Subgroups — #12253992

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610904	C₂	1	1	yes	maximal, characteristic, complemented
2	#10610898	C₂	1	1	yes	maximal, characteristic
3	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64729	C₂	1	1	yes	maximal, characteristic
5	#10602194	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601445	C₂	1	1	yes	maximal, characteristic
7	#10601442	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600147	C₂	1	2	yes	maximal, complemented
9	#10600146	C₂	1	2	yes	maximal, complemented
10	#10600127	C₂	1	2	yes	maximal, complemented
11	#10600126	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄₀	C₂²	1	1	yes	characteristic
13	#65855	—	2	1	no	—
14	#65853	—	2	1	no	—
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₅×#427	C₂²	1	1	yes	characteristic
18	C₂×#3950	C₂²	1	1	yes	characteristic
19	#65390	C₂²	1	1	yes	characteristic
20	C₂×#3944	C₂²	1	1	yes	characteristic
21	#65290	—	2	1	no	—
22	#65284	—	2	1	no	—
23	#65167	C₂²	1	2	yes	complemented
24	#65163	C₂²	1	2	yes	complemented
25	#65161	C₂²	1	2	yes	—
26	#65161	C₂²	1	2	yes	—
27	#65158	—	2	2	no	—
28	#65145	—	2	2	no	—
29	C₅×#367	C₂²	1	1	yes	characteristic, complemented
30	C₅×#367	C₂²	1	1	yes	characteristic, complemented
31	C₅×#366	C₂²	1	1	yes	characteristic, complemented
32	C₅×#366	C₂²	1	1	yes	characteristic, complemented
33	#64729	C₂²	1	2	yes	—
34	#64729	C₂²	1	2	yes	—
35	#64721	C₂²	1	2	yes	complemented
36	#64721	C₂²	1	2	yes	complemented
37	#64711	C₂²	1	2	yes	—
38	#64711	C₂²	1	2	yes	—
39	#64710	C₂²	1	2	yes	complemented
40	#64709	C₂²	1	2	yes	complemented
41	#33631	—	5	1	no	maximal, Sylow
42	C₂²×D₂₀	—	2	1	no	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#117	—	2	1	no	—
45	C₅×#117	—	2	1	no	—
46	#4072	D₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅	—	2	1	no	—
48	C₂×#681	D₄	1	1	yes	characteristic
49	C₂×D₄₀	—	2	1	no	—
50	C₂×D₄₀	—	2	2	no	—
51	C₂×D₄₀	Q₈	1	4	yes	complemented
52	C₂×D₄₀	C₂³	1	2	yes	—
53	#4039	—	2	2	no	—
54	#4039	—	2	2	no	—
55	#4036	—	2	2	no	—
56	#4034	—	2	2	no	—
57	D₅×C₂²⋊C₄	—	4	2	no	—
58	#4024	—	4	2	no	—
59	C₄×D₂₀	—	4	1	no	—
60	C₄×D₂₀	—	4	1	no	—
61	#4016	—	4	1	no	—
62	#4015	—	4	1	no	—
63	C₂×C₅×C₁₆	—	2	2	no	—
64	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
66	C₅×#107	C₂³	1	1	yes	characteristic
67	C₅×#107	C₂³	1	1	yes	characteristic
68	C₅×#107	C₂³	1	1	yes	characteristic
69	C₅×#107	C₂³	1	1	yes	characteristic
70	C₅×#106	—	2	1	no	—
71	C₅×#106	—	2	1	no	—
72	#3950	—	2	1	no	—
73	#3950	C₂³	1	2	yes	—
74	#3947	D₄	1	2	yes	—
75	C₈×Dic₅	C₂³	1	2	yes	—
76	#3944	—	2	2	no	—
77	C₂×#672	C₂³	1	2	yes	—
78	#3941	—	2	2	no	—
79	#3941	—	2	2	no	—
80	#3940	—	2	2	no	—
81	#3939	—	2	2	no	—
82	#3164	—	5	1	no	—
83	#2948	—	5	1	no	—
84	#2228	—	5	2	no	—
85	#2228	—	5	2	no	—
86	#2226	—	5	2	no	—
87	#2224	—	5	2	no	—
88	#2159	—	5	1	no	—
89	#2158	D₅	1	1	yes	characteristic, complemented
90	#2151	—	5	1	no	—
91	#2140	—	5	1	no	—
92	C₂×#357	—	5	1	no	—
93	C₂³×D₅	—	4	1	no	—
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×Dic₅	—	4	1	no	—
96	C₂×D₂₀	—	2	4	no	—
97	C₂×D₂₀	—	2	2	no	—
98	C₂×D₂₀	—	4	1	no	—
99	C₂×D₂₀	—	4	2	no	—
100	C₂×C₄×D₅	—	4	2	no	—
101	C₂×C₄×D₅	—	4	2	no	—
102	C₂×Dic₁₀	—	4	2	no	—
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
105	C₂×C₅×C₈	#41	1	2	yes	—
106	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	1	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₂²⋊C₄	—	4	1	no	—
112	C₅×C₂²⋊C₄	—	4	1	no	—
113	C₄²×C₅	—	4	1	no	—
114	C₄²×C₅	—	4	1	no	—
115	C₂²⋊Dic₅	D₈	1	2	yes	—
116	#685	—	4	2	no	—
117	#685	—	4	2	no	—
118	C₄⋊Dic₅	—	2	2	no	—
119	C₄⋊Dic₅	—	4	1	no	—
120	C₄⋊Dic₅	—	4	2	no	—
121	C₄⋊Dic₅	D₈	1	1	yes	characteristic
122	C₄⋊Dic₅	D₈	1	1	yes	characteristic
123	#683	—	4	2	no	—
124	#683	—	4	2	no	—
125	C₄×Dic₅	C₂×D₄	1	2	yes	—
126	#681	—	2	2	no	—
127	C₂×#168	C₂×D₄	1	2	yes	—
128	D₄₀	—	2	4	no	—
129	D₄₀	—	2	2	no	—
130	D₄₀	C₂×Q₈	1	4	yes	—
131	C₅×C₁₆	—	2	1	no	—
132	C₅×C₁₆	—	2	1	no	—
133	#672	—	2	2	no	—
134	C₂²×D₈	—	5	1	no	—
135	#544	—	10	1	no	—
136	#518	—	10	1	no	—
137	C₂×M₃₂	—	5	1	no	—
138	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
139	#501	—	10	2	no	—
140	#500	—	5	2	no	—
141	#499	—	5	2	no	—
142	#442	—	10	2	no	—
143	C₄×D₈	—	5	2	no	—
144	C₄×D₈	—	5	2	no	—
145	#428	—	10	1	no	—
146	#427	C₂×D₅	1	1	yes	characteristic
147	#427	C₂×D₅	1	1	yes	characteristic
148	C₂×#107	—	5	1	no	—
149	#418	—	10	1	no	—
150	#405	—	5	1	no	—
151	#367	—	5	2	no	—
152	#367	C₂×D₅	1	1	yes	characteristic, complemented
153	#367	C₂×D₅	1	1	yes	characteristic, complemented
154	#366	—	5	2	no	—
155	#366	C₂×D₅	1	1	yes	characteristic, complemented
156	#366	C₂×D₅	1	1	yes	characteristic, complemented
157	C₄⋊C₁₆	—	5	2	no	—
158	C₄⋊C₁₆	—	5	2	no	—
159	#357	—	5	2	no	—
160	#357	—	5	2	no	—
161	#357	—	5	2	no	—
162	#357	—	5	2	no	—
163	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
164	C₂²×D₅	—	4	2	no	—
165	C₂²×D₅	—	4	4	no	—
166	C₂²×D₅	—	16	1	no	—
167	C₂²×D₅	—	16	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
173	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅	#137	1	2	yes	—
175	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
176	C₂×Dic₅	—	4	2	no	—
177	C₂×Dic₅	—	16	1	no	—
178	C₂×Dic₅	C₂×D₈	1	2	yes	—
179	C₂×Dic₅	C₂×D₈	1	2	yes	—
180	D₂₀	—	2	4	no	—
181	D₂₀	—	4	2	no	—
182	D₂₀	—	4	4	no	—
183	C₄×D₅	—	16	2	no	—
184	C₄×D₅	—	16	2	no	—
185	Dic₁₀	—	16	2	no	—
186	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
187	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
188	C₅×C₈	C₂×#41	1	1	yes	characteristic
189	#168	—	2	2	no	—
190	C₂²×D₄	—	10	1	no	—
191	C₂×D₈	—	5	4	no	—
192	C₂×D₈	—	5	2	no	—
193	C₂×D₈	—	10	2	no	—
194	C₂×D₈	—	10	1	no	—
195	C₂×M₁₆	—	5	1	no	—
196	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
197	C₄⋊Q₈	—	10	2	no	—
198	C₄⋊Q₈	—	20	1	no	—
199	#125	—	10	2	no	—
200	#125	—	20	1	no	—
201	C₂²⋊Q₈	—	20	2	no	—
202	C₄×D₄	—	10	2	no	—
203	C₄×D₄	—	10	2	no	—
204	C₄×D₄	—	20	1	no	—
205	C₄×D₄	—	20	1	no	—
206	#117	—	2	1	no	—
207	#117	—	2	1	no	—
208	#117	—	5	1	no	—
209	C₂×C₄⋊C₄	—	10	1	no	—
210	C₂×C₂²⋊C₄	—	20	2	no	—
211	M₃₂	—	10	2	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	5	2	no	—
214	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
216	#107	—	5	2	no	—
217	#107	—	10	2	no	—
218	#107	—	10	1	no	—
219	#107	C₂²×D₅	1	1	yes	characteristic
220	#107	C₂²×D₅	1	1	yes	characteristic
221	#107	C₂²×D₅	1	1	yes	characteristic
222	#107	C₂²×D₅	1	1	yes	characteristic
223	#106	—	2	1	no	—
224	#106	—	2	1	no	—
225	#106	—	10	2	no	—
226	#102	—	10	2	no	—
227	#102	—	10	2	no	—
228	#97	—	5	2	no	—
229	C₄×C₈	—	5	2	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	#474	1	2	yes	—
233	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
234	C₂×D₅	—	4	4	no	—
235	C₂×D₅	—	8	2	no	—
236	C₂×D₅	—	8	2	no	—
237	C₂×D₅	—	16	4	no	—
238	C₄×C₅	—	8	1	no	—
239	C₄×C₅	—	8	1	no	—
240	C₄×C₅	—	8	1	no	—
241	C₄×C₅	—	8	1	no	—
242	C₄×C₅	#474	1	2	yes	—
243	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
244	C₄×C₅	C₂×#137	1	1	yes	characteristic
245	Dic₅	—	2	2	no	—
246	Dic₅	—	2	2	no	—
247	Dic₅	—	8	2	no	—
248	C₂⁴	—	20	1	no	—
249	C₂×Q₈	—	20	2	no	—
250	C₂×D₄	—	10	2	no	—
251	C₂×D₄	—	10	4	no	—
252	C₂×D₄	—	20	1	no	—
253	C₂×D₄	—	20	2	no	—
254	C₂²×C₄	—	20	2	no	—
255	C₂²×C₄	—	20	2	no	—
256	C₂²×C₄	—	20	1	no	—
257	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
258	D₈	—	5	4	no	—
259	D₈	—	10	2	no	—
260	D₈	—	10	4	no	—
261	M₁₆	—	10	2	no	—
262	C₂×C₈	—	5	2	no	—
263	C₂×C₈	#709	1	2	yes	—
264	C₂×C₈	D₄×D₅	1	1	yes	characteristic
265	C₂×C₈	Q₈×D₅	1	2	yes	—
266	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	5	1	no	—
272	C₄⋊C₄	—	5	1	no	—
273	C₄⋊C₄	—	10	2	no	—
274	C₄⋊C₄	—	20	2	no	—
275	C₄⋊C₄	—	20	2	no	—
276	C₄⋊C₄	—	20	2	no	—
277	C₄⋊C₄	—	20	1	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	5	2	no	—
281	C₂²⋊C₄	—	20	2	no	—
282	C₂²⋊C₄	—	20	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	5	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	10	2	no	—
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	C₂×#474	1	1	yes	characteristic
291	C₂×C₅	#3415	1	1	yes	characteristic
292	C₂×C₅	#3416	1	1	yes	characteristic
293	D₅	—	8	4	no	—
294	C₂³	—	20	2	no	—
295	C₂³	—	20	4	no	—
296	C₂³	—	80	1	no	—
297	C₂³	—	80	1	no	—
298	C₂³	D₅×D₈	1	1	yes	characteristic
299	Q₈	—	80	2	no	—
300	D₄	—	10	4	no	—
301	D₄	—	20	2	no	—
302	D₄	—	20	4	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	5	2	no	—
308	C₂×C₄	—	5	2	no	—
309	C₂×C₄	—	20	2	no	—
310	C₂×C₄	—	80	2	no	—
311	C₂×C₄	—	80	2	no	—
312	C₂×C₄	—	80	1	no	—
313	C₂×C₄	#4042	1	2	yes	—
314	C₂×C₄	D₅×D₈	1	1	yes	characteristic
315	C₂×C₄	#4064	1	2	yes	—
316	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
317	C₈	—	10	2	no	—
318	C₈	#4042	1	2	yes	—
319	C₈	C₂×#709	1	1	yes	characteristic
320	C₈	C₂×Q₈×D₅	1	1	yes	characteristic
321	C₅	#33631	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	20	4	no	—
325	C₂²	—	40	2	no	—
326	C₂²	—	40	2	no	—
327	C₂²	—	80	4	no	—
328	C₂²	#65179	1	2	yes	—
329	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	10	2	no	—
335	C₄	—	10	2	no	—
336	C₄	—	40	2	no	—
337	C₄	#65179	1	2	yes	—
338	C₄	C₂×#4042	1	1	yes	characteristic
339	C₄	C₂×#4064	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	40	4	no	—
342	C₂	C₂×#65179	1	1	yes	characteristic
343	C₂	#10613762	1	1	yes	characteristic
344	C₂	#10613763	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×#2158	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂²×D₄₀	C₂²
hypercenter	#2158	D₅
hypocenter	C₅	#33631
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
1. #10610904	C₂	semidirect
3. C₅×#2158	C₂	semidirect
5. #10602194	C₂	semidirect
7. #10601442	C₂	semidirect
8. #10600147	C₂	semidirect
9. #10600146	C₂	semidirect
10. #10600127	C₂	semidirect
11. #10600126	C₂	semidirect
23. #65167	C₂²	semidirect
24. #65163	C₂²	semidirect
29. C₅×#367	C₂²	semidirect
30. C₅×#367	C₂²	semidirect
31. C₅×#366	C₂²	semidirect
32. C₅×#366	C₂²	semidirect
35. #64721	C₂²	semidirect
36. #64721	C₂²	semidirect
39. #64710	C₂²	semidirect
40. #64709	C₂²	semidirect
51. C₂×D₄₀	Q₈	semidirect
89. #2158	D₅	semidirect
152. #367	C₂×D₅	semidirect
153. #367	C₂×D₅	semidirect
155. #366	C₂×D₅	semidirect
156. #366	C₂×D₅	semidirect
321. C₅	#33631	semidirect

Maximal subgroups

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

subgroup	index
#10610904	2
#10610898	2
C₅×#2158	2
C₂×#64729	2
#10602194	2
#10601445	2
#10601442	2
#10600147	2
#10600146	2
#10600127	2
#10600126	2
#33631	5