Subgroups — C₂×C₅×D₄×Dic₅ #13106946

Order 1600.

1088 conjugacy classes of subgroups.

The subgroup lattice has 236 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
2	C₂³×C₅×Dic₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
4	C₅×D₄×Dic₅	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₅²	C₂²	1	2	yes	—
8	C₂×C₅²×D₄	C₄	1	8	yes	complemented
9	C₂×C₅²×D₄	C₂²	1	4	yes	—
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Dic₅	—	2	8	no	—
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	8	yes	complemented
16	C₅×C₂²⋊Dic₅	C₂²	1	8	yes	complemented
17	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
18	C₄×C₅×Dic₅	C₂²	1	4	yes	complemented
19	C₂×C₄×C₅×D₄	—	5	1	no	maximal
20	C₂×D₄×Dic₅	C₅	1	1	yes	maximal, characteristic, direct product factor
21	C₂³×C₅²	—	2	4	no	—
22	C₂³×C₅²	—	2	4	no	—
23	C₂³×C₅²	C₂×C₄	1	8	yes	complemented
24	C₂³×C₅²	C₂³	1	4	yes	—
25	C₂³×C₅²	C₂³	1	1	yes	characteristic
26	C₅²×D₄	C₂×C₄	1	16	yes	complemented
27	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
28	C₂×C₄×C₅²	C₂³	1	2	yes	—
29	C₂×C₅×Dic₅	—	2	4	no	—
30	C₂×C₅×Dic₅	—	2	2	no	—
31	C₂×C₅×Dic₅	—	2	16	no	—
32	C₂×C₅×Dic₅	D₄	1	4	yes	direct product factor
33	C₂×C₅×Dic₅	C₂³	1	4	yes	—
34	C₂×C₅×Dic₅	C₂³	1	2	yes	—
35	C₂×C₅×Dic₅	C₂³	1	2	yes	—
36	C₂²×C₅×D₄	—	2	2	no	—
37	C₂²×C₅×D₄	D₅	1	1	yes	characteristic
38	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic
39	C₂³×C₄×C₅	—	5	2	no	—
40	C₂³×Dic₅	C₂×C₅	1	2	yes	complemented
41	C₄×C₅×D₄	—	5	8	no	—
42	C₂×C₅×C₄⋊C₄	—	5	1	no	—
43	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
44	C₂×C₄²×C₅	—	5	1	no	—
45	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	complemented
46	D₄×Dic₅	C₂×C₅	1	8	yes	direct product factor
47	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂²×C₅²	—	2	16	no	—
50	C₂²×C₅²	—	2	4	no	—
51	C₂²×C₅²	—	2	4	no	—
52	C₂²×C₅²	C₂²×C₄	1	2	yes	—
53	C₂²×C₅²	C₂²×C₄	1	8	yes	complemented
54	C₂²×C₅²	C₂×D₄	1	2	yes	—
55	C₂²×C₅²	#41	1	2	yes	—
56	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
57	C₄×C₅²	C₂²×C₄	1	4	yes	complemented
58	C₅×Dic₅	—	2	4	no	—
59	C₅×Dic₅	—	2	2	no	—
60	C₅×Dic₅	C₂×D₄	1	4	yes	direct product factor
61	C₂⁴×C₅	—	2	4	no	—
62	C₂⁴×C₅	C₂×D₅	1	2	yes	—
63	C₂⁴×C₅	C₂²×C₅	1	2	yes	—
64	C₂×C₅×D₄	—	2	16	no	—
65	C₂×C₅×D₄	—	2	8	no	—
66	C₂×C₅×D₄	Dic₅	1	8	yes	direct product factor
67	C₂×C₅×D₄	C₄×C₅	1	8	yes	complemented
68	C₂×C₅×D₄	C₂×D₅	1	4	yes	—
69	C₂×C₅×D₄	C₂²×C₅	1	4	yes	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	5	1	no	—
72	C₂²×C₄×C₅	—	5	8	no	—
73	C₂²×C₄×C₅	—	5	1	no	—
74	C₂²×C₄×C₅	—	5	2	no	—
75	C₂²×C₄×C₅	—	10	8	no	—
76	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
78	C₂²×Dic₅	—	2	8	no	—
79	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
80	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
81	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
82	C₂²×Dic₅	C₂²×C₅	1	8	yes	complemented
83	C₅×C₄⋊C₄	—	5	4	no	—
84	C₅×C₂²⋊C₄	—	5	8	no	—
85	C₄²×C₅	—	5	4	no	—
86	C₂²⋊Dic₅	C₂²×C₅	1	8	yes	complemented
87	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
88	C₄×Dic₅	C₂²×C₅	1	4	yes	complemented
89	C₂×C₄×D₄	—	5	1	no	Sylow
90	C₂×C₅²	—	2	8	no	—
91	C₂×C₅²	C₄×D₄	1	4	yes	complemented
92	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₅²	C₂×#41	1	1	yes	characteristic
95	C₂³×C₅	—	2	4	no	—
96	C₂³×C₅	—	2	4	no	—
97	C₂³×C₅	—	2	4	no	—
98	C₂³×C₅	—	2	4	no	—
99	C₂³×C₅	—	2	16	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	8	no	—
102	C₂³×C₅	—	4	8	no	—
103	C₂³×C₅	—	4	8	no	—
104	C₂³×C₅	C₂×Dic₅	1	8	yes	complemented
105	C₂³×C₅	C₂×C₄×C₅	1	8	yes	complemented
106	C₂³×C₅	C₂²×D₅	1	4	yes	—
107	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
108	C₂³×C₅	C₂³×C₅	1	4	yes	—
109	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
110	C₅×D₄	—	2	32	no	—
111	C₅×D₄	C₂×Dic₅	1	16	yes	direct product factor
112	C₅×D₄	C₂×C₄×C₅	1	16	yes	complemented
113	C₂×C₄×C₅	—	2	8	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	5	4	no	—
116	C₂×C₄×C₅	—	5	2	no	—
117	C₂×C₄×C₅	—	5	2	no	—
118	C₂×C₄×C₅	—	5	4	no	—
119	C₂×C₄×C₅	—	10	16	no	—
120	C₂×C₄×C₅	—	10	4	no	—
121	C₂×C₄×C₅	—	10	2	no	—
122	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	complemented
123	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	complemented
124	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
126	C₂×Dic₅	—	2	4	no	—
127	C₂×Dic₅	—	2	2	no	—
128	C₂×Dic₅	—	2	16	no	—
129	C₂×Dic₅	C₅×D₄	1	4	yes	direct product factor
130	C₂×Dic₅	C₂³×C₅	1	2	yes	—
131	C₂×Dic₅	C₂³×C₅	1	4	yes	—
132	C₂×Dic₅	C₂³×C₅	1	2	yes	—
133	C₂²×D₄	C₅×D₅	1	1	yes	characteristic
134	C₂³×C₄	—	5	2	no	—
135	C₄×D₄	—	5	8	no	—
136	C₂×C₄⋊C₄	—	5	1	no	—
137	C₂×C₂²⋊C₄	—	5	2	no	—
138	C₂×C₄²	—	5	1	no	—
139	C₅²	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₅	—	2	16	no	—
141	C₂²×C₅	—	2	4	no	—
142	C₂²×C₅	—	2	4	no	—
143	C₂²×C₅	—	2	4	no	—
144	C₂²×C₅	—	2	4	no	—
145	C₂²×C₅	—	2	16	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	—	2	4	no	—
149	C₂²×C₅	—	2	16	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	4	32	no	—
152	C₂²×C₅	—	4	8	no	—
153	C₂²×C₅	—	4	8	no	—
154	C₂²×C₅	D₄×D₅	1	2	yes	—
155	C₂²×C₅	#711	1	2	yes	—
156	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
157	C₂²×C₅	C₂²×Dic₅	1	8	yes	complemented
158	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
159	C₂²×C₅	C₂²×C₄×C₅	1	8	yes	complemented
160	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
161	C₂²×C₅	C₅×#41	1	2	yes	—
162	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
163	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
164	C₄×C₅	—	2	8	no	—
165	C₄×C₅	—	5	4	no	—
166	C₄×C₅	—	10	4	no	—
167	C₄×C₅	—	10	2	no	—
168	C₄×C₅	C₂²×Dic₅	1	4	yes	complemented
169	C₄×C₅	C₂²×C₄×C₅	1	4	yes	complemented
170	Dic₅	—	2	4	no	—
171	Dic₅	—	2	2	no	—
172	Dic₅	C₂×C₅×D₄	1	4	yes	direct product factor
173	C₂⁴	C₂×C₅×D₅	1	2	yes	—
174	C₂×D₄	C₅×Dic₅	1	8	yes	direct product factor
175	C₂×D₄	C₂×C₅×D₅	1	4	yes	—
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	—	5	1	no	—
178	C₂²×C₄	—	5	2	no	—
179	C₂²×C₄	—	5	8	no	—
180	C₂²×C₄	—	10	8	no	—
181	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
182	C₄⋊C₄	—	5	4	no	—
183	C₂²⋊C₄	—	5	8	no	—
184	C₄²	—	5	4	no	—
185	C₂×C₅	—	2	8	no	—
186	C₂×C₅	—	2	8	no	—
187	C₂×C₅	—	2	8	no	—
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	—	2	2	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	—	4	16	no	—
192	C₂×C₅	D₄×Dic₅	1	4	yes	direct product factor
193	C₂×C₅	C₄×C₅×D₄	1	4	yes	complemented
194	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
195	C₂×C₅	C₂×#711	1	1	yes	characteristic
196	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
197	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
198	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
199	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
200	C₂³	—	2	4	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×C₅×Dic₅	1	8	yes	complemented
203	C₂³	C₂²×C₅×D₅	1	4	yes	—
204	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
205	D₄	C₂×C₅×Dic₅	1	16	yes	direct product factor
206	C₂×C₄	—	5	4	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	2	no	—
209	C₂×C₄	—	5	4	no	—
210	C₂×C₄	—	10	16	no	—
211	C₂×C₄	—	10	4	no	—
212	C₂×C₄	—	10	2	no	—
213	C₂×C₄	C₂×C₅×Dic₅	1	4	yes	complemented
214	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
215	C₅	—	2	2	no	—
216	C₅	C₂×D₄×Dic₅	1	1	yes	characteristic, direct product factor
217	C₅	C₂×C₄×C₅×D₄	1	1	yes	characteristic, complemented
218	C₂²	—	2	16	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	C₅×D₄×D₅	1	2	yes	—
222	C₂²	C₅×#711	1	2	yes	—
223	C₂²	C₂²×C₅×Dic₅	1	2	yes	—
224	C₂²	C₂²×C₅×Dic₅	1	8	yes	complemented
225	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
226	C₄	—	5	4	no	—
227	C₄	—	10	4	no	—
228	C₄	—	10	2	no	—
229	C₄	C₂²×C₅×Dic₅	1	4	yes	complemented
230	C₂	—	2	8	no	—
231	C₂	C₅×D₄×Dic₅	1	4	yes	direct product factor
232	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
233	C₂	C₂×C₅×#711	1	1	yes	characteristic
234	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting, prime radical	C₂²×C₅²×D₄	C₂
socle	C₂³×C₅²	C₂³
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₂×C₄×C₅×D₄
LCS	G ▷ C₂×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
2. C₂³×C₅×Dic₅	C₂	semidirect
3. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
4. C₅×D₄×Dic₅	C₂	direct (N × H)
5. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
6. C₂×C₄×C₅×Dic₅	C₂	semidirect
8. C₂×C₅²×D₄	C₄	semidirect
15. C₂²×C₅×Dic₅	C₂²	semidirect
16. C₅×C₂²⋊Dic₅	C₂²	semidirect
17. C₅×C₄⋊Dic₅	C₂²	semidirect
18. C₄×C₅×Dic₅	C₂²	semidirect
20. C₂×D₄×Dic₅	C₅	direct (N × H)
23. C₂³×C₅²	C₂×C₄	semidirect
26. C₅²×D₄	C₂×C₄	semidirect
27. C₂×C₄×C₅²	C₂×C₄	semidirect
32. C₂×C₅×Dic₅	D₄	direct (N × H)
40. C₂³×Dic₅	C₂×C₅	semidirect
45. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
46. D₄×Dic₅	C₂×C₅	direct (N × H)
47. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
48. C₂×C₄×Dic₅	C₂×C₅	semidirect
53. C₂²×C₅²	C₂²×C₄	semidirect
57. C₄×C₅²	C₂²×C₄	semidirect
60. C₅×Dic₅	C₂×D₄	direct (N × H)
66. C₂×C₅×D₄	Dic₅	direct (N × H)
67. C₂×C₅×D₄	C₄×C₅	semidirect
82. C₂²×Dic₅	C₂²×C₅	semidirect
86. C₂²⋊Dic₅	C₂²×C₅	semidirect
87. C₄⋊Dic₅	C₂²×C₅	semidirect
88. C₄×Dic₅	C₂²×C₅	semidirect
91. C₂×C₅²	C₄×D₄	semidirect
104. C₂³×C₅	C₂×Dic₅	semidirect
105. C₂³×C₅	C₂×C₄×C₅	semidirect
111. C₅×D₄	C₂×Dic₅	direct (N × H)
112. C₅×D₄	C₂×C₄×C₅	semidirect
122. C₂×C₄×C₅	C₂×Dic₅	semidirect
123. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
129. C₂×Dic₅	C₅×D₄	direct (N × H)
139. C₅²	C₂×C₄×D₄	semidirect
157. C₂²×C₅	C₂²×Dic₅	semidirect
159. C₂²×C₅	C₂²×C₄×C₅	semidirect
168. C₄×C₅	C₂²×Dic₅	semidirect
169. C₄×C₅	C₂²×C₄×C₅	semidirect
172. Dic₅	C₂×C₅×D₄	direct (N × H)
174. C₂×D₄	C₅×Dic₅	direct (N × H)
192. C₂×C₅	D₄×Dic₅	direct (N × H)
193. C₂×C₅	C₄×C₅×D₄	semidirect
202. C₂³	C₂×C₅×Dic₅	semidirect
205. D₄	C₂×C₅×Dic₅	direct (N × H)
213. C₂×C₄	C₂×C₅×Dic₅	semidirect
216. C₅	C₂×D₄×Dic₅	direct (N × H)
217. C₅	C₂×C₄×C₅×D₄	semidirect
224. C₂²	C₂²×C₅×Dic₅	semidirect
229. C₄	C₂²×C₅×Dic₅	semidirect
231. C₂	C₅×D₄×Dic₅	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
C₂³×C₅×Dic₅	2
C₂×C₅×C₂²⋊Dic₅	2
C₅×D₄×Dic₅	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₄×C₅×Dic₅	2
C₂×C₄×C₅×D₄	5
C₂×D₄×Dic₅	5

Subgroups — C2×C5×D4×Dic5 #13106946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₄×Dic₅ #13106946