Subgroups — #12253989

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	C₅×#2948	C₂	1	1	yes	maximal, characteristic
2	#10610277	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602826	C₂	1	1	yes	maximal, characteristic
4	C₂×#64782	C₂	1	1	yes	maximal, characteristic
5	#10601681	C₂	1	2	yes	maximal, complemented
6	#10601681	C₂	1	2	yes	maximal, complemented
7	#10601680	C₂	1	2	yes	maximal, complemented
8	#10601679	C₂	1	2	yes	maximal, complemented
9	#10601442	C₂	1	1	yes	maximal, characteristic, complemented
10	#10601441	C₂	1	1	yes	maximal, characteristic, complemented
11	#10600675	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₅×#518	—	2	1	no	—
14	#65853	—	2	1	no	—
15	C₅×#442	—	2	2	no	—
16	C₄×C₅×D₈	C₂²	1	2	yes	—
17	C₄×C₅×D₈	C₂²	1	2	yes	—
18	C₅×#427	C₂²	1	1	yes	characteristic
19	C₅×#418	—	2	1	no	—
20	C₅×#405	C₂²	1	1	yes	characteristic
21	#65410	—	2	1	no	—
22	#65395	C₂²	1	1	yes	characteristic
23	C₂×#3950	C₂²	1	1	yes	characteristic
24	C₂×#3944	C₂²	1	1	yes	characteristic
25	C₂²×#672	C₂²	1	1	yes	characteristic
26	#64991	—	2	2	no	—
27	#64972	C₂²	1	2	yes	complemented
28	#64971	C₂²	1	2	yes	complemented
29	#64782	C₂²	1	2	yes	—
30	#64782	C₂²	1	2	yes	—
31	#64782	C₂²	1	2	yes	—
32	#64782	C₂²	1	2	yes	—
33	#64710	C₂²	1	1	yes	characteristic, complemented
34	#64710	C₂²	1	2	yes	complemented
35	#64710	C₂²	1	1	yes	characteristic, complemented
36	#64709	C₂²	1	1	yes	characteristic, complemented
37	#64709	C₂²	1	2	yes	complemented
38	#64709	C₂²	1	1	yes	characteristic, complemented
39	C₄⋊#672	C₂²	1	2	yes	complemented
40	C₄⋊#672	C₂²	1	2	yes	complemented
41	#33631	—	5	1	no	maximal, Sylow
42	C₂²×C₅×D₄	—	2	1	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×D₈	—	2	2	no	—
45	C₂×C₅×D₈	Q₈	1	4	yes	complemented
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₄×C₅×D₄	—	2	2	no	—
50	C₄×C₅×D₄	—	2	2	no	—
51	C₄×C₅×D₄	—	4	1	no	—
52	C₄×C₅×D₄	—	4	1	no	—
53	C₅×#117	—	2	1	no	—
54	C₅×#117	D₄	1	1	yes	characteristic
55	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
56	#4072	—	2	1	no	—
57	C₂×C₄⋊Dic₅	—	2	1	no	—
58	#4036	—	4	1	no	—
59	#4034	—	4	1	no	—
60	#4024	—	4	2	no	—
61	#4016	—	2	2	no	—
62	#4015	—	2	2	no	—
63	C₅×#107	C₂³	1	1	yes	characteristic
64	C₅×#107	C₂³	1	1	yes	characteristic
65	C₅×#106	—	2	1	no	—
66	C₅×#102	—	2	2	no	—
67	C₅×#102	—	2	2	no	—
68	C₅×#97	D₄	1	2	yes	—
69	C₄×C₅×C₈	C₂³	1	2	yes	—
70	#3950	—	2	1	no	—
71	#3950	—	2	2	no	—
72	#3950	C₂³	1	2	yes	—
73	#3950	C₂³	1	1	yes	characteristic
74	#3950	C₂³	1	1	yes	characteristic
75	#3949	—	2	1	no	—
76	#3949	—	2	2	no	—
77	#3944	—	2	2	no	—
78	C₂×#672	—	2	2	no	—
79	C₂×#672	C₂³	1	1	yes	characteristic
80	C₂×#672	C₂³	1	1	yes	characteristic
81	C₂×#672	C₂³	1	2	yes	—
82	#3164	—	5	1	no	—
83	#2948	D₅	1	1	yes	characteristic
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	—	5	1	no	—
90	#2151	—	5	1	no	—
91	#2140	—	5	1	no	—
92	C₂×#357	—	5	1	no	—
93	C₂⁴×C₅	—	4	1	no	—
94	C₂×C₅×D₄	—	2	4	no	—
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	4	1	no	—
97	C₂×C₅×D₄	—	4	2	no	—
98	C₂²×C₄×C₅	—	4	2	no	—
99	C₂²×C₄×C₅	—	4	2	no	—
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×Dic₅	—	4	1	no	—
102	C₂×Dic₁₀	—	4	2	no	—
103	C₅×D₈	—	2	4	no	—
104	C₅×D₈	—	2	2	no	—
105	C₅×D₈	C₂×Q₈	1	4	yes	—
106	C₅×M₁₆	—	2	2	no	—
107	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
115	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₅×C₂²⋊C₄	—	4	1	no	—
117	C₅×C₂²⋊C₄	—	4	2	no	—
118	C₅×C₂²⋊C₄	—	4	2	no	—
119	C₅×C₂²⋊C₄	D₈	1	2	yes	—
120	C₄²×C₅	—	4	1	no	—
121	C₄²×C₅	C₂×D₄	1	2	yes	—
122	C₂²⋊Dic₅	—	4	1	no	—
123	C₄⋊Dic₅	—	2	1	no	—
124	C₄⋊Dic₅	—	2	1	no	—
125	C₄⋊Dic₅	—	2	2	no	—
126	C₄⋊Dic₅	—	4	2	no	—
127	C₄⋊Dic₅	—	4	1	no	—
128	#683	—	4	2	no	—
129	#683	—	4	2	no	—
130	C₄×Dic₅	—	4	1	no	—
131	#672	—	2	1	no	—
132	#672	—	2	1	no	—
133	#672	—	2	2	no	—
134	C₂²×D₈	C₂×D₅	1	1	yes	characteristic
135	#544	—	10	1	no	—
136	#518	—	2	1	no	—
137	C₂×M₃₂	—	5	1	no	—
138	C₂²×C₁₆	—	5	1	no	—
139	#501	—	10	2	no	—
140	#500	—	5	2	no	—
141	#499	—	5	2	no	—
142	#442	—	2	2	no	—
143	C₄×D₈	C₂×D₅	1	2	yes	—
144	C₄×D₈	C₂×D₅	1	2	yes	—
145	#428	—	10	1	no	—
146	#427	—	5	1	no	—
147	#427	C₂×D₅	1	1	yes	characteristic
148	C₂×#107	—	5	1	no	—
149	#418	—	2	1	no	—
150	#405	C₂×D₅	1	1	yes	characteristic
151	#367	—	5	1	no	—
152	#367	—	5	1	no	—
153	#367	—	5	2	no	—
154	#366	—	5	1	no	—
155	#366	—	5	1	no	—
156	#366	—	5	2	no	—
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	—	4	2	no	—
164	C₂³×C₅	—	4	4	no	—
165	C₂³×C₅	—	16	1	no	—
166	C₂³×C₅	—	16	1	no	—
167	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
168	C₅×D₄	—	2	4	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	4	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	—	16	2	no	—
174	C₂×C₄×C₅	—	16	2	no	—
175	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
176	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
177	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
178	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₅	#137	1	2	yes	—
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×Dic₅	—	4	1	no	—
182	C₂×Dic₅	—	4	1	no	—
183	C₂×Dic₅	—	4	2	no	—
184	C₂×Dic₅	—	16	1	no	—
185	Dic₁₀	—	16	2	no	—
186	C₅×C₈	—	2	2	no	—
187	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
188	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
189	C₅×C₈	C₂×#41	1	1	yes	characteristic
190	C₂²×D₄	—	2	1	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	Dic₁₀	1	4	yes	complemented
194	C₂×D₈	C₂²×D₅	1	2	yes	—
195	C₂×M₁₆	C₂²⋊D₅	1	1	yes	characteristic
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	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	C₄×D₄	—	4	1	no	—
205	C₄×D₄	—	4	1	no	—
206	#117	—	2	1	no	—
207	#117	—	10	1	no	—
208	#117	C₂²⋊D₅	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	10	1	no	—
210	C₂×C₂²⋊C₄	—	4	2	no	—
211	M₃₂	—	10	2	no	—
212	C₂×C₁₆	—	5	1	no	—
213	C₂×C₁₆	—	5	2	no	—
214	C₂×C₁₆	—	5	1	no	—
215	C₂×C₁₆	—	10	2	no	—
216	#107	—	5	2	no	—
217	#107	—	5	1	no	—
218	#107	—	5	1	no	—
219	#107	—	10	1	no	—
220	#107	—	10	2	no	—
221	#107	C₂²×D₅	1	1	yes	characteristic
222	#107	C₂²×D₅	1	1	yes	characteristic
223	#106	—	2	1	no	—
224	#106	—	10	1	no	—
225	#106	—	10	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	#97	C₂²⋊D₅	1	2	yes	—
229	C₄×C₈	C₂²×D₅	1	2	yes	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	4	4	no	—
233	C₂²×C₅	—	8	2	no	—
234	C₂²×C₅	—	8	2	no	—
235	C₂²×C₅	—	16	4	no	—
236	C₂²×C₅	#474	1	2	yes	—
237	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	2	2	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	—	8	1	no	—
246	Dic₅	—	8	2	no	—
247	Dic₅	—	8	1	no	—
248	C₂⁴	—	4	1	no	—
249	C₂×Q₈	—	20	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	20	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
258	D₈	—	2	2	no	—
259	D₈	—	2	4	no	—
260	D₈	C₂×Dic₁₀	1	4	yes	—
261	M₁₆	—	2	2	no	—
262	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
263	C₂×C₈	#709	1	2	yes	—
264	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
265	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	10	1	no	—
270	C₄⋊C₄	—	10	2	no	—
271	C₄⋊C₄	—	10	1	no	—
272	C₄⋊C₄	—	20	1	no	—
273	C₄⋊C₄	—	20	2	no	—
274	C₄⋊C₄	—	20	2	no	—
275	C₄⋊C₄	—	20	2	no	—
276	C₄⋊C₄	#686	1	1	yes	characteristic
277	C₄⋊C₄	#686	1	1	yes	characteristic
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	20	1	no	—
282	C₂²⋊C₄	#686	1	2	yes	—
283	C₄²	—	4	1	no	—
284	C₄²	—	20	1	no	—
285	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
286	C₁₆	—	10	1	no	—
287	C₁₆	—	10	1	no	—
288	C₁₆	—	10	2	no	—
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	—	8	4	no	—
291	C₂×C₅	C₂×#474	1	1	yes	characteristic
292	C₂×C₅	#3415	1	1	yes	characteristic
293	C₂×C₅	#3416	1	1	yes	characteristic
294	C₂³	—	4	2	no	—
295	C₂³	—	4	4	no	—
296	C₂³	—	16	1	no	—
297	C₂³	—	16	1	no	—
298	C₂³	C₂×#686	1	1	yes	characteristic
299	Q₈	—	80	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	16	2	no	—
306	C₂×C₄	—	16	2	no	—
307	C₂×C₄	—	20	1	no	—
308	C₂×C₄	—	20	1	no	—
309	C₂×C₄	—	20	2	no	—
310	C₂×C₄	—	80	1	no	—
311	C₂×C₄	#4070	1	2	yes	—
312	C₂×C₄	C₂×#686	1	2	yes	—
313	C₂×C₄	C₂×#686	1	2	yes	—
314	C₂×C₄	C₂×#686	1	1	yes	characteristic
315	C₂×C₄	#4097	1	2	yes	—
316	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	#4070	1	2	yes	—
319	C₈	C₂²×Dic₁₀	1	1	yes	characteristic
320	C₈	C₂×#709	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	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#65296	1	2	yes	—
329	C₂²	C₂²×#686	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	40	2	no	—
335	C₄	—	40	1	no	—
336	C₄	—	40	1	no	—
337	C₄	#65296	1	2	yes	—
338	C₄	C₂×#4070	1	1	yes	characteristic
339	C₄	C₂×#4097	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#65296	1	1	yes	characteristic
343	C₂	#10615738	1	1	yes	characteristic
344	C₂	#10615739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2948	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂²×C₅×D₈	C₂²
hypercenter	#2948	D₅
hypocenter	C₅	#33631
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
2. #10610277	C₂	semidirect
5. #10601681	C₂	semidirect
6. #10601681	C₂	semidirect
7. #10601680	C₂	semidirect
8. #10601679	C₂	semidirect
9. #10601442	C₂	semidirect
10. #10601441	C₂	semidirect
11. #10600675	C₂	semidirect
27. #64972	C₂²	semidirect
28. #64971	C₂²	semidirect
33. #64710	C₂²	semidirect
34. #64710	C₂²	semidirect
35. #64710	C₂²	semidirect
36. #64709	C₂²	semidirect
37. #64709	C₂²	semidirect
38. #64709	C₂²	semidirect
39. C₄⋊#672	C₂²	semidirect
40. C₄⋊#672	C₂²	semidirect
45. C₂×C₅×D₈	Q₈	semidirect
193. C₂×D₈	Dic₁₀	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
C₅×#2948	2
#10610277	2
#10602826	2
C₂×#64782	2
#10601681	2
#10601680	2
#10601679	2
#10601442	2
#10601441	2
#10600675	2
#33631	5