Subgroups — C₅×#65499 #13102471

Order 1600.

704 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#683	C₂	1	2	yes	maximal, complemented
5	C₅×#3963	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅²	—	2	2	no	—
8	C₂²×C₄×C₅²	—	2	4	no	—
9	C₂²×C₄×C₅²	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
13	C₅×C₂²⋊Dic₅	—	2	2	no	—
14	C₅×C₂²⋊Dic₅	C₄	1	4	yes	complemented
15	C₅×#683	—	2	4	no	—
16	C₅×#385	—	5	1	no	maximal
17	#65499	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₂³×C₅²	—	2	2	no	—
19	C₂³×C₅²	—	2	2	no	—
20	C₂³×C₅²	C₂×C₄	1	2	yes	—
21	C₂³×C₅²	D₄	1	2	yes	—
22	C₂³×C₅²	Q₈	1	2	yes	—
23	C₂³×C₅²	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₅²	—	2	8	no	—
25	C₂×C₄×C₅²	—	2	2	no	—
26	C₂×C₄×C₅²	—	2	2	no	—
27	C₂×C₄×C₅²	D₄	1	4	yes	—
28	C₂×C₅×Dic₅	—	2	2	no	—
29	C₂×C₅×Dic₅	—	2	2	no	—
30	C₂×C₅×Dic₅	—	2	2	no	—
31	C₂×C₅×Dic₅	—	2	1	no	—
32	C₂×C₅×Dic₅	—	2	1	no	—
33	C₂×C₅×Dic₅	—	2	2	no	—
34	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
35	C₂³×C₄×C₅	—	2	2	no	—
36	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
37	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
38	C₂×C₅×C₄⋊C₄	—	5	2	no	—
39	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
40	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
41	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic
42	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic
43	C₂×#683	C₂×C₅	1	2	yes	complemented
44	C₅×#95	—	5	2	no	—
45	#3963	C₂×C₅	1	2	yes	complemented
46	C₂²×C₅²	—	2	4	no	—
47	C₂²×C₅²	—	2	2	no	—
48	C₂²×C₅²	—	2	4	no	—
49	C₂²×C₅²	—	2	2	no	—
50	C₂²×C₅²	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₅²	C₂×D₄	1	2	yes	—
54	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₅²	#41	1	2	yes	—
56	C₄×C₅²	—	2	4	no	—
57	C₅×Dic₅	—	2	4	no	—
58	C₅×Dic₅	—	4	1	no	—
59	C₅×Dic₅	—	4	1	no	—
60	C₂⁴×C₅	—	2	2	no	—
61	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
62	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	2	8	no	—
64	C₂²×C₄×C₅	—	2	16	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	5	2	no	—
71	C₂²×C₄×C₅	—	5	1	no	—
72	C₂²×C₄×C₅	—	5	1	no	—
73	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
75	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
76	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
77	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
78	C₅×C₄⋊C₄	—	10	4	no	—
79	C₅×C₂²⋊C₄	—	5	4	no	—
80	C₅×C₂²⋊C₄	—	10	2	no	—
81	C₂²⋊Dic₅	—	2	2	no	—
82	C₂²⋊Dic₅	C₄×C₅	1	4	yes	complemented
83	#683	—	2	4	no	—
84	#385	—	5	1	no	Sylow
85	C₂×C₅²	—	2	4	no	—
86	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₅²	C₄×D₄	1	2	yes	—
88	C₂×C₅²	#120	1	1	yes	characteristic
89	C₂×C₅²	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₅²	#123	1	1	yes	characteristic
91	C₂³×C₅	—	2	8	no	—
92	C₂³×C₅	—	2	8	no	—
93	C₂³×C₅	—	2	2	no	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	—	2	2	no	—
97	C₂³×C₅	—	2	4	no	—
98	C₂³×C₅	—	2	4	no	—
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	Dic₁₀	1	2	yes	—
102	C₂³×C₅	C₄×D₅	1	2	yes	—
103	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
104	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
105	C₂³×C₅	C₅×D₄	1	2	yes	—
106	C₂³×C₅	C₅×Q₈	1	2	yes	—
107	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
108	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
109	C₂×C₄×C₅	—	2	32	no	—
110	C₂×C₄×C₅	—	2	8	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	2	8	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	8	no	—
119	C₂×C₄×C₅	—	5	4	no	—
120	C₂×C₄×C₅	—	10	2	no	—
121	C₂×C₄×C₅	—	10	2	no	—
122	C₂×C₄×C₅	—	10	2	no	—
123	C₂×C₄×C₅	—	10	1	no	—
124	C₂×C₄×C₅	—	10	1	no	—
125	C₂×C₄×C₅	—	10	2	no	—
126	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
127	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
128	C₂×Dic₅	—	2	2	no	—
129	C₂×Dic₅	—	2	2	no	—
130	C₂×Dic₅	—	2	2	no	—
131	C₂×Dic₅	—	2	1	no	—
132	C₂×Dic₅	—	2	1	no	—
133	C₂×Dic₅	—	2	2	no	—
134	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	complemented
135	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	5	2	no	—
137	C₂×C₂²⋊C₄	—	5	1	no	—
138	C₂×C₂²⋊C₄	—	5	1	no	—
139	#95	—	5	2	no	—
140	C₅²	#385	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₅	—	2	8	no	—
142	C₂²×C₅	—	2	16	no	—
143	C₂²×C₅	—	2	16	no	—
144	C₂²×C₅	—	2	8	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	2	4	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	2	4	no	—
151	C₂²×C₅	—	2	4	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	8	no	—
159	C₂²×C₅	#683	1	4	yes	complemented
160	C₂²×C₅	C₅×C₄⋊C₄	1	4	yes	complemented
161	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
162	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂²×C₅	#709	1	2	yes	—
164	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
165	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
166	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
169	C₂²×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
170	C₂²×C₅	C₅×#41	1	2	yes	—
171	C₄×C₅	—	2	16	no	—
172	C₄×C₅	—	2	4	no	—
173	C₄×C₅	—	2	4	no	—
174	C₄×C₅	—	10	4	no	—
175	C₄×C₅	—	20	1	no	—
176	C₄×C₅	—	20	1	no	—
177	Dic₅	—	2	4	no	—
178	Dic₅	—	4	1	no	—
179	Dic₅	—	4	1	no	—
180	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	5	2	no	—
184	C₂²×C₄	—	5	1	no	—
185	C₂²×C₄	—	5	1	no	—
186	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
187	C₄⋊C₄	—	10	4	no	—
188	C₂²⋊C₄	—	5	4	no	—
189	C₂²⋊C₄	—	10	2	no	—
190	C₂×C₅	—	2	16	no	—
191	C₂×C₅	—	2	4	no	—
192	C₂×C₅	—	2	4	no	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	—	2	4	no	—
195	C₂×C₅	—	2	4	no	—
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	C₂×#683	1	1	yes	characteristic
199	C₂×C₅	#4070	1	2	yes	—
200	C₂×C₅	C₄×C₂²⋊D₅	1	2	yes	—
201	C₂×C₅	#4075	1	1	yes	characteristic
202	C₂×C₅	#4099	1	1	yes	characteristic
203	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₅	C₄×C₅×D₄	1	2	yes	—
205	C₂×C₅	C₅×#120	1	1	yes	characteristic
206	C₂×C₅	C₅×C₂²⋊Q₈	1	2	yes	—
207	C₂×C₅	C₅×#123	1	1	yes	characteristic
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₅×Dic₁₀	1	2	yes	—
211	C₂³	C₄×C₅×D₅	1	2	yes	—
212	C₂³	C₅×C₂²⋊D₅	1	2	yes	—
213	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	5	4	no	—
218	C₂×C₄	—	10	2	no	—
219	C₂×C₄	—	10	2	no	—
220	C₂×C₄	—	10	2	no	—
221	C₂×C₄	—	10	1	no	—
222	C₂×C₄	—	10	1	no	—
223	C₂×C₄	—	10	2	no	—
224	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
225	C₅	—	2	2	no	—
226	C₅	#65499	1	1	yes	characteristic, direct product factor
227	C₅	C₅×#385	1	1	yes	characteristic, complemented
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	C₅×#683	1	4	yes	complemented
233	C₂²	C₂×C₅×Dic₁₀	1	1	yes	characteristic
234	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
235	C₂²	C₅×#709	1	2	yes	—
236	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
237	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
238	C₄	—	2	4	no	—
239	C₄	—	10	4	no	—
240	C₄	—	20	1	no	—
241	C₄	—	20	1	no	—
242	C₂	—	2	4	no	—
243	C₂	C₂×C₅×#683	1	1	yes	characteristic
244	C₂	C₅×#4070	1	2	yes	—
245	C₂	C₄×C₅×C₂²⋊D₅	1	2	yes	—
246	C₂	C₅×#4075	1	1	yes	characteristic
247	C₂	C₅×#4099	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂⁴×C₅²	C₂²
cosocle	C₂³×C₅	C₂³×C₅
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₅×#385
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#683	C₂	semidirect
5. C₅×#3963	C₂	semidirect
14. C₅×C₂²⋊Dic₅	C₄	semidirect
17. #65499	C₅	direct (N × H)
34. C₂×C₅×Dic₅	C₂×C₄	semidirect
43. C₂×#683	C₂×C₅	semidirect
45. #3963	C₂×C₅	semidirect
50. C₂²×C₅²	C₄⋊C₄	semidirect
82. C₂²⋊Dic₅	C₄×C₅	semidirect
134. C₂×Dic₅	C₂×C₄×C₅	semidirect
140. C₅²	#385	semidirect
159. C₂²×C₅	#683	semidirect
160. C₂²×C₅	C₅×C₄⋊C₄	semidirect
226. C₅	#65499	direct (N × H)
227. C₅	C₅×#385	semidirect
232. C₂²	C₅×#683	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×#683	2
C₅×#3963	2
C₅×#385	5
#65499	5

Subgroups — C5×#65499 #13102471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65499 #13102471