Subgroups — #12086985

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610206	C₂	1	1	yes	maximal, characteristic, complemented
2	#10610201	C₂	1	1	yes	maximal, characteristic
3	#10610199	C₂	1	2	yes	maximal, complemented
4	#10610194	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598857	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598856	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598854	C₂	1	2	yes	maximal, complemented
9	#10598853	C₂	1	1	yes	maximal, characteristic, complemented
10	#10598852	C₂	1	2	yes	maximal, complemented
11	#10598851	C₂	1	1	yes	maximal, characteristic, complemented
12	#65811	C₂²	1	2	yes	—
13	#65804	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	#65405	—	2	2	no	—
18	#65404	—	2	2	no	—
19	#65403	—	2	4	no	—
20	#65402	—	2	4	no	—
21	#64988	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₀	—	2	2	no	—
23	#64986	—	2	4	no	—
24	#64984	C₂²	1	1	yes	characteristic
25	#64983	C₂²	1	2	yes	—
26	#64983	C₂²	1	2	yes	—
27	#64983	C₂²	1	2	yes	—
28	#64983	C₂²	1	2	yes	—
29	#64982	C₂²	1	2	yes	—
30	C₄×D₄₀	—	2	2	no	—
31	C₄×#677	—	2	4	no	—
32	#64971	C₂²	1	1	yes	characteristic
33	#64971	C₂²	1	1	yes	characteristic
34	#64970	C₂²	1	2	yes	—
35	#64970	C₂²	1	2	yes	—
36	#64970	C₂²	1	2	yes	—
37	#64970	C₂²	1	2	yes	—
38	#64969	C₂²	1	2	yes	—
39	C₅×C₈²	C₂²	1	1	yes	characteristic
40	C₅×C₈²	C₂²	1	1	yes	characteristic
41	C₅×C₈²	C₂²	1	1	yes	characteristic
42	C₅×C₈²	C₂²	1	1	yes	characteristic
43	#19600	—	5	1	no	maximal, Sylow
44	C₂×#709	—	2	2	no	—
45	C₂²×C₅×C₈	D₄	1	2	yes	—
46	C₂²×C₅×C₈	D₄	1	2	yes	—
47	C₂²×C₅×C₈	D₄	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	#4076	—	2	2	no	—
50	#4076	—	2	2	no	—
51	#4072	—	2	2	no	—
52	#4070	—	2	2	no	—
53	#4070	—	2	2	no	—
54	C₂×Dic₂₀	—	2	2	no	—
55	C₂×D₄₀	—	2	2	no	—
56	C₂×#677	—	2	4	no	—
57	#4021	—	2	2	no	—
58	#4020	C₂³	1	2	yes	—
59	C₄×D₂₀	—	2	2	no	—
60	C₄×D₂₀	—	2	2	no	—
61	#4016	—	2	2	no	—
62	#4015	C₂³	1	2	yes	—
63	#4015	C₂³	1	2	yes	—
64	#4015	C₂³	1	2	yes	—
65	C₄×Dic₁₀	—	2	2	no	—
66	C₄×Dic₁₀	—	2	2	no	—
67	C₄×C₅×C₈	D₄	1	2	yes	—
68	C₄×C₅×C₈	D₄	1	2	yes	—
69	C₄×C₅×C₈	D₄	1	2	yes	—
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	#3953	—	2	4	no	—
77	#3953	—	2	4	no	—
78	#3953	—	2	4	no	—
79	#3953	—	2	4	no	—
80	#3950	—	2	2	no	—
81	#3950	—	2	2	no	—
82	#3949	—	2	4	no	—
83	#3948	—	2	4	no	—
84	#3948	—	2	4	no	—
85	#3948	—	2	4	no	—
86	#3948	—	2	4	no	—
87	#3170	—	5	1	no	—
88	#3169	—	5	1	no	—
89	#3167	—	5	2	no	—
90	#3131	—	5	2	no	—
91	#1711	—	5	2	no	—
92	#1710	—	5	1	no	—
93	#1709	—	5	1	no	—
94	#1708	—	5	2	no	—
95	#1707	—	5	1	no	—
96	#1706	—	5	1	no	—
97	C₂×C₈²	D₅	1	1	yes	characteristic, complemented
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂×C₂²⋊D₅	—	4	2	no	—
102	#709	—	4	8	no	—
103	C₂×D₂₀	—	2	2	no	—
104	C₂×D₂₀	—	2	2	no	—
105	C₂×D₂₀	—	2	2	no	—
106	C₂×C₄×D₅	—	4	2	no	—
107	C₂×Dic₁₀	—	2	2	no	—
108	C₂×Dic₁₀	—	2	2	no	—
109	C₂×Dic₁₀	—	2	2	no	—
110	C₂×C₅×C₈	—	2	4	no	—
111	C₂×C₅×C₈	—	2	4	no	—
112	C₂×C₅×C₈	—	2	4	no	—
113	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
114	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₅	—	4	2	no	—
124	#685	—	4	2	no	—
125	#685	—	4	2	no	—
126	C₄⋊Dic₅	—	2	2	no	—
127	C₄⋊Dic₅	—	2	2	no	—
128	C₄⋊Dic₅	—	2	2	no	—
129	C₄⋊Dic₅	—	2	2	no	—
130	C₄⋊Dic₅	—	2	2	no	—
131	C₄⋊Dic₅	—	2	2	no	—
132	#683	—	4	2	no	—
133	#683	—	4	2	no	—
134	C₄×Dic₅	—	4	2	no	—
135	Dic₂₀	—	4	4	no	—
136	D₄₀	—	4	4	no	—
137	#677	—	4	8	no	—
138	#533	—	5	2	no	—
139	#532	—	5	2	no	—
140	#500	—	5	1	no	—
141	#500	—	5	1	no	—
142	#498	—	5	2	no	—
143	#494	—	5	1	no	—
144	#493	—	5	1	no	—
145	#492	—	5	2	no	—
146	#488	—	10	4	no	—
147	#487	—	5	2	no	—
148	#487	—	5	2	no	—
149	#487	—	5	2	no	—
150	#487	—	5	2	no	—
151	#486	—	5	2	no	—
152	#486	—	5	2	no	—
153	#486	—	5	2	no	—
154	#486	—	5	2	no	—
155	#467	—	10	2	no	—
156	#466	—	10	2	no	—
157	#465	—	10	4	no	—
158	C₄×Q₁₆	—	10	2	no	—
159	C₄×SD₁₆	—	10	4	no	—
160	C₄×D₈	—	10	2	no	—
161	#416	—	10	4	no	—
162	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
165	C₈²	C₂×D₅	1	1	yes	characteristic
166	C₈²	C₂×D₅	1	1	yes	characteristic
167	C₈²	C₂×D₅	1	1	yes	characteristic
168	C₈²	C₂×D₅	1	1	yes	characteristic
169	C₂³×C₅	#127	1	1	yes	characteristic
170	C₂²×D₅	—	4	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	#127	1	1	yes	characteristic
174	C₂×C₄×C₅	#127	1	1	yes	characteristic
175	C₂×C₄×C₅	#127	1	1	yes	characteristic
176	C₂×C₄×C₅	#135	1	4	yes	—
177	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
180	C₂²⋊D₅	—	8	4	no	—
181	C₂×Dic₅	—	4	2	no	—
182	C₂×Dic₅	—	4	2	no	—
183	C₂×Dic₅	—	4	2	no	—
184	D₂₀	—	4	4	no	—
185	D₂₀	—	4	4	no	—
186	D₂₀	—	4	4	no	—
187	C₄×D₅	—	8	4	no	—
188	Dic₁₀	—	4	4	no	—
189	Dic₁₀	—	4	4	no	—
190	Dic₁₀	—	4	4	no	—
191	C₅×C₈	—	2	2	no	—
192	C₅×C₈	—	2	2	no	—
193	C₅×C₈	—	2	2	no	—
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	#135	1	4	yes	—
196	C₅×C₈	#135	1	4	yes	—
197	C₂×#41	—	10	2	no	—
198	C₂×Q₁₆	—	10	2	no	—
199	C₂×SD₁₆	—	10	4	no	—
200	C₂×D₈	—	10	2	no	—
201	C₂²×C₈	D₂₀	1	2	yes	—
202	C₂²×C₈	D₂₀	1	2	yes	—
203	C₂²×C₈	D₂₀	1	2	yes	—
204	C₄⋊Q₈	—	5	2	no	—
205	C₄⋊Q₈	—	5	2	no	—
206	C₄⋊Q₈	—	5	2	no	—
207	#127	—	5	2	no	—
208	#125	—	10	2	no	—
209	#124	—	10	2	no	—
210	C₂²⋊Q₈	—	10	2	no	—
211	C₂²⋊Q₈	—	10	2	no	—
212	#121	—	10	2	no	—
213	#121	—	10	2	no	—
214	C₄×Q₈	—	10	2	no	—
215	C₄×Q₈	—	10	2	no	—
216	C₄×D₄	—	10	2	no	—
217	C₄×D₄	—	10	2	no	—
218	#117	—	10	2	no	—
219	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
220	#107	—	10	2	no	—
221	#107	—	10	2	no	—
222	#106	—	10	4	no	—
223	#103	—	10	4	no	—
224	#103	—	10	4	no	—
225	#103	—	10	4	no	—
226	#103	—	10	4	no	—
227	#102	—	10	4	no	—
228	#102	—	10	4	no	—
229	#102	—	10	4	no	—
230	#102	—	10	4	no	—
231	C₄×C₈	D₂₀	1	2	yes	—
232	C₄×C₈	D₂₀	1	2	yes	—
233	C₄×C₈	D₂₀	1	2	yes	—
234	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	#495	1	2	yes	—
243	C₂²×C₅	C₂×#127	1	1	yes	characteristic
244	C₂×D₅	—	8	2	no	—
245	C₂×D₅	—	8	2	no	—
246	C₂×D₅	—	8	2	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	#495	1	2	yes	—
250	C₄×C₅	C₂×#135	1	2	yes	—
251	C₄×C₅	C₂×#135	1	2	yes	—
252	C₄×C₅	C₂×#135	1	2	yes	—
253	Dic₅	—	8	2	no	—
254	Dic₅	—	8	2	no	—
255	Dic₅	—	8	2	no	—
256	#41	—	20	8	no	—
257	C₂×Q₈	—	10	2	no	—
258	C₂×Q₈	—	10	2	no	—
259	C₂×Q₈	—	10	2	no	—
260	C₂×D₄	—	10	2	no	—
261	C₂×D₄	—	10	2	no	—
262	C₂×D₄	—	10	2	no	—
263	C₂×D₄	—	20	2	no	—
264	C₂²×C₄	—	20	2	no	—
265	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
268	Q₁₆	—	20	4	no	—
269	SD₁₆	—	20	8	no	—
270	D₈	—	20	4	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	C₂×D₂₀	1	2	yes	—
275	C₂×C₈	C₂×D₂₀	1	2	yes	—
276	C₂×C₈	C₂×D₂₀	1	2	yes	—
277	C₂×C₈	C₂×D₂₀	1	2	yes	—
278	C₂×C₈	C₂×D₂₀	1	2	yes	—
279	C₂×C₈	C₂×D₂₀	1	2	yes	—
280	C₄⋊C₄	—	10	2	no	—
281	C₄⋊C₄	—	10	2	no	—
282	C₄⋊C₄	—	10	2	no	—
283	C₄⋊C₄	—	10	2	no	—
284	C₄⋊C₄	—	10	2	no	—
285	C₄⋊C₄	—	10	2	no	—
286	C₄⋊C₄	—	20	2	no	—
287	C₄⋊C₄	—	20	2	no	—
288	C₂²⋊C₄	—	20	2	no	—
289	C₂²⋊C₄	—	20	2	no	—
290	C₂²⋊C₄	—	20	2	no	—
291	C₄²	—	20	2	no	—
292	C₄²	C₂×D₂₀	1	1	yes	characteristic
293	C₄²	C₂×D₂₀	1	1	yes	characteristic
294	C₄²	C₂×D₂₀	1	1	yes	characteristic
295	C₄²	C₂³×D₅	1	1	yes	characteristic
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	C₂×#495	1	1	yes	characteristic
298	C₂×C₅	#3147	1	1	yes	characteristic
299	C₂×C₅	#3147	1	1	yes	characteristic
300	D₅	—	16	2	no	—
301	C₂³	—	20	2	no	—
302	C₂³	#4020	1	1	yes	characteristic
303	Q₈	—	20	4	no	—
304	Q₈	—	20	4	no	—
305	Q₈	—	20	4	no	—
306	D₄	—	20	4	no	—
307	D₄	—	20	4	no	—
308	D₄	—	20	4	no	—
309	D₄	—	40	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	20	2	no	—
313	C₂×C₄	—	20	2	no	—
314	C₂×C₄	—	20	2	no	—
315	C₂×C₄	—	40	4	no	—
316	C₂×C₄	#4020	1	1	yes	characteristic
317	C₂×C₄	#4020	1	1	yes	characteristic
318	C₂×C₄	#4020	1	1	yes	characteristic
319	C₂×C₄	#4050	1	4	yes	—
320	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	#4050	1	4	yes	—
328	C₈	#4050	1	4	yes	—
329	C₅	#19600	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	40	2	no	—
333	C₂²	—	40	2	no	—
334	C₂²	—	40	2	no	—
335	C₂²	#64985	1	2	yes	—
336	C₂²	C₂×#4020	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	40	2	no	—
340	C₄	—	40	2	no	—
341	C₄	—	40	2	no	—
342	C₄	#64985	1	2	yes	—
343	C₄	C₂×#4050	1	2	yes	—
344	C₄	C₂×#4050	1	2	yes	—
345	C₄	C₂×#4050	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	80	2	no	—
348	C₂	C₂×#64985	1	1	yes	characteristic
349	C₂	#10610198	1	1	yes	characteristic
350	C₂	#10610198	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4020
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×C₈²	C₂
socle	C₂²×C₅	C₂×#127
prime radical	#10610201	C₂
hypercenter	C₂×C₈²	D₅
hypocenter	C₅	#19600
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #10610206	C₂	semidirect
3. #10610199	C₂	semidirect
4. #10610194	C₂	semidirect
5. C₂×C₅×C₈²	C₂	semidirect
6. #10598857	C₂	semidirect
7. #10598856	C₂	semidirect
8. #10598854	C₂	semidirect
9. #10598853	C₂	semidirect
10. #10598852	C₂	semidirect
11. #10598851	C₂	semidirect
97. C₂×C₈²	D₅	semidirect
329. C₅	#19600	semidirect

Maximal subgroups

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

subgroup	index
#10610206	2
#10610201	2
#10610199	2
#10610194	2
C₂×C₅×C₈²	2
#10598857	2
#10598856	2
#10598854	2
#10598853	2
#10598852	2
#10598851	2
#19600	5