Subgroups — C₅×#65916 #13106736

Order 1600.

502 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4093	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4092	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4042	C₂	1	2	yes	maximal, complemented
5	C₅×#4041	C₂	1	1	yes	maximal, characteristic
6	C₅×#4040	C₂	1	2	yes	maximal, complemented
7	C₅×#4034	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4022	C₂	1	2	yes	maximal, complemented
9	C₅×#4021	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4018	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₂₀	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
17	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
18	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₄²×C₅²	C₂²	1	1	yes	characteristic
21	C₄²×C₅²	C₂²	1	2	yes	—
22	C₅×#685	C₂²	1	1	yes	characteristic
23	C₅×#685	C₂²	1	2	yes	—
24	C₅×#685	C₂²	1	2	yes	—
25	C₅×#685	C₂²	1	2	yes	—
26	C₅×#685	C₂²	1	2	yes	—
27	C₅×#685	C₂²	1	1	yes	characteristic
28	C₅×#685	C₂²	1	2	yes	—
29	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₅×#683	C₂²	1	2	yes	—
31	C₅×#683	C₂²	1	1	yes	characteristic
32	C₅×#683	C₂²	1	2	yes	—
33	C₅×#683	C₂²	1	1	yes	characteristic
34	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
35	C₅×#542	—	5	1	no	maximal
36	#65916	C₅	1	1	yes	maximal, characteristic, direct product factor
37	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
38	C₂²×C₅×D₅	C₂³	1	2	yes	—
39	C₅²×Q₈	—	2	2	no	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	2	yes	—
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅²	C₂³	1	2	yes	—
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₅×D₂₀	—	2	2	no	—
50	C₅×D₂₀	—	4	2	no	—
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	4	2	no	—
53	C₅×Dic₁₀	—	4	2	no	—
54	C₅×C₄⋊Q₈	—	5	1	no	—
55	C₅×#126	—	5	2	no	—
56	C₅×#124	—	5	1	no	—
57	C₅×#124	—	5	2	no	—
58	C₅×#123	—	5	2	no	—
59	C₅×C₂²⋊Q₈	—	5	1	no	—
60	C₅×C₂²⋊Q₈	—	5	2	no	—
61	C₅×#121	—	5	1	no	—
62	C₄×C₅×Q₈	—	2	2	no	—
63	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
64	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
65	C₄×C₅×D₄	—	5	1	no	—
66	C₅×#117	—	5	1	no	—
67	#4093	C₂×C₅	1	1	yes	characteristic, complemented
68	#4092	C₂×C₅	1	1	yes	characteristic, complemented
69	#4042	C₂×C₅	1	2	yes	complemented
70	#4041	C₂×C₅	1	1	yes	characteristic
71	#4040	C₂×C₅	1	2	yes	complemented
72	#4034	C₂×C₅	1	1	yes	characteristic, complemented
73	#4022	C₂×C₅	1	2	yes	complemented
74	#4021	C₂×C₅	1	2	yes	complemented
75	C₄×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
76	#4018	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×D₅	—	2	1	no	—
79	C₂×C₅×D₅	—	2	2	no	—
80	C₂×C₅×D₅	—	2	2	no	—
81	C₂×C₅×D₅	—	2	1	no	—
82	C₂×C₅×D₅	—	2	1	no	—
83	C₂×C₅×D₅	—	2	2	no	—
84	C₄×C₅²	—	2	2	no	—
85	C₄×C₅²	—	2	2	no	—
86	C₄×C₅²	—	2	1	no	—
87	C₄×C₅²	—	2	1	no	—
88	C₄×C₅²	#41	1	2	yes	—
89	C₅×Dic₅	—	2	1	no	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	2	no	—
92	C₅×Dic₅	—	2	1	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	—	5	2	no	—
95	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
96	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₅×D₄	—	5	1	no	—
98	C₂×C₅×D₄	—	5	2	no	—
99	C₂²×C₄×C₅	—	5	1	no	—
100	C₂²×C₄×C₅	—	5	2	no	—
101	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
102	C₂×D₂₀	C₂²×C₅	1	2	yes	—
103	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
105	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₄⋊C₄	—	5	1	no	—
109	C₅×C₄⋊C₄	—	5	1	no	—
110	C₅×C₄⋊C₄	—	5	1	no	—
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₄⋊C₄	—	5	2	no	—
113	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
114	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
115	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
116	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
117	C₅×C₂²⋊C₄	—	5	1	no	—
118	C₅×C₂²⋊C₄	—	5	1	no	—
119	C₅×C₂²⋊C₄	—	5	2	no	—
120	C₅×C₂²⋊C₄	—	5	2	no	—
121	C₅×C₂²⋊C₄	—	5	2	no	—
122	C₅×C₂²⋊C₄	—	5	2	no	—
123	C₅×C₂²⋊C₄	—	5	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	—	2	4	no	—
126	C₄²×C₅	—	5	1	no	—
127	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₅	1	2	yes	—
129	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
130	C₄²×C₅	C₂²×C₅	1	2	yes	—
131	#685	C₂²×C₅	1	1	yes	characteristic
132	#685	C₂²×C₅	1	1	yes	characteristic
133	#685	C₂²×C₅	1	2	yes	—
134	#685	C₂²×C₅	1	2	yes	—
135	#685	C₂²×C₅	1	2	yes	—
136	#685	C₂²×C₅	1	2	yes	—
137	#685	C₂²×C₅	1	2	yes	—
138	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
139	#683	C₂²×C₅	1	1	yes	characteristic
140	#683	C₂²×C₅	1	1	yes	characteristic
141	#683	C₂²×C₅	1	2	yes	—
142	#683	C₂²×C₅	1	2	yes	—
143	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
144	#542	—	5	1	no	Sylow
145	C₂×C₅²	C₂×#41	1	1	yes	characteristic
146	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
148	C₅×D₅	—	4	1	no	—
149	C₅×D₅	—	4	2	no	—
150	C₂³×C₅	—	5	1	no	—
151	C₂³×C₅	—	5	2	no	—
152	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
153	C₂²×D₅	C₂³×C₅	1	2	yes	—
154	C₅×Q₈	—	2	8	no	—
155	C₅×Q₈	—	2	2	no	—
156	C₅×Q₈	—	2	2	no	—
157	C₅×Q₈	—	20	2	no	—
158	C₅×D₄	—	10	2	no	—
159	C₅×D₄	—	20	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	5	1	no	—
166	C₂×C₄×C₅	—	5	1	no	—
167	C₂×C₄×C₅	—	5	1	no	—
168	C₂×C₄×C₅	—	5	2	no	—
169	C₂×C₄×C₅	—	10	2	no	—
170	C₂×C₄×C₅	—	20	2	no	—
171	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
173	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
175	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
178	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
179	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
181	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
182	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
183	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
184	C₂×Dic₅	C₂³×C₅	1	2	yes	—
185	D₂₀	—	2	2	no	—
186	D₂₀	—	4	2	no	—
187	C₄×D₅	—	2	2	no	—
188	C₄×D₅	—	4	2	no	—
189	Dic₁₀	—	4	2	no	—
190	C₄⋊Q₈	—	5	1	no	—
191	#126	—	5	2	no	—
192	#124	—	5	1	no	—
193	#124	—	5	2	no	—
194	#123	—	5	2	no	—
195	C₂²⋊Q₈	—	5	1	no	—
196	C₂²⋊Q₈	—	5	2	no	—
197	#121	—	5	1	no	—
198	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
199	C₄×D₄	—	5	1	no	—
200	#117	—	5	1	no	—
201	C₅²	#542	1	1	yes	characteristic, complemented, Sylow
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	10	1	no	—
204	C₂²×C₅	—	10	2	no	—
205	C₂²×C₅	—	10	1	no	—
206	C₂²×C₅	—	10	2	no	—
207	C₂²×C₅	—	10	2	no	—
208	C₂²×C₅	—	10	1	no	—
209	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
210	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	2	no	—
213	C₂×D₅	—	2	2	no	—
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	2	no	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	—	10	1	no	—
231	C₄×C₅	—	10	1	no	—
232	C₄×C₅	—	10	2	no	—
233	C₄×C₅	—	10	1	no	—
234	C₄×C₅	#709	1	2	yes	—
235	C₄×C₅	C₅×#41	1	2	yes	—
236	Dic₅	—	2	1	no	—
237	Dic₅	—	2	1	no	—
238	Dic₅	—	2	2	no	—
239	Dic₅	—	2	1	no	—
240	C₂×Q₈	—	5	2	no	—
241	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
242	C₂×D₄	—	5	1	no	—
243	C₂×D₄	—	5	2	no	—
244	C₂²×C₄	—	5	1	no	—
245	C₂²×C₄	—	5	2	no	—
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	—	5	1	no	—
248	C₄⋊C₄	—	5	1	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	5	2	no	—
251	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
252	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
253	C₂²⋊C₄	—	5	1	no	—
254	C₂²⋊C₄	—	5	1	no	—
255	C₂²⋊C₄	—	5	2	no	—
256	C₂²⋊C₄	—	5	2	no	—
257	C₂²⋊C₄	—	5	2	no	—
258	C₂²⋊C₄	—	5	2	no	—
259	C₂²⋊C₄	—	5	2	no	—
260	C₄²	—	5	1	no	—
261	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
262	C₄²	C₂×C₅×D₅	1	2	yes	—
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	20	1	no	—
267	C₂×C₅	—	20	2	no	—
268	C₂×C₅	C₂×#709	1	1	yes	characteristic
269	C₂×C₅	#4147	1	1	yes	characteristic
270	C₂×C₅	#4149	1	1	yes	characteristic
271	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
272	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
274	D₅	—	4	1	no	—
275	D₅	—	4	2	no	—
276	C₂³	—	5	1	no	—
277	C₂³	—	5	2	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	20	2	no	—
280	D₄	—	10	2	no	—
281	D₄	—	20	2	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	1	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	2	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	20	2	no	—
288	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
290	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
292	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
293	C₅	—	2	2	no	—
294	C₅	#65916	1	1	yes	characteristic, direct product factor
295	C₅	C₅×#542	1	1	yes	characteristic, complemented
296	C₂²	—	10	1	no	—
297	C₂²	—	10	2	no	—
298	C₂²	—	10	1	no	—
299	C₂²	—	10	2	no	—
300	C₂²	—	10	2	no	—
301	C₂²	—	10	1	no	—
302	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	10	1	no	—
308	C₄	—	10	1	no	—
309	C₄	—	10	2	no	—
310	C₄	—	10	1	no	—
311	C₄	C₅×#709	1	2	yes	—
312	C₂	—	20	1	no	—
313	C₂	—	20	2	no	—
314	C₂	C₂×C₅×#709	1	1	yes	characteristic
315	C₂	C₅×#4147	1	1	yes	characteristic
316	C₂	C₅×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₄×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₅×#4041	C₂
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#542
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×C₅²×Q₈	C₂	semidirect
2. C₅×#4093	C₂	semidirect
3. C₅×#4092	C₂	semidirect
4. C₅×#4042	C₂	semidirect
6. C₅×#4040	C₂	semidirect
7. C₅×#4034	C₂	semidirect
8. C₅×#4022	C₂	semidirect
9. C₅×#4021	C₂	semidirect
10. C₄×C₅×D₂₀	C₂	semidirect
11. C₅×#4018	C₂	semidirect
36. #65916	C₅	direct (N × H)
63. C₄×C₅×Q₈	D₅	semidirect
64. C₄×C₅×Q₈	C₂×C₅	semidirect
67. #4093	C₂×C₅	semidirect
68. #4092	C₂×C₅	semidirect
69. #4042	C₂×C₅	semidirect
71. #4040	C₂×C₅	semidirect
72. #4034	C₂×C₅	semidirect
73. #4022	C₂×C₅	semidirect
74. #4021	C₂×C₅	semidirect
75. C₄×D₂₀	C₂×C₅	semidirect
76. #4018	C₂×C₅	semidirect
198. C₄×Q₈	C₅×D₅	semidirect
201. C₅²	#542	semidirect
294. C₅	#65916	direct (N × H)
295. C₅	C₅×#542	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅²×Q₈	2
C₅×#4093	2
C₅×#4092	2
C₅×#4042	2
C₅×#4041	2
C₅×#4040	2
C₅×#4034	2
C₅×#4022	2
C₅×#4021	2
C₄×C₅×D₂₀	2
C₅×#4018	2
C₅×#542	5
#65916	5

Subgroups — C5×#65916 #13106736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65916 #13106736