Subgroups — C₂×C₅×#4084 #13106956

Order 1600.

1088 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
5	C₅×#4084	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₅²	C₂²	1	2	yes	—
8	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅²×D₄	—	2	4	no	—
10	C₂×C₅²×D₄	C₂²	1	4	yes	complemented
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊D₅	—	2	8	no	—
13	C₂×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
14	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	—	2	4	no	—
17	C₂×C₄×C₅×D₅	C₂²	1	4	yes	complemented
18	C₅×C₂²⋊Dic₅	C₂²	1	8	yes	complemented
19	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
20	C₂×C₅×#121	—	5	1	no	maximal
21	C₂×#4084	C₅	1	1	yes	maximal, characteristic, direct product factor
22	C₂³×C₅²	—	2	4	no	—
23	C₂³×C₅²	—	2	4	no	—
24	C₂³×C₅²	—	2	4	no	—
25	C₂³×C₅²	C₂³	1	4	yes	—
26	C₂³×C₅²	C₂³	1	1	yes	characteristic
27	C₂²×C₅×D₅	—	2	2	no	—
28	C₂²×C₅×D₅	—	2	2	no	—
29	C₂²×C₅×D₅	D₄	1	4	yes	complemented
30	C₂²×C₅×D₅	C₂³	1	2	yes	—
31	C₅²×D₄	—	2	8	no	—
32	C₂×C₄×C₅²	D₄	1	4	yes	complemented
33	C₂×C₄×C₅²	C₂³	1	2	yes	—
34	C₅×C₂²⋊D₅	—	2	16	no	—
35	C₂×C₅×Dic₅	—	2	2	no	—
36	C₂×C₅×Dic₅	—	2	4	no	—
37	C₂×C₅×Dic₅	C₂³	1	2	yes	—
38	C₂×C₅×Dic₅	C₂³	1	4	yes	—
39	C₄×C₅×D₅	—	2	8	no	—
40	C₂²×C₅×D₄	—	2	2	no	—
41	C₂²×C₅×D₄	—	5	2	no	—
42	C₂²×C₅×D₄	D₅	1	1	yes	characteristic, complemented
43	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂³×C₄×C₅	—	5	1	no	—
45	C₂²×C₂²⋊D₅	C₂×C₅	1	2	yes	complemented
46	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic, complemented
47	C₅×#121	—	5	8	no	—
48	C₂×C₅×C₄⋊C₄	—	5	1	no	—
49	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
50	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	complemented
51	#4084	C₂×C₅	1	8	yes	direct product factor
52	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂²×C₅²	—	2	4	no	—
54	C₂²×C₅²	—	2	4	no	—
55	C₂²×C₅²	—	2	4	no	—
56	C₂²×C₅²	—	4	8	no	—
57	C₂²×C₅²	C₂×D₄	1	2	yes	—
58	C₂²×C₅²	C₂×D₄	1	2	yes	—
59	C₂²×C₅²	#41	1	2	yes	—
60	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
61	C₂×C₅×D₅	—	2	8	no	—
62	C₂×C₅×D₅	—	2	2	no	—
63	C₂×C₅×D₅	—	2	2	no	—
64	C₂×C₅×D₅	C₂×D₄	1	4	yes	complemented
65	C₄×C₅²	C₂×D₄	1	4	yes	complemented
66	C₅×Dic₅	—	2	2	no	—
67	C₅×Dic₅	—	2	4	no	—
68	C₂⁴×C₅	—	2	4	no	—
69	C₂⁴×C₅	—	5	1	no	—
70	C₂⁴×C₅	C₂×D₅	1	2	yes	—
71	C₂⁴×C₅	C₂²×C₅	1	2	yes	—
72	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₅×D₄	—	2	16	no	—
74	C₂×C₅×D₄	—	2	4	no	—
75	C₂×C₅×D₄	—	2	4	no	—
76	C₂×C₅×D₄	—	2	8	no	—
77	C₂×C₅×D₄	—	5	8	no	—
78	C₂×C₅×D₄	—	10	8	no	—
79	C₂×C₅×D₄	C₂×D₅	1	4	yes	complemented
80	C₂×C₅×D₄	C₂²×C₅	1	4	yes	complemented
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	5	1	no	—
83	C₂²×C₄×C₅	—	5	2	no	—
84	C₂²×C₄×C₅	—	5	4	no	—
85	C₂²×C₄×C₅	—	10	4	no	—
86	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₂²⋊D₅	—	2	8	no	—
89	C₂×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
90	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
91	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
92	C₂×C₄×D₅	—	2	4	no	—
93	C₂×C₄×D₅	C₂²×C₅	1	4	yes	complemented
94	C₅×C₄⋊C₄	—	5	4	no	—
95	C₅×C₂²⋊C₄	—	5	8	no	—
96	C₂²⋊Dic₅	C₂²×C₅	1	8	yes	complemented
97	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
98	C₂×#121	—	5	1	no	Sylow
99	C₂×C₅²	—	4	4	no	—
100	C₂×C₅²	#121	1	4	yes	complemented
101	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₅²	C₂×#41	1	1	yes	characteristic
104	C₅×D₅	—	2	4	no	—
105	C₂³×C₅	—	2	16	no	—
106	C₂³×C₅	—	2	4	no	—
107	C₂³×C₅	—	2	4	no	—
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	—	2	4	no	—
110	C₂³×C₅	—	2	4	no	—
111	C₂³×C₅	—	2	4	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	8	no	—
114	C₂³×C₅	—	4	8	no	—
115	C₂³×C₅	—	4	8	no	—
116	C₂³×C₅	—	5	4	no	—
117	C₂³×C₅	—	5	2	no	—
118	C₂³×C₅	—	10	2	no	—
119	C₂³×C₅	—	10	2	no	—
120	C₂³×C₅	C₂²×D₅	1	4	yes	—
121	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
122	C₂³×C₅	C₂³×C₅	1	4	yes	—
123	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
124	C₂²×D₅	—	2	2	no	—
125	C₂²×D₅	—	2	2	no	—
126	C₂²×D₅	C₅×D₄	1	4	yes	complemented
127	C₂²×D₅	C₂³×C₅	1	2	yes	—
128	C₅×D₄	—	2	32	no	—
129	C₅×D₄	—	2	8	no	—
130	C₅×D₄	—	2	8	no	—
131	C₅×D₄	—	10	16	no	—
132	C₂×C₄×C₅	—	2	8	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	5	2	no	—
135	C₂×C₄×C₅	—	5	4	no	—
136	C₂×C₄×C₅	—	10	4	no	—
137	C₂×C₄×C₅	—	10	2	no	—
138	C₂×C₄×C₅	—	10	8	no	—
139	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
140	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
141	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
142	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
143	C₂²⋊D₅	—	2	16	no	—
144	C₂×Dic₅	—	2	2	no	—
145	C₂×Dic₅	—	2	4	no	—
146	C₂×Dic₅	C₂³×C₅	1	4	yes	—
147	C₂×Dic₅	C₂³×C₅	1	2	yes	—
148	C₄×D₅	—	2	8	no	—
149	C₂²×D₄	—	5	2	no	—
150	C₂²×D₄	C₅×D₅	1	1	yes	characteristic, complemented
151	C₂³×C₄	—	5	1	no	—
152	#121	—	5	8	no	—
153	C₂×C₄⋊C₄	—	5	1	no	—
154	C₂×C₂²⋊C₄	—	5	2	no	—
155	C₅²	C₂×#121	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₅	—	2	16	no	—
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	4	no	—
159	C₂²×C₅	—	2	4	no	—
160	C₂²×C₅	—	2	4	no	—
161	C₂²×C₅	—	2	4	no	—
162	C₂²×C₅	—	2	4	no	—
163	C₂²×C₅	—	2	4	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	4	8	no	—
168	C₂²×C₅	—	4	8	no	—
169	C₂²×C₅	—	4	32	no	—
170	C₂²×C₅	—	4	8	no	—
171	C₂²×C₅	—	4	8	no	—
172	C₂²×C₅	—	5	4	no	—
173	C₂²×C₅	—	10	2	no	—
174	C₂²×C₅	—	10	2	no	—
175	C₂²×C₅	—	10	8	no	—
176	C₂²×C₅	D₄×D₅	1	2	yes	—
177	C₂²×C₅	#711	1	2	yes	—
178	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
179	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
180	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
181	C₂²×C₅	C₅×#41	1	2	yes	—
182	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
183	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
184	C₂×D₅	—	2	8	no	—
185	C₂×D₅	—	2	2	no	—
186	C₂×D₅	—	2	2	no	—
187	C₂×D₅	C₂×C₅×D₄	1	4	yes	complemented
188	C₄×C₅	—	2	8	no	—
189	C₄×C₅	—	10	4	no	—
190	C₄×C₅	—	10	2	no	—
191	C₄×C₅	C₂×C₂²⋊D₅	1	4	yes	complemented
192	C₄×C₅	C₂×C₅×D₄	1	4	yes	complemented
193	Dic₅	—	2	2	no	—
194	Dic₅	—	2	4	no	—
195	C₂⁴	—	5	1	no	—
196	C₂⁴	C₂×C₅×D₅	1	2	yes	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	5	8	no	—
199	C₂×D₄	—	10	8	no	—
200	C₂×D₄	C₂×C₅×D₅	1	4	yes	complemented
201	C₂²×C₄	—	5	1	no	—
202	C₂²×C₄	—	5	2	no	—
203	C₂²×C₄	—	5	4	no	—
204	C₂²×C₄	—	10	4	no	—
205	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
206	C₄⋊C₄	—	5	4	no	—
207	C₂²⋊C₄	—	5	8	no	—
208	C₂×C₅	—	2	8	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	4	16	no	—
213	C₂×C₅	—	4	4	no	—
214	C₂×C₅	—	4	4	no	—
215	C₂×C₅	—	10	4	no	—
216	C₂×C₅	#4084	1	4	yes	direct product factor
217	C₂×C₅	C₅×#121	1	4	yes	complemented
218	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
219	C₂×C₅	C₂×#711	1	1	yes	characteristic
220	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
221	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
222	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
223	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
224	D₅	—	2	4	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	5	4	no	—
229	C₂³	—	5	2	no	—
230	C₂³	—	10	2	no	—
231	C₂³	—	10	2	no	—
232	C₂³	C₂²×C₅×D₅	1	4	yes	—
233	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
234	D₄	—	2	8	no	—
235	D₄	—	10	16	no	—
236	C₂×C₄	—	5	2	no	—
237	C₂×C₄	—	5	4	no	—
238	C₂×C₄	—	10	4	no	—
239	C₂×C₄	—	10	2	no	—
240	C₂×C₄	—	10	8	no	—
241	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
242	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
243	C₅	—	2	2	no	—
244	C₅	C₂×#4084	1	1	yes	characteristic, direct product factor
245	C₅	C₂×C₅×#121	1	1	yes	characteristic, complemented
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	5	4	no	—
251	C₂²	—	10	8	no	—
252	C₂²	—	10	2	no	—
253	C₂²	—	10	2	no	—
254	C₂²	C₅×D₄×D₅	1	2	yes	—
255	C₂²	C₅×#711	1	2	yes	—
256	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
257	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
258	C₄	—	10	4	no	—
259	C₄	—	10	2	no	—
260	C₄	C₂×C₅×C₂²⋊D₅	1	4	yes	complemented
261	C₂	—	4	4	no	—
262	C₂	—	10	4	no	—
263	C₂	C₅×#4084	1	4	yes	direct product factor
264	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
265	C₂	C₂×C₅×#711	1	1	yes	characteristic
266	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂²×C₅²×D₄	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₂×C₅×#121
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×D₄	C₂	semidirect
2. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
3. C₂²×C₄×C₅×D₅	C₂	semidirect
4. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
5. C₅×#4084	C₂	direct (N × H)
6. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
10. C₂×C₅²×D₄	C₂²	semidirect
13. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
17. C₂×C₄×C₅×D₅	C₂²	semidirect
18. C₅×C₂²⋊Dic₅	C₂²	semidirect
19. C₅×C₄⋊Dic₅	C₂²	semidirect
21. C₂×#4084	C₅	direct (N × H)
29. C₂²×C₅×D₅	D₄	semidirect
32. C₂×C₄×C₅²	D₄	semidirect
42. C₂²×C₅×D₄	D₅	semidirect
43. C₂²×C₅×D₄	C₂×C₅	semidirect
45. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
46. C₂²×C₄×D₅	C₂×C₅	semidirect
50. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
51. #4084	C₂×C₅	direct (N × H)
52. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
64. C₂×C₅×D₅	C₂×D₄	semidirect
65. C₄×C₅²	C₂×D₄	semidirect
79. C₂×C₅×D₄	C₂×D₅	semidirect
80. C₂×C₅×D₄	C₂²×C₅	semidirect
89. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
93. C₂×C₄×D₅	C₂²×C₅	semidirect
96. C₂²⋊Dic₅	C₂²×C₅	semidirect
97. C₄⋊Dic₅	C₂²×C₅	semidirect
100. C₂×C₅²	#121	semidirect
126. C₂²×D₅	C₅×D₄	semidirect
139. C₂×C₄×C₅	C₂²⋊D₅	semidirect
140. C₂×C₄×C₅	C₅×D₄	semidirect
150. C₂²×D₄	C₅×D₅	semidirect
155. C₅²	C₂×#121	semidirect
187. C₂×D₅	C₂×C₅×D₄	semidirect
191. C₄×C₅	C₂×C₂²⋊D₅	semidirect
192. C₄×C₅	C₂×C₅×D₄	semidirect
200. C₂×D₄	C₂×C₅×D₅	semidirect
216. C₂×C₅	#4084	direct (N × H)
217. C₂×C₅	C₅×#121	semidirect
241. C₂×C₄	C₅×C₂²⋊D₅	semidirect
244. C₅	C₂×#4084	direct (N × H)
245. C₅	C₂×C₅×#121	semidirect
260. C₄	C₂×C₅×C₂²⋊D₅	semidirect
263. C₂	C₅×#4084	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅²×D₄	2
C₂²×C₅×C₂²⋊D₅	2
C₂²×C₄×C₅×D₅	2
C₂×C₅×C₂²⋊Dic₅	2
C₅×#4084	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#121	5
C₂×#4084	5

Subgroups — C2×C5×#4084 #13106956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4084 #13106956