Subgroups — C₅×#66162 #13106981

Order 1600.

824 conjugacy classes of subgroups.

The subgroup lattice has 294 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	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4092	C₂	1	2	yes	maximal, complemented
4	C₅×#4084	C₂	1	2	yes	maximal, complemented
5	C₅×#4083	C₂	1	2	yes	maximal, complemented
6	C₅×#4081	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
8	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
10	C₅²×#41	—	2	4	no	—
11	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₅²×D₄	C₂²	1	2	yes	—
13	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
16	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	—
17	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₅	—	2	4	no	—
19	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
20	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
23	C₅×#685	C₂²	1	4	yes	—
24	C₅×C₄⋊Dic₅	C₂²	1	2	yes	complemented
25	C₅×#683	C₂²	1	4	yes	complemented
26	C₄×C₅×Dic₅	C₂²	1	2	yes	complemented
27	C₅×#525	—	5	1	no	maximal
28	#66162	C₅	1	1	yes	maximal, characteristic, direct product factor
29	C₂³×C₅²	C₂³	1	2	yes	—
30	C₂³×C₅²	C₂³	1	1	yes	characteristic
31	C₂²×C₅×D₅	—	2	2	no	—
32	C₂²×C₅×D₅	—	2	4	no	—
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	8	no	—
43	C₂×C₅×Dic₅	—	2	2	no	—
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₂×C₅×Dic₅	C₂³	1	4	yes	—
46	C₄×C₅×D₅	—	2	4	no	—
47	C₄×C₅×D₅	—	2	4	no	—
48	C₂×C₅×#41	—	2	2	no	—
49	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
50	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂³×C₄×C₅	—	5	1	no	—
52	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic, complemented
53	C₅×#123	—	5	2	no	—
54	C₅×C₂²⋊Q₈	—	5	2	no	—
55	C₅×#121	—	5	2	no	—
56	C₅×#120	—	5	2	no	—
57	C₄×C₅×D₄	—	5	4	no	—
58	C₅×#117	—	5	1	no	—
59	#4092	C₂×C₅	1	2	yes	complemented
60	#4084	C₂×C₅	1	2	yes	complemented
61	#4083	C₂×C₅	1	2	yes	complemented
62	#4081	C₂×C₅	1	2	yes	complemented
63	C₄×C₂²⋊D₅	C₂×C₅	1	4	yes	complemented
64	#4072	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂²×C₅²	—	2	2	no	—
66	C₂²×C₅²	—	2	2	no	—
67	C₂²×C₅²	—	2	4	no	—
68	C₂²×C₅²	C₂×D₄	1	2	yes	—
69	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
70	C₂×C₅×D₅	—	2	8	no	—
71	C₂×C₅×D₅	—	2	2	no	—
72	C₂×C₅×D₅	—	2	2	no	—
73	C₂×C₅×D₅	#41	1	4	yes	complemented
74	C₄×C₅²	—	2	2	no	—
75	C₄×C₅²	C₂×D₄	1	2	yes	—
76	C₄×C₅²	C₂×D₄	1	2	yes	—
77	C₅×Dic₅	—	2	2	no	—
78	C₅×Dic₅	—	2	4	no	—
79	C₂⁴×C₅	—	5	1	no	—
80	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
81	C₅×#41	—	2	16	no	—
82	C₅×#41	—	2	4	no	—
83	C₅×#41	—	2	4	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
86	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
87	C₂×C₅×D₄	—	2	4	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	5	4	no	—
90	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
91	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic, complemented
92	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
93	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	—	2	4	no	—
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂²×C₄×C₅	—	5	2	no	—
98	C₂²×C₄×C₅	—	5	2	no	—
99	C₂²×C₄×C₅	—	10	4	no	—
100	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₅	1	2	yes	complemented
102	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	complemented
104	C₂×C₂²⋊D₅	C₂²×C₅	1	4	yes	—
105	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₄×D₅	—	2	4	no	—
107	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
108	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
109	C₅×C₄⋊C₄	—	5	2	no	—
110	C₅×C₄⋊C₄	—	5	4	no	—
111	C₅×C₂²⋊C₄	—	5	2	no	—
112	C₅×C₂²⋊C₄	—	5	4	no	—
113	C₅×C₂²⋊C₄	—	5	4	no	—
114	C₄²×C₅	—	5	2	no	—
115	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
116	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	complemented
117	#685	C₂²×C₅	1	4	yes	—
118	C₄⋊Dic₅	C₂²×C₅	1	2	yes	complemented
119	#683	C₂²×C₅	1	4	yes	complemented
120	C₄×Dic₅	C₂²×C₅	1	2	yes	complemented
121	#525	—	5	1	no	Sylow
122	C₂×C₅²	—	2	2	no	—
123	C₂×C₅²	—	4	2	no	—
124	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₅²	C₂×#41	1	2	yes	—
126	C₅×D₅	—	2	4	no	—
127	C₂³×C₅	—	2	4	no	—
128	C₂³×C₅	—	2	2	no	—
129	C₂³×C₅	—	5	2	no	—
130	C₂³×C₅	—	10	4	no	—
131	C₂³×C₅	—	10	2	no	—
132	C₂³×C₅	C₂²×D₅	1	2	yes	—
133	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂³×C₅	C₂³×C₅	1	2	yes	—
135	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
136	C₂²×D₅	—	2	2	no	—
137	C₂²×D₅	—	2	4	no	—
138	C₂²×D₅	C₂³×C₅	1	2	yes	—
139	C₅×Q₈	—	2	8	no	—
140	C₅×Q₈	—	2	2	no	—
141	C₅×Q₈	—	2	2	no	—
142	C₅×D₄	—	2	16	no	—
143	C₅×D₄	—	2	8	no	—
144	C₅×D₄	—	2	4	no	—
145	C₅×D₄	—	2	4	no	—
146	C₅×D₄	—	2	2	no	—
147	C₅×D₄	—	2	2	no	—
148	C₅×D₄	—	10	8	no	—
149	C₂×C₄×C₅	—	2	16	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	8	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	5	4	no	—
157	C₂×C₄×C₅	—	5	2	no	—
158	C₂×C₄×C₅	—	10	2	no	—
159	C₂×C₄×C₅	—	10	4	no	—
160	C₂×C₄×C₅	—	10	4	no	—
161	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
162	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
163	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
164	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
167	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
169	C₂²⋊D₅	—	2	8	no	—
170	C₂×Dic₅	—	2	2	no	—
171	C₂×Dic₅	C₂³×C₅	1	4	yes	—
172	C₂×Dic₅	C₂³×C₅	1	2	yes	—
173	C₄×D₅	—	2	4	no	—
174	C₄×D₅	—	2	4	no	—
175	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
176	C₂³×C₄	—	5	1	no	—
177	#123	—	5	2	no	—
178	C₂²⋊Q₈	—	5	2	no	—
179	#121	—	5	2	no	—
180	#120	—	5	2	no	—
181	C₄×D₄	—	5	4	no	—
182	#117	—	5	1	no	—
183	C₅²	#525	1	1	yes	characteristic, complemented, Sylow
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	8	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	4	no	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	4	4	no	—
194	C₂²×C₅	—	4	8	no	—
195	C₂²×C₅	—	5	4	no	—
196	C₂²×C₅	—	10	8	no	—
197	C₂²×C₅	—	10	2	no	—
198	C₂²×C₅	—	10	2	no	—
199	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
200	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
201	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
202	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
203	C₂×D₅	—	2	8	no	—
204	C₂×D₅	—	2	2	no	—
205	C₂×D₅	—	2	2	no	—
206	C₂×D₅	C₅×#41	1	4	yes	complemented
207	C₄×C₅	—	2	8	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	2	4	no	—
212	C₄×C₅	—	10	4	no	—
213	C₄×C₅	—	10	2	no	—
214	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
215	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
216	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
217	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
218	Dic₅	—	2	2	no	—
219	Dic₅	—	2	4	no	—
220	C₂⁴	—	5	1	no	—
221	#41	—	2	4	no	—
222	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
223	C₂×D₄	—	5	4	no	—
224	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
225	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
226	C₂²×C₄	—	5	1	no	—
227	C₂²×C₄	—	5	2	no	—
228	C₂²×C₄	—	5	2	no	—
229	C₂²×C₄	—	10	4	no	—
230	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₅	1	2	yes	complemented
232	C₄⋊C₄	—	5	2	no	—
233	C₄⋊C₄	—	5	4	no	—
234	C₂²⋊C₄	—	5	2	no	—
235	C₂²⋊C₄	—	5	4	no	—
236	C₂²⋊C₄	—	5	4	no	—
237	C₄²	—	5	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	2	2	no	—
240	C₂×C₅	—	2	4	no	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	—	4	4	no	—
243	C₂×C₅	—	4	8	no	—
244	C₂×C₅	—	4	2	no	—
245	C₂×C₅	—	4	2	no	—
246	C₂×C₅	—	10	4	no	—
247	C₂×C₅	D₅×#41	1	2	yes	—
248	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
249	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
250	C₂×C₅	C₂×C₅×#41	1	2	yes	—
251	D₅	—	2	4	no	—
252	C₂³	—	5	2	no	—
253	C₂³	—	10	4	no	—
254	C₂³	—	10	2	no	—
255	C₂³	C₂²×C₅×D₅	1	2	yes	—
256	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	D₄	—	10	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	5	4	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	10	4	no	—
266	C₂×C₄	—	10	4	no	—
267	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
268	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
269	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
271	C₅	—	2	2	no	—
272	C₅	#66162	1	1	yes	characteristic, direct product factor
273	C₅	C₅×#525	1	1	yes	characteristic, complemented
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	5	4	no	—
278	C₂²	—	10	8	no	—
279	C₂²	—	10	2	no	—
280	C₂²	—	10	2	no	—
281	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
282	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	10	4	no	—
285	C₄	—	10	2	no	—
286	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
287	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	10	4	no	—
291	C₂	C₅×D₅×#41	1	2	yes	—
292	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×#525
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. C₂²×C₄×C₅×D₅	C₂	semidirect
3. C₅×#4092	C₂	semidirect
4. C₅×#4084	C₂	semidirect
5. C₅×#4083	C₂	semidirect
6. C₅×#4081	C₂	semidirect
7. C₄×C₅×C₂²⋊D₅	C₂	semidirect
8. C₅×#4072	C₂	semidirect
11. C₂×C₅²×Q₈	C₂²	semidirect
13. C₂×C₅²×D₄	C₂²	semidirect
15. C₂²×C₄×C₅²	C₂²	semidirect
22. C₅×C₂²⋊Dic₅	C₂²	semidirect
24. C₅×C₄⋊Dic₅	C₂²	semidirect
25. C₅×#683	C₂²	semidirect
26. C₄×C₅×Dic₅	C₂²	semidirect
28. #66162	C₅	direct (N × H)
38. C₂×C₄×C₅²	D₄	semidirect
49. C₂×C₅×#41	D₅	semidirect
50. C₂×C₅×#41	C₂×C₅	semidirect
52. C₂²×C₄×D₅	C₂×C₅	semidirect
59. #4092	C₂×C₅	semidirect
60. #4084	C₂×C₅	semidirect
61. #4083	C₂×C₅	semidirect
62. #4081	C₂×C₅	semidirect
63. C₄×C₂²⋊D₅	C₂×C₅	semidirect
64. #4072	C₂×C₅	semidirect
73. C₂×C₅×D₅	#41	semidirect
85. C₂×C₅×Q₈	C₂×D₅	semidirect
86. C₂×C₅×Q₈	C₂²×C₅	semidirect
91. C₂×C₅×D₄	C₂×D₅	semidirect
93. C₂×C₅×D₄	C₂²×C₅	semidirect
101. C₂²×C₄×C₅	C₂×D₅	semidirect
103. C₂²×C₄×C₅	C₂²×C₅	semidirect
116. C₂²⋊Dic₅	C₂²×C₅	semidirect
118. C₄⋊Dic₅	C₂²×C₅	semidirect
119. #683	C₂²×C₅	semidirect
120. C₄×Dic₅	C₂²×C₅	semidirect
161. C₂×C₄×C₅	C₂²⋊D₅	semidirect
162. C₂×C₄×C₅	C₅×D₄	semidirect
175. C₂×#41	C₅×D₅	semidirect
183. C₅²	#525	semidirect
206. C₂×D₅	C₅×#41	semidirect
222. C₂×Q₈	C₂×C₅×D₅	semidirect
225. C₂×D₄	C₂×C₅×D₅	semidirect
231. C₂²×C₄	C₂×C₅×D₅	semidirect
267. C₂×C₄	C₅×C₂²⋊D₅	semidirect
272. C₅	#66162	direct (N × H)
273. C₅	C₅×#525	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
C₂²×C₄×C₅×D₅	2
C₅×#4092	2
C₅×#4084	2
C₅×#4083	2
C₅×#4081	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4072	2
C₅×#525	5
#66162	5

Subgroups — C5×#66162 #13106981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66162 #13106981