Subgroups — C₅×#65912 #13106732

Order 1600.

514 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#4043	C₂	1	1	yes	maximal, characteristic
5	C₅×#4042	C₂	1	2	yes	maximal
6	C₅×#4035	C₂	1	2	yes	maximal, complemented
7	C₅×#4034	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4018	C₂	1	2	yes	maximal
9	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
10	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₁₀	C₂	1	2	yes	maximal, complemented
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
17	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
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	2	yes	—
23	C₅×#685	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₅×#683	C₂²	1	2	yes	—
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	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
32	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
33	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
34	C₄×C₅×Dic₅	C₂²	1	2	yes	—
35	C₅×#533	—	5	1	no	maximal
36	#65912	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	2	yes	—
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	1	yes	characteristic
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₄×C₅×D₅	—	2	2	no	—
50	C₄×C₅×D₅	—	2	2	no	—
51	C₄×C₅×D₅	Q₈	1	4	yes	complemented
52	C₅×Dic₁₀	—	2	2	no	—
53	C₅×Dic₁₀	—	2	4	no	—
54	C₅×C₄⋊Q₈	—	5	1	no	—
55	C₅×C₄⋊Q₈	—	5	1	no	—
56	C₅×#125	—	5	2	no	—
57	C₅×C₂²⋊Q₈	—	5	1	no	—
58	C₅×C₂²⋊Q₈	—	5	2	no	—
59	C₅×C₂²⋊Q₈	—	5	1	no	—
60	C₄×C₅×Q₈	—	2	2	no	—
61	C₄×C₅×Q₈	—	5	2	no	—
62	C₄×C₅×Q₈	—	5	1	no	—
63	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
64	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
65	C₅×#117	—	5	2	no	—
66	C₂×C₄²×C₅	—	5	1	no	—
67	#4092	C₂×C₅	1	1	yes	characteristic
68	#4090	C₂×C₅	1	1	yes	characteristic, complemented
69	#4043	C₂×C₅	1	1	yes	characteristic
70	#4042	C₂×C₅	1	2	yes	—
71	#4035	C₂×C₅	1	2	yes	complemented
72	#4034	C₂×C₅	1	1	yes	characteristic, complemented
73	#4018	C₂×C₅	1	2	yes	—
74	C₄²×D₅	C₂×C₅	1	1	yes	characteristic
75	C₄×Dic₁₀	C₂×C₅	1	2	yes	complemented
76	C₄×Dic₁₀	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	1	no	—
80	C₂×C₅×D₅	C₂×Q₈	1	2	yes	—
81	C₄×C₅²	—	2	2	no	—
82	C₄×C₅²	—	2	2	no	—
83	C₄×C₅²	—	2	1	no	—
84	C₄×C₅²	C₂×Q₈	1	2	yes	—
85	C₄×C₅²	#41	1	2	yes	—
86	C₅×Dic₅	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	—	2	1	no	—
89	C₅×Dic₅	C₂×Q₈	1	2	yes	—
90	C₅×Dic₅	#41	1	2	yes	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	5	2	no	—
93	C₂×C₅×Q₈	—	5	1	no	—
94	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
95	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂²×C₄×C₅	—	5	1	no	—
98	C₂²×C₄×C₅	—	5	1	no	—
99	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
102	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
103	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	5	2	no	—
107	C₅×C₄⋊C₄	—	5	2	no	—
108	C₅×C₄⋊C₄	—	5	2	no	—
109	C₅×C₄⋊C₄	—	5	2	no	—
110	C₅×C₄⋊C₄	—	5	2	no	—
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₄⋊C₄	—	5	1	no	—
113	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
114	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
115	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
116	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
117	C₅×C₂²⋊C₄	—	5	2	no	—
118	C₅×C₂²⋊C₄	—	5	2	no	—
119	C₄²×C₅	—	2	2	no	—
120	C₄²×C₅	—	2	4	no	—
121	C₄²×C₅	—	5	1	no	—
122	C₄²×C₅	—	5	1	no	—
123	C₄²×C₅	—	5	1	no	—
124	C₄²×C₅	—	5	2	no	—
125	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₅	1	2	yes	—
127	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
128	C₄²×C₅	C₂²×C₅	1	2	yes	—
129	#685	C₂²×C₅	1	2	yes	—
130	#685	C₂²×C₅	1	2	yes	—
131	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
132	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
133	#683	C₂²×C₅	1	2	yes	—
134	#683	C₂²×C₅	1	2	yes	—
135	#683	C₂²×C₅	1	2	yes	—
136	#683	C₂²×C₅	1	2	yes	—
137	#683	C₂²×C₅	1	2	yes	—
138	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
139	C₄×Dic₅	C₂²×C₅	1	2	yes	—
140	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
141	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
142	#533	—	5	1	no	Sylow
143	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₅²	C₂×#41	1	1	yes	characteristic
145	C₂×C₅²	C₂×#41	1	1	yes	characteristic
146	C₅×D₅	—	2	2	no	—
147	C₂³×C₅	—	5	1	no	—
148	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
149	C₅×Q₈	—	2	8	no	—
150	C₅×Q₈	—	2	2	no	—
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	—	10	4	no	—
153	C₅×Q₈	—	10	2	no	—
154	C₂×C₄×C₅	—	2	2	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	—	2	4	no	—
159	C₂×C₄×C₅	—	5	1	no	—
160	C₂×C₄×C₅	—	5	1	no	—
161	C₂×C₄×C₅	—	5	4	no	—
162	C₂×C₄×C₅	—	5	1	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	5	2	no	—
165	C₂×C₄×C₅	—	10	2	no	—
166	C₂×C₄×C₅	—	10	2	no	—
167	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
169	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
170	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
174	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
175	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
177	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
178	C₂×Dic₅	C₂³×C₅	1	2	yes	—
179	C₂×Dic₅	C₂³×C₅	1	2	yes	—
180	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
181	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
182	C₄×D₅	—	2	2	no	—
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	C₅×Q₈	1	4	yes	complemented
185	Dic₁₀	—	2	2	no	—
186	Dic₁₀	—	2	4	no	—
187	C₄⋊Q₈	—	5	1	no	—
188	C₄⋊Q₈	—	5	1	no	—
189	#125	—	5	2	no	—
190	C₂²⋊Q₈	—	5	1	no	—
191	C₂²⋊Q₈	—	5	2	no	—
192	C₂²⋊Q₈	—	5	1	no	—
193	C₄×Q₈	—	5	2	no	—
194	C₄×Q₈	—	5	1	no	—
195	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
196	#117	—	5	2	no	—
197	C₂×C₄²	—	5	1	no	—
198	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	5	2	no	—
201	C₂²×C₅	—	10	1	no	—
202	C₂²×C₅	—	10	1	no	—
203	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
204	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
205	C₂×D₅	—	2	1	no	—
206	C₂×D₅	—	2	1	no	—
207	C₂×D₅	C₂×C₅×Q₈	1	2	yes	—
208	C₄×C₅	—	2	8	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	4	no	—
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	4	2	no	—
218	C₄×C₅	—	4	4	no	—
219	C₄×C₅	—	5	2	no	—
220	C₄×C₅	—	5	2	no	—
221	C₄×C₅	—	10	1	no	—
222	C₄×C₅	—	10	2	no	—
223	C₄×C₅	—	10	2	no	—
224	C₄×C₅	#709	1	2	yes	—
225	C₄×C₅	Q₈×D₅	1	2	yes	—
226	C₄×C₅	C₂×C₅×Q₈	1	2	yes	—
227	C₄×C₅	C₅×#41	1	2	yes	—
228	Dic₅	—	2	2	no	—
229	Dic₅	—	2	2	no	—
230	Dic₅	—	2	1	no	—
231	Dic₅	C₂×C₅×Q₈	1	2	yes	—
232	Dic₅	C₅×#41	1	2	yes	—
233	C₂×Q₈	—	5	2	no	—
234	C₂×Q₈	—	5	1	no	—
235	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	1	no	—
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	5	2	no	—
242	C₄⋊C₄	—	5	2	no	—
243	C₄⋊C₄	—	5	2	no	—
244	C₄⋊C₄	—	5	2	no	—
245	C₄⋊C₄	—	5	1	no	—
246	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
247	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
248	C₂²⋊C₄	—	5	2	no	—
249	C₂²⋊C₄	—	5	2	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	5	1	no	—
252	C₄²	—	5	1	no	—
253	C₄²	—	5	2	no	—
254	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
255	C₄²	C₂×C₅×D₅	1	2	yes	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	10	2	no	—
260	C₂×C₅	C₂×#709	1	1	yes	characteristic
261	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
262	C₂×C₅	D₅×#41	1	1	yes	characteristic
263	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
264	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
265	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
266	D₅	—	2	2	no	—
267	C₂³	—	5	1	no	—
268	Q₈	—	2	2	no	—
269	Q₈	—	10	4	no	—
270	Q₈	—	10	2	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	4	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	5	2	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
281	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
282	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
284	C₅	—	2	2	no	—
285	C₅	#65912	1	1	yes	characteristic, direct product factor
286	C₅	C₅×#533	1	1	yes	characteristic, complemented
287	C₂²	—	5	2	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	5	2	no	—
295	C₄	—	5	2	no	—
296	C₄	—	10	1	no	—
297	C₄	—	10	2	no	—
298	C₄	—	10	2	no	—
299	C₄	C₅×#709	1	2	yes	—
300	C₄	C₅×Q₈×D₅	1	2	yes	—
301	C₂	—	10	2	no	—
302	C₂	C₂×C₅×#709	1	1	yes	characteristic
303	C₂	C₂×C₅×Q₈×D₅	1	1	yes	characteristic
304	C₂	C₅×D₅×#41	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	C₄×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₂²×C₅×D₅	C₂³
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#533
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×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×#4035	C₂	semidirect
7. C₅×#4034	C₂	semidirect
10. C₄×C₅×Dic₁₀	C₂	semidirect
11. C₄×C₅×Dic₁₀	C₂	semidirect
36. #65912	C₅	direct (N × H)
51. C₄×C₅×D₅	Q₈	semidirect
63. C₄×C₅×Q₈	D₅	semidirect
64. C₄×C₅×Q₈	C₂×C₅	semidirect
68. #4090	C₂×C₅	semidirect
71. #4035	C₂×C₅	semidirect
72. #4034	C₂×C₅	semidirect
75. C₄×Dic₁₀	C₂×C₅	semidirect
76. C₄×Dic₁₀	C₂×C₅	semidirect
184. C₄×D₅	C₅×Q₈	semidirect
195. C₄×Q₈	C₅×D₅	semidirect
198. C₅²	#533	semidirect
285. C₅	#65912	direct (N × H)
286. C₅	C₅×#533	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×#4043	2
C₅×#4042	2
C₅×#4035	2
C₅×#4034	2
C₅×#4018	2
C₄²×C₅×D₅	2
C₄×C₅×Dic₁₀	2
C₅×#533	5
#65912	5

Subgroups — C5×#65912 #13106732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65912 #13106732