Subgroups — #12256251

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611474	C₂	1	1	yes	maximal, characteristic, complemented
2	#10611465	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602296	C₂	1	2	yes	maximal, complemented
5	#10601707	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601703	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600116	C₂	1	2	yes	maximal, complemented
8	#10600021	C₂	1	2	yes	maximal, complemented
9	#10600013	C₂	1	2	yes	maximal, complemented
10	#10599851	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#4065	C₂²	1	2	yes	—
13	#65918	—	2	1	no	—
14	#65911	—	2	1	no	—
15	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
16	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
18	#65477	C₂²	1	2	yes	—
19	C₂×#3960	C₂²	1	2	yes	—
20	#65316	—	2	1	no	—
21	#65309	—	2	1	no	—
22	C₂×D₅×C₁₆	C₂²	1	2	yes	—
23	#65172	C₂²	1	2	yes	—
24	#65169	C₂²	1	2	yes	—
25	#65103	—	2	2	no	—
26	#65100	—	2	2	no	—
27	#64984	C₂²	1	1	yes	characteristic
28	C₈×D₂₀	C₂²	1	1	yes	characteristic
29	#64971	C₂²	1	1	yes	characteristic
30	C₅×#366	C₂²	1	1	yes	characteristic
31	C₅×#358	C₂²	1	1	yes	characteristic
32	C₅×#358	C₂²	1	1	yes	characteristic
33	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
34	#64784	C₂²	1	2	yes	—
35	#64729	C₂²	1	2	yes	—
36	#64729	C₂²	1	2	yes	—
37	#64727	C₂²	1	2	yes	—
38	#64711	C₂²	1	2	yes	—
39	#64710	C₂²	1	2	yes	—
40	C₄⋊#672	C₂²	1	1	yes	characteristic
41	#33871	—	5	1	no	maximal, Sylow
42	C₂×#713	—	2	2	no	—
43	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
45	C₄×C₅×Q₈	—	2	1	no	—
46	C₄×C₅×Q₈	—	2	1	no	—
47	#4092	—	2	2	no	—
48	C₂×#688	—	2	2	no	—
49	#4065	—	2	4	no	—
50	#4065	—	4	2	no	—
51	C₂×D₄₀	—	2	1	no	—
52	C₂×D₄₀	C₂³	1	2	yes	—
53	C₂×C₈×D₅	C₂³	1	2	yes	—
54	#4044	—	4	2	no	—
55	#4041	—	2	2	no	—
56	#4041	—	4	2	no	—
57	#4038	—	2	2	no	—
58	#4021	—	4	1	no	—
59	#4020	—	2	1	no	—
60	#4020	—	4	1	no	—
61	C₄×D₂₀	—	4	1	no	—
62	C₄×D₂₀	D₄	1	1	yes	characteristic
63	#4015	—	2	1	no	—
64	C₄×Dic₁₀	—	4	1	no	—
65	C₅×Q₃₂	—	2	2	no	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
68	C₅×#107	C₂³	1	1	yes	characteristic
69	C₅×#107	C₂³	1	1	yes	characteristic
70	C₅×#103	—	2	1	no	—
71	C₅×#103	—	2	1	no	—
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	#3967	—	2	2	no	—
74	#3960	—	4	2	no	—
75	#3952	D₄	1	2	yes	—
76	#3950	—	2	1	no	—
77	#3950	C₂³	1	2	yes	—
78	C₂×#672	C₂³	1	2	yes	—
79	#3941	—	2	2	no	—
80	#3940	—	2	2	no	—
81	C₄⋊#168	D₄	1	1	yes	characteristic
82	D₅×C₁₆	—	4	2	no	—
83	#3342	—	5	1	no	—
84	#3287	—	5	1	no	—
85	#2240	—	5	1	no	—
86	#2234	—	5	2	no	—
87	#2225	—	5	1	no	—
88	#2217	—	5	2	no	—
89	#2207	—	5	1	no	—
90	#2189	—	5	2	no	—
91	C₄×Q₃₂	D₅	1	1	yes	characteristic, complemented
92	D₄×C₁₆	—	5	1	no	—
93	#2138	—	5	2	no	—
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	#713	—	4	4	no	—
97	#713	—	8	2	no	—
98	C₂×D₂₀	—	2	2	no	—
99	C₂×D₂₀	—	4	2	no	—
100	C₂×D₂₀	—	4	1	no	—
101	C₂×D₂₀	—	4	2	no	—
102	C₂×D₂₀	—	4	1	no	—
103	C₂×D₂₀	D₈	1	1	yes	characteristic
104	C₂×C₄×D₅	—	4	2	no	—
105	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
106	C₂×Dic₁₀	—	4	1	no	—
107	C₅×Q₁₆	—	2	1	no	—
108	C₅×Q₁₆	—	2	1	no	—
109	C₅×Q₁₆	—	2	1	no	—
110	C₅×Q₁₆	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	4	1	no	—
116	C₅×C₄⋊C₄	—	4	1	no	—
117	C₄²×C₅	—	4	1	no	—
118	C₄²×C₅	—	4	1	no	—
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	#688	—	4	4	no	—
121	#685	—	4	2	no	—
122	#685	—	4	2	no	—
123	#685	—	4	2	no	—
124	#685	D₈	1	2	yes	—
125	C₄⋊Dic₅	—	2	2	no	—
126	C₄⋊Dic₅	—	4	1	no	—
127	C₄⋊Dic₅	D₈	1	1	yes	characteristic
128	#683	—	4	2	no	—
129	C₄×Dic₅	—	4	2	no	—
130	C₂×#168	C₂×D₄	1	2	yes	—
131	D₄₀	—	2	2	no	—
132	D₄₀	—	2	2	no	—
133	D₄₀	—	8	1	no	—
134	C₈×D₅	—	2	2	no	—
135	C₈×D₅	—	2	2	no	—
136	C₅×C₁₆	—	2	1	no	—
137	C₅×C₁₆	—	2	1	no	—
138	#672	—	2	2	no	—
139	C₂×#135	—	5	2	no	—
140	#556	—	10	1	no	—
141	#550	—	10	1	no	—
142	C₂×Q₃₂	C₂×D₅	1	1	yes	characteristic
143	C₂×SD₃₂	—	5	2	no	—
144	C₂²×C₁₆	—	5	2	no	—
145	#500	—	5	1	no	—
146	#493	—	5	1	no	—
147	#485	—	10	2	no	—
148	#475	—	10	1	no	—
149	#467	—	5	2	no	—
150	#466	—	5	2	no	—
151	#460	—	10	1	no	—
152	#449	—	10	2	no	—
153	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
154	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
155	C₈×D₄	—	5	1	no	—
156	#427	—	5	2	no	—
157	#367	—	5	2	no	—
158	#366	C₂×D₅	1	1	yes	characteristic
159	C₄⋊C₁₆	—	5	1	no	—
160	#358	—	5	2	no	—
161	#358	C₂×D₅	1	1	yes	characteristic
162	#358	C₂×D₅	1	1	yes	characteristic
163	#357	—	5	2	no	—
164	#357	—	5	2	no	—
165	#357	—	5	2	no	—
166	C₂²⋊C₁₆	—	5	2	no	—
167	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
168	C₂²×D₅	—	4	2	no	—
169	C₂²×D₅	C₂×D₈	1	2	yes	—
170	C₅×Q₈	—	2	2	no	—
171	C₅×Q₈	—	4	1	no	—
172	C₅×Q₈	—	4	1	no	—
173	C₅×Q₈	—	4	1	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×Dic₅	—	4	2	no	—
182	C₂×Dic₅	C₂×D₈	1	2	yes	—
183	D₂₀	—	2	2	no	—
184	D₂₀	—	4	2	no	—
185	D₂₀	—	4	2	no	—
186	D₂₀	—	8	4	no	—
187	D₂₀	—	8	2	no	—
188	D₂₀	—	16	1	no	—
189	D₂₀	—	16	2	no	—
190	C₄×D₅	—	2	2	no	—
191	C₄×D₅	—	2	2	no	—
192	C₄×D₅	—	8	4	no	—
193	Dic₁₀	—	8	2	no	—
194	C₅×C₈	—	2	1	no	—
195	C₅×C₈	C₂×#41	1	1	yes	characteristic
196	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
197	#168	—	2	2	no	—
198	C₂×#41	—	10	2	no	—
199	#135	—	10	4	no	—
200	#135	—	20	2	no	—
201	C₂×Q₁₆	C₂²×D₅	1	1	yes	characteristic
202	C₂×Q₁₆	C₂²×D₅	1	1	yes	characteristic
203	C₂×SD₁₆	—	10	2	no	—
204	C₂×D₈	—	5	2	no	—
205	C₂×D₈	—	10	1	no	—
206	C₂²×C₈	—	5	2	no	—
207	C₄⋊Q₈	—	10	1	no	—
208	#127	—	10	1	no	—
209	#127	—	20	1	no	—
210	#126	—	20	2	no	—
211	#124	—	20	1	no	—
212	C₂²⋊Q₈	—	10	2	no	—
213	#121	—	10	2	no	—
214	#121	—	20	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	20	1	no	—
218	C₄×D₄	—	5	1	no	—
219	C₄×D₄	—	20	1	no	—
220	#117	—	10	2	no	—
221	Q₃₂	—	2	2	no	—
222	SD₃₂	—	20	2	no	—
223	C₂×C₁₆	—	5	2	no	—
224	C₂×C₁₆	—	20	2	no	—
225	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
227	#107	—	5	2	no	—
228	#107	—	10	1	no	—
229	#107	C₂²×D₅	1	1	yes	characteristic
230	#107	C₂²×D₅	1	1	yes	characteristic
231	#106	—	10	2	no	—
232	C₄⋊C₈	—	5	1	no	—
233	#103	—	2	1	no	—
234	#103	—	2	1	no	—
235	#103	—	10	2	no	—
236	#102	—	10	2	no	—
237	C₂²⋊C₈	—	5	2	no	—
238	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
239	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
240	C₂×D₅	—	2	2	no	—
241	C₂×D₅	—	2	2	no	—
242	C₂×D₅	—	2	2	no	—
243	C₂×D₅	—	8	2	no	—
244	C₂×D₅	—	8	2	no	—
245	C₂×D₅	—	8	2	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	4	2	no	—
248	C₄×C₅	—	8	1	no	—
249	C₄×C₅	—	8	1	no	—
250	C₄×C₅	—	8	1	no	—
251	C₄×C₅	#508	1	2	yes	—
252	C₄×C₅	#546	1	1	yes	characteristic
253	C₄×C₅	#577	1	1	yes	characteristic
254	Dic₅	—	2	2	no	—
255	Dic₅	—	8	2	no	—
256	#41	—	20	4	no	—
257	#41	—	40	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	20	1	no	—
261	C₂×D₄	—	5	1	no	—
262	C₂×D₄	—	10	2	no	—
263	C₂×D₄	—	20	2	no	—
264	C₂×D₄	—	20	2	no	—
265	C₂×D₄	—	20	1	no	—
266	C₂×D₄	—	20	1	no	—
267	C₂²×C₄	—	5	2	no	—
268	C₂²×C₄	—	20	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	#709	1	2	yes	—
273	SD₁₆	—	20	4	no	—
274	D₈	—	10	2	no	—
275	D₈	—	10	2	no	—
276	D₈	—	40	1	no	—
277	C₂×C₈	—	5	2	no	—
278	C₂×C₈	—	10	2	no	—
279	C₂×C₈	—	10	2	no	—
280	C₂×C₈	D₄×D₅	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	5	1	no	—
287	C₄⋊C₄	—	10	2	no	—
288	C₄⋊C₄	—	20	2	no	—
289	C₄⋊C₄	—	20	1	no	—
290	C₂²⋊C₄	—	5	2	no	—
291	C₂²⋊C₄	—	20	2	no	—
292	C₂²⋊C₄	—	20	2	no	—
293	C₂²⋊C₄	—	20	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	20	2	no	—
297	C₄²	D₄×D₅	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	1	no	—
300	C₁₆	—	10	2	no	—
301	C₂×C₅	#3294	1	1	yes	characteristic
302	C₂×C₅	C₂×#508	1	1	yes	characteristic
303	C₂×C₅	#3416	1	1	yes	characteristic
304	D₅	—	4	2	no	—
305	D₅	—	16	2	no	—
306	C₂³	—	5	2	no	—
307	C₂³	—	20	2	no	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	40	2	no	—
313	D₄	—	10	2	no	—
314	D₄	—	20	2	no	—
315	D₄	—	20	2	no	—
316	D₄	—	40	4	no	—
317	D₄	—	40	2	no	—
318	D₄	—	80	1	no	—
319	D₄	—	80	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	5	2	no	—
325	C₂×C₄	—	10	2	no	—
326	C₂×C₄	—	10	2	no	—
327	C₂×C₄	—	20	2	no	—
328	C₂×C₄	—	40	4	no	—
329	C₂×C₄	D₅×D₈	1	1	yes	characteristic
330	C₂×C₄	D₅×D₈	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	10	2	no	—
334	C₈	C₂×#709	1	1	yes	characteristic
335	C₈	#4149	1	1	yes	characteristic
336	C₅	#33871	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	10	2	no	—
338	C₂²	—	10	2	no	—
339	C₂²	—	10	2	no	—
340	C₂²	—	40	2	no	—
341	C₂²	—	40	2	no	—
342	C₂²	—	40	2	no	—
343	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	10	2	no	—
350	C₄	—	40	2	no	—
351	C₄	#65208	1	2	yes	—
352	C₄	#65884	1	1	yes	characteristic
353	C₄	#66104	1	1	yes	characteristic
354	C₂	—	20	2	no	—
355	C₂	—	80	2	no	—
356	C₂	#10611339	1	1	yes	characteristic
357	C₂	C₂×#65208	1	1	yes	characteristic
358	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	C₄×C₅×Q₃₂	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	#10611465	C₂
hypercenter	C₄×Q₃₂	D₅
hypocenter	C₅	#33871
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #10611474	C₂	semidirect
3. C₄×C₅×Q₃₂	C₂	semidirect
4. #10602296	C₂	semidirect
5. #10601707	C₂	semidirect
6. #10601703	C₂	semidirect
7. #10600116	C₂	semidirect
8. #10600021	C₂	semidirect
9. #10600013	C₂	semidirect
10. #10599851	C₂	semidirect
11. C₁₆×D₂₀	C₂	semidirect
91. C₄×Q₃₂	D₅	semidirect
336. C₅	#33871	semidirect

Maximal subgroups

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

subgroup	index
#10611474	2
#10611465	2
C₄×C₅×Q₃₂	2
#10602296	2
#10601707	2
#10601703	2
#10600116	2
#10600021	2
#10600013	2
#10599851	2
C₁₆×D₂₀	2
#33871	5