Subgroups — #405×D₂₅₁ #44558775

Order 32128.

592 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#117×D₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#30631645	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631644	C₂	1	2	yes	maximal, complemented
6	#30631643	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631642	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631641	C₂	1	2	yes	maximal, complemented
9	#97×D₂₅₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
11	#405×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×#16860794	C₂²	1	1	yes	characteristic
14	C₂×#16860793	C₂²	1	2	yes	—
15	M₁₆×D₂₅₁	C₄	1	8	yes	complemented
16	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₅₁	C₄	1	8	yes	complemented
18	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
20	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
23	#22548674	C₂²	1	1	yes	characteristic
24	#22548673	C₄	1	2	yes	—
25	#22548672	C₂²	1	2	yes	—
26	#22548671	C₄	1	2	yes	—
27	C₄⋊C₄×D₂₅₁	C₄	1	2	yes	—
28	C₂²⋊C₄×D₂₅₁	C₄	1	2	yes	—
29	C₄²×D₂₅₁	C₂²	1	2	yes	—
30	#22548626	C₂²	1	2	yes	complemented
31	#22548625	C₂²	1	2	yes	complemented
32	#22548625	C₂²	1	2	yes	complemented
33	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	complemented
34	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
35	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
38	#117×C₂₅₁	C₂²	1	1	yes	characteristic
39	#97×C₂₅₁	C₂²	1	2	yes	complemented
40	C₄×C₈×C₂₅₁	C₂²	1	2	yes	complemented
41	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
44	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
50	#16860794	C₂×C₄	1	4	yes	complemented
51	#16860793	C₂×C₄	1	8	yes	complemented
52	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	complemented
53	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
55	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
56	C₈×D₂₅₁	C₂×C₄	1	4	yes	—
57	C₈×D₂₅₁	C₂×C₄	1	4	yes	—
58	C₈×D₂₅₁	C₂×C₄	1	8	yes	—
59	C₈×D₂₅₁	C₂×C₄	1	4	yes	complemented
60	C₈×D₂₅₁	C₂×C₄	1	4	yes	complemented
61	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
62	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
63	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
64	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
65	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
66	C₄×Dic₂₅₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
68	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
70	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
71	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
74	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
75	C₄²×C₂₅₁	C₂³	1	2	yes	—
76	C₂²×D₂₅₁	—	2	4	no	—
77	C₂²×D₂₅₁	C₄²	1	4	yes	—
78	C₂²×D₂₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₂₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₂₅₁	C₄²	1	4	yes	—
81	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
82	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₂₅₁	—	2	8	no	—
85	C₄×D₂₅₁	C₄²	1	4	yes	—
86	C₄×D₂₅₁	C₄²	1	4	yes	—
87	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
88	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
89	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
90	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
91	C₂₅₁⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂₅₁⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
100	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
101	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
102	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
103	C₂²×C₂₅₁	—	2	2	no	—
104	C₂²×C₂₅₁	C₂×C₄²	1	2	yes	—
105	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₂₅₁	—	2	2	no	—
107	C₂×D₂₅₁	—	2	8	no	—
108	C₂×D₂₅₁	C₂×C₄²	1	2	yes	—
109	C₂×D₂₅₁	C₂×C₄²	1	2	yes	—
110	C₂×D₂₅₁	#131	1	4	yes	—
111	C₄×C₂₅₁	—	2	4	no	—
112	C₄×C₂₅₁	C₂×C₄²	1	2	yes	—
113	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
115	Dic₂₅₁	—	2	4	no	—
116	Dic₂₅₁	C₂×C₄²	1	2	yes	—
117	Dic₂₅₁	C₂×C₄²	1	2	yes	—
118	C₂×C₂₅₁	—	2	2	no	—
119	C₂×C₂₅₁	C₂²×C₄²	1	1	yes	characteristic
120	C₂×C₂₅₁	C₂×#131	1	2	yes	—
121	D₂₅₁	—	2	2	no	—
122	D₂₅₁	#405	1	4	yes	direct product factor
123	C₂₅₁	C₂×#405	1	1	yes	characteristic, complemented, Sylow
124	C₂×#405	—	251	1	no	maximal, Sylow
125	C₂²×M₁₆	—	251	1	no	—
126	C₂³×C₈	—	251	1	no	—
127	C₂×#117	—	251	1	no	—
128	#405	—	251	1	no	—
129	#405	—	251	1	no	—
130	#405	—	251	1	no	—
131	#405	—	251	2	no	—
132	#405	—	251	2	no	—
133	#405	D₂₅₁	1	1	yes	characteristic, direct product factor
134	C₂×#97	—	251	2	no	—
135	C₂×C₄×C₈	—	251	2	no	—
136	C₂³×C₄	—	251	1	no	—
137	C₂×M₁₆	—	251	1	no	—
138	C₂×M₁₆	—	251	8	no	—
139	C₂×M₁₆	—	251	2	no	—
140	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
141	C₂²×C₈	—	251	1	no	—
142	C₂²×C₈	—	251	8	no	—
143	C₂²×C₈	—	251	1	no	—
144	C₂²×C₈	—	251	1	no	—
145	C₂²×C₈	—	251	2	no	—
146	C₂²×C₈	—	251	1	no	—
147	C₂²×C₈	—	251	1	no	—
148	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
149	#117	—	251	1	no	—
150	#117	—	251	2	no	—
151	#117	—	251	2	no	—
152	#117	—	251	2	no	—
153	#117	C₂×D₂₅₁	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	251	2	no	—
155	C₂×C₂²⋊C₄	—	251	2	no	—
156	C₂×C₄²	—	251	2	no	—
157	#97	—	251	2	no	—
158	#97	—	251	2	no	—
159	#97	—	251	2	no	—
160	#97	C₂×D₂₅₁	1	2	yes	complemented
161	C₄×C₈	—	251	2	no	—
162	C₄×C₈	—	251	2	no	—
163	C₄×C₈	—	251	2	no	—
164	C₄×C₈	C₂×D₂₅₁	1	2	yes	complemented
165	C₂⁴	—	251	1	no	—
166	C₂²×C₄	—	251	4	no	—
167	C₂²×C₄	—	251	1	no	—
168	C₂²×C₄	—	251	1	no	—
169	C₂²×C₄	—	251	1	no	—
170	C₂²×C₄	—	251	4	no	—
171	C₂²×C₄	—	251	4	no	—
172	C₂²×C₄	—	251	1	no	—
173	C₂²×C₄	—	251	1	no	—
174	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
175	M₁₆	—	251	4	no	—
176	M₁₆	—	251	8	no	—
177	M₁₆	C₄×D₂₅₁	1	4	yes	complemented
178	C₂×C₈	—	251	4	no	—
179	C₂×C₈	—	251	1	no	—
180	C₂×C₈	—	251	2	no	—
181	C₂×C₈	—	251	1	no	—
182	C₂×C₈	—	251	4	no	—
183	C₂×C₈	—	251	4	no	—
184	C₂×C₈	—	251	8	no	—
185	C₂×C₈	—	251	4	no	—
186	C₂×C₈	—	251	4	no	—
187	C₂×C₈	C₄×D₂₅₁	1	4	yes	complemented
188	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
189	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
190	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
191	C₄⋊C₄	—	251	4	no	—
192	C₄⋊C₄	—	251	2	no	—
193	C₄⋊C₄	C₄×D₂₅₁	1	2	yes	—
194	C₂²⋊C₄	—	251	2	no	—
195	C₂²⋊C₄	—	251	4	no	—
196	C₂²⋊C₄	C₄×D₂₅₁	1	2	yes	—
197	C₄²	—	251	4	no	—
198	C₄²	—	251	2	no	—
199	C₄²	C₂²×D₂₅₁	1	2	yes	—
200	C₂³	—	502	4	no	—
201	C₂³	—	251	1	no	—
202	C₂³	—	251	4	no	—
203	C₂³	—	251	1	no	—
204	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
205	C₂×C₄	—	502	8	no	—
206	C₂×C₄	—	251	4	no	—
207	C₂×C₄	—	251	1	no	—
208	C₂×C₄	—	251	1	no	—
209	C₂×C₄	—	251	4	no	—
210	C₂×C₄	—	251	2	no	—
211	C₂×C₄	—	251	2	no	—
212	C₂×C₄	—	251	2	no	—
213	C₂×C₄	—	251	2	no	—
214	C₂×C₄	—	251	4	no	—
215	C₂×C₄	—	251	4	no	—
216	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
221	C₈	—	251	2	no	—
222	C₈	—	251	2	no	—
223	C₈	—	251	4	no	—
224	C₈	C₂×C₄×D₂₅₁	1	2	yes	—
225	C₈	C₂×C₄×D₂₅₁	1	2	yes	—
226	C₈	C₂×C₄×D₂₅₁	1	4	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	502	8	no	—
229	C₂²	—	502	2	no	—
230	C₂²	—	251	2	no	—
231	C₂²	—	251	4	no	—
232	C₂²	—	251	2	no	—
233	C₂²	C₄²×D₂₅₁	1	2	yes	—
234	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	502	4	no	—
237	C₄	—	251	2	no	—
238	C₄	—	251	2	no	—
239	C₄	C₄²×D₂₅₁	1	2	yes	—
240	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	502	2	no	—
244	C₂	—	251	4	no	—
245	C₂	C₂×C₄²×D₂₅₁	1	1	yes	characteristic
246	C₂	#131×D₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₅₁
commutator	C₂×C₂₅₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#405×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂³×C₄
prime radical	C₂³×D₂₅₁	C₂×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#405	D₂₅₁
hypocenter	C₂₅₁	C₂×#405
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #30631645	C₂	semidirect
5. #30631644	C₂	semidirect
6. #30631643	C₂	semidirect
7. #30631642	C₂	semidirect
8. #30631641	C₂	semidirect
9. #97×D₂₅₁	C₂	semidirect
10. C₄×C₈×D₂₅₁	C₂	semidirect
11. #405×C₂₅₁	C₂	semidirect
15. M₁₆×D₂₅₁	C₄	semidirect
17. C₂×C₈×D₂₅₁	C₄	semidirect
30. #22548626	C₂²	semidirect
31. #22548625	C₂²	semidirect
32. #22548625	C₂²	semidirect
33. C₄×C₂₅₁⋊C₈	C₂²	semidirect
34. C₈×Dic₂₅₁	C₂²	semidirect
35. C₈×Dic₂₅₁	C₂²	semidirect
39. #97×C₂₅₁	C₂²	semidirect
40. C₄×C₈×C₂₅₁	C₂²	semidirect
50. #16860794	C₂×C₄	semidirect
51. #16860793	C₂×C₄	semidirect
52. C₂×C₂₅₁⋊C₈	C₂×C₄	semidirect
59. C₈×D₂₅₁	C₂×C₄	semidirect
60. C₈×D₂₅₁	C₂×C₄	semidirect
68. M₁₆×C₂₅₁	C₂×C₄	semidirect
69. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
122. D₂₅₁	#405	direct (N × H)
123. C₂₅₁	C₂×#405	semidirect
133. #405	D₂₅₁	direct (N × H)
160. #97	C₂×D₂₅₁	semidirect
164. C₄×C₈	C₂×D₂₅₁	semidirect
177. M₁₆	C₄×D₂₅₁	semidirect
187. C₂×C₈	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₅₁	2
C₂²×C₈×D₂₅₁	2
#117×D₂₅₁	2
#30631645	2
#30631644	2
#30631643	2
#30631642	2
#30631641	2
#97×D₂₅₁	2
C₄×C₈×D₂₅₁	2
#405×C₂₅₁	2
C₂×#405	251

Subgroups — #405×D251 #44558775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #405×D₂₅₁ #44558775