Subgroups — C₂×C₅×#4025 #13106637

Order 1600.

624 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₂²⋊C₄	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×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4025	C₂	1	8	yes	maximal, direct product factor
9	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₅²×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
18	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
19	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
20	C₅×#683	C₂²	1	4	yes	complemented
21	C₅×#683	C₂²	1	4	yes	complemented
22	C₄×C₅×Dic₅	C₂²	1	4	yes	complemented
23	C₂×C₅×#126	—	5	1	no	maximal
24	C₂×#4025	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	—	2	2	no	—
26	C₂³×C₅²	—	2	2	no	—
27	C₂³×C₅²	—	2	2	no	—
28	C₂³×C₅²	C₂³	1	2	yes	—
29	C₂³×C₅²	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅²	—	2	2	no	—
31	C₂×C₄×C₅²	—	2	2	no	—
32	C₂×C₄×C₅²	C₂³	1	2	yes	—
33	C₂×C₄×C₅²	C₂³	1	2	yes	—
34	C₂×C₅×Dic₅	—	2	2	no	—
35	C₂×C₅×Dic₅	—	2	2	no	—
36	C₂×C₅×Dic₅	—	2	2	no	—
37	C₂×C₅×Dic₅	—	2	2	no	—
38	C₂×C₅×Dic₅	C₂³	1	2	yes	—
39	C₂×C₅×Dic₅	C₂³	1	2	yes	—
40	C₂×C₅×Dic₅	C₂³	1	2	yes	—
41	C₂×C₅×Dic₅	C₂³	1	2	yes	—
42	C₅×#126	—	5	8	no	—
43	C₂×C₅×C₄⋊C₄	—	5	1	no	—
44	C₂×C₅×C₄⋊C₄	—	5	1	no	—
45	C₂×C₅×C₄⋊C₄	—	5	1	no	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
48	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
49	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic
50	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
51	C₂×C₄²×C₅	—	5	1	no	—
52	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic
53	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic
54	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
58	#4025	C₂×C₅	1	8	yes	direct product factor
59	C₂²×C₅²	—	2	2	no	—
60	C₂²×C₅²	—	2	2	no	—
61	C₂²×C₅²	—	2	2	no	—
62	C₂²×C₅²	—	4	4	no	—
63	C₂²×C₅²	#41	1	2	yes	—
64	C₂²×C₅²	#41	1	2	yes	—
65	C₂²×C₅²	#41	1	2	yes	—
66	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
67	C₄×C₅²	—	2	2	no	—
68	C₄×C₅²	—	2	2	no	—
69	C₅×Dic₅	—	2	2	no	—
70	C₅×Dic₅	—	2	2	no	—
71	C₅×Dic₅	—	2	2	no	—
72	C₅×Dic₅	—	2	2	no	—
73	C₂⁴×C₅	—	2	2	no	—
74	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
75	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	—	5	1	no	—
79	C₂²×C₄×C₅	—	5	1	no	—
80	C₂²×C₄×C₅	—	5	1	no	—
81	C₂²×C₄×C₅	—	5	1	no	—
82	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
86	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
87	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
88	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
89	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	5	4	no	—
91	C₅×C₄⋊C₄	—	5	4	no	—
92	C₅×C₄⋊C₄	—	5	4	no	—
93	C₅×C₂²⋊C₄	—	2	8	no	—
94	C₅×C₂²⋊C₄	—	5	4	no	—
95	C₅×C₂²⋊C₄	—	5	4	no	—
96	C₅×C₂²⋊C₄	C₂×D₅	1	4	yes	—
97	C₅×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
98	C₄²×C₅	—	5	4	no	—
99	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
100	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
101	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
102	#683	C₂²×C₅	1	4	yes	complemented
103	#683	C₂²×C₅	1	4	yes	complemented
104	C₄×Dic₅	C₂²×C₅	1	4	yes	complemented
105	C₂×#126	—	5	1	no	Sylow
106	C₂×C₅²	—	4	2	no	—
107	C₂×C₅²	#126	1	4	yes	complemented
108	C₂×C₅²	C₂×#41	1	1	yes	characteristic
109	C₂×C₅²	C₂×#41	1	1	yes	characteristic
110	C₂×C₅²	C₂×#41	1	1	yes	characteristic
111	C₂³×C₅	—	2	8	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	2	4	no	—
120	C₂³×C₅	—	4	4	no	—
121	C₂³×C₅	—	4	4	no	—
122	C₂³×C₅	C₂²×D₅	1	2	yes	—
123	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
124	C₂³×C₅	C₂³×C₅	1	2	yes	—
125	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	5	2	no	—
135	C₂×C₄×C₅	—	5	2	no	—
136	C₂×C₄×C₅	—	5	2	no	—
137	C₂×C₄×C₅	—	5	2	no	—
138	C₂×C₄×C₅	—	10	2	no	—
139	C₂×C₄×C₅	—	10	2	no	—
140	C₂×C₄×C₅	—	10	2	no	—
141	C₂×C₄×C₅	—	10	2	no	—
142	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
143	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
144	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
145	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
146	C₂×Dic₅	—	2	2	no	—
147	C₂×Dic₅	—	2	2	no	—
148	C₂×Dic₅	—	2	2	no	—
149	C₂×Dic₅	—	2	2	no	—
150	C₂×Dic₅	C₂³×C₅	1	2	yes	—
151	C₂×Dic₅	C₂³×C₅	1	2	yes	—
152	C₂×Dic₅	C₂³×C₅	1	2	yes	—
153	C₂×Dic₅	C₂³×C₅	1	2	yes	—
154	#126	—	5	8	no	—
155	C₂×C₄⋊C₄	—	5	1	no	—
156	C₂×C₄⋊C₄	—	5	1	no	—
157	C₂×C₄⋊C₄	—	5	1	no	—
158	C₂×C₂²⋊C₄	—	5	1	no	—
159	C₂×C₂²⋊C₄	—	5	1	no	—
160	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic
161	C₂×C₄²	—	5	1	no	—
162	C₅²	C₂×#126	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₅	—	2	8	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	4	no	—
171	C₂²×C₅	—	2	4	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	4	4	no	—
175	C₂²×C₅	—	4	4	no	—
176	C₂²×C₅	—	4	16	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	4	4	no	—
179	C₂²×C₅	#709	1	2	yes	—
180	C₂²×C₅	#711	1	2	yes	—
181	C₂²×C₅	#711	1	2	yes	—
182	C₂²×C₅	C₅×#41	1	2	yes	—
183	C₂²×C₅	C₅×#41	1	2	yes	—
184	C₂²×C₅	C₅×#41	1	2	yes	—
185	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
186	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	4	no	—
192	C₄×C₅	—	4	4	no	—
193	C₄×C₅	—	10	2	no	—
194	C₄×C₅	—	10	2	no	—
195	C₄×C₅	—	10	2	no	—
196	C₄×C₅	—	10	2	no	—
197	Dic₅	—	2	2	no	—
198	Dic₅	—	2	2	no	—
199	Dic₅	—	2	2	no	—
200	Dic₅	—	2	2	no	—
201	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
202	C₂²×C₄	—	5	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
208	C₄⋊C₄	—	5	4	no	—
209	C₄⋊C₄	—	5	4	no	—
210	C₄⋊C₄	—	5	4	no	—
211	C₂²⋊C₄	—	5	4	no	—
212	C₂²⋊C₄	—	5	4	no	—
213	C₂²⋊C₄	C₂×C₅×D₅	1	4	yes	—
214	C₄²	—	5	4	no	—
215	C₂×C₅	—	2	2	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	2	8	no	—
219	C₂×C₅	—	4	8	no	—
220	C₂×C₅	—	4	2	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	#4025	1	4	yes	direct product factor
223	C₂×C₅	C₅×#126	1	4	yes	complemented
224	C₂×C₅	C₂×#709	1	1	yes	characteristic
225	C₂×C₅	C₂×#711	1	1	yes	characteristic
226	C₂×C₅	C₂×#711	1	1	yes	characteristic
227	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
228	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
229	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₂²×C₅×D₅	1	2	yes	—
234	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	5	2	no	—
239	C₂×C₄	—	5	2	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	—	10	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
246	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
247	C₅	—	2	2	no	—
248	C₅	C₂×#4025	1	1	yes	characteristic, direct product factor
249	C₅	C₂×C₅×#126	1	1	yes	characteristic, complemented
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₅×#709	1	2	yes	—
255	C₂²	C₅×#711	1	2	yes	—
256	C₂²	C₅×#711	1	2	yes	—
257	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	10	2	no	—
261	C₄	—	10	2	no	—
262	C₄	—	10	2	no	—
263	C₄	—	10	2	no	—
264	C₂	—	4	2	no	—
265	C₂	C₅×#4025	1	4	yes	direct product factor
266	C₂	C₂×C₅×#709	1	1	yes	characteristic
267	C₂	C₂×C₅×#711	1	1	yes	characteristic
268	C₂	C₂×C₅×#711	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂⁴×C₅²	C₂²
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₂×C₅×#126
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
5. C₂×C₅×#683	C₂	semidirect
6. C₂×C₅×#683	C₂	semidirect
7. C₂×C₄×C₅×Dic₅	C₂	semidirect
8. C₅×#4025	C₂	direct (N × H)
19. C₅×C₄⋊Dic₅	C₂²	semidirect
20. C₅×#683	C₂²	semidirect
21. C₅×#683	C₂²	semidirect
22. C₄×C₅×Dic₅	C₂²	semidirect
24. C₂×#4025	C₅	direct (N × H)
54. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
55. C₂×#683	C₂×C₅	semidirect
56. C₂×#683	C₂×C₅	semidirect
57. C₂×C₄×Dic₅	C₂×C₅	semidirect
58. #4025	C₂×C₅	direct (N × H)
101. C₄⋊Dic₅	C₂²×C₅	semidirect
102. #683	C₂²×C₅	semidirect
103. #683	C₂²×C₅	semidirect
104. C₄×Dic₅	C₂²×C₅	semidirect
107. C₂×C₅²	#126	semidirect
162. C₅²	C₂×#126	semidirect
222. C₂×C₅	#4025	direct (N × H)
223. C₂×C₅	C₅×#126	semidirect
248. C₅	C₂×#4025	direct (N × H)
249. C₅	C₂×C₅×#126	semidirect
265. C₂	C₅×#4025	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#683	2
C₂×C₄×C₅×Dic₅	2
C₅×#4025	2
C₂×C₅×#126	5
C₂×#4025	5

Subgroups — C2×C5×#4025 #13106637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4025 #13106637