Subgroups — #12005119

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65224	C₂	1	2	yes	maximal
3	C₂×#64771	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₅	—	2	4	no	—
7	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
8	C₂×C₄×C₅×C₈	—	2	4	no	—
9	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
10	#65224	—	2	4	no	—
11	#65224	C₄	1	8	yes	complemented
12	#64771	—	2	8	no	—
13	#10841	—	5	1	no	maximal, Sylow
14	C₂³×C₄×C₅	D₄	1	2	yes	—
15	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
16	C₂³×Dic₅	—	2	4	no	—
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₅×C₈	D₄	1	4	yes	—
21	C₂²×C₅×C₈	Q₈	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	D₄	1	4	yes	—
25	C₂×C₄²×C₅	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅	—	2	8	no	—
27	C₂×C₄⋊Dic₅	—	2	8	no	—
28	C₂×C₄⋊Dic₅	—	2	16	no	—
29	C₂×C₄⋊Dic₅	—	2	4	no	—
30	C₂×C₄⋊Dic₅	—	4	8	no	—
31	C₂×C₄⋊Dic₅	—	4	8	no	—
32	C₂×C₄⋊Dic₅	C₂×C₄	1	8	yes	—
33	C₄×C₅×C₈	—	2	8	no	—
34	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
35	C₂×#384	—	5	2	no	—
36	C₂×#340	—	5	4	no	—
37	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
39	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅	D₈	1	4	yes	—
41	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
42	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₅	#41	1	4	yes	—
50	C₂²×Dic₅	—	2	8	no	—
51	C₂²×Dic₅	—	4	4	no	—
52	C₂²×Dic₅	—	4	4	no	—
53	C₂²×Dic₅	—	4	8	no	—
54	C₂²×Dic₅	—	4	8	no	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	—	2	4	no	—
57	C₂×C₅×C₈	—	2	8	no	—
58	C₂×C₅×C₈	—	2	8	no	—
59	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	complemented
60	C₄²×C₅	C₄²	1	4	yes	—
61	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
62	C₄²×C₅	C₄⋊C₄	1	8	yes	—
63	C₄⋊Dic₅	—	2	16	no	—
64	C₄⋊Dic₅	—	4	32	no	—
65	C₄⋊Dic₅	—	4	8	no	—
66	C₂³×C₈	C₂×D₅	1	2	yes	—
67	C₂²×C₄⋊C₄	—	5	4	no	—
68	C₂²×C₄⋊C₄	—	10	4	no	—
69	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
70	C₂×C₄×C₈	—	2	4	no	—
71	C₂×C₄×C₈	Dic₅	1	4	yes	complemented
72	#384	—	5	8	no	—
73	#384	—	10	4	no	—
74	#340	—	10	8	no	—
75	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₅	#124	1	2	yes	—
78	C₂³×C₅	#127	1	1	yes	characteristic
79	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
80	C₂³×C₅	C₂×D₈	1	2	yes	—
81	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
82	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₅	#95	1	8	yes	—
84	C₂×C₄×C₅	#102	1	16	yes	—
85	C₂×C₄×C₅	#103	1	16	yes	—
86	C₂×C₄×C₅	#106	1	8	yes	—
87	C₂×C₄×C₅	#107	1	8	yes	—
88	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅	#117	1	2	yes	—
93	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₅	#121	1	4	yes	—
96	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
97	C₂×C₄×C₅	#125	1	2	yes	—
98	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
99	C₂×Dic₅	—	4	8	no	—
100	C₂×Dic₅	—	4	16	no	—
101	C₂×Dic₅	—	8	8	no	—
102	C₂×Dic₅	—	8	4	no	—
103	C₂×Dic₅	—	8	4	no	—
104	C₅×C₈	—	2	8	no	—
105	C₂³×C₄	—	10	4	no	—
106	C₂³×C₄	D₂₀	1	2	yes	—
107	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	—	2	4	no	—
110	C₂²×C₈	Dic₁₀	1	4	yes	—
111	C₂²×C₈	D₂₀	1	4	yes	—
112	C₂²×C₈	C₂×Dic₅	1	4	yes	—
113	C₂×C₄⋊C₄	—	5	8	no	—
114	C₂×C₄⋊C₄	—	10	8	no	—
115	C₂×C₄⋊C₄	—	10	8	no	—
116	C₂×C₄⋊C₄	—	10	16	no	—
117	C₂×C₄⋊C₄	—	10	4	no	—
118	C₂×C₄⋊C₄	—	20	8	no	—
119	C₂×C₄⋊C₄	—	20	8	no	—
120	C₂×C₄²	Dic₁₀	1	2	yes	—
121	C₂×C₄²	C₄×D₅	1	4	yes	—
122	C₂×C₄²	C₂×Dic₅	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₅	1	4	yes	—
124	C₄×C₈	—	2	8	no	—
125	C₂²×C₅	C₂×#95	1	1	yes	characteristic
126	C₂²×C₅	#382	1	1	yes	characteristic
127	C₂²×C₅	#384	1	1	yes	characteristic
128	C₂²×C₅	#390	1	2	yes	—
129	C₂²×C₅	#391	1	2	yes	—
130	C₂²×C₅	C₂×#102	1	4	yes	—
131	C₂²×C₅	C₂×#103	1	4	yes	—
132	C₂²×C₅	C₂×#106	1	2	yes	—
133	C₂²×C₅	C₂×#107	1	2	yes	—
134	C₂²×C₅	#486	1	4	yes	—
135	C₂²×C₅	#487	1	4	yes	—
136	C₂²×C₅	#492	1	2	yes	—
137	C₂²×C₅	#493	1	1	yes	characteristic
138	C₂²×C₅	#494	1	1	yes	characteristic
139	C₂²×C₅	#498	1	2	yes	—
140	C₂²×C₅	#500	1	2	yes	—
141	C₄×C₅	#340	1	16	yes	complemented
142	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
143	C₄×C₅	#380	1	2	yes	—
144	C₄×C₅	#389	1	4	yes	—
145	Dic₅	—	8	8	no	—
146	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
147	C₂²×C₄	—	10	8	no	—
148	C₂²×C₄	—	20	4	no	—
149	C₂²×C₄	—	20	4	no	—
150	C₂²×C₄	—	20	8	no	—
151	C₂²×C₄	—	20	8	no	—
152	C₂²×C₄	#677	1	8	yes	—
153	C₂²×C₄	D₄₀	1	4	yes	—
154	C₂²×C₄	Dic₂₀	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
156	C₂²×C₄	#685	1	8	yes	—
157	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
158	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
160	C₂²×C₄	C₂×D₂₀	1	2	yes	—
161	C₂²×C₄	#709	1	4	yes	—
162	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	C₄⋊Dic₅	1	8	yes	complemented
169	C₄⋊C₄	—	10	16	no	—
170	C₄⋊C₄	—	20	32	no	—
171	C₄⋊C₄	—	20	8	no	—
172	C₄²	C₄×Dic₅	1	4	yes	—
173	C₄²	#683	1	8	yes	—
174	C₄²	C₂²⋊Dic₅	1	4	yes	—
175	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
176	C₂×C₅	C₂×#340	1	2	yes	—
177	C₂×C₅	#1834	1	2	yes	—
178	C₂×C₅	#1844	1	1	yes	characteristic
179	C₂×C₅	#1845	1	1	yes	characteristic
180	C₂×C₅	#1956	1	2	yes	—
181	C₂×C₅	#1957	1	2	yes	—
182	C₂×C₅	#1990	1	4	yes	—
183	C₂³	#4015	1	1	yes	characteristic
184	C₂³	#4020	1	1	yes	characteristic
185	C₂³	#4021	1	2	yes	—
186	C₂³	C₂×#677	1	4	yes	—
187	C₂³	C₂×D₄₀	1	2	yes	—
188	C₂³	C₂×Dic₂₀	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
190	C₂³	C₂×#685	1	2	yes	—
191	C₂×C₄	—	20	16	no	—
192	C₂×C₄	—	20	8	no	—
193	C₂×C₄	—	40	4	no	—
194	C₂×C₄	—	40	8	no	—
195	C₂×C₄	—	40	4	no	—
196	C₂×C₄	#3948	1	16	yes	—
197	C₂×C₄	#3949	1	8	yes	—
198	C₂×C₄	#3950	1	8	yes	—
199	C₂×C₄	#3953	1	16	yes	—
200	C₂×C₄	#3963	1	8	yes	—
201	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
202	C₂×C₄	#4015	1	2	yes	—
203	C₂×C₄	#4016	1	2	yes	—
204	C₂×C₄	C₄×D₂₀	1	4	yes	—
205	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
206	C₂×C₄	C₂×#683	1	2	yes	—
207	C₂×C₄	#4070	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
209	C₂×C₄	#4072	1	2	yes	—
210	C₂×C₄	#4076	1	4	yes	—
211	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
212	C₈	—	2	8	no	—
213	C₅	#10841	1	1	yes	characteristic, complemented, Sylow
214	C₂²	#64969	1	2	yes	—
215	C₂²	#64970	1	4	yes	—
216	C₂²	#64971	1	2	yes	—
217	C₂²	#64982	1	2	yes	—
218	C₂²	#64983	1	4	yes	—
219	C₂²	#64984	1	1	yes	characteristic
220	C₂²	#64988	1	1	yes	characteristic
221	C₂²	#65218	1	2	yes	—
222	C₂²	#65224	1	1	yes	characteristic
223	C₂²	#65225	1	1	yes	characteristic
224	C₂²	#65228	1	2	yes	—
225	C₂²	C₂×#3948	1	4	yes	—
226	C₂²	C₂×#3949	1	2	yes	—
227	C₂²	C₂×#3950	1	2	yes	—
228	C₂²	C₂×#3953	1	4	yes	—
229	C₂²	C₂×#3963	1	1	yes	characteristic
230	C₄	—	40	8	no	—
231	C₄	#64771	1	16	yes	complemented
232	C₄	#65217	1	4	yes	—
233	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
234	C₄	#65501	1	2	yes	—
235	C₂	#10600555	1	2	yes	—
236	C₂	#10600557	1	1	yes	characteristic
237	C₂	#10600558	1	1	yes	characteristic
238	C₂	#10600564	1	2	yes	—
239	C₂	#10600565	1	2	yes	—
240	C₂	#10600570	1	4	yes	—
241	C₂	#10602328	1	1	yes	characteristic
242	C₂	C₂×#64771	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#10841
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₅×C₈	C₄	semidirect
11. #65224	C₄	semidirect
59. C₂×C₅×C₈	C₄⋊C₄	semidirect
71. C₂×C₄×C₈	Dic₅	semidirect
141. C₄×C₅	#340	semidirect
168. C₂×C₈	C₄⋊Dic₅	semidirect
213. C₅	#10841	semidirect
231. C₄	#64771	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×#65224	2
C₂×#64771	2
#10841	5