Subgroups — C₂×C₄×D₅×Dic₅ #13105929

Order 1600.

1104 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₅×Dic₅	C₂	1	2	yes	maximal
3	C₂×C₄×#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₅²	C₂	1	2	yes	maximal, complemented
6	C₄×D₅×Dic₅	C₂	1	8	yes	maximal, direct product factor
7	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×#1039	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₅	C₄	1	8	yes	complemented
13	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
14	C₂×D₅×Dic₅	C₄	1	16	yes	direct product factor
15	C₂×D₅×Dic₅	C₂²	1	8	yes	—
16	C₄×#1039	C₂²	1	4	yes	complemented
17	C₄×C₅×Dic₅	C₂²	1	4	yes	complemented
18	Dic₅²	C₂²	1	8	yes	complemented
19	C₂²×C₄×Dic₅	—	5	1	no	maximal
20	C₂×C₄²×D₅	—	5	1	no	maximal
21	C₂³×C₅²	C₂³	1	1	yes	characteristic
22	C₂²×C₅×D₅	C₂×C₄	1	8	yes	—
23	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
24	C₂²×C₅×D₅	C₂³	1	2	yes	—
25	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
26	C₂×C₄×C₅²	C₂³	1	2	yes	—
27	C₂×#1039	C₂×C₄	1	8	yes	complemented
28	C₂×#1039	C₂³	1	4	yes	—
29	C₂×C₅×Dic₅	C₂×C₄	1	8	yes	complemented
30	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
31	C₂×C₅×Dic₅	C₂³	1	4	yes	—
32	C₂×C₅×Dic₅	C₂³	1	2	yes	—
33	C₄×C₅×D₅	C₂×C₄	1	16	yes	complemented
34	D₅×Dic₅	C₂×C₄	1	32	yes	direct product factor
35	C₂³×C₄×C₅	—	5	1	no	—
36	C₂³×Dic₅	—	5	2	no	—
37	C₂²×C₄×D₅	—	5	2	no	—
38	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
39	C₂×C₄²×C₅	—	5	1	no	—
40	C₂×C₄×Dic₅	—	5	1	no	—
41	C₂×C₄×Dic₅	—	5	1	no	—
42	C₂×C₄×Dic₅	—	5	2	no	—
43	C₂×C₄×Dic₅	—	5	2	no	—
44	C₂×C₄×Dic₅	—	5	8	no	—
45	C₂×C₄×Dic₅	—	10	2	no	—
46	C₂×C₄×Dic₅	D₅	1	1	yes	characteristic, direct product factor
47	C₄²×D₅	—	5	8	no	—
48	C₂²×C₅²	C₂²×C₄	1	4	yes	—
49	C₂²×C₅²	C₂²×C₄	1	2	yes	—
50	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
51	C₂×C₅×D₅	C₄²	1	16	yes	complemented
52	C₂×C₅×D₅	C₂²×C₄	1	8	yes	—
53	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
54	C₄×C₅²	C₂²×C₄	1	4	yes	complemented
55	#1039	C₂²×C₄	1	8	yes	complemented
56	C₅×Dic₅	C₂²×C₄	1	8	yes	complemented
57	C₅×Dic₅	C₂²×C₄	1	4	yes	complemented
58	C₂⁴×C₅	—	5	1	no	—
59	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	5	2	no	—
62	C₂²×C₄×C₅	—	5	1	no	—
63	C₂²×C₄×C₅	—	5	8	no	—
64	C₂²×C₄×C₅	—	5	4	no	—
65	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
67	C₂²×Dic₅	—	5	2	no	—
68	C₂²×Dic₅	—	5	2	no	—
69	C₂²×Dic₅	—	5	16	no	—
70	C₂²×Dic₅	—	5	8	no	—
71	C₂²×Dic₅	—	10	4	no	—
72	C₂²×Dic₅	C₂×D₅	1	2	yes	—
73	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
74	C₂×C₄×D₅	—	5	16	no	—
75	C₂×C₄×D₅	—	5	8	no	—
76	C₂×C₄×D₅	Dic₅	1	8	yes	direct product factor
77	C₂×C₄×D₅	C₂×D₅	1	4	yes	—
78	C₄²×C₅	—	5	4	no	—
79	C₄×Dic₅	—	5	4	no	—
80	C₄×Dic₅	—	5	4	no	—
81	C₄×Dic₅	—	5	8	no	—
82	C₄×Dic₅	—	5	8	no	—
83	C₄×Dic₅	—	10	8	no	—
84	C₄×Dic₅	C₂×D₅	1	4	yes	direct product factor
85	C₂²×C₄²	—	25	1	no	Sylow
86	C₂×C₅²	C₂×C₄²	1	4	yes	complemented
87	C₂×C₅²	C₂³×C₄	1	2	yes	—
88	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
89	C₅×D₅	C₂×C₄²	1	8	yes	complemented
90	C₂³×C₅	—	2	2	no	—
91	C₂³×C₅	—	5	8	no	—
92	C₂³×C₅	—	5	4	no	—
93	C₂³×C₅	—	5	2	no	—
94	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
95	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
96	C₂²×D₅	C₄×D₅	1	8	yes	—
97	C₂²×D₅	C₂×Dic₅	1	4	yes	—
98	C₂²×D₅	C₂²×D₅	1	2	yes	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	5	8	no	—
102	C₂×C₄×C₅	—	5	4	no	—
103	C₂×C₄×C₅	—	5	2	no	—
104	C₂×C₄×C₅	—	5	4	no	—
105	C₂×C₄×C₅	—	5	16	no	—
106	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
107	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	complemented
108	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
109	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
110	C₂×Dic₅	—	5	8	no	—
111	C₂×Dic₅	—	5	8	no	—
112	C₂×Dic₅	—	5	4	no	—
113	C₂×Dic₅	—	5	4	no	—
114	C₂×Dic₅	—	5	32	no	—
115	C₂×Dic₅	—	10	16	no	—
116	C₂×Dic₅	—	10	8	no	—
117	C₂×Dic₅	C₄×D₅	1	8	yes	direct product factor
118	C₂×Dic₅	C₂×Dic₅	1	4	yes	complemented
119	C₂×Dic₅	C₂²×D₅	1	4	yes	—
120	C₂×Dic₅	C₂²×D₅	1	2	yes	—
121	C₄×D₅	—	5	32	no	—
122	C₄×D₅	C₂×Dic₅	1	16	yes	direct product factor
123	C₂³×C₄	—	5	1	no	—
124	C₂³×C₄	—	25	2	no	—
125	C₂×C₄²	—	5	1	no	—
126	C₂×C₄²	—	25	8	no	—
127	C₂×C₄²	—	25	2	no	—
128	C₂×C₄²	—	25	1	no	—
129	C₅²	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₅	—	2	8	no	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	5	8	no	—
134	C₂²×C₅	—	5	4	no	—
135	C₂²×C₅	—	5	16	no	—
136	C₂²×C₅	C₂×C₄×D₅	1	4	yes	—
137	C₂²×C₅	C₂×C₄×D₅	1	4	yes	—
138	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
139	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
140	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
141	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
142	C₂×D₅	C₄×Dic₅	1	16	yes	direct product factor
143	C₂×D₅	C₂×C₄×D₅	1	8	yes	—
144	C₂×D₅	C₂²×Dic₅	1	4	yes	—
145	C₄×C₅	—	2	8	no	—
146	C₄×C₅	—	5	8	no	—
147	C₄×C₅	—	5	4	no	—
148	C₄×C₅	C₂×C₄×D₅	1	4	yes	complemented
149	C₄×C₅	C₂²×Dic₅	1	4	yes	complemented
150	Dic₅	—	5	8	no	—
151	Dic₅	—	5	8	no	—
152	Dic₅	—	10	16	no	—
153	Dic₅	C₂×C₄×D₅	1	8	yes	direct product factor
154	Dic₅	C₂²×Dic₅	1	4	yes	complemented
155	C₂⁴	—	5	1	no	—
156	C₂²×C₄	—	5	2	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	5	8	no	—
159	C₂²×C₄	—	5	4	no	—
160	C₂²×C₄	—	25	8	no	—
161	C₂²×C₄	—	25	16	no	—
162	C₂²×C₄	—	25	2	no	—
163	C₂²×C₄	D₅²	1	1	yes	characteristic
164	C₄²	—	5	4	no	—
165	C₄²	—	25	8	no	—
166	C₄²	—	25	4	no	—
167	C₂×C₅	—	2	4	no	—
168	C₂×C₅	—	2	2	no	—
169	C₂×C₅	—	2	8	no	—
170	C₂×C₅	—	5	8	no	—
171	C₂×C₅	C₄²×D₅	1	4	yes	complemented
172	C₂×C₅	C₂×C₄×Dic₅	1	4	yes	complemented
173	C₂×C₅	C₂²×C₄×D₅	1	2	yes	—
174	C₂×C₅	C₂²×C₄×D₅	1	2	yes	—
175	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
176	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
177	D₅	C₂×C₄×Dic₅	1	8	yes	direct product factor
178	C₂³	—	5	8	no	—
179	C₂³	—	5	4	no	—
180	C₂³	—	5	2	no	—
181	C₂³	C₂×D₅²	1	1	yes	characteristic
182	C₂×C₄	—	5	8	no	—
183	C₂×C₄	—	5	4	no	—
184	C₂×C₄	—	5	4	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	5	16	no	—
187	C₂×C₄	—	25	32	no	—
188	C₂×C₄	—	25	8	no	—
189	C₂×C₄	—	25	4	no	—
190	C₂×C₄	D₅×Dic₅	1	4	yes	direct product factor
191	C₂×C₄	C₂×D₅²	1	2	yes	—
192	C₅	—	2	2	no	—
193	C₅	C₂×C₄²×D₅	1	1	yes	characteristic, complemented
194	C₅	C₂²×C₄×Dic₅	1	1	yes	characteristic, complemented
195	C₂²	—	5	8	no	—
196	C₂²	—	5	4	no	—
197	C₂²	—	5	16	no	—
198	C₂²	C₄×D₅²	1	4	yes	—
199	C₂²	C₂×D₅×Dic₅	1	2	yes	—
200	C₂²	C₂²×D₅²	1	1	yes	characteristic
201	C₄	—	5	8	no	—
202	C₄	—	5	4	no	—
203	C₄	—	25	8	no	—
204	C₄	C₂×D₅×Dic₅	1	4	yes	direct product factor
205	C₂	—	5	8	no	—
206	C₂	C₄×D₅×Dic₅	1	4	yes	direct product factor
207	C₂	C₂×C₄×D₅²	1	2	yes	—
208	C₂	C₂²×D₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₅²
commutator, hypocenter	C₅²	C₂²×C₄²
Frattini	C₂²	C₂²×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
cosocle	C₂²×C₅²	C₂⁴
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#1039	C₂	semidirect
4. C₂×C₄×C₅×Dic₅	C₂	semidirect
5. C₂×Dic₅²	C₂	semidirect
6. C₄×D₅×Dic₅	C₂	direct (N × H)
12. C₂×C₄×C₅×D₅	C₄	semidirect
14. C₂×D₅×Dic₅	C₄	direct (N × H)
16. C₄×#1039	C₂²	semidirect
17. C₄×C₅×Dic₅	C₂²	semidirect
18. Dic₅²	C₂²	semidirect
25. C₂×C₄×C₅²	C₂×C₄	semidirect
27. C₂×#1039	C₂×C₄	semidirect
29. C₂×C₅×Dic₅	C₂×C₄	semidirect
30. C₂×C₅×Dic₅	C₂×C₄	semidirect
33. C₄×C₅×D₅	C₂×C₄	semidirect
34. D₅×Dic₅	C₂×C₄	direct (N × H)
46. C₂×C₄×Dic₅	D₅	direct (N × H)
51. C₂×C₅×D₅	C₄²	semidirect
54. C₄×C₅²	C₂²×C₄	semidirect
55. #1039	C₂²×C₄	semidirect
56. C₅×Dic₅	C₂²×C₄	semidirect
57. C₅×Dic₅	C₂²×C₄	semidirect
76. C₂×C₄×D₅	Dic₅	direct (N × H)
84. C₄×Dic₅	C₂×D₅	direct (N × H)
86. C₂×C₅²	C₂×C₄²	semidirect
89. C₅×D₅	C₂×C₄²	semidirect
106. C₂×C₄×C₅	C₄×D₅	semidirect
107. C₂×C₄×C₅	C₂×Dic₅	semidirect
117. C₂×Dic₅	C₄×D₅	direct (N × H)
118. C₂×Dic₅	C₂×Dic₅	semidirect
122. C₄×D₅	C₂×Dic₅	direct (N × H)
129. C₅²	C₂²×C₄²	semidirect
142. C₂×D₅	C₄×Dic₅	direct (N × H)
148. C₄×C₅	C₂×C₄×D₅	semidirect
149. C₄×C₅	C₂²×Dic₅	semidirect
153. Dic₅	C₂×C₄×D₅	direct (N × H)
154. Dic₅	C₂²×Dic₅	semidirect
171. C₂×C₅	C₄²×D₅	semidirect
172. C₂×C₅	C₂×C₄×Dic₅	semidirect
177. D₅	C₂×C₄×Dic₅	direct (N × H)
190. C₂×C₄	D₅×Dic₅	direct (N × H)
193. C₅	C₂×C₄²×D₅	semidirect
194. C₅	C₂²×C₄×Dic₅	semidirect
204. C₄	C₂×D₅×Dic₅	direct (N × H)
206. C₂	C₄×D₅×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×C₄×C₅×D₅	2
C₂²×D₅×Dic₅	2
C₂×C₄×#1039	2
C₂×C₄×C₅×Dic₅	2
C₂×Dic₅²	2
C₄×D₅×Dic₅	2
C₂²×C₄×Dic₅	5
C₂×C₄²×D₅	5

Subgroups — C2×C4×D5×Dic5 #13105929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₅×Dic₅ #13105929