Subgroups — #578×D₂₅₁ #44565127

Order 32128.

1408 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×D₂₅₁	C₂	1	2	yes	maximal, complemented
2	#30632504	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632503	C₂	1	6	yes	maximal, complemented
4	#30632502	C₂	1	2	yes	maximal, complemented
5	#30632501	C₂	1	3	yes	maximal, complemented
6	#30632500	C₂	1	3	yes	maximal, complemented
7	#137×D₂₅₁	C₂	1	6	yes	maximal, complemented
8	#135×D₂₅₁	C₂	1	3	yes	maximal, complemented
9	C₂×Q₁₆×D₂₅₁	C₂	1	3	yes	maximal, complemented
10	#131×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#578×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#22548763	—	2	2	no	—
13	#22548763	C₂²	1	6	yes	complemented
14	#22548762	—	2	6	no	—
15	#22548762	C₂²	1	2	yes	complemented
16	#22548762	C₂²	1	6	yes	complemented
17	#41×D₂₅₁	—	2	6	no	—
18	#41×D₂₅₁	C₂²	1	6	yes	—
19	#41×D₂₅₁	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₂₅₁	—	2	2	no	—
21	C₂×Q₈×D₂₅₁	C₂²	1	6	yes	—
22	#22548748	C₂²	1	6	yes	complemented
23	#22548747	C₂²	1	12	yes	complemented
24	#22548746	C₂²	1	6	yes	complemented
25	#22548745	C₂²	1	6	yes	complemented
26	#22548744	C₂²	1	12	yes	complemented
27	#22548738	C₂²	1	3	yes	complemented
28	#22548737	C₂²	1	3	yes	complemented
29	#22548736	C₂²	1	6	yes	complemented
30	#22548735	C₂²	1	6	yes	complemented
31	#22548734	C₂²	1	3	yes	complemented
32	C₂×Dic₁₀₀₄	C₂²	1	3	yes	complemented
33	C₂×#16860800	C₂²	1	6	yes	complemented
34	Q₁₆×D₂₅₁	C₂²	1	3	yes	—
35	Q₁₆×D₂₅₁	C₂²	1	6	yes	—
36	Q₁₆×D₂₅₁	C₂²	1	6	yes	complemented
37	Q₁₆×D₂₅₁	C₂²	1	6	yes	complemented
38	SD₁₆×D₂₅₁	C₂²	1	6	yes	—
39	D₈×D₂₅₁	C₂²	1	1	yes	characteristic
40	#22548723	C₂²	1	3	yes	complemented
41	#22548722	C₂²	1	1	yes	characteristic, complemented
42	#22548721	C₂²	1	3	yes	complemented
43	M₁₆×D₂₅₁	C₂²	1	3	yes	—
44	C₂×C₈×D₂₅₁	C₂²	1	3	yes	—
45	2₋¹⁺⁴×C₂₅₁	C₂²	1	2	yes	complemented
46	#137×C₂₅₁	C₂²	1	6	yes	complemented
47	#135×C₂₅₁	C₂²	1	3	yes	complemented
48	C₂×Q₁₆×C₂₅₁	C₂²	1	3	yes	complemented
49	#131×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
50	#16860811	—	2	6	no	—
51	#16860811	D₄	1	1	yes	characteristic
52	#16860811	C₂³	1	6	yes	—
53	#16860810	—	2	6	no	—
54	#16860810	—	2	12	no	—
55	#16860810	—	2	6	no	—
56	#16860810	C₂³	1	6	yes	—
57	#16860810	C₂³	1	3	yes	—
58	#16860809	—	2	12	no	—
59	#16860809	—	2	6	no	—
60	#16860809	—	2	6	no	—
61	#16860809	D₄	1	3	yes	—
62	#16860809	C₂³	1	6	yes	—
63	C₂×Dic₅₀₂	—	2	6	no	—
64	C₂×Dic₅₀₂	—	2	6	no	—
65	C₂×Dic₅₀₂	C₂³	1	6	yes	—
66	Q₈×D₂₅₁	—	2	6	no	—
67	Q₈×D₂₅₁	—	2	6	no	—
68	Q₈×D₂₅₁	—	2	6	no	—
69	Q₈×D₂₅₁	—	2	2	no	—
70	Q₈×D₂₅₁	D₄	1	1	yes	characteristic
71	Q₈×D₂₅₁	C₂³	1	12	yes	—
72	Q₈×D₂₅₁	C₂³	1	6	yes	—
73	D₄×D₂₅₁	—	2	6	no	—
74	D₄×D₂₅₁	D₄	1	3	yes	—
75	D₄×D₂₅₁	C₂³	1	2	yes	—
76	C₂×C₄×D₂₅₁	—	2	6	no	—
77	C₂×C₄×D₂₅₁	—	2	6	no	—
78	C₂×C₄×D₂₅₁	C₂³	1	3	yes	—
79	Dic₁₀₀₄	C₂³	1	3	yes	—
80	Dic₁₀₀₄	C₂³	1	6	yes	—
81	#16860800	C₂³	1	6	yes	—
82	#16860800	C₂³	1	12	yes	—
83	#16860799	C₂³	1	6	yes	—
84	#16860798	C₂³	1	6	yes	—
85	#16860797	C₂³	1	6	yes	—
86	D₂₀₀₈	C₂³	1	1	yes	characteristic
87	#16860795	C₂³	1	2	yes	—
88	#16860794	C₂³	1	3	yes	—
89	#16860793	C₂³	1	6	yes	—
90	C₂×C₂₅₁⋊C₈	C₂³	1	3	yes	—
91	C₈×D₂₅₁	C₂³	1	3	yes	—
92	C₈×D₂₅₁	C₂³	1	3	yes	—
93	C₈×D₂₅₁	C₂³	1	1	yes	characteristic
94	C₈×D₂₅₁	C₂³	1	3	yes	—
95	#41×C₂₅₁	—	2	6	no	—
96	#41×C₂₅₁	C₂³	1	6	yes	—
97	#41×C₂₅₁	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₂₅₁	—	2	2	no	—
99	C₂×Q₈×C₂₅₁	C₂³	1	6	yes	—
100	Q₁₆×C₂₅₁	C₂³	1	3	yes	—
101	Q₁₆×C₂₅₁	C₂³	1	6	yes	—
102	SD₁₆×C₂₅₁	C₂³	1	6	yes	—
103	D₈×C₂₅₁	C₂³	1	1	yes	characteristic
104	M₁₆×C₂₅₁	C₂³	1	3	yes	—
105	C₂×C₈×C₂₅₁	C₂³	1	3	yes	—
106	C₂²×D₂₅₁	—	2	2	no	—
107	C₂²×D₂₅₁	C₂×D₄	1	3	yes	—
108	Dic₅₀₂	—	2	6	no	—
109	Dic₅₀₂	—	2	12	no	—
110	Dic₅₀₂	—	2	6	no	—
111	Dic₅₀₂	C₂×D₄	1	3	yes	—
112	Dic₅₀₂	C₂⁴	1	6	yes	—
113	D₁₀₀₄	—	2	6	no	—
114	D₁₀₀₄	C₂×D₄	1	3	yes	—
115	D₁₀₀₄	C₂⁴	1	2	yes	—
116	C₂²⋊D₂₅₁	—	2	6	no	—
117	C₂²⋊D₂₅₁	—	2	6	no	—
118	C₂²⋊D₂₅₁	—	2	2	no	—
119	C₂²⋊D₂₅₁	C₂×D₄	1	6	yes	—
120	C₂×Dic₂₅₁	—	2	6	no	—
121	C₂×Dic₂₅₁	—	2	6	no	—
122	C₂×Dic₂₅₁	C₂×D₄	1	3	yes	—
123	C₄×D₂₅₁	—	2	6	no	—
124	C₄×D₂₅₁	—	2	12	no	—
125	C₄×D₂₅₁	—	2	6	no	—
126	C₄×D₂₅₁	—	2	6	no	—
127	C₄×D₂₅₁	C₂×D₄	1	3	yes	—
128	C₄×D₂₅₁	C₂×D₄	1	3	yes	—
129	C₄×D₂₅₁	C₂⁴	1	1	yes	characteristic
130	C₄×D₂₅₁	C₂⁴	1	3	yes	—
131	C₂₅₁⋊C₈	C₂⁴	1	1	yes	characteristic
132	C₂₅₁⋊C₈	C₂⁴	1	3	yes	—
133	Q₈×C₂₅₁	—	2	6	no	—
134	Q₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
135	Q₈×C₂₅₁	C₂⁴	1	6	yes	—
136	D₄×C₂₅₁	—	2	6	no	—
137	D₄×C₂₅₁	C₂×D₄	1	3	yes	—
138	D₄×C₂₅₁	C₂⁴	1	2	yes	—
139	C₂×C₄×C₂₅₁	—	2	6	no	—
140	C₂×C₄×C₂₅₁	—	2	6	no	—
141	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
142	C₈×C₂₅₁	C₂⁴	1	1	yes	characteristic
143	C₈×C₂₅₁	C₂⁴	1	3	yes	—
144	C₂²×C₂₅₁	—	2	2	no	—
145	C₂²×C₂₅₁	C₂²×D₄	1	3	yes	—
146	C₂×D₂₅₁	—	2	6	no	—
147	C₂×D₂₅₁	—	2	2	no	—
148	C₂×D₂₅₁	—	4	4	no	—
149	C₂×D₂₅₁	C₂²×D₄	1	3	yes	—
150	C₂×D₂₅₁	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₂₅₁	—	2	6	no	—
152	C₄×C₂₅₁	C₂²×D₄	1	3	yes	—
153	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
154	Dic₂₅₁	—	2	6	no	—
155	Dic₂₅₁	C₂²×D₄	1	3	yes	—
156	Dic₂₅₁	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₂₅₁	—	2	3	no	—
158	C₂×C₂₅₁	—	4	2	no	—
159	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
160	D₂₅₁	—	2	3	no	—
161	D₂₅₁	—	4	2	no	—
162	D₂₅₁	#578	1	2	yes	direct product factor
163	C₂₅₁	C₂×#578	1	1	yes	characteristic, complemented, Sylow
164	C₂×#578	—	251	1	no	maximal, Sylow
165	C₂×2₋¹⁺⁴	—	251	2	no	—
166	#578	—	251	2	no	—
167	#578	—	251	6	no	—
168	#578	—	251	3	no	—
169	#578	—	251	3	no	—
170	#578	—	251	1	no	—
171	#578	D₂₅₁	1	1	yes	characteristic, direct product factor
172	C₂×#137	—	251	6	no	—
173	C₂×#135	—	251	3	no	—
174	C₂²×Q₁₆	—	251	3	no	—
175	C₂×#131	—	251	1	no	—
176	2₋¹⁺⁴	—	502	2	no	—
177	2₋¹⁺⁴	—	502	6	no	—
178	2₋¹⁺⁴	—	251	6	no	—
179	2₋¹⁺⁴	—	251	2	no	—
180	2₋¹⁺⁴	—	251	6	no	—
181	2₋¹⁺⁴	C₂×D₂₅₁	1	2	yes	complemented
182	C₂×#41	—	502	6	no	—
183	C₂×#41	—	251	6	no	—
184	C₂×#41	—	251	1	no	—
185	C₂²×Q₈	—	502	2	no	—
186	C₂²×Q₈	—	251	6	no	—
187	#137	—	251	6	no	—
188	#137	—	251	6	no	—
189	#137	—	251	12	no	—
190	#137	—	251	12	no	—
191	#137	—	251	6	no	—
192	#137	C₂×D₂₅₁	1	6	yes	complemented
193	#135	—	251	6	no	—
194	#135	—	251	3	no	—
195	#135	—	251	6	no	—
196	#135	—	251	3	no	—
197	#135	—	251	3	no	—
198	#135	C₂×D₂₅₁	1	3	yes	complemented
199	C₂×Q₁₆	—	251	6	no	—
200	C₂×Q₁₆	—	251	3	no	—
201	C₂×Q₁₆	—	251	6	no	—
202	C₂×Q₁₆	—	251	6	no	—
203	C₂×Q₁₆	—	251	6	no	—
204	C₂×Q₁₆	—	251	3	no	—
205	C₂×Q₁₆	C₂×D₂₅₁	1	3	yes	complemented
206	C₂×SD₁₆	—	251	6	no	—
207	C₂×D₈	—	251	1	no	—
208	#131	—	251	1	no	—
209	#131	—	251	3	no	—
210	#131	—	251	3	no	—
211	#131	C₂×D₂₅₁	1	1	yes	characteristic, complemented
212	C₂×M₁₆	—	251	3	no	—
213	C₂²×C₈	—	251	3	no	—
214	#41	—	2	6	no	—
215	#41	—	502	12	no	—
216	#41	—	502	6	no	—
217	#41	—	502	6	no	—
218	#41	—	502	12	no	—
219	#41	—	502	6	no	—
220	#41	—	502	6	no	—
221	#41	—	502	6	no	—
222	#41	—	251	6	no	—
223	#41	—	251	6	no	—
224	#41	—	251	6	no	—
225	#41	—	251	1	no	—
226	#41	—	251	3	no	—
227	#41	—	251	3	no	—
228	#41	C₂²×D₂₅₁	1	6	yes	—
229	#41	C₂²×D₂₅₁	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	502	6	no	—
232	C₂×Q₈	—	502	6	no	—
233	C₂×Q₈	—	502	6	no	—
234	C₂×Q₈	—	502	2	no	—
235	C₂×Q₈	—	502	6	no	—
236	C₂×Q₈	—	502	6	no	—
237	C₂×Q₈	—	251	6	no	—
238	C₂×Q₈	—	251	12	no	—
239	C₂×Q₈	—	251	6	no	—
240	C₂×Q₈	—	251	1	no	—
241	C₂×Q₈	C₂²×D₂₅₁	1	6	yes	—
242	C₂×D₄	—	502	6	no	—
243	C₂×D₄	—	251	2	no	—
244	C₂×D₄	—	251	3	no	—
245	C₂²×C₄	—	502	6	no	—
246	C₂²×C₄	—	502	6	no	—
247	C₂²×C₄	—	251	3	no	—
248	Q₁₆	—	251	6	no	—
249	Q₁₆	—	251	12	no	—
250	Q₁₆	—	251	3	no	—
251	Q₁₆	—	251	6	no	—
252	Q₁₆	C₂²×D₂₅₁	1	3	yes	—
253	Q₁₆	C₂²×D₂₅₁	1	6	yes	—
254	SD₁₆	—	251	6	no	—
255	SD₁₆	—	251	6	no	—
256	SD₁₆	—	251	6	no	—
257	SD₁₆	C₂²×D₂₅₁	1	6	yes	—
258	D₈	—	251	2	no	—
259	D₈	—	251	1	no	—
260	D₈	C₂²×D₂₅₁	1	1	yes	characteristic
261	M₁₆	—	251	3	no	—
262	M₁₆	—	251	6	no	—
263	M₁₆	C₂²×D₂₅₁	1	3	yes	—
264	C₂×C₈	—	251	3	no	—
265	C₂×C₈	—	251	1	no	—
266	C₂×C₈	—	251	3	no	—
267	C₂×C₈	—	251	3	no	—
268	C₂×C₈	—	251	3	no	—
269	C₂×C₈	C₂²×D₂₅₁	1	3	yes	—
270	C₂³	—	502	2	no	—
271	C₂³	—	251	3	no	—
272	Q₈	—	2	6	no	—
273	Q₈	—	502	12	no	—
274	Q₈	—	502	6	no	—
275	Q₈	—	502	6	no	—
276	Q₈	—	251	6	no	—
277	Q₈	—	251	3	no	—
278	Q₈	D₄×D₂₅₁	1	1	yes	characteristic
279	Q₈	C₂³×D₂₅₁	1	6	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	502	6	no	—
282	D₄	—	502	6	no	—
283	D₄	—	502	6	no	—
284	D₄	—	502	2	no	—
285	D₄	—	251	6	no	—
286	D₄	—	251	3	no	—
287	D₄	—	251	2	no	—
288	D₄	D₄×D₂₅₁	1	3	yes	—
289	D₄	C₂³×D₂₅₁	1	2	yes	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	2	6	no	—
292	C₂×C₄	—	502	6	no	—
293	C₂×C₄	—	502	12	no	—
294	C₂×C₄	—	502	6	no	—
295	C₂×C₄	—	502	6	no	—
296	C₂×C₄	—	502	6	no	—
297	C₂×C₄	—	502	6	no	—
298	C₂×C₄	—	251	3	no	—
299	C₂×C₄	—	251	3	no	—
300	C₂×C₄	—	251	3	no	—
301	C₂×C₄	—	251	1	no	—
302	C₂×C₄	—	251	3	no	—
303	C₂×C₄	C₂³×D₂₅₁	1	3	yes	—
304	C₈	—	251	1	no	—
305	C₈	—	251	3	no	—
306	C₈	C₂³×D₂₅₁	1	1	yes	characteristic
307	C₈	C₂³×D₂₅₁	1	3	yes	—
308	C₂²	—	2	2	no	—
309	C₂²	—	1004	4	no	—
310	C₂²	—	502	6	no	—
311	C₂²	—	502	2	no	—
312	C₂²	—	251	3	no	—
313	C₂²	—	251	1	no	—
314	C₂²	C₂×D₄×D₂₅₁	1	3	yes	—
315	C₄	—	2	6	no	—
316	C₄	—	502	6	no	—
317	C₄	—	251	3	no	—
318	C₄	—	251	1	no	—
319	C₄	C₂×D₄×D₂₅₁	1	3	yes	—
320	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
321	C₂	—	2	3	no	—
322	C₂	—	4	2	no	—
323	C₂	—	1004	2	no	—
324	C₂	—	502	3	no	—
325	C₂	—	251	2	no	—
326	C₂	C₂²×D₄×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₂₅₁
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	#578×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂³×D₄
hypercenter	#578	D₂₅₁
hypocenter	C₂₅₁	C₂×#578
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×D₂₅₁	C₂	semidirect
2. #30632504	C₂	semidirect
3. #30632503	C₂	semidirect
4. #30632502	C₂	semidirect
5. #30632501	C₂	semidirect
6. #30632500	C₂	semidirect
7. #137×D₂₅₁	C₂	semidirect
8. #135×D₂₅₁	C₂	semidirect
9. C₂×Q₁₆×D₂₅₁	C₂	semidirect
10. #131×D₂₅₁	C₂	semidirect
11. #578×C₂₅₁	C₂	semidirect
13. #22548763	C₂²	semidirect
15. #22548762	C₂²	semidirect
16. #22548762	C₂²	semidirect
22. #22548748	C₂²	semidirect
23. #22548747	C₂²	semidirect
24. #22548746	C₂²	semidirect
25. #22548745	C₂²	semidirect
26. #22548744	C₂²	semidirect
27. #22548738	C₂²	semidirect
28. #22548737	C₂²	semidirect
29. #22548736	C₂²	semidirect
30. #22548735	C₂²	semidirect
31. #22548734	C₂²	semidirect
32. C₂×Dic₁₀₀₄	C₂²	semidirect
33. C₂×#16860800	C₂²	semidirect
36. Q₁₆×D₂₅₁	C₂²	semidirect
37. Q₁₆×D₂₅₁	C₂²	semidirect
40. #22548723	C₂²	semidirect
41. #22548722	C₂²	semidirect
42. #22548721	C₂²	semidirect
45. 2₋¹⁺⁴×C₂₅₁	C₂²	semidirect
46. #137×C₂₅₁	C₂²	semidirect
47. #135×C₂₅₁	C₂²	semidirect
48. C₂×Q₁₆×C₂₅₁	C₂²	semidirect
49. #131×C₂₅₁	C₂²	semidirect
162. D₂₅₁	#578	direct (N × H)
163. C₂₅₁	C₂×#578	semidirect
171. #578	D₂₅₁	direct (N × H)
181. 2₋¹⁺⁴	C₂×D₂₅₁	semidirect
192. #137	C₂×D₂₅₁	semidirect
198. #135	C₂×D₂₅₁	semidirect
205. C₂×Q₁₆	C₂×D₂₅₁	semidirect
211. #131	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×D₂₅₁	2
#30632504	2
#30632503	2
#30632502	2
#30632501	2
#30632500	2
#137×D₂₅₁	2
#135×D₂₅₁	2
C₂×Q₁₆×D₂₅₁	2
#131×D₂₅₁	2
#578×C₂₅₁	2
C₂×#578	251

Subgroups — #578×D251 #44565127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #578×D₂₅₁ #44565127