Subgroups — C₂×C₄×C₅×Dic₁₀ #13106621

Order 1600.

704 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#683	C₂	1	2	yes	maximal
5	C₂×C₄×C₅×Dic₅	C₂	1	2	yes	maximal
6	C₄×C₅×Dic₁₀	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
12	C₂×C₅×Dic₁₀	C₄	1	8	yes	direct product factor
13	C₂×C₅×Dic₁₀	C₂²	1	4	yes	—
14	C₄²×C₅²	C₂²	1	4	yes	—
15	C₅×C₄⋊Dic₅	C₂²	1	4	yes	—
16	C₅×#683	C₂²	1	8	yes	—
17	C₄×C₅×Dic₅	C₂²	1	8	yes	—
18	C₂×C₄×C₅×Q₈	—	5	1	no	maximal
19	C₂×C₄×Dic₁₀	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₂³×C₅²	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅²	—	2	2	no	—
22	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₅²	Q₈	1	4	yes	complemented
24	C₂×C₄×C₅²	C₂³	1	2	yes	—
25	C₂×C₄×C₅²	C₂³	1	2	yes	—
26	C₂×C₄×C₅²	C₂³	1	2	yes	—
27	C₂×C₅×Dic₅	—	2	4	no	—
28	C₂×C₅×Dic₅	C₂×C₄	1	8	yes	—
29	C₂×C₅×Dic₅	C₂³	1	4	yes	—
30	C₂×C₅×Dic₅	C₂³	1	4	yes	—
31	C₅×Dic₁₀	C₂×C₄	1	16	yes	direct product factor
32	C₂²×C₅×Q₈	—	5	1	no	—
33	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic
34	C₄×C₅×Q₈	—	5	8	no	—
35	C₂×C₅×C₄⋊C₄	—	5	2	no	—
36	C₂×C₅×C₄⋊C₄	—	5	1	no	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	—	5	2	no	—
39	C₂×C₄²×C₅	D₅	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic
42	C₂×#683	C₂×C₅	1	2	yes	—
43	C₂×C₄×Dic₅	C₂×C₅	1	2	yes	—
44	C₄×Dic₁₀	C₂×C₅	1	8	yes	direct product factor
45	C₂²×C₅²	C₂²×C₄	1	2	yes	—
46	C₂²×C₅²	C₂×Q₈	1	2	yes	—
47	C₂²×C₅²	#41	1	2	yes	—
48	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
49	C₄×C₅²	—	2	2	no	—
50	C₄×C₅²	C₂²×C₄	1	4	yes	—
51	C₄×C₅²	C₂×Q₈	1	4	yes	complemented
52	C₅×Dic₅	—	2	4	no	—
53	C₅×Dic₅	C₂²×C₄	1	8	yes	—
54	C₂×C₅×Q₈	—	5	8	no	—
55	C₂×C₅×Q₈	—	5	4	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	5	2	no	—
60	C₂²×C₄×C₅	—	5	2	no	—
61	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
67	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
68	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
69	C₂×Dic₁₀	C₄×C₅	1	8	yes	direct product factor
70	C₂×Dic₁₀	C₂²×C₅	1	4	yes	—
71	C₅×C₄⋊C₄	—	5	8	no	—
72	C₅×C₄⋊C₄	—	5	4	no	—
73	C₄²×C₅	—	2	8	no	—
74	C₄²×C₅	—	5	8	no	—
75	C₄²×C₅	C₂×D₅	1	4	yes	—
76	C₄²×C₅	C₂²×C₅	1	4	yes	—
77	C₄⋊Dic₅	C₂²×C₅	1	4	yes	—
78	#683	C₂²×C₅	1	8	yes	—
79	C₄×Dic₅	C₂²×C₅	1	8	yes	—
80	C₂×C₄×Q₈	—	5	1	no	Sylow
81	C₂×C₅²	C₄×Q₈	1	4	yes	complemented
82	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₅²	C₂×#41	1	1	yes	characteristic
85	C₂³×C₅	—	2	2	no	—
86	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
87	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
88	C₅×Q₈	—	5	16	no	—
89	C₂×C₄×C₅	—	2	8	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	8	no	—
93	C₂×C₄×C₅	—	2	8	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	5	8	no	—
98	C₂×C₄×C₅	—	5	4	no	—
99	C₂×C₄×C₅	—	5	4	no	—
100	C₂×C₄×C₅	—	10	4	no	—
101	C₂×C₄×C₅	Dic₁₀	1	4	yes	direct product factor
102	C₂×C₄×C₅	C₄×D₅	1	4	yes	—
103	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₄×C₅	C₅×Q₈	1	4	yes	complemented
105	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
106	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
107	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
108	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
109	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
111	C₂×Dic₅	—	2	4	no	—
112	C₂×Dic₅	C₂×C₄×C₅	1	8	yes	—
113	C₂×Dic₅	C₂³×C₅	1	4	yes	—
114	C₂×Dic₅	C₂³×C₅	1	4	yes	—
115	Dic₁₀	C₂×C₄×C₅	1	16	yes	direct product factor
116	C₂²×Q₈	—	5	1	no	—
117	C₄×Q₈	—	5	8	no	—
118	C₂×C₄⋊C₄	—	5	2	no	—
119	C₂×C₄⋊C₄	—	5	1	no	—
120	C₂×C₄²	—	5	2	no	—
121	C₂×C₄²	C₅×D₅	1	1	yes	characteristic
122	C₅²	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₅	—	2	4	no	—
124	C₂²×C₅	—	2	4	no	—
125	C₂²×C₅	—	2	4	no	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	C₂×Dic₁₀	1	2	yes	—
128	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
129	C₂²×C₅	#709	1	2	yes	—
130	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
131	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
132	C₂²×C₅	C₅×#41	1	2	yes	—
133	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
134	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
135	C₄×C₅	—	2	8	no	—
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	—	2	2	no	—
138	C₄×C₅	—	2	8	no	—
139	C₄×C₅	—	2	8	no	—
140	C₄×C₅	—	5	8	no	—
141	C₄×C₅	—	10	4	no	—
142	C₄×C₅	C₂×Dic₁₀	1	4	yes	direct product factor
143	C₄×C₅	C₂×C₄×D₅	1	4	yes	—
144	C₄×C₅	C₂²×C₄×C₅	1	4	yes	—
145	C₄×C₅	C₂×C₅×Q₈	1	4	yes	complemented
146	Dic₅	—	2	4	no	—
147	Dic₅	C₂²×C₄×C₅	1	8	yes	—
148	C₂×Q₈	—	5	8	no	—
149	C₂×Q₈	—	5	4	no	—
150	C₂²×C₄	—	5	2	no	—
151	C₂²×C₄	—	5	2	no	—
152	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
155	C₄⋊C₄	—	5	8	no	—
156	C₄⋊C₄	—	5	4	no	—
157	C₄²	—	5	8	no	—
158	C₄²	C₂×C₅×D₅	1	4	yes	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	2	2	no	—
162	C₂×C₅	—	2	8	no	—
163	C₂×C₅	C₄×Dic₁₀	1	4	yes	direct product factor
164	C₂×C₅	C₄×C₅×Q₈	1	4	yes	complemented
165	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
166	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
167	C₂×C₅	C₂×#709	1	1	yes	characteristic
168	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
169	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
170	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
171	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
172	Q₈	—	5	16	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	5	8	no	—
175	C₂×C₄	—	5	4	no	—
176	C₂×C₄	—	5	4	no	—
177	C₂×C₄	—	10	4	no	—
178	C₂×C₄	C₅×Dic₁₀	1	4	yes	direct product factor
179	C₂×C₄	C₄×C₅×D₅	1	4	yes	—
180	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
181	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
182	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
183	C₅	—	2	2	no	—
184	C₅	C₂×C₄×Dic₁₀	1	1	yes	characteristic, direct product factor
185	C₅	C₂×C₄×C₅×Q₈	1	1	yes	characteristic, complemented
186	C₂²	C₂×C₅×Dic₁₀	1	2	yes	—
187	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
188	C₂²	C₅×#709	1	2	yes	—
189	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	5	8	no	—
192	C₄	—	10	4	no	—
193	C₄	C₂×C₅×Dic₁₀	1	4	yes	direct product factor
194	C₄	C₂×C₄×C₅×D₅	1	4	yes	—
195	C₂	C₄×C₅×Dic₁₀	1	4	yes	direct product factor
196	C₂	C₂²×C₅×Dic₁₀	1	1	yes	characteristic
197	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
198	C₂	C₂×C₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle, prime radical	C₂³×C₅²	C₂³
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₂×C₄×C₅×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
6. C₄×C₅×Dic₁₀	C₂	direct (N × H)
12. C₂×C₅×Dic₁₀	C₄	direct (N × H)
19. C₂×C₄×Dic₁₀	C₅	direct (N × H)
23. C₂×C₄×C₅²	Q₈	semidirect
31. C₅×Dic₁₀	C₂×C₄	direct (N × H)
44. C₄×Dic₁₀	C₂×C₅	direct (N × H)
51. C₄×C₅²	C₂×Q₈	semidirect
69. C₂×Dic₁₀	C₄×C₅	direct (N × H)
81. C₂×C₅²	C₄×Q₈	semidirect
101. C₂×C₄×C₅	Dic₁₀	direct (N × H)
104. C₂×C₄×C₅	C₅×Q₈	semidirect
115. Dic₁₀	C₂×C₄×C₅	direct (N × H)
122. C₅²	C₂×C₄×Q₈	semidirect
142. C₄×C₅	C₂×Dic₁₀	direct (N × H)
145. C₄×C₅	C₂×C₅×Q₈	semidirect
163. C₂×C₅	C₄×Dic₁₀	direct (N × H)
164. C₂×C₅	C₄×C₅×Q₈	semidirect
178. C₂×C₄	C₅×Dic₁₀	direct (N × H)
184. C₅	C₂×C₄×Dic₁₀	direct (N × H)
185. C₅	C₂×C₄×C₅×Q₈	semidirect
193. C₄	C₂×C₅×Dic₁₀	direct (N × H)
195. C₂	C₄×C₅×Dic₁₀	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×C₅×Dic₁₀	2
C₂×C₄²×C₅²	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#683	2
C₂×C₄×C₅×Dic₅	2
C₄×C₅×Dic₁₀	2
C₂×C₄×C₅×Q₈	5
C₂×C₄×Dic₁₀	5

Subgroups — C2×C4×C5×Dic10 #13106621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×Dic₁₀ #13106621