Subgroups — #44548629

Order 32128.

608 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₂²×Dic₂₅₁⋊C₄	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	C₂×Dic₂₅₁⋊C₄	—	2	4	no	—
11	C₂×Dic₂₅₁⋊C₄	—	2	4	no	—
12	C₂×Dic₂₅₁⋊C₄	—	2	4	no	—
13	C₂₅₁⋊#95	—	2	4	no	—
14	C₂₅₁⋊#95	C₄	1	4	yes	complemented
15	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
18	#95×C₂₅₁	—	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	C₂²×Dic₂₅₁	D₄	1	4	yes	—
32	C₂²×Dic₂₅₁	Q₈	1	4	yes	—
33	C₄⋊Dic₂₅₁	—	2	8	no	—
34	Dic₂₅₁⋊C₄	—	2	16	no	—
35	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₅₁	—	2	2	no	—
37	C₂²×C₄×C₂₅₁	—	2	2	no	—
38	C₂²×C₄×C₂₅₁	—	2	1	no	—
39	C₂²×C₄×C₂₅₁	—	2	1	no	—
40	C₂²×C₄×C₂₅₁	—	2	2	no	—
41	C₂²×C₄×C₂₅₁	—	2	2	no	—
42	C₂²×C₄×C₂₅₁	—	2	2	no	—
43	C₂²×C₄×C₂₅₁	—	2	2	no	—
44	C₂²×C₄×C₂₅₁	—	2	1	no	—
45	C₂²×C₄×C₂₅₁	—	2	1	no	—
46	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
48	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
49	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
50	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
51	C₂×Dic₂₅₁	—	2	4	no	—
52	C₂×Dic₂₅₁	—	2	4	no	—
53	C₂×Dic₂₅₁	—	2	4	no	—
54	C₂×Dic₂₅₁	—	2	4	no	—
55	C₂×Dic₂₅₁	—	2	4	no	—
56	C₂×Dic₂₅₁	—	2	4	no	—
57	C₂×Dic₂₅₁	—	2	4	no	—
58	C₂×Dic₂₅₁	—	2	4	no	—
59	C₂×Dic₂₅₁	—	2	4	no	—
60	C₂×Dic₂₅₁	—	4	2	no	—
61	C₂×Dic₂₅₁	—	4	2	no	—
62	C₂×Dic₂₅₁	—	4	2	no	—
63	C₂×Dic₂₅₁	—	4	2	no	—
64	C₂×Dic₂₅₁	C₄⋊C₄	1	8	yes	complemented
65	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
66	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
67	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
68	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₂₅₁	C₂×Q₈	1	2	yes	—
71	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
72	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₂₅₁	#41	1	2	yes	—
74	C₂³×C₂₅₁	#41	1	1	yes	characteristic
75	C₂³×C₂₅₁	#41	1	1	yes	characteristic
76	C₂×C₄×C₂₅₁	—	2	4	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	—	2	2	no	—
82	C₂×C₄×C₂₅₁	—	2	4	no	—
83	C₂×C₄×C₂₅₁	—	2	2	no	—
84	C₂×C₄×C₂₅₁	—	4	1	no	—
85	C₂×C₄×C₂₅₁	—	4	2	no	—
86	C₂×C₄×C₂₅₁	—	4	1	no	—
87	C₂×C₄×C₂₅₁	—	4	2	no	—
88	C₂×C₄×C₂₅₁	—	4	1	no	—
89	C₂×C₄×C₂₅₁	—	4	1	no	—
90	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	complemented
91	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
94	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
95	C₂²×C₂₅₁	C₄×Q₈	1	1	yes	characteristic
96	C₂²×C₂₅₁	#120	1	1	yes	characteristic
97	C₂²×C₂₅₁	#120	1	2	yes	—
98	C₂²×C₂₅₁	#121	1	2	yes	—
99	C₂²×C₂₅₁	#121	1	1	yes	characteristic
100	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
101	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
102	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
103	C₂²×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
105	C₂²×C₂₅₁	#123	1	1	yes	characteristic
106	C₂²×C₂₅₁	#123	1	2	yes	—
107	C₂²×C₂₅₁	#125	1	2	yes	—
108	C₂²×C₂₅₁	#125	1	1	yes	characteristic
109	C₂²×C₂₅₁	#127	1	1	yes	characteristic
110	C₂²×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
112	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
113	C₂²×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
114	C₄×C₂₅₁	—	2	4	no	—
115	C₄×C₂₅₁	—	4	2	no	—
116	C₄×C₂₅₁	—	4	2	no	—
117	Dic₂₅₁	—	2	8	no	—
118	Dic₂₅₁	—	4	4	no	—
119	C₂×C₂₅₁	#384	1	1	yes	characteristic
120	C₂×C₂₅₁	#385	1	1	yes	characteristic
121	C₂×C₂₅₁	#385	1	2	yes	—
122	C₂×C₂₅₁	#389	1	2	yes	—
123	C₂×C₂₅₁	#389	1	1	yes	characteristic
124	C₂×C₂₅₁	#392	1	1	yes	characteristic
125	C₂×C₂₅₁	#395	1	1	yes	characteristic
126	C₂×C₂₅₁	#395	1	2	yes	—
127	C₂×C₂₅₁	#398	1	1	yes	characteristic
128	C₂×C₂₅₁	#398	1	2	yes	—
129	C₂×C₂₅₁	#399	1	1	yes	characteristic
130	C₂₅₁	#1444	1	1	yes	characteristic, complemented, Sylow
131	#1444	—	251	1	no	maximal, Sylow
132	C₂²×C₄⋊C₄	—	251	2	no	—
133	C₂²×C₄⋊C₄	—	251	1	no	—
134	C₂×#95	—	251	2	no	—
135	C₂×#95	—	251	1	no	—
136	C₂×#95	D₂₅₁	1	1	yes	characteristic
137	C₂³×C₄	—	251	2	no	—
138	C₂³×C₄	—	251	2	no	—
139	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
140	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
141	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	502	4	no	—
143	C₂×C₄⋊C₄	—	502	4	no	—
144	C₂×C₄⋊C₄	—	502	4	no	—
145	C₂×C₄⋊C₄	—	502	2	no	—
146	C₂×C₄⋊C₄	—	502	4	no	—
147	#95	—	2	2	no	—
148	#95	—	502	4	no	—
149	#95	—	251	4	no	—
150	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	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	—	502	2	no	—
169	C₂²×C₄	—	502	2	no	—
170	C₂²×C₄	—	502	2	no	—
171	C₂²×C₄	—	251	4	no	—
172	C₂²×C₄	—	251	4	no	—
173	C₂²×C₄	—	251	4	no	—
174	C₂²×C₄	—	251	4	no	—
175	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
176	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
177	C₂²×C₄	D₁₀₀₄	1	2	yes	—
178	C₂²×C₄	Dic₅₀₂	1	2	yes	—
179	C₂²×C₄	Dic₅₀₂	1	2	yes	—
180	C₄⋊C₄	—	502	16	no	—
181	C₄⋊C₄	—	502	8	no	—
182	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
183	C₂³	D₄×D₂₅₁	1	2	yes	—
184	C₂³	D₄×D₂₅₁	1	2	yes	—
185	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
186	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
187	C₂³	Q₈×D₂₅₁	1	2	yes	—
188	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
189	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
190	C₂³	#16860809	1	2	yes	—
191	C₂³	#16860810	1	1	yes	characteristic
192	C₂³	#16860810	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	1004	2	no	—
208	C₂×C₄	—	1004	2	no	—
209	C₂×C₄	—	1004	2	no	—
210	C₂×C₄	—	1004	2	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	—	502	4	no	—
216	C₂×C₄	—	502	4	no	—
217	C₂×C₄	—	502	4	no	—
218	C₂×C₄	—	502	4	no	—
219	C₂×C₄	—	502	4	no	—
220	C₂×C₄	—	251	8	no	—
221	C₂×C₄	Dic₂₅₁⋊C₄	1	4	yes	complemented
222	C₂²	C₄⋊C₄×D₂₅₁	1	2	yes	—
223	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
224	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
225	C₂²	C₄×D₁₀₀₄	1	1	yes	characteristic
226	C₂²	C₄×Dic₅₀₂	1	1	yes	characteristic
227	C₂²	#22548683	1	1	yes	characteristic
228	C₂²	#22548685	1	2	yes	—
229	C₂²	#22548688	1	2	yes	—
230	C₂²	#22548689	1	1	yes	characteristic
231	C₂²	#22548692	1	2	yes	—
232	C₂²	#22548692	1	2	yes	—
233	C₂²	#22548693	1	2	yes	—
234	C₂²	#22548694	1	1	yes	characteristic
235	C₂²	#22548695	1	2	yes	—
236	C₂²	#22548696	1	1	yes	characteristic
237	C₂²	#22548697	1	2	yes	—
238	C₂²	#22548705	1	2	yes	—
239	C₂²	#22548707	1	1	yes	characteristic
240	C₂²	#22548711	1	1	yes	characteristic
241	C₂²	#22548713	1	1	yes	characteristic
242	C₂²	#22548714	1	2	yes	—
243	C₂²	#22548714	1	2	yes	—
244	C₂²	#22548715	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	1004	4	no	—
249	C₄	—	502	8	no	—
250	C₂	#30631556	1	1	yes	characteristic
251	C₂	#30631559	1	1	yes	characteristic
252	C₂	#30631561	1	2	yes	—
253	C₂	#30631584	1	2	yes	—
254	C₂	#30631585	1	1	yes	characteristic
255	C₂	#30631596	1	1	yes	characteristic
256	C₂	#30631607	1	1	yes	characteristic
257	C₂	#30631608	1	2	yes	—
258	C₂	#30631617	1	1	yes	characteristic
259	C₂	#30631621	1	2	yes	—
260	C₂	#30631623	1	1	yes	characteristic

	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	#1444
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
14. C₂₅₁⋊#95	C₄	semidirect
64. C₂×Dic₂₅₁	C₄⋊C₄	semidirect
90. C₂×C₄×C₂₅₁	C₄⋊C₄	semidirect
130. C₂₅₁	#1444	semidirect
221. C₂×C₄	Dic₂₅₁⋊C₄	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
C₂²×Dic₂₅₁⋊C₄	2
C₂×C₂₅₁⋊#95	2
C₂×#95×C₂₅₁	2
#1444	251