Subgroups — C₅×#65921 #13106742

Order 1600.

626 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
3	C₅×#4082	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4081	C₂	1	2	yes	maximal, complemented
5	C₅×#4081	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×D₄×Dic₅	C₂	1	2	yes	maximal, complemented
7	C₅×#4025	C₂	1	2	yes	maximal, complemented
8	C₅×#4024	C₂	1	2	yes	maximal, complemented
9	C₅×#4023	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
11	C₂×C₅²×D₄	C₂²	1	2	yes	—
12	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
14	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
17	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
24	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
25	C₅×C₄⋊Dic₅	C₂²	1	2	yes	complemented
26	C₅×#683	C₂²	1	2	yes	complemented
27	C₅×#683	C₂²	1	2	yes	complemented
28	C₅×#683	C₂²	1	2	yes	complemented
29	C₄×C₅×Dic₅	C₂²	1	2	yes	complemented
30	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
31	C₅×#551	—	5	1	no	maximal
32	#65921	C₅	1	1	yes	maximal, characteristic, direct product factor
33	C₂³×C₅²	—	2	4	no	—
34	C₂³×C₅²	—	4	1	no	—
35	C₂³×C₅²	C₂³	1	1	yes	characteristic
36	C₂³×C₅²	C₂³	1	1	yes	characteristic
37	C₂³×C₅²	C₂³	1	2	yes	—
38	C₅²×D₄	—	2	4	no	—
39	C₅²×D₄	—	4	1	no	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	2	yes	—
42	C₂×C₅×Dic₅	—	2	4	no	—
43	C₂×C₅×Dic₅	—	4	2	no	—
44	C₂×C₅×Dic₅	—	4	1	no	—
45	C₂×C₅×Dic₅	C₂³	1	2	yes	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₅×Dic₁₀	—	4	1	no	—
50	C₅×#126	—	5	2	no	—
51	C₅×#124	—	5	1	no	—
52	C₅×#123	—	5	2	no	—
53	C₅×#123	—	5	1	no	—
54	C₅×C₂²⋊Q₈	—	5	2	no	—
55	C₅×#120	—	2	2	no	—
56	C₅×#120	D₅	1	1	yes	characteristic
57	C₅×#120	C₂×C₅	1	1	yes	characteristic
58	C₄×C₅×D₄	—	5	2	no	—
59	C₅×#117	—	5	2	no	—
60	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
61	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	complemented
62	#4082	C₂×C₅	1	1	yes	characteristic, complemented
63	#4081	C₂×C₅	1	1	yes	characteristic, complemented
64	#4081	C₂×C₅	1	2	yes	complemented
65	D₄×Dic₅	C₂×C₅	1	2	yes	complemented
66	#4025	C₂×C₅	1	2	yes	complemented
67	#4024	C₂×C₅	1	2	yes	complemented
68	#4023	C₂×C₅	1	2	yes	complemented
69	C₂²×C₅²	—	2	2	no	—
70	C₂²×C₅²	—	2	2	no	—
71	C₂²×C₅²	—	2	2	no	—
72	C₂²×C₅²	—	2	2	no	—
73	C₂²×C₅²	—	2	1	no	—
74	C₂²×C₅²	—	2	1	no	—
75	C₂²×C₅²	—	4	4	no	—
76	C₂²×C₅²	#41	1	4	yes	—
77	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
78	C₄×C₅²	—	2	2	no	—
79	C₄×C₅²	—	2	1	no	—
80	C₅×Dic₅	—	2	2	no	—
81	C₅×Dic₅	—	2	2	no	—
82	C₅×Dic₅	—	2	2	no	—
83	C₅×Dic₅	—	2	2	no	—
84	C₂⁴×C₅	—	2	2	no	—
85	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
86	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₅×Q₈	—	5	1	no	—
88	C₂×C₅×D₄	—	2	4	no	—
89	C₂×C₅×D₄	—	2	2	no	—
90	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
91	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
92	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
93	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
94	C₂²×C₄×C₅	—	5	2	no	—
95	C₂²×C₄×C₅	—	5	2	no	—
96	C₂²×C₄×C₅	—	5	1	no	—
97	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
98	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
99	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
100	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
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×C₂²⋊C₄	—	2	4	no	—
106	C₅×C₂²⋊C₄	—	2	2	no	—
107	C₅×C₂²⋊C₄	—	5	2	no	—
108	C₅×C₂²⋊C₄	—	5	2	no	—
109	C₅×C₂²⋊C₄	—	5	2	no	—
110	C₅×C₂²⋊C₄	—	5	2	no	—
111	C₅×C₂²⋊C₄	—	5	2	no	—
112	C₅×C₂²⋊C₄	—	5	1	no	—
113	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	—
114	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
115	C₅×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
116	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
117	C₄²×C₅	—	5	1	no	—
118	C₄²×C₅	—	5	2	no	—
119	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
120	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
121	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
122	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
123	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	complemented
124	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
125	C₄⋊Dic₅	C₂²×C₅	1	2	yes	complemented
126	#683	C₂²×C₅	1	2	yes	complemented
127	#683	C₂²×C₅	1	2	yes	complemented
128	#683	C₂²×C₅	1	2	yes	complemented
129	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
130	C₄×Dic₅	C₂²×C₅	1	2	yes	complemented
131	#551	—	5	1	no	Sylow
132	C₂×C₅²	—	2	4	no	—
133	C₂×C₅²	—	4	1	no	—
134	C₂×C₅²	—	4	1	no	—
135	C₂×C₅²	C₂×#41	1	2	yes	—
136	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂³×C₅	—	2	4	no	—
138	C₂³×C₅	—	2	4	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	2	4	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	—	4	8	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	—	4	1	no	—
145	C₂³×C₅	—	4	4	no	—
146	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
147	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
148	C₂³×C₅	C₂²×D₅	1	2	yes	—
149	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
150	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
151	C₂³×C₅	C₂³×C₅	1	2	yes	—
152	C₅×Q₈	—	20	1	no	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	4	8	no	—
156	C₅×D₄	—	4	1	no	—
157	C₅×D₄	—	4	1	no	—
158	C₅×D₄	—	4	4	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	5	2	no	—
162	C₂×C₄×C₅	—	5	2	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	5	2	no	—
165	C₂×C₄×C₅	—	10	4	no	—
166	C₂×C₄×C₅	—	20	1	no	—
167	C₂×C₄×C₅	—	20	2	no	—
168	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
169	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
171	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
172	C₂×Dic₅	—	2	4	no	—
173	C₂×Dic₅	—	4	2	no	—
174	C₂×Dic₅	—	4	1	no	—
175	C₂×Dic₅	C₂³×C₅	1	2	yes	—
176	C₂×Dic₅	C₂³×C₅	1	2	yes	—
177	C₂×Dic₅	C₂³×C₅	1	2	yes	—
178	C₂×Dic₅	C₂³×C₅	1	2	yes	—
179	Dic₁₀	—	4	1	no	—
180	#126	—	5	2	no	—
181	#124	—	5	1	no	—
182	#123	—	5	2	no	—
183	#123	—	5	1	no	—
184	C₂²⋊Q₈	—	5	2	no	—
185	#120	C₅×D₅	1	1	yes	characteristic
186	C₄×D₄	—	5	2	no	—
187	#117	—	5	2	no	—
188	C₂×C₂²⋊C₄	—	5	2	no	—
189	C₅²	#551	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	8	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	4	4	no	—
206	C₂²×C₅	—	4	4	no	—
207	C₂²×C₅	—	4	4	no	—
208	C₂²×C₅	—	4	4	no	—
209	C₂²×C₅	—	4	2	no	—
210	C₂²×C₅	—	4	4	no	—
211	C₂²×C₅	—	4	4	no	—
212	C₂²×C₅	—	4	16	no	—
213	C₂²×C₅	#711	1	4	yes	—
214	C₂²×C₅	C₅×#41	1	4	yes	—
215	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
216	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	4	4	no	—
222	C₄×C₅	—	4	2	no	—
223	C₄×C₅	—	10	2	no	—
224	C₄×C₅	—	10	2	no	—
225	C₄×C₅	—	10	2	no	—
226	C₄×C₅	—	10	2	no	—
227	Dic₅	—	2	2	no	—
228	Dic₅	—	2	2	no	—
229	Dic₅	—	2	2	no	—
230	Dic₅	—	2	2	no	—
231	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
232	C₂×Q₈	—	5	1	no	—
233	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
234	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
235	C₂²×C₄	—	5	2	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₄⋊C₄	—	5	2	no	—
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	5	2	no	—
242	C₂²⋊C₄	—	5	2	no	—
243	C₂²⋊C₄	—	5	2	no	—
244	C₂²⋊C₄	—	5	2	no	—
245	C₂²⋊C₄	—	5	2	no	—
246	C₂²⋊C₄	—	5	2	no	—
247	C₂²⋊C₄	—	5	1	no	—
248	C₂²⋊C₄	C₂×C₅×D₅	1	2	yes	—
249	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
250	C₄²	—	5	1	no	—
251	C₄²	—	5	2	no	—
252	C₂×C₅	—	2	4	no	—
253	C₂×C₅	—	2	4	no	—
254	C₂×C₅	—	2	4	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	4	8	no	—
257	C₂×C₅	—	4	4	no	—
258	C₂×C₅	—	4	1	no	—
259	C₂×C₅	—	4	1	no	—
260	C₂×C₅	—	4	1	no	—
261	C₂×C₅	—	4	1	no	—
262	C₂×C₅	—	8	2	no	—
263	C₂×C₅	C₂×#711	1	2	yes	—
264	C₂×C₅	#4144	1	1	yes	characteristic
265	C₂×C₅	C₂×C₅×#41	1	2	yes	—
266	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
267	C₂³	—	2	4	no	—
268	C₂³	—	4	1	no	—
269	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
270	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
271	C₂³	C₂²×C₅×D₅	1	2	yes	—
272	Q₈	—	20	1	no	—
273	D₄	—	2	4	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	5	2	no	—
277	C₂×C₄	—	5	2	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	10	4	no	—
280	C₂×C₄	—	20	1	no	—
281	C₂×C₄	—	20	2	no	—
282	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
283	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
284	C₅	—	2	2	no	—
285	C₅	#65921	1	1	yes	characteristic, direct product factor
286	C₅	C₅×#551	1	1	yes	characteristic, complemented
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	4	4	no	—
294	C₂²	C₅×#711	1	4	yes	—
295	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	10	2	no	—
299	C₄	—	10	2	no	—
300	C₄	—	10	2	no	—
301	C₄	—	10	2	no	—
302	C₂	—	2	4	no	—
303	C₂	—	4	1	no	—
304	C₂	—	4	1	no	—
305	C₂	C₂×C₅×#711	1	2	yes	—
306	C₂	C₅×#4144	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting, prime radical	C₅²×#120	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₅×#120	D₅
hypocenter	C₅	C₅×#551
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#120

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
3. C₅×#4082	C₂	semidirect
4. C₅×#4081	C₂	semidirect
5. C₅×#4081	C₂	semidirect
6. C₅×D₄×Dic₅	C₂	semidirect
7. C₅×#4025	C₂	semidirect
8. C₅×#4024	C₂	semidirect
9. C₅×#4023	C₂	semidirect
15. C₂²×C₅×Dic₅	C₂²	semidirect
16. C₂×C₅×Dic₁₀	C₂²	semidirect
23. C₅×C₂²⋊Dic₅	C₂²	semidirect
24. C₅×C₂²⋊Dic₅	C₂²	semidirect
25. C₅×C₄⋊Dic₅	C₂²	semidirect
26. C₅×#683	C₂²	semidirect
27. C₅×#683	C₂²	semidirect
28. C₅×#683	C₂²	semidirect
29. C₄×C₅×Dic₅	C₂²	semidirect
30. C₄×C₅×Dic₅	C₂²	semidirect
32. #65921	C₅	direct (N × H)
61. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
62. #4082	C₂×C₅	semidirect
63. #4081	C₂×C₅	semidirect
64. #4081	C₂×C₅	semidirect
65. D₄×Dic₅	C₂×C₅	semidirect
66. #4025	C₂×C₅	semidirect
67. #4024	C₂×C₅	semidirect
68. #4023	C₂×C₅	semidirect
99. C₂²×Dic₅	C₂²×C₅	semidirect
100. C₂×Dic₁₀	C₂²×C₅	semidirect
123. C₂²⋊Dic₅	C₂²×C₅	semidirect
124. C₂²⋊Dic₅	C₂²×C₅	semidirect
125. C₄⋊Dic₅	C₂²×C₅	semidirect
126. #683	C₂²×C₅	semidirect
127. #683	C₂²×C₅	semidirect
128. #683	C₂²×C₅	semidirect
129. C₄×Dic₅	C₂²×C₅	semidirect
130. C₄×Dic₅	C₂²×C₅	semidirect
189. C₅²	#551	semidirect
285. C₅	#65921	direct (N × H)
286. C₅	C₅×#551	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#120	2
C₂×C₅×C₂²⋊Dic₅	2
C₅×#4082	2
C₅×#4081	2
C₅×D₄×Dic₅	2
C₅×#4025	2
C₅×#4024	2
C₅×#4023	2
C₅×#551	5
#65921	5

Subgroups — C5×#65921 #13106742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65921 #13106742