Subgroups — #12252657

Order 1280.

718 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	C₂×D₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#10611319	C₂	1	2	yes	maximal, complemented
3	C₄×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601678	C₂	1	2	yes	maximal, complemented
6	#10600118	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600110	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599983	C₂	1	2	yes	maximal, complemented
9	#10599970	C₂	1	2	yes	maximal, complemented
10	C₄×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#10599844	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₅×D₈	C₂²	1	2	yes	—
13	#65879	—	2	2	no	—
14	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₈	C₂²	1	2	yes	complemented
16	D₈×Dic₅	C₂²	1	2	yes	complemented
17	C₂×#3958	C₂²	1	2	yes	—
18	#65303	—	2	2	no	—
19	D₅×D₁₆	C₄	1	8	yes	complemented
20	C₂×D₈₀	C₂²	1	1	yes	characteristic
21	C₂×#3930	C₂²	1	1	yes	characteristic
22	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
23	#65169	C₂²	1	2	yes	—
24	#65161	C₂²	1	2	yes	complemented
25	#65074	—	2	2	no	—
26	#65059	—	2	2	no	—
27	C₄×D₄₀	C₂²	1	2	yes	complemented
28	#64978	C₂²	1	1	yes	characteristic
29	C₅×#366	C₂²	1	1	yes	characteristic, complemented
30	C₅×#357	C₂²	1	2	yes	complemented
31	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
32	#64782	C₂²	1	2	yes	complemented
33	#64729	C₂²	1	2	yes	complemented
34	#64724	C₂²	1	1	yes	characteristic, complemented
35	#64722	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₅	C₂²	1	1	yes	characteristic, complemented
37	#64711	C₂²	1	2	yes	complemented
38	#64710	C₂²	1	2	yes	complemented
39	#64681	C₂²	1	1	yes	characteristic, complemented
40	#33456	—	5	1	no	maximal, Sylow
41	C₂×D₄×D₅	—	2	2	no	—
42	C₂×C₅×D₈	C₂³	1	2	yes	—
43	C₄×C₅×D₄	—	2	2	no	—
44	D₄×Dic₅	—	2	2	no	—
45	C₂×#686	—	2	2	no	—
46	C₄×C₂²⋊D₅	—	4	2	no	—
47	C₂×#685	—	4	2	no	—
48	D₅×D₈	—	2	4	no	—
49	D₅×D₈	C₂×C₄	1	8	yes	complemented
50	C₂×D₄₀	C₂³	1	2	yes	—
51	C₂×#676	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
53	#4039	—	2	2	no	—
54	#4038	—	2	2	no	—
55	#4027	—	4	2	no	—
56	D₅×C₂²⋊C₄	—	4	2	no	—
57	C₄×D₂₀	—	2	2	no	—
58	#4018	D₄	1	1	yes	characteristic
59	C₅×D₁₆	C₂×C₄	1	4	yes	complemented
60	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#102	—	2	2	no	—
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	#3964	—	2	2	no	—
66	#3958	C₂×C₄	1	8	yes	complemented
67	#3953	—	2	2	no	—
68	#3950	C₂³	1	2	yes	—
69	#3947	D₄	1	1	yes	characteristic
70	C₈×Dic₅	C₂³	1	1	yes	characteristic
71	C₂×#672	C₂³	1	1	yes	characteristic
72	C₂×#672	C₂³	1	1	yes	characteristic
73	#3941	—	2	2	no	—
74	#3940	—	2	2	no	—
75	#3935	D₄	1	1	yes	characteristic
76	D₈₀	C₂×C₄	1	4	yes	complemented
77	#3930	—	2	2	no	—
78	D₅×C₁₆	C₂×C₄	1	4	yes	complemented
79	C₂²×D₁₆	—	5	1	no	—
80	#2948	—	5	2	no	—
81	#2177	—	5	2	no	—
82	#2177	—	5	2	no	—
83	C₄×D₁₆	—	5	1	no	—
84	C₄×D₁₆	—	5	1	no	—
85	C₄×D₁₆	—	5	1	no	—
86	C₄×D₁₆	D₅	1	1	yes	characteristic, complemented
87	#2158	—	5	1	no	—
88	#2140	—	5	2	no	—
89	#2108	—	5	1	no	—
90	C₂³×D₅	—	4	2	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂×C₂²⋊D₅	—	4	2	no	—
94	C₂²×Dic₅	—	4	2	no	—
95	D₄×D₅	—	2	8	no	—
96	D₄×D₅	—	4	4	no	—
97	C₂×D₂₀	—	2	2	no	—
98	C₂×C₄×D₅	—	4	2	no	—
99	C₂×C₄×D₅	—	4	2	no	—
100	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
101	C₅×D₈	—	2	2	no	—
102	C₅×D₈	C₂²×C₄	1	4	yes	—
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂²⋊Dic₅	—	4	2	no	—
109	#686	—	2	8	no	—
110	#685	—	4	2	no	—
111	#685	—	4	2	no	—
112	#685	—	4	2	no	—
113	#685	D₈	1	1	yes	characteristic
114	#685	D₈	1	1	yes	characteristic
115	C₄⋊Dic₅	—	2	2	no	—
116	#683	D₈	1	1	yes	characteristic
117	#683	D₈	1	1	yes	characteristic
118	C₄×Dic₅	—	4	2	no	—
119	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
120	C₂×#168	C₂×D₄	1	1	yes	characteristic
121	C₂×#168	C₂×D₄	1	1	yes	characteristic
122	D₄₀	—	2	2	no	—
123	D₄₀	C₂²×C₄	1	4	yes	—
124	#676	—	2	2	no	—
125	C₈×D₅	C₂²×C₄	1	2	yes	—
126	C₈×D₅	#41	1	2	yes	—
127	C₅×C₁₆	—	2	1	no	—
128	C₅×C₁₆	C₂²×C₄	1	2	yes	—
129	#672	—	2	1	no	—
130	#672	C₂²×C₄	1	2	yes	—
131	C₂²×D₈	—	5	2	no	—
132	#518	—	10	2	no	—
133	C₂×D₁₆	—	5	2	no	—
134	C₂×D₁₆	—	5	8	no	—
135	C₂×D₁₆	—	5	1	no	—
136	C₂×D₁₆	C₂×D₅	1	1	yes	characteristic
137	C₂×M₃₂	—	5	1	no	—
138	C₂²×C₁₆	—	5	1	no	—
139	#442	—	10	2	no	—
140	#442	—	10	2	no	—
141	C₄×D₈	—	5	2	no	—
142	C₄×D₈	—	5	2	no	—
143	C₄×D₈	—	5	2	no	—
144	C₄×D₈	C₂×D₅	1	2	yes	complemented
145	#427	—	5	2	no	—
146	#418	—	10	2	no	—
147	#405	—	5	1	no	—
148	#367	—	5	2	no	—
149	#366	—	5	1	no	—
150	#366	C₂×D₅	1	1	yes	characteristic, complemented
151	#357	—	5	2	no	—
152	#357	—	5	2	no	—
153	#357	—	5	2	no	—
154	#357	C₂×D₅	1	2	yes	complemented
155	#346	—	5	1	no	—
156	#346	—	5	1	no	—
157	C₄×C₁₆	—	5	1	no	—
158	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
159	C₂³×C₅	—	4	2	no	—
160	C₂²×D₅	—	4	2	no	—
161	C₂²×D₅	—	4	8	no	—
162	C₂²×D₅	—	16	2	no	—
163	C₂²×D₅	—	16	2	no	—
164	C₂²×D₅	C₂×D₈	1	1	yes	characteristic
165	C₅×D₄	—	2	4	no	—
166	C₅×D₄	—	4	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	16	2	no	—
169	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂²⋊D₅	—	4	8	no	—
173	C₂×Dic₅	—	4	2	no	—
174	C₂×Dic₅	—	16	2	no	—
175	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
176	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
177	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
178	D₂₀	—	2	4	no	—
179	D₂₀	—	4	2	no	—
180	C₄×D₅	—	16	2	no	—
181	C₄×D₅	—	16	2	no	—
182	C₄×D₅	C₄×D₄	1	2	yes	—
183	C₄×D₅	#135	1	2	yes	—
184	C₅×C₈	—	2	1	no	—
185	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
186	C₅×C₈	C₂×#41	1	1	yes	characteristic
187	#168	—	2	1	no	—
188	#168	C₄×D₄	1	2	yes	—
189	C₂²×D₄	—	10	2	no	—
190	C₂×D₈	—	5	8	no	—
191	C₂×D₈	—	5	2	no	—
192	C₂×D₈	—	10	4	no	—
193	C₂×D₈	—	10	2	no	—
194	C₂×D₈	C₂²×D₅	1	2	yes	—
195	C₂×M₁₆	—	5	1	no	—
196	C₂²×C₈	—	5	1	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	10	2	no	—
199	C₄×D₄	—	10	2	no	—
200	C₄×D₄	—	10	2	no	—
201	C₄×D₄	—	20	2	no	—
202	C₄×D₄	—	20	2	no	—
203	#117	—	5	1	no	—
204	#117	—	10	2	no	—
205	C₂×C₂²⋊C₄	—	20	2	no	—
206	C₂×C₂²⋊C₄	—	20	2	no	—
207	D₁₆	—	5	8	no	—
208	D₁₆	—	5	4	no	—
209	D₁₆	C₄×D₅	1	4	yes	complemented
210	M₃₂	—	10	2	no	—
211	C₂×C₁₆	—	5	1	no	—
212	C₂×C₁₆	—	5	4	no	—
213	C₂×C₁₆	—	5	1	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	C₂²×D₅	1	2	yes	—
218	#106	—	10	2	no	—
219	#102	—	2	2	no	—
220	#102	—	10	2	no	—
221	#102	—	10	2	no	—
222	#102	—	10	2	no	—
223	#97	—	5	1	no	—
224	#97	—	5	1	no	—
225	C₄×C₈	—	5	1	no	—
226	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
227	C₂²×C₅	—	4	4	no	—
228	C₂²×C₅	—	8	2	no	—
229	C₂²×C₅	—	8	2	no	—
230	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
231	C₂×D₅	—	2	1	no	—
232	C₂×D₅	—	2	1	no	—
233	C₂×D₅	—	4	4	no	—
234	C₂×D₅	—	8	2	no	—
235	C₂×D₅	—	8	2	no	—
236	C₂×D₅	—	16	8	no	—
237	C₂×D₅	C₄×D₈	1	2	yes	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	8	2	no	—
241	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
242	C₄×C₅	C₂×#135	1	1	yes	characteristic
243	Dic₅	—	2	1	no	—
244	Dic₅	—	2	1	no	—
245	Dic₅	—	8	2	no	—
246	Dic₅	C₄×D₈	1	2	yes	—
247	C₂⁴	—	20	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	10	8	no	—
250	C₂×D₄	—	10	2	no	—
251	C₂×D₄	—	20	4	no	—
252	C₂×D₄	—	20	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	20	2	no	—
256	C₂²×C₄	—	20	2	no	—
257	C₂²×C₄	—	20	2	no	—
258	D₈	—	2	2	no	—
259	D₈	—	5	4	no	—
260	D₈	—	10	8	no	—
261	D₈	—	10	2	no	—
262	D₈	C₂×C₄×D₅	1	4	yes	—
263	M₁₆	—	10	2	no	—
264	C₂×C₈	—	5	1	no	—
265	C₂×C₈	—	5	2	no	—
266	C₂×C₈	—	5	2	no	—
267	C₂×C₈	—	5	1	no	—
268	C₂×C₈	D₄×D₅	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	5	1	no	—
272	C₄⋊C₄	—	5	1	no	—
273	C₄⋊C₄	—	10	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	5	1	no	—
276	C₂²⋊C₄	—	5	1	no	—
277	C₂²⋊C₄	—	20	2	no	—
278	C₂²⋊C₄	—	20	2	no	—
279	C₂²⋊C₄	—	20	2	no	—
280	C₂²⋊C₄	—	20	2	no	—
281	C₄²	—	5	1	no	—
282	C₄²	—	20	2	no	—
283	C₄²	D₄×D₅	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	5	2	no	—
286	C₁₆	—	10	1	no	—
287	C₁₆	C₂×C₄×D₅	1	2	yes	—
288	C₂×C₅	—	8	4	no	—
289	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
290	C₂×C₅	#3415	1	1	yes	characteristic
291	C₂×C₅	#3415	1	1	yes	characteristic
292	D₅	—	2	2	no	—
293	D₅	—	8	4	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	5	1	no	—
296	C₂³	—	20	8	no	—
297	C₂³	—	20	2	no	—
298	C₂³	—	80	2	no	—
299	C₂³	—	80	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	4	2	no	—
302	D₄	—	10	4	no	—
303	D₄	—	20	8	no	—
304	D₄	—	20	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	5	1	no	—
307	C₂×C₄	—	5	2	no	—
308	C₂×C₄	—	5	2	no	—
309	C₂×C₄	—	5	1	no	—
310	C₂×C₄	—	5	1	no	—
311	C₂×C₄	—	16	2	no	—
312	C₂×C₄	—	20	2	no	—
313	C₂×C₄	—	80	2	no	—
314	C₂×C₄	—	80	2	no	—
315	C₂×C₄	—	80	2	no	—
316	C₂×C₄	D₅×D₈	1	1	yes	characteristic
317	C₂×C₄	D₅×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	5	2	no	—
321	C₈	—	10	1	no	—
322	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
323	C₈	D₅×#41	1	1	yes	characteristic
324	C₅	#33456	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	4	4	no	—
326	C₂²	—	5	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	10	1	no	—
330	C₂²	—	10	1	no	—
331	C₂²	—	20	4	no	—
332	C₂²	—	40	2	no	—
333	C₂²	—	40	2	no	—
334	C₂²	—	80	8	no	—
335	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	5	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	10	1	no	—
341	C₄	—	10	1	no	—
342	C₄	—	40	2	no	—
343	C₄	C₄×D₄×D₅	1	1	yes	characteristic
344	C₄	D₅×#135	1	1	yes	characteristic
345	C₂	—	8	4	no	—
346	C₂	—	10	2	no	—
347	C₂	—	40	4	no	—
348	C₂	C₄×D₅×D₈	1	1	yes	characteristic
349	C₂	#10613750	1	1	yes	characteristic
350	C₂	#10613762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅×D₈
commutator	C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₄×C₅×D₁₆	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂×D₅×D₁₆	C₂
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×D₁₆	D₅
hypocenter	C₅	#33456
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. #10611319	C₂	semidirect
3. C₄×C₅×D₁₆	C₂	semidirect
4. Dic₅×D₁₆	C₂	semidirect
5. #10601678	C₂	semidirect
6. #10600118	C₂	semidirect
7. #10600110	C₂	semidirect
8. #10599983	C₂	semidirect
9. #10599970	C₂	semidirect
10. C₄×D₈₀	C₂	semidirect
11. #10599844	C₂	semidirect
15. C₄×C₅×D₈	C₂²	semidirect
16. D₈×Dic₅	C₂²	semidirect
19. D₅×D₁₆	C₄	semidirect
24. #65161	C₂²	semidirect
27. C₄×D₄₀	C₂²	semidirect
29. C₅×#366	C₂²	semidirect
30. C₅×#357	C₂²	semidirect
31. C₄×C₅×C₁₆	C₂²	semidirect
32. #64782	C₂²	semidirect
33. #64729	C₂²	semidirect
34. #64724	C₂²	semidirect
35. #64722	C₂²	semidirect
36. C₁₆×Dic₅	C₂²	semidirect
37. #64711	C₂²	semidirect
38. #64710	C₂²	semidirect
39. #64681	C₂²	semidirect
49. D₅×D₈	C₂×C₄	semidirect
59. C₅×D₁₆	C₂×C₄	semidirect
66. #3958	C₂×C₄	semidirect
76. D₈₀	C₂×C₄	semidirect
78. D₅×C₁₆	C₂×C₄	semidirect
86. C₄×D₁₆	D₅	semidirect
144. C₄×D₈	C₂×D₅	semidirect
150. #366	C₂×D₅	semidirect
154. #357	C₂×D₅	semidirect
158. C₄×C₁₆	C₂×D₅	semidirect
209. D₁₆	C₄×D₅	semidirect
324. C₅	#33456	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×D₁₆	2
#10611319	2
C₄×C₅×D₁₆	2
Dic₅×D₁₆	2
#10601678	2
#10600118	2
#10600110	2
#10599983	2
#10599970	2
C₄×D₈₀	2
#10599844	2
#33456	5