—
52	C₂×C₅×#41	—	5	1	no	—
53	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
54	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#709	C₂×C₅	1	1	yes	characteristic, complemented
56	C₅×C₄⋊Q₈	—	5	1	no	—
57	C₅×#127	—	5	1	no	—
58	C₅×#124	—	5	1	no	—
59	C₅×#124	—	5	1	no	—
60	C₅×#121	—	5	4	no	—
61	C₄×C₅×D₄	—	5	4	no	—
62	C₂×C₄²×C₅	—	5	1	no	—
63	#4093	C₂×C₅	1	1	yes	characteristic, complemented
64	#4090	C₂×C₅	1	1	yes	characteristic, complemented
65	#4086	C₂×C₅	1	1	yes	characteristic, complemented
66	#4085	C₂×C₅	1	4	yes	complemented
67	#4082	C₂×C₅	1	1	yes	characteristic, complemented
68	C₄×C₂²⋊D₅	C₂×C₅	1	4	yes	complemented
69	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂²×C₅²	—	2	2	no	—
71	C₂²×C₅²	—	2	2	no	—
72	C₂²×C₅²	—	2	4	no	—
73	C₂²×C₅²	C₂×D₄	1	2	yes	—
74	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×D₅	—	2	2	no	—
76	C₂×C₅×D₅	—	2	4	no	—
77	C₄×C₅²	—	2	2	no	—
78	C₄×C₅²	C₂×D₄	1	2	yes	—
79	C₄×C₅²	C₂×D₄	1	2	yes	—
80	C₅×Dic₅	—	2	2	no	—
81	C₅×Dic₅	—	2	2	no	—
82	C₅×Dic₅	#41	1	4	yes	complemented
83	C₅×#41	—	2	16	no	—
84	C₅×#41	—	2	4	no	—
85	C₅×#41	—	2	4	no	—
86	C₅×#41	—	10	4	no	—
87	C₂×C₅×Q₈	—	2	2	no	—
88	C₂×C₅×Q₈	—	5	1	no	—
89	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
90	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₅×D₄	—	2	4	no	—
92	C₂×C₅×D₄	—	2	2	no	—
93	C₂×C₅×D₄	—	5	4	no	—
94	C₂×C₅×D₄	—	5	2	no	—
95	C₂×C₅×D₄	—	5	1	no	—
96	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
97	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
98	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
100	C₂²×C₄×C₅	—	2	4	no	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	—	5	2	no	—
103	C₂²×C₄×C₅	—	5	2	no	—
104	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
105	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
107	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
109	C₂×C₂²⋊D₅	C₂²×C₅	1	4	yes	—
110	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
111	#709	—	2	4	no	—
112	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
114	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	5	4	no	—
116	C₅×C₂²⋊C₄	—	5	4	no	—
117	C₅×C₂²⋊C₄	—	5	4	no	—
118	C₄²×C₅	—	5	1	no	—
119	C₄²×C₅	—	5	1	no	—
120	C₄²×C₅	—	5	2	no	—
121	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
122	#685	C₂²×C₅	1	4	yes	—
123	#683	C₂²×C₅	1	4	yes	—
124	C₄×Dic₅	C₂²×C₅	1	2	yes	—
125	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
126	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
127	#532	—	5	1	no	Sylow
128	C₂×C₅²	—	2	2	no	—
129	C₂×C₅²	—	4	2	no	—
130	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₅²	C₂×#41	1	2	yes	—
132	C₅×D₅	—	4	2	no	—
133	C₂³×C₅	—	2	4	no	—
134	C₂³×C₅	—	2	2	no	—
135	C₂³×C₅	—	5	2	no	—
136	C₂³×C₅	C₂²×D₅	1	2	yes	—
137	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
138	C₂³×C₅	C₂³×C₅	1	2	yes	—
139	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
140	C₂²×D₅	C₂³×C₅	1	2	yes	—
141	C₅×Q₈	—	2	8	no	—
142	C₅×Q₈	—	2	2	no	—
143	C₅×Q₈	—	2	2	no	—
144	C₅×Q₈	—	10	2	no	—
145	C₅×D₄	—	2	16	no	—
146	C₅×D₄	—	2	8	no	—
147	C₅×D₄	—	2	4	no	—
148	C₅×D₄	—	2	4	no	—
149	C₅×D₄	—	2	2	no	—
150	C₅×D₄	—	2	2	no	—
151	C₅×D₄	—	10	8	no	—
152	C₅×D₄	—	10	4	no	—
153	C₅×D₄	—	10	2	no	—
154	C₂×C₄×C₅	—	2	16	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	8	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	5	2	no	—
162	C₂×C₄×C₅	—	5	2	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	10	4	no	—
165	C₂×C₄×C₅	—	10	4	no	—
166	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
167	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
168	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
169	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
172	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
174	C₂²⋊D₅	—	2	4	no	—
175	C₂²⋊D₅	—	2	8	no	—
176	C₂×Dic₅	—	2	4	no	—
177	C₂×Dic₅	C₂³×C₅	1	2	yes	—
178	C₂×Dic₅	C₂³×C₅	1	2	yes	—
179	C₂×Dic₅	C₂³×C₅	1	2	yes	—
180	D₂₀	—	2	2	no	—
181	C₄×D₅	—	2	4	no	—
182	Dic₁₀	—	2	2	no	—
183	C₂×#41	—	5	1	no	—
184	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
185	C₄⋊Q₈	—	5	1	no	—
186	#127	—	5	1	no	—
187	#124	—	5	1	no	—
188	#124	—	5	1	no	—
189	#121	—	5	4	no	—
190	C₄×D₄	—	5	4	no	—
191	C₂×C₄²	—	5	1	no	—
192	C₅²	#532	1	1	yes	characteristic, complemented, Sylow
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	8	no	—
196	C₂²×C₅	—	2	4	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	—	2	4	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	4	4	no	—
203	C₂²×C₅	—	4	8	no	—
204	C₂²×C₅	—	10	4	no	—
205	C₂²×C₅	—	10	2	no	—
206	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
207	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
208	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
209	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
210	C₂×D₅	—	2	2	no	—
211	C₂×D₅	—	2	4	no	—
212	C₄×C₅	—	2	8	no	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	4	no	—
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	5	4	no	—
218	C₄×C₅	—	10	2	no	—
219	C₄×C₅	—	10	2	no	—
220	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
221	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
222	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
223	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	C₅×#41	1	4	yes	complemented
227	#41	—	2	4	no	—
228	#41	—	10	4	no	—
229	C₂×Q₈	—	5	1	no	—
230	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
231	C₂×D₄	—	5	4	no	—
232	C₂×D₄	—	5	2	no	—
233	C₂×D₄	—	5	1	no	—
234	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
235	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
239	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
240	C₄⋊C₄	—	5	4	no	—
241	C₂²⋊C₄	—	5	4	no	—
242	C₂²⋊C₄	—	5	4	no	—
243	C₄²	—	5	2	no	—
244	C₄²	—	5	1	no	—
245	C₄²	—	5	1	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	2	4	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	4	4	no	—
251	C₂×C₅	—	4	8	no	—
252	C₂×C₅	—	4	2	no	—
253	C₂×C₅	—	4	2	no	—
254	C₂×C₅	—	20	2	no	—
255	C₂×C₅	D₅×#41	1	2	yes	—
256	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
257	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₂×C₅	C₂×C₅×#41	1	2	yes	—
259	D₅	—	4	2	no	—
260	C₂³	—	5	2	no	—
261	C₂³	C₂²×C₅×D₅	1	2	yes	—
262	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	Q₈	—	10	2	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	—	10	8	no	—
268	D₄	—	10	4	no	—
269	D₄	—	10	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	10	4	no	—
275	C₂×C₄	—	10	4	no	—
276	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
277	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
278	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
280	C₅	—	2	2	no	—
281	C₅	#66166	1	1	yes	characteristic, direct product factor
282	C₅	C₅×#532	1	1	yes	characteristic, complemented
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	10	4	no	—
287	C₂²	—	10	2	no	—
288	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
289	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	5	4	no	—
292	C₄	—	10	2	no	—
293	C₄	—	10	2	no	—
294	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
295	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	20	2	no	—
299	C₂	C₅×D₅×#41	1	2	yes	—
300	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₅×#532
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₅×#4093	C₂	semidirect
4. C₅×#4090	C₂	semidirect
5. C₅×#4086	C₂	semidirect
6. C₅×#4085	C₂	semidirect
7. C₅×#4082	C₂	semidirect
8. C₄×C₅×C₂²⋊D₅	C₂	semidirect
9. C₂×C₄×C₅×Dic₅	C₂	semidirect
30. #66166	C₅	direct (N × H)
38. C₂×C₄×C₅²	D₄	semidirect
53. C₂×C₅×#41	D₅	semidirect
54. C₂×C₅×#41	C₂×C₅	semidirect
55. C₂×#709	C₂×C₅	semidirect
63. #4093	C₂×C₅	semidirect
64. #4090	C₂×C₅	semidirect
65. #4086	C₂×C₅	semidirect
66. #4085	C₂×C₅	semidirect
67. #4082	C₂×C₅	semidirect
68. C₄×C₂²⋊D₅	C₂×C₅	semidirect
69. C₂×C₄×Dic₅	C₂×C₅	semidirect
82. C₅×Dic₅	#41	semidirect
166. C₂×C₄×C₅	C₂²⋊D₅	semidirect
167. C₂×C₄×C₅	C₅×D₄	semidirect
184. C₂×#41	C₅×D₅	semidirect
192. C₅²	#532	semidirect
226. Dic₅	C₅×#41	semidirect
276. C₂×C₄	C₅×C₂²⋊D₅	semidirect
281. C₅	#66166	direct (N × H)
282. C₅	C₅×#532	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂×C₅×#709	2
C₅×#4093	2
C₅×#4090	2
C₅×#4086	2
C₅×#4085	2
C₅×#4082	2
C₄×C₅×C₂²⋊D₅	2
C₂×C₄×C₅×Dic₅	2
C₅×#532	5
#66166	5

Subgroups — C5×#66166 #13106985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66166 #13106985