Subgroups — C₅×#66125 #13106945

Order 1600.

896 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4099	C₂	1	2	yes	maximal, complemented
4	C₅×#4076	C₂	1	2	yes	maximal, complemented
5	C₅×#4075	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
7	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4070	C₂	1	2	yes	maximal, complemented
9	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	—	2	4	no	—
11	C₂²×C₄×C₅²	C₂²	1	2	yes	—
12	C₂²×C₄×C₅²	C₂²	1	2	yes	—
13	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	—
16	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₅×#709	—	2	4	no	—
18	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄×C₅×D₅	C₂²	1	2	yes	complemented
20	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
21	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
22	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₅×#685	C₂²	1	4	yes	complemented
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	#66125	C₅	1	1	yes	maximal, characteristic, direct product factor
29	C₂³×C₅²	—	2	2	no	—
30	C₂³×C₅²	—	2	4	no	—
31	C₂³×C₅²	C₂³	1	1	yes	characteristic
32	C₂³×C₅²	C₂³	1	2	yes	—
33	C₂²×C₅×D₅	C₂³	1	2	yes	—
34	C₂×C₄×C₅²	—	2	2	no	—
35	C₂×C₄×C₅²	—	2	4	no	—
36	C₂×C₄×C₅²	—	2	4	no	—
37	C₂×C₄×C₅²	D₄	1	4	yes	complemented
38	C₂×C₄×C₅²	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₅×C₂²⋊D₅	—	2	4	no	—
42	C₅×C₂²⋊D₅	—	2	8	no	—
43	C₂×C₅×Dic₅	C₂³	1	2	yes	—
44	C₂×C₅×Dic₅	C₂³	1	4	yes	—
45	C₅×D₂₀	—	2	2	no	—
46	C₄×C₅×D₅	—	2	4	no	—
47	C₅×Dic₁₀	—	2	2	no	—
48	C₂×C₅×#41	—	5	1	no	—
49	C₂³×C₄×C₅	—	2	2	no	—
50	C₂³×C₄×C₅	D₅	1	1	yes	characteristic, complemented
51	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#709	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	#4099	C₂×C₅	1	2	yes	complemented
60	#4076	C₂×C₅	1	2	yes	complemented
61	#4075	C₂×C₅	1	2	yes	complemented
62	C₄×C₂²⋊D₅	C₂×C₅	1	4	yes	complemented
63	#4072	C₂×C₅	1	1	yes	characteristic, complemented
64	#4070	C₂×C₅	1	2	yes	complemented
65	C₂²×C₅²	—	2	8	no	—
66	C₂²×C₅²	—	2	2	no	—
67	C₂²×C₅²	—	2	2	no	—
68	C₂²×C₅²	—	2	2	no	—
69	C₂²×C₅²	C₂×D₄	1	2	yes	—
70	C₂²×C₅²	#41	1	4	yes	complemented
71	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
72	C₂×C₅×D₅	—	2	2	no	—
73	C₂×C₅×D₅	—	2	4	no	—
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	—	2	2	no	—
80	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
81	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
82	C₅×#41	—	10	4	no	—
83	C₂×C₅×Q₈	—	5	1	no	—
84	C₂×C₅×D₄	—	5	4	no	—
85	C₂×C₅×D₄	—	5	2	no	—
86	C₂×C₅×D₄	—	5	1	no	—
87	C₂²×C₄×C₅	—	2	16	no	—
88	C₂²×C₄×C₅	—	2	4	no	—
89	C₂²×C₄×C₅	—	2	4	no	—
90	C₂²×C₄×C₅	—	2	4	no	—
91	C₂²×C₄×C₅	—	2	4	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	5	2	no	—
95	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
96	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
98	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
100	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
101	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
104	C₂×C₂²⋊D₅	C₂²×C₅	1	4	yes	—
105	#709	—	2	4	no	—
106	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
107	C₂×C₄×D₅	C₂²×C₅	1	2	yes	complemented
108	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
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	4	yes	—
116	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
117	#685	C₂²×C₅	1	4	yes	complemented
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	—	2	4	no	—
124	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₅²	C₂×#41	1	2	yes	—
126	C₅×D₅	—	4	2	no	—
127	C₂³×C₅	—	2	8	no	—
128	C₂³×C₅	—	2	16	no	—
129	C₂³×C₅	—	2	4	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	2	4	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	1	yes	characteristic
139	C₂³×C₅	C₂³×C₅	1	2	yes	—
140	C₂²×D₅	C₂³×C₅	1	2	yes	—
141	C₅×Q₈	—	10	2	no	—
142	C₅×D₄	—	10	8	no	—
143	C₅×D₄	—	10	4	no	—
144	C₅×D₄	—	10	2	no	—
145	C₂×C₄×C₅	—	2	8	no	—
146	C₂×C₄×C₅	—	2	16	no	—
147	C₂×C₄×C₅	—	2	16	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	8	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	5	4	no	—
159	C₂×C₄×C₅	—	5	2	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	4	no	—
170	C₂²⋊D₅	—	2	8	no	—
171	C₂×Dic₅	C₂³×C₅	1	4	yes	—
172	C₂×Dic₅	C₂³×C₅	1	2	yes	—
173	D₂₀	—	2	2	no	—
174	C₄×D₅	—	2	4	no	—
175	Dic₁₀	—	2	2	no	—
176	C₂×#41	—	5	1	no	—
177	C₂³×C₄	C₅×D₅	1	1	yes	characteristic, complemented
178	#123	—	5	2	no	—
179	C₂²⋊Q₈	—	5	2	no	—
180	#121	—	5	2	no	—
181	#120	—	5	2	no	—
182	C₄×D₄	—	5	4	no	—
183	#117	—	5	1	no	—
184	C₅²	#525	1	1	yes	characteristic, complemented, Sylow
185	C₂²×C₅	—	2	8	no	—
186	C₂²×C₅	—	2	32	no	—
187	C₂²×C₅	—	2	8	no	—
188	C₂²×C₅	—	2	8	no	—
189	C₂²×C₅	—	2	8	no	—
190	C₂²×C₅	—	2	8	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	8	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	10	4	no	—
201	C₂²×C₅	—	10	2	no	—
202	C₂²×C₅	#709	1	4	yes	complemented
203	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
204	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
205	C₂²×C₅	C₅×#41	1	4	yes	complemented
206	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
207	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
208	C₂×D₅	—	2	2	no	—
209	C₂×D₅	—	2	4	no	—
210	C₄×C₅	—	2	8	no	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	4	no	—
214	C₄×C₅	—	2	4	no	—
215	C₄×C₅	—	10	4	no	—
216	C₄×C₅	—	10	2	no	—
217	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
218	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
219	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
220	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
221	Dic₅	—	2	2	no	—
222	Dic₅	—	2	4	no	—
223	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
224	#41	—	10	4	no	—
225	C₂×Q₈	—	5	1	no	—
226	C₂×D₄	—	5	4	no	—
227	C₂×D₄	—	5	2	no	—
228	C₂×D₄	—	5	1	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
232	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
233	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
235	C₄⋊C₄	—	5	2	no	—
236	C₄⋊C₄	—	5	4	no	—
237	C₂²⋊C₄	—	5	2	no	—
238	C₂²⋊C₄	—	5	4	no	—
239	C₂²⋊C₄	—	5	4	no	—
240	C₄²	—	5	2	no	—
241	C₂×C₅	—	2	16	no	—
242	C₂×C₅	—	2	8	no	—
243	C₂×C₅	—	2	4	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	—	2	2	no	—
246	C₂×C₅	—	2	4	no	—
247	C₂×C₅	—	2	4	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	20	2	no	—
250	C₂×C₅	C₂×#709	1	2	yes	—
251	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
252	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
253	C₂×C₅	C₂×C₅×#41	1	2	yes	—
254	D₅	—	4	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	5	2	no	—
258	C₂³	C₂²×C₅×D₅	1	2	yes	—
259	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
260	Q₈	—	10	2	no	—
261	D₄	—	10	8	no	—
262	D₄	—	10	4	no	—
263	D₄	—	10	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	5	4	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	10	4	no	—
270	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
271	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
272	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
274	C₅	—	2	2	no	—
275	C₅	#66125	1	1	yes	characteristic, direct product factor
276	C₅	C₅×#525	1	1	yes	characteristic, complemented
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	10	4	no	—
282	C₂²	—	10	2	no	—
283	C₂²	C₅×#709	1	4	yes	complemented
284	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
285	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	10	4	no	—
288	C₄	—	10	2	no	—
289	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
290	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	—	2	2	no	—
293	C₂	—	20	2	no	—
294	C₂	C₂×C₅×#709	1	2	yes	—
295	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	C₂³×C₄×C₅²	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₅×#525
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄×C₅²	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₅×#4099	C₂	semidirect
4. C₅×#4076	C₂	semidirect
5. C₅×#4075	C₂	semidirect
6. C₄×C₅×C₂²⋊D₅	C₂	semidirect
7. C₅×#4072	C₂	semidirect
8. C₅×#4070	C₂	semidirect
18. C₂×C₅×D₂₀	C₂²	semidirect
19. C₂×C₄×C₅×D₅	C₂²	semidirect
20. C₂×C₅×Dic₁₀	C₂²	semidirect
23. C₅×#685	C₂²	semidirect
24. C₅×C₄⋊Dic₅	C₂²	semidirect
25. C₅×#683	C₂²	semidirect
26. C₄×C₅×Dic₅	C₂²	semidirect
28. #66125	C₅	direct (N × H)
37. C₂×C₄×C₅²	D₄	semidirect
50. C₂³×C₄×C₅	D₅	semidirect
51. C₂³×C₄×C₅	C₂×C₅	semidirect
52. C₂×#709	C₂×C₅	semidirect
59. #4099	C₂×C₅	semidirect
60. #4076	C₂×C₅	semidirect
61. #4075	C₂×C₅	semidirect
62. C₄×C₂²⋊D₅	C₂×C₅	semidirect
63. #4072	C₂×C₅	semidirect
64. #4070	C₂×C₅	semidirect
70. C₂²×C₅²	#41	semidirect
106. C₂×D₂₀	C₂²×C₅	semidirect
107. C₂×C₄×D₅	C₂²×C₅	semidirect
108. C₂×Dic₁₀	C₂²×C₅	semidirect
117. #685	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
177. C₂³×C₄	C₅×D₅	semidirect
184. C₅²	#525	semidirect
202. C₂²×C₅	#709	semidirect
205. C₂²×C₅	C₅×#41	semidirect
270. C₂×C₄	C₅×C₂²⋊D₅	semidirect
275. C₅	#66125	direct (N × H)
276. C₅	C₅×#525	semidirect
283. C₂²	C₅×#709	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂×C₅×#709	2
C₅×#4099	2
C₅×#4076	2
C₅×#4075	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4072	2
C₅×#4070	2
C₅×#525	5
#66125	5

Subgroups — C5×#66125 #13106945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66125 #13106945