Subgroups — #419×D₂₅₁ #44558868

Order 32128.

752 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#117×D₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#30631697	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631696	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631695	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631694	C₂	1	4	yes	maximal, complemented
8	#103×D₂₅₁	C₂	1	4	yes	maximal, complemented
9	#419×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₅₀₂	—	2	1	no	—
11	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
12	C₂×Q₈×D₂₅₁	—	2	4	no	—
13	C₂×Q₈×D₂₅₁	—	2	1	no	—
14	C₂×Q₈×D₂₅₁	—	2	1	no	—
15	C₂×Q₈×D₂₅₁	C₄	1	8	yes	complemented
16	C₂×Q₈×D₂₅₁	C₂²	1	2	yes	—
17	C₂×Q₈×D₂₅₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₂₅₁	—	2	1	no	—
19	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
20	C₂×#16860794	C₂²	1	1	yes	characteristic
21	C₂×#16860793	C₂²	1	2	yes	—
22	M₁₆×D₂₅₁	—	2	4	no	—
23	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
24	#22548674	C₂²	1	1	yes	characteristic
25	#22548673	C₂²	1	2	yes	—
26	#22548672	—	2	1	no	—
27	#22548671	—	2	1	no	—
28	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×D₂₅₁	—	2	1	no	—
30	C₄²×D₂₅₁	—	2	1	no	—
31	#22548640	C₂²	1	4	yes	complemented
32	#22548639	C₂²	1	4	yes	complemented
33	#22548638	C₂²	1	4	yes	complemented
34	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
36	#117×C₂₅₁	C₂²	1	1	yes	characteristic
37	#103×C₂₅₁	C₂²	1	4	yes	complemented
38	C₂³×D₂₅₁	D₄	1	1	yes	characteristic
39	C₂×Dic₅₀₂	—	2	8	no	—
40	C₂×Dic₅₀₂	—	2	2	no	—
41	C₂×Dic₅₀₂	—	2	2	no	—
42	C₂×Dic₅₀₂	—	2	1	no	—
43	C₂×Dic₅₀₂	—	2	2	no	—
44	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₅₀₂	C₂³	1	2	yes	—
46	Q₈×D₂₅₁	—	2	8	no	—
47	Q₈×D₂₅₁	—	2	8	no	—
48	Q₈×D₂₅₁	—	2	4	no	—
49	Q₈×D₂₅₁	—	2	4	no	—
50	Q₈×D₂₅₁	C₂×C₄	1	8	yes	—
51	Q₈×D₂₅₁	C₂×C₄	1	8	yes	complemented
52	C₂²×Dic₂₅₁	—	2	1	no	—
53	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
54	C₂×C₄×D₂₅₁	—	2	8	no	—
55	C₂×C₄×D₂₅₁	—	2	2	no	—
56	C₂×C₄×D₂₅₁	—	2	2	no	—
57	C₂×C₄×D₂₅₁	—	2	2	no	—
58	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
59	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
60	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
61	C₂×C₄×D₂₅₁	D₄	1	4	yes	—
62	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
64	#16860794	—	2	2	no	—
65	#16860793	—	2	4	no	—
66	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
67	C₈×D₂₅₁	—	2	4	no	—
68	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
69	Dic₂₅₁⋊C₄	—	2	2	no	—
70	C₂²⋊Dic₂₅₁	—	2	1	no	—
71	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
72	C₄×Dic₂₅₁	—	2	2	no	—
73	C₄×Dic₂₅₁	—	2	1	no	—
74	C₂×Q₈×C₂₅₁	—	2	2	no	—
75	C₂×Q₈×C₂₅₁	—	2	1	no	—
76	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
77	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₂₅₁	—	2	1	no	—
79	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
80	M₁₆×C₂₅₁	—	2	2	no	—
81	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
82	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
84	C₄²×C₂₅₁	—	2	1	no	—
85	C₂²×D₂₅₁	—	2	4	no	—
86	C₂²×D₂₅₁	C₂²⋊C₄	1	4	yes	—
87	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
89	Dic₅₀₂	—	2	4	no	—
90	Dic₅₀₂	—	2	2	no	—
91	Dic₅₀₂	—	2	8	no	—
92	Dic₅₀₂	—	2	8	no	—
93	Dic₅₀₂	C₂²×C₄	1	4	yes	—
94	C₂×Dic₂₅₁	—	2	4	no	—
95	C₂×Dic₂₅₁	—	2	2	no	—
96	C₂×Dic₂₅₁	—	2	2	no	—
97	C₂×Dic₂₅₁	C₂²⋊C₄	1	4	yes	—
98	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
100	C₄×D₂₅₁	—	2	8	no	—
101	C₄×D₂₅₁	—	2	8	no	—
102	C₄×D₂₅₁	—	4	4	no	—
103	C₄×D₂₅₁	C₂²⋊C₄	1	4	yes	—
104	C₄×D₂₅₁	C₂²⋊C₄	1	4	yes	—
105	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
106	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
107	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
108	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
109	C₂₅₁⋊C₈	—	2	2	no	—
110	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₂₅₁	—	2	4	no	—
112	Q₈×C₂₅₁	—	2	2	no	—
113	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	—
114	C₂×C₄×C₂₅₁	—	2	4	no	—
115	C₂×C₄×C₂₅₁	—	2	2	no	—
116	C₂×C₄×C₂₅₁	—	2	2	no	—
117	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
121	C₈×C₂₅₁	—	2	2	no	—
122	C₂²×C₂₅₁	—	2	2	no	—
123	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×D₂₅₁	—	2	2	no	—
126	C₂×D₂₅₁	—	2	8	no	—
127	C₂×D₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×D₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×D₂₅₁	#137	1	4	yes	—
130	C₄×C₂₅₁	—	2	4	no	—
131	C₄×C₂₅₁	—	4	2	no	—
132	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
133	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
135	Dic₂₅₁	—	2	4	no	—
136	Dic₂₅₁	—	4	2	no	—
137	Dic₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
138	Dic₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₂₅₁	—	2	2	no	—
140	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₂₅₁	C₂×#137	1	2	yes	—
142	D₂₅₁	—	2	2	no	—
143	D₂₅₁	#419	1	4	yes	direct product factor
144	C₂₅₁	C₂×#419	1	1	yes	characteristic, complemented, Sylow
145	C₂×#419	—	251	1	no	maximal, Sylow
146	C₂³×Q₈	—	251	1	no	—
147	C₂²×M₁₆	—	251	1	no	—
148	C₂×#117	—	251	1	no	—
149	#419	—	251	1	no	—
150	#419	—	251	1	no	—
151	#419	—	251	1	no	—
152	#419	—	251	4	no	—
153	#419	D₂₅₁	1	1	yes	characteristic, direct product factor
154	C₂×#103	—	251	4	no	—
155	C₂²×Q₈	—	502	1	no	—
156	C₂²×Q₈	—	502	1	no	—
157	C₂²×Q₈	—	502	1	no	—
158	C₂²×Q₈	—	502	4	no	—
159	C₂²×Q₈	—	251	8	no	—
160	C₂²×Q₈	—	251	2	no	—
161	C₂²×Q₈	—	251	2	no	—
162	C₂²×Q₈	—	251	1	no	—
163	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
164	C₂³×C₄	—	502	1	no	—
165	C₂³×C₄	—	251	1	no	—
166	C₂×M₁₆	—	502	4	no	—
167	C₂×M₁₆	—	251	1	no	—
168	C₂×M₁₆	—	251	2	no	—
169	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
170	C₂²×C₈	—	251	2	no	—
171	#117	—	502	1	no	—
172	#117	—	502	1	no	—
173	#117	—	251	1	no	—
174	#117	—	251	2	no	—
175	#117	C₂×D₂₅₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	251	2	no	—
177	C₂×C₂²⋊C₄	—	502	1	no	—
178	C₂×C₄²	—	502	1	no	—
179	#103	—	251	4	no	—
180	#103	—	251	4	no	—
181	#103	—	251	4	no	—
182	#103	C₂×D₂₅₁	1	4	yes	complemented
183	C₂⁴	—	251	1	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	502	2	no	—
187	C₂×Q₈	—	502	2	no	—
188	C₂×Q₈	—	502	8	no	—
189	C₂×Q₈	—	502	4	no	—
190	C₂×Q₈	—	502	4	no	—
191	C₂×Q₈	—	502	8	no	—
192	C₂×Q₈	—	502	8	no	—
193	C₂×Q₈	—	502	1	no	—
194	C₂×Q₈	—	502	2	no	—
195	C₂×Q₈	—	251	8	no	—
196	C₂×Q₈	—	251	8	no	—
197	C₂×Q₈	—	251	4	no	—
198	C₂×Q₈	—	251	2	no	—
199	C₂×Q₈	C₄×D₂₅₁	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	502	2	no	—
203	C₂²×C₄	—	502	2	no	—
204	C₂²×C₄	—	502	2	no	—
205	C₂²×C₄	—	502	8	no	—
206	C₂²×C₄	—	502	1	no	—
207	C₂²×C₄	—	251	1	no	—
208	C₂²×C₄	—	251	4	no	—
209	C₂²×C₄	—	251	1	no	—
210	C₂²×C₄	—	251	1	no	—
211	C₂²×C₄	—	251	4	no	—
212	C₂²×C₄	—	251	1	no	—
213	C₂²×C₄	—	251	1	no	—
214	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	502	4	no	—
217	M₁₆	—	502	2	no	—
218	C₂×C₈	—	502	4	no	—
219	C₂×C₈	—	251	2	no	—
220	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
221	C₄⋊C₄	—	502	2	no	—
222	C₄⋊C₄	—	251	2	no	—
223	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	502	2	no	—
226	C₂²⋊C₄	—	502	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	502	2	no	—
229	C₄²	—	502	1	no	—
230	C₂³	—	502	4	no	—
231	C₂³	—	251	1	no	—
232	C₂³	—	251	4	no	—
233	C₂³	—	251	1	no	—
234	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	4	no	—
237	Q₈	—	502	8	no	—
238	Q₈	—	502	8	no	—
239	Q₈	—	502	2	no	—
240	Q₈	—	502	4	no	—
241	Q₈	—	251	4	no	—
242	Q₈	C₂×C₄×D₂₅₁	1	4	yes	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	1004	4	no	—
247	C₂×C₄	—	502	8	no	—
248	C₂×C₄	—	502	8	no	—
249	C₂×C₄	—	502	2	no	—
250	C₂×C₄	—	502	2	no	—
251	C₂×C₄	—	502	4	no	—
252	C₂×C₄	—	251	1	no	—
253	C₂×C₄	—	251	1	no	—
254	C₂×C₄	—	251	4	no	—
255	C₂×C₄	—	251	2	no	—
256	C₂×C₄	—	251	2	no	—
257	C₂×C₄	—	251	2	no	—
258	C₂×C₄	—	251	2	no	—
259	C₂×C₄	—	251	4	no	—
260	C₂×C₄	—	251	4	no	—
261	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
262	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
263	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	502	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	502	8	no	—
269	C₂²	—	502	2	no	—
270	C₂²	—	251	2	no	—
271	C₂²	—	251	4	no	—
272	C₂²	—	251	2	no	—
273	C₂²	C₂²⋊C₄×D₂₅₁	1	2	yes	—
274	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	4	2	no	—
277	C₄	—	1004	2	no	—
278	C₄	—	502	4	no	—
279	C₄	—	251	2	no	—
280	C₄	—	251	2	no	—
281	C₄	C₂²⋊C₄×D₂₅₁	1	2	yes	—
282	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
283	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	502	2	no	—
286	C₂	—	251	4	no	—
287	C₂	C₂×C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
288	C₂	#137×D₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#419×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂³×D₂₅₁	D₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#419	D₂₅₁
hypocenter	C₂₅₁	C₂×#419
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
4. #30631697	C₂	semidirect
5. #30631696	C₂	semidirect
6. #30631695	C₂	semidirect
7. #30631694	C₂	semidirect
8. #103×D₂₅₁	C₂	semidirect
9. #419×C₂₅₁	C₂	semidirect
15. C₂×Q₈×D₂₅₁	C₄	semidirect
31. #22548640	C₂²	semidirect
32. #22548639	C₂²	semidirect
33. #22548638	C₂²	semidirect
37. #103×C₂₅₁	C₂²	semidirect
44. C₂×Dic₅₀₂	C₂×C₄	semidirect
51. Q₈×D₂₅₁	C₂×C₄	semidirect
76. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
143. D₂₅₁	#419	direct (N × H)
144. C₂₅₁	C₂×#419	semidirect
153. #419	D₂₅₁	direct (N × H)
182. #103	C₂×D₂₅₁	semidirect
199. C₂×Q₈	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₂₅₁	2
C₂×M₁₆×D₂₅₁	2
#117×D₂₅₁	2
#30631697	2
#30631696	2
#30631695	2
#30631694	2
#103×D₂₅₁	2
#419×C₂₅₁	2
C₂×#419	251

Subgroups — #419×D251 #44558868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #419×D₂₅₁ #44558868