Subgroups — D₅×#2252 #12254774

Order 1280.

558 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#500	C₂	1	2	yes	maximal
2	C₅×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602177	C₂	1	2	yes	maximal, complemented
4	D₅×#366	C₂	1	4	yes	maximal
5	#10600113	C₂	1	4	yes	maximal, complemented
6	C₄×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
7	#10599837	C₂	1	1	yes	maximal, characteristic, complemented
8	D₅×C₄⋊Q₈	—	2	2	no	—
9	C₅×#500	C₂²	1	2	yes	—
10	#65367	—	2	2	no	—
11	C₂×D₅×C₁₆	C₂²	1	2	yes	—
12	D₅×#107	—	2	2	no	—
13	D₅×#107	C₂²	1	4	yes	—
14	#65163	—	2	2	no	—
15	#65163	C₂²	1	4	yes	—
16	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
17	#64971	C₂²	1	2	yes	—
18	C₅×#366	C₂²	1	4	yes	—
19	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	#64724	C₂²	1	4	yes	—
21	C₁₆×Dic₅	C₂²	1	2	yes	—
22	#64709	C₂²	1	8	yes	—
23	C₄×#672	C₂²	1	1	yes	characteristic
24	C₂×#2252	—	5	1	no	maximal, Sylow
25	C₂×Q₈×D₅	—	4	2	no	—
26	C₅×C₄⋊Q₈	—	2	2	no	—
27	#4090	—	4	2	no	—
28	C₂×C₈×D₅	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
30	D₅×C₄⋊C₄	—	2	4	no	—
31	D₅×C₄⋊C₄	—	4	2	no	—
32	#4034	—	2	4	no	—
33	#4034	—	4	2	no	—
34	C₄²×D₅	D₄	1	1	yes	characteristic
35	#4015	—	2	2	no	—
36	C₂×C₅×C₁₆	C₂³	1	2	yes	—
37	C₅×#107	—	2	2	no	—
38	C₅×#107	C₂³	1	4	yes	—
39	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
40	#3950	—	2	2	no	—
41	#3950	C₂³	1	4	yes	—
42	C₈×Dic₅	D₄	1	1	yes	characteristic
43	C₈×Dic₅	C₂³	1	1	yes	characteristic
44	C₂×#672	C₂³	1	2	yes	—
45	#3939	—	2	4	no	—
46	#3939	—	2	4	no	—
47	C₄×#168	D₄	1	1	yes	characteristic
48	D₅×C₁₆	Q₈	1	8	yes	complemented
49	C₂×#500	—	5	2	no	—
50	#2252	—	5	2	no	—
51	#2252	—	5	4	no	—
52	#2252	—	5	1	no	—
53	#2252	D₅	1	1	yes	characteristic, direct product factor
54	C₂×#366	—	5	4	no	—
55	C₂×C₄×C₁₆	—	5	1	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	Q₈×D₅	—	8	8	no	—
58	C₂×C₄×D₅	—	4	4	no	—
59	C₂×C₄×D₅	D₈	1	1	yes	characteristic
60	C₂×C₄×D₅	D₈	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₁₀	—	4	2	no	—
63	C₂×Dic₁₀	—	4	4	no	—
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
69	C₄⋊Dic₅	—	2	4	no	—
70	C₄⋊Dic₅	—	4	2	no	—
71	#683	—	4	4	no	—
72	#683	—	4	4	no	—
73	C₄×Dic₅	D₈	1	1	yes	characteristic
74	C₄×Dic₅	D₈	1	1	yes	characteristic
75	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
76	C₂×#168	C₂×D₄	1	1	yes	characteristic
77	C₂×#168	C₂×D₄	1	1	yes	characteristic
78	C₈×D₅	Q₁₆	1	4	yes	—
79	C₈×D₅	C₂×Q₈	1	4	yes	—
80	C₅×C₁₆	C₂×Q₈	1	4	yes	complemented
81	#672	C₂×Q₈	1	4	yes	complemented
82	C₂×C₄⋊Q₈	—	10	2	no	—
83	C₂²×C₁₆	—	5	2	no	—
84	#500	—	5	4	no	—
85	#500	—	5	2	no	—
86	#500	—	10	2	no	—
87	#500	—	10	2	no	—
88	#500	C₂×D₅	1	2	yes	—
89	C₂×#107	—	5	4	no	—
90	C₂×#107	—	10	2	no	—
91	C₂×C₄×C₈	—	5	1	no	—
92	#366	—	5	4	no	—
93	#366	—	5	8	no	—
94	#366	C₂×D₅	1	4	yes	—
95	C₄×C₁₆	—	5	2	no	—
96	C₄×C₁₆	—	5	1	no	—
97	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
98	C₂²×D₅	C₂×D₈	1	1	yes	characteristic
99	C₅×Q₈	—	8	4	no	—
100	C₂×C₄×C₅	—	4	4	no	—
101	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₅	—	4	4	no	—
105	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
106	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
107	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
108	C₄×D₅	—	8	8	no	—
109	C₄×D₅	D₁₆	1	4	yes	—
110	C₄×D₅	Q₃₂	1	4	yes	—
111	C₄×D₅	C₄⋊Q₈	1	2	yes	—
112	C₄×D₅	C₂×Q₁₆	1	2	yes	—
113	Dic₁₀	—	8	4	no	—
114	Dic₁₀	—	8	8	no	—
115	C₅×C₈	C₂×Q₁₆	1	2	yes	—
116	C₅×C₈	C₂²×Q₈	1	2	yes	—
117	#168	C₄⋊Q₈	1	2	yes	—
118	#168	C₂×Q₁₆	1	2	yes	—
119	C₂²×Q₈	—	20	2	no	—
120	C₂²×C₈	—	5	1	no	—
121	C₂²×C₈	—	5	1	no	—
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	10	4	no	—
124	C₄⋊Q₈	—	10	2	no	—
125	C₄⋊Q₈	—	20	2	no	—
126	C₄⋊Q₈	—	20	2	no	—
127	C₂×C₄⋊C₄	—	10	4	no	—
128	C₂×C₄⋊C₄	—	20	2	no	—
129	C₂×C₄²	—	5	1	no	—
130	C₂×C₁₆	—	5	2	no	—
131	C₂×C₁₆	—	5	8	no	—
132	C₂×C₁₆	C₂²×D₅	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	5	4	no	—
135	#107	—	10	4	no	—
136	#107	—	10	4	no	—
137	#107	—	10	2	no	—
138	#107	C₂²×D₅	1	4	yes	—
139	C₄×C₈	—	5	1	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
143	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₅	#500	1	2	yes	—
145	C₂×D₅	C₂×D₁₆	1	2	yes	—
146	C₂×D₅	C₂×Q₃₂	1	2	yes	—
147	C₄×C₅	—	8	4	no	—
148	C₄×C₅	C₂×D₁₆	1	2	yes	—
149	C₄×C₅	C₂×Q₃₂	1	2	yes	—
150	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
152	Dic₅	—	8	4	no	—
153	Dic₅	#500	1	2	yes	—
154	Dic₅	C₂×D₁₆	1	2	yes	—
155	Dic₅	C₂×Q₃₂	1	2	yes	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	20	4	no	—
158	C₂×Q₈	—	20	2	no	—
159	C₂×Q₈	—	40	8	no	—
160	C₂²×C₄	—	5	1	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	1	no	—
163	C₂²×C₄	—	20	4	no	—
164	C₂×C₈	—	5	1	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	4	no	—
167	C₂×C₈	—	5	4	no	—
168	C₂×C₈	D₄×D₅	1	1	yes	characteristic
169	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	10	4	no	—
173	C₄⋊C₄	—	20	4	no	—
174	C₄⋊C₄	—	20	4	no	—
175	C₄⋊C₄	—	20	2	no	—
176	C₄²	—	5	1	no	—
177	C₄²	—	5	1	no	—
178	C₄²	—	5	1	no	—
179	C₄²	D₄×D₅	1	1	yes	characteristic
180	C₁₆	—	5	4	no	—
181	C₁₆	Q₈×D₅	1	4	yes	complemented
182	C₂×C₅	C₂×#500	1	1	yes	characteristic
183	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
184	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
185	D₅	#2252	1	4	yes	direct product factor
186	C₂³	—	5	1	no	—
187	Q₈	—	8	4	no	—
188	Q₈	—	40	8	no	—
189	Q₈	—	40	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	—	5	1	no	—
193	C₂×C₄	—	5	1	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	4	no	—
197	C₂×C₄	—	5	4	no	—
198	C₂×C₄	—	20	4	no	—
199	C₂×C₄	—	40	8	no	—
200	C₂×C₄	D₅×D₈	1	1	yes	characteristic
201	C₂×C₄	D₅×D₈	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
203	C₈	—	5	2	no	—
204	C₈	—	5	2	no	—
205	C₈	D₅×Q₁₆	1	2	yes	—
206	C₈	C₂×Q₈×D₅	1	2	yes	—
207	C₅	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	5	2	no	—
209	C₂²	—	5	2	no	—
210	C₂²	—	5	2	no	—
211	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
212	C₄	—	5	2	no	—
213	C₄	—	5	2	no	—
214	C₄	—	5	2	no	—
215	C₄	—	8	4	no	—
216	C₄	—	40	4	no	—
217	C₄	D₅×D₁₆	1	2	yes	—
218	C₄	D₅×Q₃₂	1	2	yes	—
219	C₄	D₅×C₄⋊Q₈	1	1	yes	characteristic
220	C₄	C₂×D₅×Q₁₆	1	1	yes	characteristic
221	C₂	—	5	4	no	—
222	C₂	D₅×#500	1	1	yes	characteristic
223	C₂	C₂×D₅×D₁₆	1	1	yes	characteristic
224	C₂	C₂×D₅×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅×D₈
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2252	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂²×D₅	C₂×D₈
hypercenter	#2252	D₅
hypocenter	C₅	C₂×#2252
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. C₅×#2252	C₂	semidirect
3. #10602177	C₂	semidirect
5. #10600113	C₂	semidirect
7. #10599837	C₂	semidirect
48. D₅×C₁₆	Q₈	semidirect
53. #2252	D₅	direct (N × H)
80. C₅×C₁₆	C₂×Q₈	semidirect
81. #672	C₂×Q₈	semidirect
181. C₁₆	Q₈×D₅	semidirect
185. D₅	#2252	direct (N × H)
207. C₅	C₂×#2252	semidirect

Maximal subgroups

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

subgroup	index
D₅×#500	2
C₅×#2252	2
#10602177	2
D₅×#366	2
#10600113	2
C₄×D₅×C₁₆	2
#10599837	2
C₂×#2252	5

Subgroups — D5×#2252 #12254774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#2252 #12254774