Subgroups — #559×D₂₅₁ #44563949

Order 32128.

1240 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632423	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632422	C₂	1	2	yes	maximal, complemented
3	#30632421	C₂	1	4	yes	maximal, complemented
4	#30632420	C₂	1	4	yes	maximal, complemented
5	#30632419	C₂	1	4	yes	maximal, complemented
6	#127×D₂₅₁	C₂	1	1	yes	maximal, characteristic
7	#124×D₂₅₁	C₂	1	4	yes	maximal, complemented
8	#123×D₂₅₁	C₂	1	4	yes	maximal, complemented
9	C₄×Q₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₂₅₁	C₂	1	4	yes	maximal, complemented
11	#559×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₅₁	C₂²	1	2	yes	—
13	C₂×D₄×D₂₅₁	C₂²	1	4	yes	—
14	C₂×D₄×D₂₅₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₅₁	C₂²	1	4	yes	—
16	#22548712	C₂²	1	1	yes	characteristic
17	#22548711	C₂²	1	4	yes	—
18	#22548711	C₂²	1	2	yes	—
19	#22548704	C₂²	1	4	yes	complemented
20	#22548703	C₂²	1	4	yes	complemented
21	#22548702	C₂²	1	4	yes	complemented
22	#22548701	C₂²	1	8	yes	complemented
23	#22548701	C₂²	1	8	yes	complemented
24	#22548700	C₂²	1	8	yes	complemented
25	#22548699	C₂²	1	4	yes	complemented
26	#22548698	C₂²	1	4	yes	complemented
27	#22548697	C₂²	1	8	yes	complemented
28	#22548696	C₂²	1	4	yes	complemented
29	Q₈×Dic₂₅₁	C₂²	1	2	yes	complemented
30	C₄×Dic₅₀₂	C₂²	1	4	yes	complemented
31	C₄×Dic₅₀₂	C₂²	1	2	yes	complemented
32	#22548680	C₂²	1	2	yes	complemented
33	#22548680	C₂²	1	4	yes	complemented
34	C₄×D₁₀₀₄	C₂²	1	4	yes	complemented
35	D₄×Dic₂₅₁	C₂²	1	4	yes	complemented
36	#22548677	C₂²	1	4	yes	complemented
37	C₄×C₂²⋊D₂₅₁	C₂²	1	8	yes	complemented
38	(C₂²⋊D₂₅₁)⋊C₄	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
40	C₄⋊C₄×D₂₅₁	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₂₅₁	C₂²	1	4	yes	—
42	C₂²⋊C₄×D₂₅₁	C₂²	1	8	yes	—
43	C₄²×D₂₅₁	C₂²	1	4	yes	—
44	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
45	#127×C₂₅₁	C₂²	1	1	yes	characteristic
46	#124×C₂₅₁	C₂²	1	4	yes	complemented
47	#123×C₂₅₁	C₂²	1	4	yes	complemented
48	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	complemented
49	C₄×D₄×C₂₅₁	C₂²	1	4	yes	complemented
50	C₂³×D₂₅₁	C₂³	1	4	yes	—
51	C₂×Dic₅₀₂	C₂³	1	4	yes	—
52	C₂×Dic₅₀₂	C₂³	1	2	yes	—
53	Q₈×D₂₅₁	—	2	8	no	—
54	C₂×D₁₀₀₄	C₂³	1	4	yes	—
55	C₂×D₁₀₀₄	C₂³	1	2	yes	—
56	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₂₅₁	C₂³	1	4	yes	—
58	D₄×D₂₅₁	—	2	16	no	—
59	D₄×D₂₅₁	—	4	4	no	—
60	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
61	C₂×C₄×D₂₅₁	—	4	8	no	—
62	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
63	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
65	C₂×C₄×D₂₅₁	C₂³	1	8	yes	—
66	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
67	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
68	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
69	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
70	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
71	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
72	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
73	C₂²⋊Dic₂₅₁	C₂³	1	4	yes	—
74	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	—
77	C₄×Dic₂₅₁	C₂³	1	4	yes	—
78	C₄×Dic₂₅₁	C₂³	1	4	yes	—
79	C₄×Dic₂₅₁	C₂³	1	4	yes	—
80	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
81	C₄×Dic₂₅₁	C₂³	1	2	yes	—
82	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
83	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
84	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
85	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
86	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
87	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₂₅₁	C₂³	1	4	yes	—
90	C₄²×C₂₅₁	C₂³	1	4	yes	—
91	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
92	C₂²×D₂₅₁	—	2	16	no	—
93	C₂²×D₂₅₁	—	2	8	no	—
94	C₂²×D₂₅₁	C₂⁴	1	4	yes	—
95	C₂²×D₂₅₁	C₂⁴	1	1	yes	characteristic
96	Dic₅₀₂	—	2	4	no	—
97	Dic₅₀₂	—	2	8	no	—
98	D₁₀₀₄	—	2	8	no	—
99	D₁₀₀₄	—	4	2	no	—
100	C₂²⋊D₂₅₁	—	2	16	no	—
101	C₂²⋊D₂₅₁	—	4	4	no	—
102	C₂×Dic₂₅₁	—	4	4	no	—
103	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
104	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
105	C₂×Dic₂₅₁	C₂⁴	1	1	yes	characteristic
106	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
107	C₄×D₂₅₁	—	2	8	no	—
108	C₄×D₂₅₁	—	2	8	no	—
109	C₄×D₂₅₁	—	2	2	no	—
110	C₄×D₂₅₁	—	4	8	no	—
111	C₄×D₂₅₁	#41	1	8	yes	—
112	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
113	Q₈×C₂₅₁	—	2	4	no	—
114	D₄×C₂₅₁	—	2	8	no	—
115	D₄×C₂₅₁	—	4	2	no	—
116	C₂×C₄×C₂₅₁	—	4	4	no	—
117	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
118	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
119	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
121	C₂²×C₂₅₁	—	2	8	no	—
122	C₂²×C₂₅₁	—	2	4	no	—
123	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
124	C₂×D₂₅₁	—	2	8	no	—
125	C₂×D₂₅₁	—	2	4	no	—
126	C₂×D₂₅₁	—	4	16	no	—
127	C₂×D₂₅₁	C₂×#41	1	4	yes	—
128	C₂×D₂₅₁	H(5,𝔽₂)	1	2	yes	—
129	C₄×C₂₅₁	—	2	4	no	—
130	C₄×C₂₅₁	—	2	4	no	—
131	C₄×C₂₅₁	—	2	1	no	—
132	C₄×C₂₅₁	C₂×#41	1	4	yes	—
133	Dic₂₅₁	—	2	4	no	—
134	Dic₂₅₁	—	2	4	no	—
135	Dic₂₅₁	—	2	1	no	—
136	Dic₂₅₁	C₂×#41	1	4	yes	—
137	C₂×C₂₅₁	—	4	4	no	—
138	C₂×C₂₅₁	C₂²×#41	1	2	yes	—
139	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₂₅₁	—	4	4	no	—
141	D₂₅₁	#559	1	4	yes	direct product factor
142	C₂₅₁	C₂×#559	1	1	yes	characteristic, complemented, Sylow
143	C₂×#559	—	251	1	no	maximal, Sylow
144	#559	—	251	1	no	—
145	#559	—	251	4	no	—
146	#559	—	251	4	no	—
147	#559	—	251	2	no	—
148	#559	—	251	4	no	—
149	#559	D₂₅₁	1	1	yes	characteristic, direct product factor
150	C₂×#127	—	251	1	no	—
151	C₂×#124	—	251	4	no	—
152	C₂×#123	—	251	4	no	—
153	C₂×C₄×Q₈	—	251	2	no	—
154	C₂×C₄×D₄	—	251	4	no	—
155	C₂²×Q₈	—	251	2	no	—
156	C₂²×D₄	—	251	2	no	—
157	C₂²×D₄	—	251	4	no	—
158	C₂³×C₄	—	251	4	no	—
159	#127	—	251	4	no	—
160	#127	—	251	2	no	—
161	#127	—	251	1	no	—
162	#127	C₂×D₂₅₁	1	1	yes	characteristic
163	#124	—	251	4	no	—
164	#124	—	251	4	no	—
165	#124	—	251	8	no	—
166	#124	—	251	8	no	—
167	#124	—	251	4	no	—
168	#124	C₂×D₂₅₁	1	4	yes	complemented
169	#123	—	251	4	no	—
170	#123	—	251	4	no	—
171	#123	—	251	8	no	—
172	#123	—	251	8	no	—
173	#123	—	251	4	no	—
174	#123	C₂×D₂₅₁	1	4	yes	complemented
175	C₄×Q₈	—	251	2	no	—
176	C₄×Q₈	—	251	4	no	—
177	C₄×Q₈	—	251	2	no	—
178	C₄×Q₈	—	251	2	no	—
179	C₄×Q₈	—	251	4	no	—
180	C₄×Q₈	C₂×D₂₅₁	1	2	yes	complemented
181	C₄×D₄	—	251	4	no	—
182	C₄×D₄	—	251	8	no	—
183	C₄×D₄	—	251	4	no	—
184	C₄×D₄	—	251	8	no	—
185	C₄×D₄	—	251	4	no	—
186	C₄×D₄	C₂×D₂₅₁	1	4	yes	complemented
187	C₂×C₄⋊C₄	—	251	2	no	—
188	C₂×C₄⋊C₄	—	251	4	no	—
189	C₂×C₂²⋊C₄	—	251	8	no	—
190	C₂×C₂²⋊C₄	—	251	4	no	—
191	C₂×C₄²	—	251	4	no	—
192	C₂×C₄²	—	251	1	no	—
193	C₂⁴	—	251	4	no	—
194	C₂×Q₈	—	502	8	no	—
195	C₂×Q₈	—	251	2	no	—
196	C₂×Q₈	—	251	4	no	—
197	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
198	C₂×D₄	—	1004	4	no	—
199	C₂×D₄	—	502	16	no	—
200	C₂×D₄	—	251	4	no	—
201	C₂×D₄	—	251	8	no	—
202	C₂×D₄	—	251	2	no	—
203	C₂×D₄	—	251	4	no	—
204	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
205	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
206	C₂²×C₄	—	1004	8	no	—
207	C₂²×C₄	—	251	4	no	—
208	C₂²×C₄	—	251	8	no	—
209	C₂²×C₄	—	251	4	no	—
210	C₂²×C₄	—	251	4	no	—
211	C₂²×C₄	—	251	1	no	—
212	C₂²×C₄	—	251	2	no	—
213	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
214	C₄⋊C₄	—	251	4	no	—
215	C₄⋊C₄	—	251	4	no	—
216	C₄⋊C₄	—	251	4	no	—
217	C₄⋊C₄	—	251	2	no	—
218	C₄⋊C₄	—	251	4	no	—
219	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
220	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
221	C₂²⋊C₄	—	251	8	no	—
222	C₂²⋊C₄	—	251	4	no	—
223	C₂²⋊C₄	—	251	8	no	—
224	C₂²⋊C₄	—	251	8	no	—
225	C₂²⋊C₄	—	251	8	no	—
226	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
227	C₂²⋊C₄	C₂²×D₂₅₁	1	4	yes	—
228	C₄²	—	251	4	no	—
229	C₄²	—	251	4	no	—
230	C₄²	—	251	4	no	—
231	C₄²	—	251	2	no	—
232	C₄²	—	251	1	no	—
233	C₄²	C₂²×D₂₅₁	1	4	yes	—
234	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
235	C₂³	—	502	8	no	—
236	C₂³	—	502	16	no	—
237	C₂³	—	251	4	no	—
238	C₂³	—	251	1	no	—
239	C₂³	C₂³×D₂₅₁	1	4	yes	—
240	Q₈	—	2	4	no	—
241	Q₈	—	502	8	no	—
242	Q₈	—	502	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	1004	2	no	—
246	D₄	—	1004	4	no	—
247	D₄	—	502	16	no	—
248	D₄	—	502	8	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	1004	8	no	—
251	C₂×C₄	—	1004	4	no	—
252	C₂×C₄	—	502	8	no	—
253	C₂×C₄	—	502	2	no	—
254	C₂×C₄	—	502	8	no	—
255	C₂×C₄	—	251	4	no	—
256	C₂×C₄	—	251	4	no	—
257	C₂×C₄	—	251	1	no	—
258	C₂×C₄	—	251	2	no	—
259	C₂×C₄	—	251	8	no	—
260	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
261	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
262	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	1004	16	no	—
267	C₂²	—	502	8	no	—
268	C₂²	—	502	4	no	—
269	C₂²	—	251	4	no	—
270	C₂²	—	251	2	no	—
271	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	1	no	—
275	C₄	—	502	4	no	—
276	C₄	—	502	4	no	—
277	C₄	—	502	1	no	—
278	C₄	—	251	4	no	—
279	C₄	#41×D₂₅₁	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	1004	4	no	—
282	C₂	—	251	4	no	—
283	C₂	C₂×#41×D₂₅₁	1	2	yes	—
284	C₂	H(5,𝔽₂)×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁵
Fitting	#559×C₂₅₁	C₂
prime radical	#127×D₂₅₁	C₂
hypercenter	#559	D₂₅₁
hypocenter	C₂₅₁	C₂×#559
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #30632423	C₂	semidirect
2. #30632422	C₂	semidirect
3. #30632421	C₂	semidirect
4. #30632420	C₂	semidirect
5. #30632419	C₂	semidirect
7. #124×D₂₅₁	C₂	semidirect
8. #123×D₂₅₁	C₂	semidirect
9. C₄×Q₈×D₂₅₁	C₂	semidirect
10. C₄×D₄×D₂₅₁	C₂	semidirect
11. #559×C₂₅₁	C₂	semidirect
19. #22548704	C₂²	semidirect
20. #22548703	C₂²	semidirect
21. #22548702	C₂²	semidirect
22. #22548701	C₂²	semidirect
23. #22548701	C₂²	semidirect
24. #22548700	C₂²	semidirect
25. #22548699	C₂²	semidirect
26. #22548698	C₂²	semidirect
27. #22548697	C₂²	semidirect
28. #22548696	C₂²	semidirect
29. Q₈×Dic₂₅₁	C₂²	semidirect
30. C₄×Dic₅₀₂	C₂²	semidirect
31. C₄×Dic₅₀₂	C₂²	semidirect
32. #22548680	C₂²	semidirect
33. #22548680	C₂²	semidirect
34. C₄×D₁₀₀₄	C₂²	semidirect
35. D₄×Dic₂₅₁	C₂²	semidirect
36. #22548677	C₂²	semidirect
37. C₄×C₂²⋊D₂₅₁	C₂²	semidirect
38. (C₂²⋊D₂₅₁)⋊C₄	C₂²	semidirect
46. #124×C₂₅₁	C₂²	semidirect
47. #123×C₂₅₁	C₂²	semidirect
48. C₄×Q₈×C₂₅₁	C₂²	semidirect
49. C₄×D₄×C₂₅₁	C₂²	semidirect
141. D₂₅₁	#559	direct (N × H)
142. C₂₅₁	C₂×#559	semidirect
149. #559	D₂₅₁	direct (N × H)
168. #124	C₂×D₂₅₁	semidirect
174. #123	C₂×D₂₅₁	semidirect
180. C₄×Q₈	C₂×D₂₅₁	semidirect
186. C₄×D₄	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632423	2
#30632422	2
#30632421	2
#30632420	2
#30632419	2
#127×D₂₅₁	2
#124×D₂₅₁	2
#123×D₂₅₁	2
C₄×Q₈×D₂₅₁	2
C₄×D₄×D₂₅₁	2
#559×C₂₅₁	2
C₂×#559	251

Subgroups — #559×D251 #44563949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #559×D₂₅₁ #44563949