Subgroups — #12087265

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611044	C₂	1	2	yes	maximal
2	#10610237	C₂	1	2	yes	maximal
3	#10610206	C₂	1	1	yes	maximal, characteristic
4	C₂×#64971	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	#65804	C₂²	1	2	yes	—
10	C₂×#4015	C₂²	1	2	yes	—
11	C₅×#431	C₂²	1	1	yes	characteristic
12	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65410	—	2	4	no	—
15	#65405	—	2	2	no	—
16	C₂×#3950	—	2	2	no	—
17	#65138	—	2	4	no	—
18	#65028	—	2	4	no	—
19	#64991	—	2	4	no	—
20	#64988	C₂²	1	1	yes	characteristic
21	#64988	C₂²	1	4	yes	—
22	C₄×Dic₂₀	—	2	2	no	—
23	#64971	—	2	2	no	—
24	#64971	C₂²	1	1	yes	characteristic
25	#64971	C₂²	1	4	yes	—
26	#64971	C₂²	1	2	yes	—
27	#64971	C₂²	1	2	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	#19633	—	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	2	no	—
44	C₂×Dic₂₀	—	2	8	no	—
45	#4016	—	2	2	no	—
46	#4015	—	2	2	no	—
47	#4015	—	2	2	no	—
48	#4015	—	2	2	no	—
49	#4015	C₂³	1	4	yes	—
50	#4015	C₂³	1	2	yes	—
51	#4015	C₂³	1	2	yes	—
52	C₄×Dic₁₀	—	2	2	no	—
53	C₄×Dic₁₀	—	2	2	no	—
54	C₅×C₄⋊C₈	D₄	1	2	yes	—
55	C₅×C₂²⋊C₈	D₄	1	2	yes	—
56	C₅×#97	D₄	1	2	yes	—
57	C₄×C₅×C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂³	1	2	yes	—
60	C₄×C₅×C₈	C₂³	1	2	yes	—
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	#3950	—	2	4	no	—
63	#3950	—	2	4	no	—
64	#3950	—	2	2	no	—
65	#3950	—	2	8	no	—
66	#3949	—	2	8	no	—
67	#3948	—	2	8	no	—
68	#3948	—	2	4	no	—
69	#3209	—	5	2	no	—
70	#3170	—	5	1	no	—
71	#3165	—	5	2	no	—
72	C₂×#500	—	5	1	no	—
73	#1723	—	5	4	no	—
74	#1713	—	5	2	no	—
75	#1707	—	5	2	no	—
76	C₈×M₁₆	D₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	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×Dic₅	—	4	2	no	—
81	C₂×Dic₁₀	—	2	4	no	—
82	C₂×Dic₁₀	—	2	2	no	—
83	C₂×Dic₁₀	—	2	2	no	—
84	C₂×Dic₁₀	—	4	2	no	—
85	C₂×Dic₁₀	—	8	4	no	—
86	C₅×M₁₆	Q₁₆	1	8	yes	complemented
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
91	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂⁴	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	4	2	no	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	—	2	2	no	—
99	C₄⋊Dic₅	—	2	2	no	—
100	C₄⋊Dic₅	—	2	4	no	—
101	C₄⋊Dic₅	—	2	4	no	—
102	C₄⋊Dic₅	—	2	2	no	—
103	C₄⋊Dic₅	—	4	2	no	—
104	C₄⋊Dic₅	—	4	2	no	—
105	#683	—	4	2	no	—
106	#683	—	4	2	no	—
107	C₄×Dic₅	—	4	2	no	—
108	Dic₂₀	—	4	4	no	—
109	Dic₂₀	—	8	8	no	—
110	#533	—	5	2	no	—
111	C₂×C₄⋊Q₈	—	5	2	no	—
112	#501	—	10	4	no	—
113	#500	—	5	2	no	—
114	#500	—	5	2	no	—
115	#500	—	5	4	no	—
116	#500	—	5	1	no	—
117	#500	—	10	2	no	—
118	#498	—	5	4	no	—
119	#494	—	5	1	no	—
120	#494	—	5	4	no	—
121	#487	—	5	2	no	—
122	#487	—	5	4	no	—
123	#467	—	10	2	no	—
124	#462	—	10	4	no	—
125	#451	—	10	4	no	—
126	C₄×Q₁₆	—	10	2	no	—
127	#431	C₂×D₅	1	1	yes	characteristic
128	#428	—	10	4	no	—
129	C₂×#107	—	10	2	no	—
130	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
132	#322	C₂×D₅	1	2	yes	—
133	C₈²	C₂×D₅	1	2	yes	—
134	C₂³×C₅	#127	1	1	yes	characteristic
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	#127	1	1	yes	characteristic
138	C₂×C₄×C₅	#127	1	1	yes	characteristic
139	C₂×C₄×C₅	#127	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
141	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₅	—	4	2	no	—
145	C₂×Dic₅	—	4	2	no	—
146	C₂×Dic₅	—	4	4	no	—
147	C₂×Dic₅	—	8	4	no	—
148	Dic₁₀	—	4	4	no	—
149	Dic₁₀	—	4	4	no	—
150	Dic₁₀	—	4	8	no	—
151	Dic₁₀	—	8	4	no	—
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	2	4	no	—
154	C₅×C₈	C₂×Q₁₆	1	8	yes	—
155	C₅×C₈	#137	1	4	yes	—
156	C₂²×Q₈	—	10	2	no	—
157	C₂×Q₁₆	—	10	2	no	—
158	C₂×Q₁₆	—	10	8	no	—
159	C₂×M₁₆	D₂₀	1	2	yes	—
160	C₂²×C₈	D₂₀	1	2	yes	—
161	C₄⋊Q₈	—	5	4	no	—
162	C₄⋊Q₈	—	5	2	no	—
163	C₄⋊Q₈	—	5	2	no	—
164	C₄⋊Q₈	—	10	2	no	—
165	C₄⋊Q₈	—	10	2	no	—
166	C₄⋊Q₈	—	10	2	no	—
167	#125	—	10	2	no	—
168	C₂²⋊Q₈	—	10	2	no	—
169	C₂²⋊Q₈	—	10	2	no	—
170	C₄×Q₈	—	10	2	no	—
171	C₄×Q₈	—	10	2	no	—
172	#117	—	10	2	no	—
173	C₂×C₄⋊C₄	—	10	2	no	—
174	C₂×C₄⋊C₄	—	10	2	no	—
175	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
176	#107	—	10	4	no	—
177	#107	—	10	4	no	—
178	#107	—	10	8	no	—
179	#107	—	10	2	no	—
180	#106	—	10	8	no	—
181	C₄⋊C₈	D₂₀	1	2	yes	—
182	#103	—	10	4	no	—
183	#103	—	10	8	no	—
184	C₂²⋊C₈	D₂₀	1	2	yes	—
185	#97	D₂₀	1	2	yes	—
186	C₄×C₈	D₂₀	1	2	yes	—
187	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₅	1	2	yes	—
189	C₄×C₈	C₂²×D₅	1	2	yes	—
190	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	#494	1	2	yes	—
194	C₂²×C₅	C₂×#127	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	#494	1	2	yes	—
198	C₄×C₅	C₂²×Q₁₆	1	2	yes	—
199	C₄×C₅	C₂×#137	1	2	yes	—
200	C₄×C₅	#576	1	2	yes	—
201	Dic₅	—	8	2	no	—
202	Dic₅	—	8	4	no	—
203	Dic₅	—	8	2	no	—
204	C₂×Q₈	—	10	2	no	—
205	C₂×Q₈	—	10	4	no	—
206	C₂×Q₈	—	10	2	no	—
207	C₂×Q₈	—	20	2	no	—
208	C₂×Q₈	—	40	4	no	—
209	C₂²×C₄	—	20	2	no	—
210	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
213	Q₁₆	—	20	4	no	—
214	Q₁₆	—	40	8	no	—
215	M₁₆	Dic₂₀	1	8	yes	complemented
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	C₂×D₂₀	1	2	yes	—
218	C₂×C₈	C₂×D₂₀	1	4	yes	—
219	C₂×C₈	C₂×D₂₀	1	4	yes	—
220	C₂×C₈	C₂×D₂₀	1	2	yes	—
221	C₄⋊C₄	—	10	2	no	—
222	C₄⋊C₄	—	10	2	no	—
223	C₄⋊C₄	—	10	2	no	—
224	C₄⋊C₄	—	10	4	no	—
225	C₄⋊C₄	—	10	4	no	—
226	C₄⋊C₄	—	10	2	no	—
227	C₄⋊C₄	—	20	2	no	—
228	C₄⋊C₄	—	20	2	no	—
229	C₄⋊C₄	—	20	2	no	—
230	C₄⋊C₄	—	20	2	no	—
231	C₂²⋊C₄	—	20	2	no	—
232	C₄²	—	20	2	no	—
233	C₄²	C₂×D₂₀	1	1	yes	characteristic
234	C₄²	C₂×D₂₀	1	1	yes	characteristic
235	C₄²	C₂×D₂₀	1	1	yes	characteristic
236	C₄²	C₂³×D₅	1	1	yes	characteristic
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	C₂×#494	1	1	yes	characteristic
239	C₂×C₅	#3152	1	1	yes	characteristic
240	C₂×C₅	#3155	1	1	yes	characteristic
241	C₂³	#4020	1	1	yes	characteristic
242	Q₈	—	20	4	no	—
243	Q₈	—	20	4	no	—
244	Q₈	—	20	8	no	—
245	Q₈	—	40	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	20	2	no	—
249	C₂×C₄	—	20	2	no	—
250	C₂×C₄	—	20	4	no	—
251	C₂×C₄	—	40	4	no	—
252	C₂×C₄	#4020	1	1	yes	characteristic
253	C₂×C₄	#4020	1	1	yes	characteristic
254	C₂×C₄	#4020	1	1	yes	characteristic
255	C₂×C₄	C₂×Dic₂₀	1	4	yes	—
256	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	4	no	—
261	C₈	C₂×Dic₂₀	1	8	yes	—
262	C₈	#4055	1	4	yes	—
263	C₅	#19633	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#64988	1	2	yes	—
267	C₂²	C₂×#4020	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	40	2	no	—
271	C₄	—	40	4	no	—
272	C₄	—	40	2	no	—
273	C₄	#64988	1	2	yes	—
274	C₄	C₂²×Dic₂₀	1	2	yes	—
275	C₄	C₂×#4055	1	2	yes	—
276	C₄	#66086	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₂×#64988	1	1	yes	characteristic
279	C₂	#10610269	1	1	yes	characteristic
280	C₂	#10610296	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	#19633
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
86. C₅×M₁₆	Q₁₆	semidirect
215. M₁₆	Dic₂₀	semidirect
263. C₅	#19633	semidirect

Maximal subgroups

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

subgroup	index
#10611044	2
#10610237	2
#10610206	2
C₂×#64971	2
C₅×C₈×M₁₆	2
#10598892	2
#10598858	2
#10598856	2
#19633	5