Subgroups — #44548621

Order 32128.

520 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal
3	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal
4	C₂×C₂₅₁⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×#95×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
7	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₅₁	—	2	4	no	—
9	C₂×C₄⋊Dic₂₅₁	—	2	2	no	—
10	C₂₅₁⋊#95	—	2	4	no	—
11	C₂₅₁⋊#95	—	2	4	no	—
12	C₂₅₁⋊#95	C₄	1	4	yes	complemented
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
16	#95×C₂₅₁	—	2	2	no	—
17	C₂²×Dic₂₅₁	—	2	2	no	—
18	C₂²×Dic₂₅₁	—	2	2	no	—
19	C₂²×Dic₂₅₁	—	2	2	no	—
20	C₂²×Dic₂₅₁	—	2	2	no	—
21	C₂²×Dic₂₅₁	—	2	2	no	—
22	C₂²×Dic₂₅₁	—	2	2	no	—
23	C₂²×Dic₂₅₁	—	2	2	no	—
24	C₂²×Dic₂₅₁	—	2	2	no	—
25	C₂²×Dic₂₅₁	—	2	2	no	—
26	C₂²×Dic₂₅₁	—	2	2	no	—
27	C₂²×Dic₂₅₁	—	2	2	no	—
28	C₂²×Dic₂₅₁	—	2	2	no	—
29	C₂²×Dic₂₅₁	C₂×C₄	1	4	yes	—
30	C₂²×Dic₂₅₁	D₄	1	4	yes	—
31	C₄⋊Dic₂₅₁	—	2	8	no	—
32	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₅₁	—	2	2	no	—
34	C₂²×C₄×C₂₅₁	—	2	2	no	—
35	C₂²×C₄×C₂₅₁	—	2	1	no	—
36	C₂²×C₄×C₂₅₁	—	2	1	no	—
37	C₂²×C₄×C₂₅₁	—	2	2	no	—
38	C₂²×C₄×C₂₅₁	—	2	2	no	—
39	C₂²×C₄×C₂₅₁	—	2	2	no	—
40	C₂²×C₄×C₂₅₁	—	2	2	no	—
41	C₂²×C₄×C₂₅₁	—	2	1	no	—
42	C₂²×C₄×C₂₅₁	—	2	1	no	—
43	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
45	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
46	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
47	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
48	C₂×Dic₂₅₁	—	2	4	no	—
49	C₂×Dic₂₅₁	—	2	4	no	—
50	C₂×Dic₂₅₁	—	2	4	no	—
51	C₂×Dic₂₅₁	—	2	4	no	—
52	C₂×Dic₂₅₁	—	2	4	no	—
53	C₂×Dic₂₅₁	—	2	4	no	—
54	C₂×Dic₂₅₁	—	4	2	no	—
55	C₂×Dic₂₅₁	—	4	2	no	—
56	C₂×Dic₂₅₁	—	4	2	no	—
57	C₂×Dic₂₅₁	—	4	2	no	—
58	C₂×Dic₂₅₁	—	4	2	no	—
59	C₂×Dic₂₅₁	—	4	2	no	—
60	C₂×Dic₂₅₁	—	4	2	no	—
61	C₂×Dic₂₅₁	—	4	2	no	—
62	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
63	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
64	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
67	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
68	C₂³×C₂₅₁	#41	1	2	yes	—
69	C₂³×C₂₅₁	#41	1	2	yes	—
70	C₂³×C₂₅₁	#41	1	1	yes	characteristic
71	C₂³×C₂₅₁	#41	1	1	yes	characteristic
72	C₂³×C₂₅₁	#41	1	2	yes	—
73	C₂×C₄×C₂₅₁	—	2	4	no	—
74	C₂×C₄×C₂₅₁	—	2	2	no	—
75	C₂×C₄×C₂₅₁	—	2	4	no	—
76	C₂×C₄×C₂₅₁	—	2	2	no	—
77	C₂×C₄×C₂₅₁	—	2	2	no	—
78	C₂×C₄×C₂₅₁	—	2	4	no	—
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	4	no	—
81	C₂×C₄×C₂₅₁	—	4	1	no	—
82	C₂×C₄×C₂₅₁	—	4	1	no	—
83	C₂×C₄×C₂₅₁	—	4	2	no	—
84	C₂×C₄×C₂₅₁	—	4	1	no	—
85	C₂×C₄×C₂₅₁	—	4	1	no	—
86	C₂×C₄×C₂₅₁	—	4	2	no	—
87	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
88	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₅₁	#117	1	2	yes	—
90	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
91	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
92	C₂²×C₂₅₁	C₄×Q₈	1	1	yes	characteristic
93	C₂²×C₂₅₁	#120	1	1	yes	characteristic
94	C₂²×C₂₅₁	#121	1	1	yes	characteristic
95	C₂²×C₂₅₁	#121	1	2	yes	—
96	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
97	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂²×C₂₅₁	#123	1	1	yes	characteristic
100	C₂²×C₂₅₁	#123	1	2	yes	—
101	C₂²×C₂₅₁	#123	1	2	yes	—
102	C₂²×C₂₅₁	#124	1	2	yes	—
103	C₂²×C₂₅₁	#124	1	1	yes	characteristic
104	C₂²×C₂₅₁	#124	1	1	yes	characteristic
105	C₂²×C₂₅₁	#125	1	2	yes	—
106	C₂²×C₂₅₁	#125	1	2	yes	—
107	C₂²×C₂₅₁	#125	1	1	yes	characteristic
108	C₂²×C₂₅₁	#126	1	2	yes	—
109	C₂²×C₂₅₁	#126	1	2	yes	—
110	C₂²×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₂₅₁	—	2	4	no	—
112	C₄×C₂₅₁	—	4	2	no	—
113	C₄×C₂₅₁	—	4	2	no	—
114	Dic₂₅₁	—	4	4	no	—
115	Dic₂₅₁	—	4	4	no	—
116	C₂×C₂₅₁	#382	1	1	yes	characteristic
117	C₂×C₂₅₁	#385	1	1	yes	characteristic
118	C₂×C₂₅₁	#387	1	2	yes	—
119	C₂×C₂₅₁	#388	1	2	yes	—
120	C₂×C₂₅₁	#389	1	1	yes	characteristic
121	C₂×C₂₅₁	#393	1	1	yes	characteristic
122	C₂×C₂₅₁	#394	1	1	yes	characteristic
123	C₂×C₂₅₁	#397	1	1	yes	characteristic
124	C₂×C₂₅₁	#398	1	2	yes	—
125	C₂×C₂₅₁	#400	1	1	yes	characteristic
126	C₂×C₂₅₁	#400	1	2	yes	—
127	C₂₅₁	#1442	1	1	yes	characteristic, complemented, Sylow
128	#1442	—	251	1	no	maximal, Sylow
129	C₂²×C₄⋊C₄	—	251	1	no	—
130	C₂×#95	—	251	2	no	—
131	C₂×#95	—	251	2	no	—
132	C₂×#95	—	251	1	no	—
133	C₂×#95	D₂₅₁	1	1	yes	characteristic
134	C₂³×C₄	—	251	2	no	—
135	C₂³×C₄	—	251	2	no	—
136	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
137	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
138	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	502	4	no	—
140	C₂×C₄⋊C₄	—	502	2	no	—
141	#95	—	2	2	no	—
142	#95	—	502	4	no	—
143	#95	—	502	4	no	—
144	#95	—	251	4	no	—
145	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	502	2	no	—
157	C₂²×C₄	—	502	2	no	—
158	C₂²×C₄	—	502	2	no	—
159	C₂²×C₄	—	502	2	no	—
160	C₂²×C₄	—	502	2	no	—
161	C₂²×C₄	—	502	2	no	—
162	C₂²×C₄	—	502	2	no	—
163	C₂²×C₄	—	502	2	no	—
164	C₂²×C₄	—	502	2	no	—
165	C₂²×C₄	—	502	2	no	—
166	C₂²×C₄	—	502	2	no	—
167	C₂²×C₄	—	502	2	no	—
168	C₂²×C₄	—	251	4	no	—
169	C₂²×C₄	—	251	4	no	—
170	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
171	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
172	C₂²×C₄	D₁₀₀₄	1	2	yes	—
173	C₂²×C₄	Dic₅₀₂	1	2	yes	—
174	C₂²×C₄	Dic₅₀₂	1	2	yes	—
175	C₄⋊C₄	—	502	8	no	—
176	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
177	C₂³	D₄×D₂₅₁	1	2	yes	—
178	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
179	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
180	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
181	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
182	C₂³	#16860809	1	2	yes	—
183	C₂³	#16860809	1	2	yes	—
184	C₂³	#16860810	1	1	yes	characteristic
185	C₂³	#16860810	1	1	yes	characteristic
186	C₂³	#16860811	1	2	yes	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	1004	2	no	—
202	C₂×C₄	—	1004	2	no	—
203	C₂×C₄	—	1004	2	no	—
204	C₂×C₄	—	1004	2	no	—
205	C₂×C₄	—	1004	2	no	—
206	C₂×C₄	—	1004	2	no	—
207	C₂×C₄	—	1004	2	no	—
208	C₂×C₄	—	1004	2	no	—
209	C₂×C₄	—	502	4	no	—
210	C₂×C₄	—	502	4	no	—
211	C₂×C₄	—	502	4	no	—
212	C₂×C₄	—	502	4	no	—
213	C₂×C₄	—	502	4	no	—
214	C₂×C₄	—	502	4	no	—
215	C₂×C₄	Dic₂₅₁⋊C₄	1	4	yes	—
216	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
217	C₂²	#22548673	1	2	yes	—
218	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
219	C₂²	C₄×D₁₀₀₄	1	1	yes	characteristic
220	C₂²	C₄×Dic₅₀₂	1	1	yes	characteristic
221	C₂²	#22548683	1	1	yes	characteristic
222	C₂²	#22548689	1	1	yes	characteristic
223	C₂²	#22548690	1	2	yes	—
224	C₂²	#22548692	1	2	yes	—
225	C₂²	#22548692	1	2	yes	—
226	C₂²	#22548694	1	1	yes	characteristic
227	C₂²	#22548696	1	1	yes	characteristic
228	C₂²	#22548699	1	2	yes	—
229	C₂²	#22548700	1	2	yes	—
230	C₂²	#22548701	1	2	yes	—
231	C₂²	#22548702	1	1	yes	characteristic
232	C₂²	#22548704	1	1	yes	characteristic
233	C₂²	#22548706	1	2	yes	—
234	C₂²	#22548706	1	2	yes	—
235	C₂²	#22548707	1	1	yes	characteristic
236	C₂²	#22548708	1	2	yes	—
237	C₂²	#22548710	1	2	yes	—
238	C₂²	#22548715	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	1004	4	no	—
243	C₄	—	1004	4	no	—
244	C₂	#30631552	1	1	yes	characteristic
245	C₂	#30631559	1	1	yes	characteristic
246	C₂	#30631574	1	2	yes	—
247	C₂	#30631579	1	2	yes	—
248	C₂	#30631585	1	1	yes	characteristic
249	C₂	#30631599	1	1	yes	characteristic
250	C₂	#30631604	1	1	yes	characteristic
251	C₂	#30631612	1	1	yes	characteristic
252	C₂	#30631619	1	2	yes	—
253	C₂	#30631626	1	1	yes	characteristic
254	C₂	#30631629	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₅₁
commutator	C₂³×C₂₅₁	C₂²×C₄
Fitting	C₂×#95×C₂₅₁	C₂
socle, prime radical, cosocle	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂×#95	D₂₅₁
hypocenter	C₂₅₁	#1442
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
12. C₂₅₁⋊#95	C₄	semidirect
127. C₂₅₁	#1442	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₂₅₁	2
C₂×C₂₅₁⋊#95	2
C₂×#95×C₂₅₁	2
#1442	251