Subgroups — C₅×#65917 #13106737

Order 1600.

454 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	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4092	C₂	1	1	yes	maximal, characteristic
3	C₅×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4044	C₂	1	2	yes	maximal
5	C₅×#4043	C₂	1	1	yes	maximal, characteristic
6	C₅×#4036	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4035	C₂	1	2	yes	maximal, complemented
8	C₅×#4022	C₂	1	2	yes	maximal
9	C₅×#4018	C₂	1	1	yes	maximal, characteristic
10	C₅×#4016	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
15	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
16	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₄²×C₅²	C₂²	1	1	yes	characteristic
18	C₄²×C₅²	C₂²	1	2	yes	—
19	C₅×#685	C₂²	1	2	yes	—
20	C₅×#685	C₂²	1	1	yes	characteristic
21	C₅×#685	C₂²	1	2	yes	—
22	C₅×#685	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₅×#683	C₂²	1	1	yes	characteristic
27	C₅×#683	C₂²	1	2	yes	—
28	C₅×#683	C₂²	1	2	yes	—
29	C₅×#683	C₂²	1	2	yes	—
30	C₅×#683	C₂²	1	2	yes	—
31	C₅×#683	C₂²	1	2	yes	—
32	C₅×#683	C₂²	1	1	yes	characteristic
33	C₄×C₅×Dic₅	C₂²	1	2	yes	—
34	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
35	C₅×#541	—	5	1	no	maximal
36	#65917	C₅	1	1	yes	maximal, characteristic, direct product factor
37	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
38	C₅²×Q₈	—	2	2	no	—
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	2	yes	—
42	C₂×C₄×C₅²	C₂³	1	2	yes	—
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₄×C₅×D₅	—	2	2	no	—
50	C₅×Dic₁₀	—	2	2	no	—
51	C₅×C₄⋊Q₈	—	5	1	no	—
52	C₅×#126	—	5	2	no	—
53	C₅×#126	—	5	2	no	—
54	C₅×#125	—	5	2	no	—
55	C₅×#125	—	5	1	no	—
56	C₅×#125	—	5	2	no	—
57	C₅×C₂²⋊Q₈	—	5	1	no	—
58	C₅×C₂²⋊Q₈	—	5	1	no	—
59	C₄×C₅×Q₈	—	2	2	no	—
60	C₄×C₅×Q₈	—	5	1	no	—
61	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
62	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
63	C₅×#117	—	5	1	no	—
64	#4092	C₂×C₅	1	1	yes	characteristic
65	#4090	C₂×C₅	1	1	yes	characteristic, complemented
66	#4044	C₂×C₅	1	2	yes	—
67	#4043	C₂×C₅	1	1	yes	characteristic
68	#4036	C₂×C₅	1	1	yes	characteristic, complemented
69	#4035	C₂×C₅	1	2	yes	complemented
70	#4022	C₂×C₅	1	2	yes	—
71	#4018	C₂×C₅	1	1	yes	characteristic
72	#4016	C₂×C₅	1	2	yes	complemented
73	C₄×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×D₅	—	2	1	no	—
76	C₂×C₅×D₅	—	2	1	no	—
77	C₂×C₅×D₅	—	2	1	no	—
78	C₄×C₅²	—	2	2	no	—
79	C₄×C₅²	—	2	1	no	—
80	C₄×C₅²	—	2	1	no	—
81	C₄×C₅²	—	2	2	no	—
82	C₄×C₅²	#41	1	2	yes	—
83	C₅×Dic₅	—	2	2	no	—
84	C₅×Dic₅	—	2	1	no	—
85	C₅×Dic₅	—	2	2	no	—
86	C₅×Dic₅	—	2	1	no	—
87	C₅×Dic₅	—	2	1	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×Q₈	—	5	1	no	—
90	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
91	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₂²×C₄×C₅	—	5	1	no	—
93	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
94	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	5	1	no	—
98	C₅×C₄⋊C₄	—	5	1	no	—
99	C₅×C₄⋊C₄	—	5	1	no	—
100	C₅×C₄⋊C₄	—	5	2	no	—
101	C₅×C₄⋊C₄	—	5	2	no	—
102	C₅×C₄⋊C₄	—	5	2	no	—
103	C₅×C₄⋊C₄	—	5	2	no	—
104	C₅×C₄⋊C₄	—	5	2	no	—
105	C₅×C₄⋊C₄	—	5	2	no	—
106	C₅×C₄⋊C₄	—	5	2	no	—
107	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
108	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
109	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
110	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
111	C₅×C₂²⋊C₄	—	5	1	no	—
112	C₅×C₂²⋊C₄	—	5	1	no	—
113	C₅×C₂²⋊C₄	—	5	2	no	—
114	C₅×C₂²⋊C₄	—	5	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	2	4	no	—
117	C₄²×C₅	—	5	1	no	—
118	C₄²×C₅	—	5	2	no	—
119	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₅	1	2	yes	—
121	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
122	C₄²×C₅	C₂²×C₅	1	2	yes	—
123	#685	C₂²×C₅	1	1	yes	characteristic
124	#685	C₂²×C₅	1	1	yes	characteristic
125	#685	C₂²×C₅	1	2	yes	—
126	#685	C₂²×C₅	1	2	yes	—
127	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
128	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
129	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
130	#683	C₂²×C₅	1	2	yes	—
131	#683	C₂²×C₅	1	1	yes	characteristic
132	#683	C₂²×C₅	1	2	yes	—
133	#683	C₂²×C₅	1	2	yes	—
134	#683	C₂²×C₅	1	1	yes	characteristic
135	#683	C₂²×C₅	1	2	yes	—
136	#683	C₂²×C₅	1	2	yes	—
137	C₄×Dic₅	C₂²×C₅	1	2	yes	—
138	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
139	#541	—	5	1	no	Sylow
140	C₂×C₅²	C₂×#41	1	1	yes	characteristic
141	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
143	C₅×D₅	—	4	1	no	—
144	C₂³×C₅	—	5	1	no	—
145	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
146	C₅×Q₈	—	2	8	no	—
147	C₅×Q₈	—	2	2	no	—
148	C₅×Q₈	—	2	2	no	—
149	C₅×Q₈	—	10	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	5	1	no	—
156	C₂×C₄×C₅	—	5	2	no	—
157	C₂×C₄×C₅	—	5	1	no	—
158	C₂×C₄×C₅	—	5	2	no	—
159	C₂×C₄×C₅	—	5	1	no	—
160	C₂×C₄×C₅	—	10	2	no	—
161	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
164	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
169	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
170	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
171	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
172	C₂×Dic₅	C₂³×C₅	1	2	yes	—
173	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
174	C₂×Dic₅	C₂³×C₅	1	2	yes	—
175	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
176	C₄×D₅	—	2	2	no	—
177	Dic₁₀	—	2	2	no	—
178	C₄⋊Q₈	—	5	1	no	—
179	#126	—	5	2	no	—
180	#126	—	5	2	no	—
181	#125	—	5	2	no	—
182	#125	—	5	1	no	—
183	#125	—	5	2	no	—
184	C₂²⋊Q₈	—	5	1	no	—
185	C₂²⋊Q₈	—	5	1	no	—
186	C₄×Q₈	—	5	1	no	—
187	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
188	#117	—	5	1	no	—
189	C₅²	#541	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	10	1	no	—
192	C₂²×C₅	—	10	1	no	—
193	C₂²×C₅	—	10	1	no	—
194	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
195	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
196	C₂×D₅	—	2	1	no	—
197	C₂×D₅	—	2	1	no	—
198	C₂×D₅	—	2	1	no	—
199	C₄×C₅	—	2	4	no	—
200	C₄×C₅	—	2	8	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	4	no	—
210	C₄×C₅	—	4	2	no	—
211	C₄×C₅	—	4	4	no	—
212	C₄×C₅	—	10	1	no	—
213	C₄×C₅	—	10	2	no	—
214	C₄×C₅	—	10	1	no	—
215	C₄×C₅	—	10	2	no	—
216	C₄×C₅	—	10	1	no	—
217	C₄×C₅	#709	1	2	yes	—
218	C₄×C₅	C₅×#41	1	2	yes	—
219	Dic₅	—	2	2	no	—
220	Dic₅	—	2	1	no	—
221	Dic₅	—	2	2	no	—
222	Dic₅	—	2	1	no	—
223	Dic₅	—	2	1	no	—
224	C₂×Q₈	—	5	1	no	—
225	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
226	C₂²×C₄	—	5	1	no	—
227	C₄⋊C₄	—	5	1	no	—
228	C₄⋊C₄	—	5	1	no	—
229	C₄⋊C₄	—	5	1	no	—
230	C₄⋊C₄	—	5	2	no	—
231	C₄⋊C₄	—	5	2	no	—
232	C₄⋊C₄	—	5	2	no	—
233	C₄⋊C₄	—	5	2	no	—
234	C₄⋊C₄	—	5	2	no	—
235	C₄⋊C₄	—	5	2	no	—
236	C₄⋊C₄	—	5	2	no	—
237	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
239	C₂²⋊C₄	—	5	1	no	—
240	C₂²⋊C₄	—	5	1	no	—
241	C₂²⋊C₄	—	5	2	no	—
242	C₂²⋊C₄	—	5	2	no	—
243	C₄²	—	5	1	no	—
244	C₄²	—	5	2	no	—
245	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
246	C₄²	C₂×C₅×D₅	1	2	yes	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	20	1	no	—
251	C₂×C₅	C₂×#709	1	1	yes	characteristic
252	C₂×C₅	#4147	1	1	yes	characteristic
253	C₂×C₅	#4150	1	1	yes	characteristic
254	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
255	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
256	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
257	D₅	—	4	1	no	—
258	C₂³	—	5	1	no	—
259	Q₈	—	2	2	no	—
260	Q₈	—	10	2	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	5	2	no	—
265	C₂×C₄	—	5	1	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
270	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
271	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
272	C₅	—	2	2	no	—
273	C₅	#65917	1	1	yes	characteristic, direct product factor
274	C₅	C₅×#541	1	1	yes	characteristic, complemented
275	C₂²	—	10	1	no	—
276	C₂²	—	10	1	no	—
277	C₂²	—	10	1	no	—
278	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	10	1	no	—
284	C₄	—	10	2	no	—
285	C₄	—	10	1	no	—
286	C₄	—	10	2	no	—
287	C₄	—	10	1	no	—
288	C₄	C₅×#709	1	2	yes	—
289	C₂	—	20	1	no	—
290	C₂	C₂×C₅×#709	1	1	yes	characteristic
291	C₂	C₅×#4147	1	1	yes	characteristic
292	C₂	C₅×#4150	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	C₂²×C₅×D₅	C₂³
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#541
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
3. C₅×#4090	C₂	semidirect
6. C₅×#4036	C₂	semidirect
7. C₅×#4035	C₂	semidirect
10. C₅×#4016	C₂	semidirect
11. C₄×C₅×Dic₁₀	C₂	semidirect
36. #65917	C₅	direct (N × H)
61. C₄×C₅×Q₈	D₅	semidirect
62. C₄×C₅×Q₈	C₂×C₅	semidirect
65. #4090	C₂×C₅	semidirect
68. #4036	C₂×C₅	semidirect
69. #4035	C₂×C₅	semidirect
72. #4016	C₂×C₅	semidirect
73. C₄×Dic₁₀	C₂×C₅	semidirect
187. C₄×Q₈	C₅×D₅	semidirect
189. C₅²	#541	semidirect
273. C₅	#65917	direct (N × H)
274. C₅	C₅×#541	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×#4092	2
C₅×#4090	2
C₅×#4044	2
C₅×#4043	2
C₅×#4036	2
C₅×#4035	2
C₅×#4022	2
C₅×#4018	2
C₅×#4016	2
C₄×C₅×Dic₁₀	2
C₅×#541	5
#65917	5

Subgroups — C5×#65917 #13106737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65917 #13106737