Subgroups — C₄×C₅×C₂²⋊Dic₅ #13102470

Order 1600.

752 conjugacy classes of subgroups.

The subgroup lattice has 204 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	2	yes	maximal
3	C₂×C₄×C₅×Dic₅	C₂	1	2	yes	maximal, complemented
4	C₅×#3963	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₅²	—	2	4	no	—
7	C₂²×C₄×C₅²	C₄	1	4	yes	complemented
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	4	yes	—
11	C₅×C₂²⋊Dic₅	C₄	1	8	yes	direct product factor
12	C₄×C₅×Dic₅	—	2	4	no	—
13	C₄×C₅×C₂²⋊C₄	—	5	1	no	maximal
14	C₄×C₂²⋊Dic₅	C₅	1	1	yes	maximal, characteristic, direct product factor
15	C₂³×C₅²	—	2	2	no	—
16	C₂³×C₅²	—	2	2	no	—
17	C₂³×C₅²	C₂×C₄	1	4	yes	—
18	C₂³×C₅²	C₂×C₄	1	2	yes	—
19	C₂³×C₅²	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅²	—	2	2	no	—
21	C₂×C₄×C₅²	—	2	8	no	—
22	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₅²	D₄	1	4	yes	—
25	C₂×C₅×Dic₅	—	2	4	no	—
26	C₂×C₅×Dic₅	—	2	4	no	—
27	C₂×C₅×Dic₅	C₂×C₄	1	8	yes	complemented
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
31	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
32	C₂×C₄²×C₅	—	5	2	no	—
33	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	—
34	C₂×C₄×Dic₅	C₂×C₅	1	2	yes	complemented
35	C₅×#95	—	5	2	no	—
36	#3963	C₂×C₅	1	2	yes	complemented
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	—	2	2	no	—
39	C₂²×C₅²	—	2	8	no	—
40	C₂²×C₅²	C₄²	1	4	yes	complemented
41	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₅²	C₂²×C₄	1	2	yes	—
43	C₂²×C₅²	C₂×D₄	1	2	yes	—
44	C₂²×C₅²	#41	1	2	yes	—
45	C₄×C₅²	—	2	2	no	—
46	C₄×C₅²	C₂²⋊C₄	1	4	yes	complemented
47	C₅×Dic₅	—	2	8	no	—
48	C₂⁴×C₅	—	2	2	no	—
49	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
50	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	16	no	—
52	C₂²×C₄×C₅	—	2	4	no	—
53	C₂²×C₄×C₅	—	2	4	no	—
54	C₂²×C₄×C₅	—	2	8	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	5	4	no	—
58	C₂²×C₄×C₅	Dic₅	1	4	yes	complemented
59	C₂²×C₄×C₅	C₄×C₅	1	4	yes	complemented
60	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
64	C₂²×Dic₅	C₂²×C₅	1	4	yes	—
65	C₅×C₂²⋊C₄	—	5	8	no	—
66	C₄²×C₅	—	10	4	no	—
67	C₂²⋊Dic₅	C₄×C₅	1	8	yes	direct product factor
68	C₄×Dic₅	—	2	4	no	—
69	C₄×C₂²⋊C₄	—	5	1	no	Sylow
70	C₂×C₅²	—	2	4	no	—
71	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅²	#117	1	1	yes	characteristic
74	C₂×C₅²	C₄×D₄	1	4	yes	—
75	C₂³×C₅	—	2	8	no	—
76	C₂³×C₅	—	2	8	no	—
77	C₂³×C₅	—	2	2	no	—
78	C₂³×C₅	—	2	2	no	—
79	C₂³×C₅	—	2	2	no	—
80	C₂³×C₅	—	2	2	no	—
81	C₂³×C₅	—	2	8	no	—
82	C₂³×C₅	—	2	4	no	—
83	C₂³×C₅	—	2	2	no	—
84	C₂³×C₅	C₄×D₅	1	4	yes	—
85	C₂³×C₅	C₂×Dic₅	1	2	yes	—
86	C₂³×C₅	C₂×C₄×C₅	1	4	yes	—
87	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
88	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
89	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
90	C₂×C₄×C₅	—	2	8	no	—
91	C₂×C₄×C₅	—	2	32	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	8	no	—
95	C₂×C₄×C₅	—	2	8	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	—	5	8	no	—
100	C₂×C₄×C₅	—	10	4	no	—
101	C₂×C₄×C₅	—	10	4	no	—
102	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
103	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
104	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
105	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
106	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
107	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
108	C₂×Dic₅	—	2	4	no	—
109	C₂×Dic₅	—	2	4	no	—
110	C₂×Dic₅	C₂×C₄×C₅	1	8	yes	complemented
111	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	5	2	no	—
113	C₂×C₄²	—	5	2	no	—
114	#95	—	5	2	no	—
115	C₅²	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₅	—	2	8	no	—
117	C₂²×C₅	—	2	8	no	—
118	C₂²×C₅	—	2	32	no	—
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	—	2	2	no	—
121	C₂²×C₅	—	2	2	no	—
122	C₂²×C₅	—	2	8	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	8	no	—
125	C₂²×C₅	—	2	4	no	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	—	2	8	no	—
130	C₂²×C₅	C₄×Dic₅	1	4	yes	complemented
131	C₂²×C₅	C₄²×C₅	1	4	yes	complemented
132	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
133	C₂²×C₅	#709	1	2	yes	—
134	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
135	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
136	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
137	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
139	C₂²×C₅	C₅×#41	1	2	yes	—
140	C₄×C₅	—	2	8	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	8	no	—
144	C₄×C₅	—	10	8	no	—
145	C₄×C₅	C₂²⋊Dic₅	1	4	yes	direct product factor
146	C₄×C₅	C₅×C₂²⋊C₄	1	4	yes	complemented
147	Dic₅	—	2	8	no	—
148	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	5	4	no	—
151	C₂²×C₄	C₅×Dic₅	1	4	yes	complemented
152	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
154	C₂²⋊C₄	—	5	8	no	—
155	C₄²	—	10	4	no	—
156	C₂×C₅	—	2	16	no	—
157	C₂×C₅	—	2	4	no	—
158	C₂×C₅	—	2	4	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	2	8	no	—
162	C₂×C₅	—	2	2	no	—
163	C₂×C₅	C₂×C₄×Dic₅	1	1	yes	characteristic
164	C₂×C₅	#4072	1	1	yes	characteristic
165	C₂×C₅	C₄×C₂²⋊D₅	1	4	yes	—
166	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
167	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
168	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₅	C₅×#117	1	1	yes	characteristic
170	C₂×C₅	C₄×C₅×D₄	1	4	yes	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₄×C₅×D₅	1	4	yes	—
174	C₂³	C₂×C₅×Dic₅	1	2	yes	—
175	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	5	8	no	—
179	C₂×C₄	—	10	4	no	—
180	C₂×C₄	—	10	4	no	—
181	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
182	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
183	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
184	C₅	—	2	2	no	—
185	C₅	C₄×C₂²⋊Dic₅	1	1	yes	characteristic, direct product factor
186	C₅	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented
187	C₂²	—	2	2	no	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	C₄×C₅×Dic₅	1	4	yes	complemented
191	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
192	C₂²	C₅×#709	1	2	yes	—
193	C₂²	C₂²×C₅×Dic₅	1	1	yes	characteristic
194	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	10	8	no	—
197	C₄	C₅×C₂²⋊Dic₅	1	4	yes	direct product factor
198	C₂	—	2	4	no	—
199	C₂	C₂×C₄×C₅×Dic₅	1	1	yes	characteristic
200	C₂	C₅×#4072	1	1	yes	characteristic
201	C₂	C₄×C₅×C₂²⋊D₅	1	4	yes	—
202	C₂	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂×C₄²×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	C₄×C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₅×Dic₅	C₂	semidirect
4. C₅×#3963	C₂	semidirect
7. C₂²×C₄×C₅²	C₄	semidirect
11. C₅×C₂²⋊Dic₅	C₄	direct (N × H)
14. C₄×C₂²⋊Dic₅	C₅	direct (N × H)
27. C₂×C₅×Dic₅	C₂×C₄	semidirect
34. C₂×C₄×Dic₅	C₂×C₅	semidirect
36. #3963	C₂×C₅	semidirect
40. C₂²×C₅²	C₄²	semidirect
46. C₄×C₅²	C₂²⋊C₄	semidirect
58. C₂²×C₄×C₅	Dic₅	semidirect
59. C₂²×C₄×C₅	C₄×C₅	semidirect
67. C₂²⋊Dic₅	C₄×C₅	direct (N × H)
110. C₂×Dic₅	C₂×C₄×C₅	semidirect
115. C₅²	C₄×C₂²⋊C₄	semidirect
130. C₂²×C₅	C₄×Dic₅	semidirect
131. C₂²×C₅	C₄²×C₅	semidirect
145. C₄×C₅	C₂²⋊Dic₅	direct (N × H)
146. C₄×C₅	C₅×C₂²⋊C₄	semidirect
151. C₂²×C₄	C₅×Dic₅	semidirect
185. C₅	C₄×C₂²⋊Dic₅	direct (N × H)
186. C₅	C₄×C₅×C₂²⋊C₄	semidirect
190. C₂²	C₄×C₅×Dic₅	semidirect
197. C₄	C₅×C₂²⋊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
C₂³×C₄×C₅²	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₄×C₅×Dic₅	2
C₅×#3963	2
C₄×C₅×C₂²⋊C₄	5
C₄×C₂²⋊Dic₅	5

Subgroups — C4×C5×C22⋊Dic5 #13102470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₂²⋊Dic₅ #13102470