Subgroups — #12087257

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611003	C₂	1	2	yes	maximal
2	#10610237	C₂	1	2	yes	maximal
3	#10610206	C₂	1	1	yes	maximal, characteristic
4	C₂×#64988	C₂	1	1	yes	maximal, characteristic
5	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
6	#10598892	C₂	1	4	yes	maximal, complemented
7	#10598858	C₂	1	2	yes	maximal, complemented
8	#10598856	C₂	1	2	yes	maximal, complemented
9	C₂²×Dic₂₀	—	2	2	no	—
10	#65804	C₂²	1	2	yes	—
11	C₂×#4015	C₂²	1	2	yes	—
12	C₅×#431	C₂²	1	1	yes	characteristic
13	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65410	—	2	4	no	—
16	#65405	—	2	2	no	—
17	#65139	—	2	4	no	—
18	#65011	—	2	4	no	—
19	#64991	—	2	4	no	—
20	#64988	—	2	2	no	—
21	#64988	C₂²	1	2	yes	—
22	#64988	C₂²	1	2	yes	—
23	#64988	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₀	—	2	2	no	—
25	#64971	C₂²	1	1	yes	characteristic
26	#64971	C₂²	1	4	yes	—
27	#64971	C₂²	1	4	yes	—
28	#64970	C₂²	1	2	yes	—
29	#64970	C₂²	1	4	yes	—
30	#64969	C₂²	1	4	yes	—
31	C₅×#322	C₂²	1	2	yes	—
32	C₅×C₈²	C₂²	1	2	yes	—
33	#19632	—	5	1	no	maximal, Sylow
34	C₂²×Dic₁₀	—	2	2	no	—
35	C₂×C₅×M₁₆	D₄	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	#4072	—	2	2	no	—
39	C₂×C₄⋊Dic₅	—	2	2	no	—
40	C₂×C₄⋊Dic₅	—	2	2	no	—
41	#4070	—	2	2	no	—
42	#4070	—	2	2	no	—
43	C₂×Dic₂₀	—	2	4	no	—
44	C₂×Dic₂₀	—	2	4	no	—
45	C₂×Dic₂₀	—	2	2	no	—
46	C₂×Dic₂₀	—	4	8	no	—
47	#4016	—	2	2	no	—
48	#4015	—	2	2	no	—
49	#4015	—	2	2	no	—
50	#4015	—	2	2	no	—
51	#4015	C₂³	1	4	yes	—
52	#4015	C₂³	1	2	yes	—
53	#4015	C₂³	1	2	yes	—
54	C₄×Dic₁₀	—	2	2	no	—
55	C₄×Dic₁₀	—	2	2	no	—
56	C₅×C₄⋊C₈	D₄	1	2	yes	—
57	C₅×C₂²⋊C₈	D₄	1	2	yes	—
58	C₅×#97	D₄	1	2	yes	—
59	C₄×C₅×C₈	D₄	1	2	yes	—
60	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	2	yes	—
62	C₄×C₅×C₈	C₂³	1	2	yes	—
63	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
64	#3950	—	2	8	no	—
65	#3950	—	2	2	no	—
66	#3950	—	2	8	no	—
67	#3949	—	2	8	no	—
68	#3948	—	2	8	no	—
69	#3948	—	2	4	no	—
70	#3230	—	5	2	no	—
71	#3170	—	5	1	no	—
72	#3165	—	5	2	no	—
73	C₂×#494	—	5	1	no	—
74	#1723	—	5	4	no	—
75	#1713	—	5	2	no	—
76	#1707	—	5	2	no	—
77	C₈×M₁₆	D₅	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₅	—	4	2	no	—
82	C₂×Dic₁₀	—	2	4	no	—
83	C₂×Dic₁₀	—	2	2	no	—
84	C₂×Dic₁₀	—	2	2	no	—
85	C₂×Dic₁₀	—	4	8	no	—
86	C₂×Dic₁₀	—	4	2	no	—
87	C₅×M₁₆	Q₁₆	1	8	yes	complemented
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
91	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
92	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₅	—	4	2	no	—
98	C₄⋊Dic₅	—	2	2	no	—
99	C₄⋊Dic₅	—	2	4	no	—
100	C₄⋊Dic₅	—	2	2	no	—
101	C₄⋊Dic₅	—	2	4	no	—
102	C₄⋊Dic₅	—	2	2	no	—
103	C₄⋊Dic₅	—	2	2	no	—
104	C₄⋊Dic₅	—	4	2	no	—
105	C₄⋊Dic₅	—	4	2	no	—
106	#683	—	4	2	no	—
107	#683	—	4	2	no	—
108	C₄×Dic₅	—	4	2	no	—
109	Dic₂₀	—	4	8	no	—
110	Dic₂₀	—	4	4	no	—
111	Dic₂₀	—	4	8	no	—
112	C₂²×Q₁₆	—	10	2	no	—
113	#533	—	5	2	no	—
114	C₂×C₄⋊Q₈	—	5	2	no	—
115	#501	—	10	4	no	—
116	#500	—	5	4	no	—
117	#500	—	5	1	no	—
118	#500	—	5	4	no	—
119	#498	—	5	4	no	—
120	#494	—	5	2	no	—
121	#494	—	5	2	no	—
122	#494	—	5	1	no	—
123	#494	—	10	2	no	—
124	#487	—	5	2	no	—
125	#487	—	5	4	no	—
126	#484	—	10	4	no	—
127	#477	—	10	4	no	—
128	#467	—	10	2	no	—
129	C₄×Q₁₆	—	10	2	no	—
130	#431	C₂×D₅	1	1	yes	characteristic
131	#428	—	10	4	no	—
132	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
134	#322	C₂×D₅	1	2	yes	—
135	C₈²	C₂×D₅	1	2	yes	—
136	C₂³×C₅	#127	1	1	yes	characteristic
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	#127	1	1	yes	characteristic
140	C₂×C₄×C₅	#127	1	1	yes	characteristic
141	C₂×C₄×C₅	#127	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₅	—	4	2	no	—
147	C₂×Dic₅	—	4	2	no	—
148	C₂×Dic₅	—	4	4	no	—
149	C₂×Dic₅	—	8	4	no	—
150	Dic₁₀	—	4	8	no	—
151	Dic₁₀	—	4	4	no	—
152	Dic₁₀	—	4	4	no	—
153	Dic₁₀	—	8	4	no	—
154	C₅×C₈	—	2	2	no	—
155	C₅×C₈	—	2	4	no	—
156	C₅×C₈	C₂×Q₁₆	1	8	yes	complemented
157	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
158	C₂²×Q₈	—	10	2	no	—
159	C₂×Q₁₆	—	10	4	no	—
160	C₂×Q₁₆	—	10	4	no	—
161	C₂×Q₁₆	—	10	2	no	—
162	C₂×Q₁₆	—	20	8	no	—
163	C₂×M₁₆	D₂₀	1	2	yes	—
164	C₂²×C₈	D₂₀	1	2	yes	—
165	C₄⋊Q₈	—	5	4	no	—
166	C₄⋊Q₈	—	5	2	no	—
167	C₄⋊Q₈	—	5	2	no	—
168	C₄⋊Q₈	—	10	2	no	—
169	C₄⋊Q₈	—	10	2	no	—
170	C₄⋊Q₈	—	10	2	no	—
171	#125	—	10	2	no	—
172	C₂²⋊Q₈	—	10	2	no	—
173	C₂²⋊Q₈	—	10	2	no	—
174	C₄×Q₈	—	10	2	no	—
175	C₄×Q₈	—	10	2	no	—
176	#117	—	10	2	no	—
177	C₂×C₄⋊C₄	—	10	2	no	—
178	C₂×C₄⋊C₄	—	10	2	no	—
179	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
180	#107	—	10	8	no	—
181	#107	—	10	2	no	—
182	#107	—	10	8	no	—
183	#106	—	10	8	no	—
184	C₄⋊C₈	D₂₀	1	2	yes	—
185	#103	—	10	4	no	—
186	#103	—	10	8	no	—
187	C₂²⋊C₈	D₂₀	1	2	yes	—
188	#97	D₂₀	1	2	yes	—
189	C₄×C₈	D₂₀	1	2	yes	—
190	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅	1	2	yes	—
192	C₄×C₈	C₂²×D₅	1	2	yes	—
193	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	#494	1	2	yes	—
197	C₂²×C₅	C₂×#127	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	#494	1	2	yes	—
201	C₄×C₅	C₂²×Q₁₆	1	2	yes	—
202	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
203	C₄×C₅	#578	1	2	yes	—
204	Dic₅	—	8	4	no	—
205	Dic₅	—	8	2	no	—
206	Dic₅	—	8	2	no	—
207	C₂×Q₈	—	10	2	no	—
208	C₂×Q₈	—	10	2	no	—
209	C₂×Q₈	—	10	4	no	—
210	C₂×Q₈	—	20	8	no	—
211	C₂×Q₈	—	20	2	no	—
212	C₂²×C₄	—	20	2	no	—
213	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
216	Q₁₆	—	20	8	no	—
217	Q₁₆	—	20	8	no	—
218	Q₁₆	—	20	4	no	—
219	M₁₆	Dic₂₀	1	8	yes	complemented
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	C₂×D₂₀	1	2	yes	—
222	C₂×C₈	C₂×D₂₀	1	4	yes	—
223	C₂×C₈	C₂×D₂₀	1	4	yes	—
224	C₂×C₈	C₂×D₂₀	1	2	yes	—
225	C₄⋊C₄	—	10	4	no	—
226	C₄⋊C₄	—	10	2	no	—
227	C₄⋊C₄	—	10	2	no	—
228	C₄⋊C₄	—	10	4	no	—
229	C₄⋊C₄	—	10	2	no	—
230	C₄⋊C₄	—	10	2	no	—
231	C₄⋊C₄	—	20	2	no	—
232	C₄⋊C₄	—	20	2	no	—
233	C₄⋊C₄	—	20	2	no	—
234	C₄⋊C₄	—	20	2	no	—
235	C₂²⋊C₄	—	20	2	no	—
236	C₄²	—	20	2	no	—
237	C₄²	C₂×D₂₀	1	1	yes	characteristic
238	C₄²	C₂×D₂₀	1	1	yes	characteristic
239	C₄²	C₂×D₂₀	1	1	yes	characteristic
240	C₄²	C₂³×D₅	1	1	yes	characteristic
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	C₂×#494	1	1	yes	characteristic
243	C₂×C₅	#3151	1	1	yes	characteristic
244	C₂×C₅	#3156	1	1	yes	characteristic
245	C₂³	#4020	1	1	yes	characteristic
246	Q₈	—	20	8	no	—
247	Q₈	—	20	4	no	—
248	Q₈	—	20	4	no	—
249	Q₈	—	40	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	20	2	no	—
253	C₂×C₄	—	20	2	no	—
254	C₂×C₄	—	20	4	no	—
255	C₂×C₄	—	40	4	no	—
256	C₂×C₄	#4020	1	1	yes	characteristic
257	C₂×C₄	#4020	1	1	yes	characteristic
258	C₂×C₄	#4020	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₂₀	1	4	yes	—
260	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	C₂×Dic₂₀	1	8	yes	complemented
266	C₈	#4054	1	4	yes	—
267	C₅	#19632	1	1	yes	characteristic, complemented, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	#64988	1	2	yes	—
271	C₂²	C₂×#4020	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	40	4	no	—
275	C₄	—	40	2	no	—
276	C₄	—	40	2	no	—
277	C₄	#64988	1	2	yes	—
278	C₄	C₂²×Dic₂₀	1	2	yes	—
279	C₄	C₂×#4054	1	2	yes	—
280	C₄	#66087	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	C₂×#64988	1	1	yes	characteristic
283	C₂	#10610261	1	1	yes	characteristic
284	C₂	#10610307	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4020
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×M₁₆	C₂
socle	C₂²×C₅	C₂×#127
prime radical	C₂³×C₅	#127
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#19632
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
6. #10598892	C₂	semidirect
7. #10598858	C₂	semidirect
8. #10598856	C₂	semidirect
87. C₅×M₁₆	Q₁₆	semidirect
156. C₅×C₈	C₂×Q₁₆	semidirect
219. M₁₆	Dic₂₀	semidirect
265. C₈	C₂×Dic₂₀	semidirect
267. C₅	#19632	semidirect

Maximal subgroups

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

subgroup	index
#10611003	2
#10610237	2
#10610206	2
C₂×#64988	2
C₅×C₈×M₁₆	2
#10598892	2
#10598858	2
#10598856	2
#19632	5