Subgroups — C₅×#65914 #13106734

Order 1600.

538 conjugacy classes of subgroups.

The subgroup lattice has 322 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×#4044	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×#4036	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×D₂₀	C₂	1	2	yes	maximal, complemented
10	C₅×#4018	C₂	1	2	yes	maximal, complemented
11	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₂₀	C₂²	1	2	yes	—
14	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
19	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄²×C₅²	C₂²	1	1	yes	characteristic
22	C₄²×C₅²	C₂²	1	2	yes	—
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	2	yes	—
28	C₅×C₄⋊Dic₅	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	2	yes	—
32	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
33	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
34	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
35	C₅×#529	—	5	1	no	maximal
36	#65914	C₅	1	1	yes	maximal, characteristic, direct product factor
37	C₂²×C₅×D₅	C₂³	1	2	yes	—
38	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
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	2	yes	—
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₅×D₂₀	—	2	4	no	—
50	C₅×D₂₀	—	2	2	no	—
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	2	2	no	—
53	C₄×C₅×D₅	—	2	2	no	—
54	C₄×C₅×D₅	—	2	4	no	—
55	C₅×#126	—	5	2	no	—
56	C₅×#125	—	5	1	no	—
57	C₅×#124	—	5	1	no	—
58	C₅×#123	—	5	2	no	—
59	C₅×C₂²⋊Q₈	—	5	1	no	—
60	C₅×#121	—	5	1	no	—
61	C₄×C₅×Q₈	—	2	2	no	—
62	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
63	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
64	C₄×C₅×D₄	—	5	1	no	—
65	C₄×C₅×D₄	—	5	2	no	—
66	C₅×#117	—	5	2	no	—
67	C₂×C₄²×C₅	—	5	1	no	—
68	#4093	C₂×C₅	1	1	yes	characteristic, complemented
69	#4092	C₂×C₅	1	1	yes	characteristic, complemented
70	#4044	C₂×C₅	1	2	yes	complemented
71	#4041	C₂×C₅	1	1	yes	characteristic
72	#4040	C₂×C₅	1	2	yes	complemented
73	#4036	C₂×C₅	1	1	yes	characteristic, complemented
74	C₄×D₂₀	C₂×C₅	1	2	yes	complemented
75	C₄×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
76	#4018	C₂×C₅	1	2	yes	complemented
77	C₄²×D₅	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
79	C₂×C₅×D₅	—	2	1	no	—
80	C₂×C₅×D₅	—	2	1	no	—
81	C₂×C₅×D₅	—	2	2	no	—
82	C₂×C₅×D₅	—	2	2	no	—
83	C₂×C₅×D₅	—	2	2	no	—
84	C₂×C₅×D₅	#41	1	2	yes	—
85	C₄×C₅²	—	2	2	no	—
86	C₄×C₅²	—	2	1	no	—
87	C₄×C₅²	—	2	2	no	—
88	C₄×C₅²	#41	1	2	yes	—
89	C₄×C₅²	#41	1	2	yes	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₅×Dic₅	—	2	2	no	—
93	C₅×Dic₅	—	2	1	no	—
94	C₂×C₅×Q₈	—	2	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	2	no	—
98	C₂×C₅×D₄	—	5	1	no	—
99	C₂²×C₄×C₅	—	5	1	no	—
100	C₂²×C₄×C₅	—	5	1	no	—
101	C₂²×C₄×C₅	—	5	1	no	—
102	C₂²×C₄×C₅	—	5	2	no	—
103	C₂×D₂₀	C₂²×C₅	1	2	yes	—
104	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₄⋊C₄	—	5	2	no	—
113	C₅×C₄⋊C₄	—	5	1	no	—
114	C₅×C₄⋊C₄	—	5	2	no	—
115	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
116	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
117	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
118	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
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	—	5	1	no	—
128	C₄²×C₅	—	5	1	no	—
129	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
130	C₄²×C₅	C₂×D₅	1	2	yes	—
131	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
132	C₄²×C₅	C₂²×C₅	1	2	yes	—
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	2	yes	—
139	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
140	#683	C₂²×C₅	1	2	yes	—
141	#683	C₂²×C₅	1	2	yes	—
142	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
143	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
144	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
145	#529	—	5	1	no	Sylow
146	C₂×C₅²	C₂×#41	1	1	yes	characteristic
147	C₂×C₅²	C₂×#41	1	1	yes	characteristic
148	C₂×C₅²	C₂×#41	1	1	yes	characteristic
149	C₅×D₅	—	2	2	no	—
150	C₅×D₅	—	4	2	no	—
151	C₂³×C₅	—	5	1	no	—
152	C₂³×C₅	—	5	2	no	—
153	C₂²×D₅	C₂³×C₅	1	2	yes	—
154	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
155	C₅×Q₈	—	2	8	no	—
156	C₅×Q₈	—	2	2	no	—
157	C₅×Q₈	—	2	2	no	—
158	C₅×D₄	—	10	4	no	—
159	C₅×D₄	—	10	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	—	10	2	no	—
171	C₂×C₄×C₅	—	10	2	no	—
172	C₂×C₄×C₅	—	10	4	no	—
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	2	yes	—
176	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
180	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
181	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
183	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
184	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
185	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
186	C₂×Dic₅	C₂³×C₅	1	2	yes	—
187	D₂₀	—	2	4	no	—
188	D₂₀	—	2	2	no	—
189	C₄×D₅	—	2	2	no	—
190	C₄×D₅	—	2	2	no	—
191	C₄×D₅	—	2	2	no	—
192	C₄×D₅	—	2	4	no	—
193	#126	—	5	2	no	—
194	#125	—	5	1	no	—
195	#124	—	5	1	no	—
196	#123	—	5	2	no	—
197	C₂²⋊Q₈	—	5	1	no	—
198	#121	—	5	1	no	—
199	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
200	C₄×D₄	—	5	1	no	—
201	C₄×D₄	—	5	2	no	—
202	#117	—	5	2	no	—
203	C₂×C₄²	—	5	1	no	—
204	C₅²	#529	1	1	yes	characteristic, complemented, Sylow
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	5	2	no	—
207	C₂²×C₅	—	10	1	no	—
208	C₂²×C₅	—	10	1	no	—
209	C₂²×C₅	—	10	2	no	—
210	C₂²×C₅	—	10	2	no	—
211	C₂²×C₅	—	10	2	no	—
212	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
213	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	C₅×#41	1	2	yes	—
220	C₄×C₅	—	2	8	no	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	2	4	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	4	no	—
231	C₄×C₅	—	10	1	no	—
232	C₄×C₅	—	10	1	no	—
233	C₄×C₅	—	10	2	no	—
234	C₄×C₅	—	10	1	no	—
235	C₄×C₅	#709	1	2	yes	—
236	C₄×C₅	#713	1	2	yes	—
237	C₄×C₅	C₅×#41	1	2	yes	—
238	C₄×C₅	C₅×#41	1	2	yes	—
239	Dic₅	—	2	1	no	—
240	Dic₅	—	2	1	no	—
241	Dic₅	—	2	2	no	—
242	Dic₅	—	2	1	no	—
243	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
244	C₂×D₄	—	5	2	no	—
245	C₂×D₄	—	5	1	no	—
246	C₂²×C₄	—	5	1	no	—
247	C₂²×C₄	—	5	1	no	—
248	C₂²×C₄	—	5	1	no	—
249	C₂²×C₄	—	5	2	no	—
250	C₄⋊C₄	—	5	2	no	—
251	C₄⋊C₄	—	5	2	no	—
252	C₄⋊C₄	—	5	1	no	—
253	C₄⋊C₄	—	5	2	no	—
254	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
255	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
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	C₂²⋊C₄	—	5	2	no	—
261	C₄²	—	5	1	no	—
262	C₄²	—	5	1	no	—
263	C₄²	—	5	1	no	—
264	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
265	C₄²	C₂×C₅×D₅	1	2	yes	—
266	C₂×C₅	—	2	2	no	—
267	C₂×C₅	—	2	2	no	—
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	—	10	2	no	—
270	C₂×C₅	—	20	2	no	—
271	C₂×C₅	C₂×#709	1	1	yes	characteristic
272	C₂×C₅	C₂×#713	1	1	yes	characteristic
273	C₂×C₅	D₅×#41	1	1	yes	characteristic
274	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
275	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
276	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
277	D₅	—	2	2	no	—
278	D₅	—	4	2	no	—
279	C₂³	—	5	1	no	—
280	C₂³	—	5	2	no	—
281	Q₈	—	2	2	no	—
282	D₄	—	10	4	no	—
283	D₄	—	10	2	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	10	2	no	—
289	C₂×C₄	—	10	2	no	—
290	C₂×C₄	—	10	2	no	—
291	C₂×C₄	—	10	4	no	—
292	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
294	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
295	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
297	C₅	—	2	2	no	—
298	C₅	#65914	1	1	yes	characteristic, direct product factor
299	C₅	C₅×#529	1	1	yes	characteristic, complemented
300	C₂²	—	5	2	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	1	no	—
303	C₂²	—	10	2	no	—
304	C₂²	—	10	2	no	—
305	C₂²	—	10	2	no	—
306	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	1	no	—
312	C₄	—	10	2	no	—
313	C₄	—	10	1	no	—
314	C₄	C₅×#709	1	2	yes	—
315	C₄	C₅×#713	1	2	yes	—
316	C₂	—	10	2	no	—
317	C₂	—	20	2	no	—
318	C₂	C₂×C₅×#709	1	1	yes	characteristic
319	C₂	C₂×C₅×#713	1	1	yes	characteristic
320	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	C₅×#4041	C₂
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#529
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
2. C₅×#4093	C₂	semidirect
3. C₅×#4092	C₂	semidirect
4. C₅×#4044	C₂	semidirect
6. C₅×#4040	C₂	semidirect
7. C₅×#4036	C₂	semidirect
8. C₄×C₅×D₂₀	C₂	semidirect
9. C₄×C₅×D₂₀	C₂	semidirect
10. C₅×#4018	C₂	semidirect
11. C₄²×C₅×D₅	C₂	semidirect
36. #65914	C₅	direct (N × H)
62. C₄×C₅×Q₈	D₅	semidirect
63. C₄×C₅×Q₈	C₂×C₅	semidirect
68. #4093	C₂×C₅	semidirect
69. #4092	C₂×C₅	semidirect
70. #4044	C₂×C₅	semidirect
72. #4040	C₂×C₅	semidirect
73. #4036	C₂×C₅	semidirect
74. C₄×D₂₀	C₂×C₅	semidirect
75. C₄×D₂₀	C₂×C₅	semidirect
76. #4018	C₂×C₅	semidirect
77. C₄²×D₅	C₂×C₅	semidirect
199. C₄×Q₈	C₅×D₅	semidirect
204. C₅²	#529	semidirect
298. C₅	#65914	direct (N × H)
299. C₅	C₅×#529	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×#4044	2
C₅×#4041	2
C₅×#4040	2
C₅×#4036	2
C₄×C₅×D₂₀	2
C₅×#4018	2
C₄²×C₅×D₅	2
C₅×#529	5
#65914	5

Subgroups — C5×#65914 #13106734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65914 #13106734