Subgroups — #539×D₂₅₁ #44563709

Order 32128.

1144 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632286	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632285	C₂	1	2	yes	maximal, complemented
3	#30632284	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632283	C₂	1	2	yes	maximal, complemented
5	#30632282	C₂	1	1	yes	maximal, characteristic, complemented
6	#30632281	C₂	1	2	yes	maximal, complemented
7	#30632280	C₂	1	2	yes	maximal, complemented
8	#30632279	C₂	1	2	yes	maximal, complemented
9	#30632278	C₂	1	2	yes	maximal, complemented
10	#126×D₂₅₁	C₂	1	2	yes	maximal, complemented
11	#125×D₂₅₁	C₂	1	2	yes	maximal, complemented
12	#123×D₂₅₁	C₂	1	2	yes	maximal, complemented
13	#123×D₂₅₁	C₂	1	2	yes	maximal, complemented
14	C₂²⋊Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²⋊Q₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
16	#121×D₂₅₁	C₂	1	1	yes	maximal, characteristic
17	C₄×D₄×D₂₅₁	C₂	1	2	yes	maximal, complemented
18	C₂×C₄⋊C₄×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
19	#539×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×D₄×D₂₅₁	C₂²	1	2	yes	—
22	C₂×D₄×D₂₅₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	—
27	#22548710	C₂²	1	2	yes	complemented
28	#22548710	C₂²	1	2	yes	complemented
29	#22548710	C₂²	1	2	yes	complemented
30	#22548709	C₂²	1	2	yes	complemented
31	#22548708	C₂²	1	2	yes	complemented
32	#22548708	C₂²	1	2	yes	complemented
33	#22548708	C₂²	1	2	yes	complemented
34	#22548707	C₂²	1	2	yes	complemented
35	#22548706	C₂²	1	2	yes	complemented
36	#22548706	C₂²	1	2	yes	complemented
37	#22548705	C₂²	1	2	yes	complemented
38	#22548705	C₂²	1	2	yes	complemented
39	#22548705	C₂²	1	2	yes	complemented
40	#22548705	C₂²	1	2	yes	complemented
41	#22548700	C₂²	1	2	yes	complemented
42	#22548700	C₂²	1	2	yes	complemented
43	#22548700	C₂²	1	2	yes	complemented
44	#22548700	C₂²	1	2	yes	complemented
45	#22548699	C₂²	1	2	yes	complemented
46	#22548699	C₂²	1	2	yes	complemented
47	#22548698	C₂²	1	2	yes	complemented
48	#22548698	C₂²	1	2	yes	complemented
49	#22548697	C₂²	1	2	yes	complemented
50	#22548697	C₂²	1	2	yes	complemented
51	#22548697	C₂²	1	2	yes	complemented
52	#22548697	C₂²	1	2	yes	complemented
53	#22548696	C₂²	1	2	yes	complemented
54	#22548696	C₂²	1	2	yes	complemented
55	#22548695	C₂²	1	1	yes	characteristic, complemented
56	#22548695	C₂²	1	2	yes	complemented
57	#22548694	C₂²	1	2	yes	complemented
58	#22548694	C₂²	1	1	yes	characteristic, complemented
59	#22548693	C₂²	1	2	yes	complemented
60	#22548693	C₂²	1	2	yes	complemented
61	#22548693	C₂²	1	2	yes	complemented
62	#22548692	C₂²	1	2	yes	complemented
63	#22548692	C₂²	1	2	yes	complemented
64	#22548692	C₂²	1	2	yes	complemented
65	#22548691	C₂²	1	1	yes	characteristic, complemented
66	#22548691	C₂²	1	2	yes	complemented
67	#22548690	C₂²	1	1	yes	characteristic
68	#22548689	C₂²	1	1	yes	characteristic
69	#22548688	C₂²	1	2	yes	—
70	#22548687	C₂²	1	2	yes	—
71	#22548686	C₂²	1	1	yes	characteristic
72	C₄×D₁₀₀₄	C₂²	1	2	yes	complemented
73	D₄×Dic₂₅₁	C₂²	1	2	yes	complemented
74	#22548677	C₂²	1	2	yes	complemented
75	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	complemented
76	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	complemented
77	(C₂²⋊D₂₅₁)⋊C₄	C₂²	1	2	yes	complemented
78	(C₂²⋊D₂₅₁)⋊C₄	C₂²	1	2	yes	complemented
79	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
80	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic, complemented
81	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic, complemented
82	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
83	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
84	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
85	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
86	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
87	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
88	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
89	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
90	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	complemented
91	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	complemented
92	C₂²⋊C₄×D₂₅₁	C₂²	1	2	yes	—
93	C₂²⋊C₄×D₂₅₁	C₂²	1	2	yes	—
94	C₂²⋊C₄×D₂₅₁	C₂²	1	2	yes	—
95	C₂²⋊C₄×D₂₅₁	C₂²	1	2	yes	—
96	C₂²⋊C₄×D₂₅₁	C₂²	1	2	yes	—
97	C₄²×D₂₅₁	C₂²	1	2	yes	—
98	#126×C₂₅₁	C₂²	1	2	yes	complemented
99	#125×C₂₅₁	C₂²	1	2	yes	complemented
100	#123×C₂₅₁	C₂²	1	2	yes	complemented
101	#123×C₂₅₁	C₂²	1	2	yes	complemented
102	C₂²⋊Q₈×C₂₅₁	C₂²	1	2	yes	complemented
103	C₂²⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
104	#121×C₂₅₁	C₂²	1	1	yes	characteristic
105	C₄×D₄×C₂₅₁	C₂²	1	2	yes	complemented
106	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
107	C₂³×D₂₅₁	C₂³	1	2	yes	—
108	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
109	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
110	C₂×Dic₅₀₂	C₂³	1	2	yes	—
111	Q₈×D₂₅₁	—	4	2	no	—
112	C₂×D₁₀₀₄	C₂³	1	2	yes	—
113	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊D₂₅₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊D₂₅₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊D₂₅₁	C₂³	1	2	yes	—
117	D₄×D₂₅₁	—	2	8	no	—
118	D₄×D₂₅₁	—	4	2	no	—
119	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
120	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
121	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
122	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
123	C₂×C₄×D₂₅₁	—	2	4	no	—
124	C₂×C₄×D₂₅₁	—	4	2	no	—
125	C₂×C₄×D₂₅₁	—	4	4	no	—
126	C₂×C₄×D₂₅₁	—	4	2	no	—
127	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
128	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
129	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
130	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
131	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
132	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
133	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
134	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
135	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
136	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
137	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
138	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
139	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
140	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
141	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
142	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
143	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
144	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
145	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
146	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
147	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
148	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
149	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
150	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
151	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
152	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
153	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
154	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
155	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
156	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
157	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
158	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
159	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
160	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
161	Dic₂₅₁⋊C₄	C₂³	1	2	yes	—
162	C₂²⋊Dic₂₅₁	C₂³	1	2	yes	—
163	C₂²⋊Dic₂₅₁	C₂³	1	2	yes	—
164	C₂²⋊Dic₂₅₁	C₂³	1	2	yes	—
165	C₂²⋊Dic₂₅₁	C₂³	1	2	yes	—
166	C₂²⋊Dic₂₅₁	C₂³	1	2	yes	—
167	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
168	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
169	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
170	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
171	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
172	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
173	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
174	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
175	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
176	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
177	C₄×Dic₂₅₁	C₂³	1	2	yes	—
178	C₄×Dic₂₅₁	C₂³	1	2	yes	—
179	C₄×Dic₂₅₁	C₂³	1	2	yes	—
180	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
181	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
182	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
183	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
184	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
185	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
186	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
187	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
188	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
189	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
190	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
191	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
192	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
193	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
194	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
195	C₂²⋊C₄×C₂₅₁	C₂³	1	2	yes	—
196	C₂²⋊C₄×C₂₅₁	C₂³	1	2	yes	—
197	C₂²⋊C₄×C₂₅₁	C₂³	1	2	yes	—
198	C₂²⋊C₄×C₂₅₁	C₂³	1	2	yes	—
199	C₂²⋊C₄×C₂₅₁	C₂³	1	2	yes	—
200	C₄²×C₂₅₁	C₂³	1	2	yes	—
201	C₂²×D₂₅₁	—	2	2	no	—
202	C₂²×D₂₅₁	—	2	2	no	—
203	C₂²×D₂₅₁	—	2	4	no	—
204	C₂²×D₂₅₁	—	2	4	no	—
205	C₂²×D₂₅₁	—	2	4	no	—
206	C₂²×D₂₅₁	#41	1	4	yes	—
207	C₂²×D₂₅₁	C₂⁴	1	2	yes	—
208	C₂²×D₂₅₁	C₂⁴	1	1	yes	characteristic
209	C₂²×D₂₅₁	C₂⁴	1	1	yes	characteristic
210	Dic₅₀₂	—	4	1	no	—
211	Dic₅₀₂	—	4	2	no	—
212	D₁₀₀₄	—	2	4	no	—
213	D₁₀₀₄	—	4	1	no	—
214	C₂²⋊D₂₅₁	—	2	4	no	—
215	C₂²⋊D₂₅₁	—	2	4	no	—
216	C₂²⋊D₂₅₁	—	4	2	no	—
217	C₂×Dic₂₅₁	—	2	2	no	—
218	C₂×Dic₂₅₁	—	4	1	no	—
219	C₂×Dic₂₅₁	—	4	2	no	—
220	C₂×Dic₂₅₁	—	4	1	no	—
221	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
222	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
223	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
224	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
225	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
226	C₂×Dic₂₅₁	C₂⁴	1	1	yes	characteristic
227	C₂×Dic₂₅₁	C₂⁴	1	1	yes	characteristic
228	C₄×D₂₅₁	—	2	4	no	—
229	C₄×D₂₅₁	—	2	4	no	—
230	C₄×D₂₅₁	—	2	4	no	—
231	C₄×D₂₅₁	—	2	4	no	—
232	C₄×D₂₅₁	—	2	2	no	—
233	C₄×D₂₅₁	—	2	4	no	—
234	C₄×D₂₅₁	—	2	2	no	—
235	C₄×D₂₅₁	—	2	4	no	—
236	C₄×D₂₅₁	—	4	2	no	—
237	C₄×D₂₅₁	—	4	2	no	—
238	C₄×D₂₅₁	—	4	1	no	—
239	C₄×D₂₅₁	—	4	2	no	—
240	C₄×D₂₅₁	—	4	1	no	—
241	C₂³×C₂₅₁	C₂⁴	1	2	yes	—
242	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
243	Q₈×C₂₅₁	—	4	1	no	—
244	D₄×C₂₅₁	—	2	4	no	—
245	D₄×C₂₅₁	—	4	1	no	—
246	C₂×C₄×C₂₅₁	—	2	2	no	—
247	C₂×C₄×C₂₅₁	—	4	1	no	—
248	C₂×C₄×C₂₅₁	—	4	2	no	—
249	C₂×C₄×C₂₅₁	—	4	1	no	—
250	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
251	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
252	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
253	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
254	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
255	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
256	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
257	C₂²×C₂₅₁	—	2	1	no	—
258	C₂²×C₂₅₁	—	2	1	no	—
259	C₂²×C₂₅₁	—	2	2	no	—
260	C₂²×C₂₅₁	—	2	2	no	—
261	C₂²×C₂₅₁	—	2	2	no	—
262	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
263	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
264	C₂×D₂₅₁	—	2	1	no	—
265	C₂×D₂₅₁	—	2	1	no	—
266	C₂×D₂₅₁	—	2	8	no	—
267	C₂×D₂₅₁	—	2	2	no	—
268	C₂×D₂₅₁	—	2	2	no	—
269	C₂×D₂₅₁	—	2	2	no	—
270	C₂×D₂₅₁	—	4	8	no	—
271	C₂×D₂₅₁	C₂×#41	1	2	yes	—
272	C₂×D₂₅₁	C₂×#41	1	2	yes	—
273	C₂×D₂₅₁	H(5,𝔽₂)	1	2	yes	—
274	C₂×D₂₅₁	2₋¹⁺⁴	1	2	yes	—
275	C₄×C₂₅₁	—	2	2	no	—
276	C₄×C₂₅₁	—	2	2	no	—
277	C₄×C₂₅₁	—	2	2	no	—
278	C₄×C₂₅₁	—	2	1	no	—
279	C₄×C₂₅₁	—	2	2	no	—
280	C₄×C₂₅₁	—	2	1	no	—
281	C₄×C₂₅₁	—	2	2	no	—
282	Dic₂₅₁	—	2	2	no	—
283	Dic₂₅₁	—	2	2	no	—
284	Dic₂₅₁	—	2	2	no	—
285	Dic₂₅₁	—	2	1	no	—
286	Dic₂₅₁	—	2	2	no	—
287	Dic₂₅₁	—	2	1	no	—
288	Dic₂₅₁	—	2	2	no	—
289	C₂×C₂₅₁	—	2	2	no	—
290	C₂×C₂₅₁	—	4	2	no	—
291	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
292	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
293	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
294	D₂₅₁	—	2	2	no	—
295	D₂₅₁	—	4	2	no	—
296	D₂₅₁	#539	1	4	yes	direct product factor
297	C₂₅₁	C₂×#539	1	1	yes	characteristic, complemented, Sylow
298	C₂×#539	—	251	1	no	maximal, Sylow
299	#539	—	251	2	no	—
300	#539	—	251	2	no	—
301	#539	—	251	2	no	—
302	#539	—	251	2	no	—
303	#539	—	251	2	no	—
304	#539	—	251	1	no	—
305	#539	—	251	1	no	—
306	#539	—	251	2	no	—
307	#539	—	251	1	no	—
308	#539	D₂₅₁	1	1	yes	characteristic, direct product factor
309	C₂×#126	—	251	2	no	—
310	C₂×#125	—	251	2	no	—
311	C₂×#123	—	251	2	no	—
312	C₂×#123	—	251	2	no	—
313	C₂×C₂²⋊Q₈	—	251	1	no	—
314	C₂×C₂²⋊Q₈	—	251	2	no	—
315	C₂×#121	—	251	1	no	—
316	C₂×C₄×D₄	—	251	2	no	—
317	C₂²×C₄⋊C₄	—	251	1	no	—
318	C₂²×Q₈	—	251	1	no	—
319	C₂²×D₄	—	251	1	no	—
320	C₂²×D₄	—	251	2	no	—
321	C₂³×C₄	—	251	1	no	—
322	C₂³×C₄	—	251	1	no	—
323	C₂³×C₄	—	251	2	no	—
324	C₂³×C₄	—	251	1	no	—
325	#126	—	251	2	no	—
326	#126	—	251	2	no	—
327	#126	—	251	2	no	—
328	#126	—	251	2	no	—
329	#126	—	251	2	no	—
330	#126	—	251	2	no	—
331	#126	—	251	2	no	—
332	#126	C₂×D₂₅₁	1	2	yes	complemented
333	#125	—	251	2	no	—
334	#125	—	251	2	no	—
335	#125	—	251	2	no	—
336	#125	—	251	2	no	—
337	#125	—	251	2	no	—
338	#125	—	251	2	no	—
339	#125	—	251	2	no	—
340	#125	C₂×D₂₅₁	1	2	yes	complemented
341	#123	—	251	2	no	—
342	#123	—	251	2	no	—
343	#123	—	251	2	no	—
344	#123	—	251	2	no	—
345	#123	—	251	2	no	—
346	#123	—	251	2	no	—
347	#123	—	251	2	no	—
348	#123	—	251	2	no	—
349	#123	—	251	2	no	—
350	#123	—	251	2	no	—
351	#123	—	251	2	no	—
352	#123	—	251	2	no	—
353	#123	—	251	2	no	—
354	#123	—	251	2	no	—
355	#123	C₂×D₂₅₁	1	2	yes	complemented
356	#123	C₂×D₂₅₁	1	2	yes	complemented
357	C₂²⋊Q₈	—	251	1	no	—
358	C₂²⋊Q₈	—	251	2	no	—
359	C₂²⋊Q₈	—	251	1	no	—
360	C₂²⋊Q₈	—	251	2	no	—
361	C₂²⋊Q₈	—	251	2	no	—
362	C₂²⋊Q₈	—	251	2	no	—
363	C₂²⋊Q₈	—	251	2	no	—
364	C₂²⋊Q₈	—	251	1	no	—
365	C₂²⋊Q₈	—	251	2	no	—
366	C₂²⋊Q₈	—	251	2	no	—
367	C₂²⋊Q₈	—	251	2	no	—
368	C₂²⋊Q₈	—	251	2	no	—
369	C₂²⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic, complemented
370	C₂²⋊Q₈	C₂×D₂₅₁	1	2	yes	complemented
371	#121	—	251	1	no	—
372	#121	—	251	2	no	—
373	#121	—	251	1	no	—
374	#121	—	251	1	no	—
375	#121	—	251	2	no	—
376	#121	C₂×D₂₅₁	1	1	yes	characteristic
377	C₄×D₄	—	251	2	no	—
378	C₄×D₄	—	251	2	no	—
379	C₄×D₄	—	251	2	no	—
380	C₄×D₄	—	251	2	no	—
381	C₄×D₄	—	251	2	no	—
382	C₄×D₄	—	251	2	no	—
383	C₄×D₄	—	251	2	no	—
384	C₄×D₄	C₂×D₂₅₁	1	2	yes	complemented
385	C₂×C₄⋊C₄	—	251	1	no	—
386	C₂×C₄⋊C₄	—	251	1	no	—
387	C₂×C₄⋊C₄	—	251	1	no	—
388	C₂×C₄⋊C₄	—	251	2	no	—
389	C₂×C₄⋊C₄	—	251	2	no	—
390	C₂×C₄⋊C₄	—	251	2	no	—
391	C₂×C₄⋊C₄	—	251	2	no	—
392	C₂×C₄⋊C₄	—	251	2	no	—
393	C₂×C₄⋊C₄	—	251	1	no	—
394	C₂×C₄⋊C₄	—	251	2	no	—
395	C₂×C₄⋊C₄	—	251	2	no	—
396	C₂×C₄⋊C₄	—	251	1	no	—
397	C₂×C₄⋊C₄	—	251	2	no	—
398	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic, complemented
399	C₂×C₂²⋊C₄	—	251	2	no	—
400	C₂×C₂²⋊C₄	—	251	2	no	—
401	C₂×C₂²⋊C₄	—	251	2	no	—
402	C₂×C₂²⋊C₄	—	251	2	no	—
403	C₂×C₂²⋊C₄	—	251	2	no	—
404	C₂×C₄²	—	251	2	no	—
405	C₂⁴	—	251	2	no	—
406	C₂⁴	—	251	1	no	—
407	C₂×Q₈	—	1004	2	no	—
408	C₂×Q₈	—	251	1	no	—
409	C₂×Q₈	—	251	2	no	—
410	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
411	C₂×D₄	—	1004	2	no	—
412	C₂×D₄	—	502	8	no	—
413	C₂×D₄	—	251	2	no	—
414	C₂×D₄	—	251	2	no	—
415	C₂×D₄	—	251	2	no	—
416	C₂×D₄	—	251	1	no	—
417	C₂×D₄	—	251	2	no	—
418	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
419	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
420	C₂²×C₄	—	1004	2	no	—
421	C₂²×C₄	—	1004	4	no	—
422	C₂²×C₄	—	1004	2	no	—
423	C₂²×C₄	—	502	4	no	—
424	C₂²×C₄	—	251	1	no	—
425	C₂²×C₄	—	251	1	no	—
426	C₂²×C₄	—	251	2	no	—
427	C₂²×C₄	—	251	2	no	—
428	C₂²×C₄	—	251	1	no	—
429	C₂²×C₄	—	251	1	no	—
430	C₂²×C₄	—	251	2	no	—
431	C₂²×C₄	—	251	2	no	—
432	C₂²×C₄	—	251	1	no	—
433	C₂²×C₄	—	251	2	no	—
434	C₂²×C₄	—	251	2	no	—
435	C₂²×C₄	—	251	2	no	—
436	C₂²×C₄	—	251	2	no	—
437	C₂²×C₄	—	251	2	no	—
438	C₂²×C₄	—	251	1	no	—
439	C₂²×C₄	—	251	1	no	—
440	C₂²×C₄	—	251	2	no	—
441	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
442	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
443	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
444	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
445	C₄⋊C₄	—	251	2	no	—
446	C₄⋊C₄	—	251	2	no	—
447	C₄⋊C₄	—	251	2	no	—
448	C₄⋊C₄	—	251	2	no	—
449	C₄⋊C₄	—	251	2	no	—
450	C₄⋊C₄	—	251	2	no	—
451	C₄⋊C₄	—	251	2	no	—
452	C₄⋊C₄	—	251	2	no	—
453	C₄⋊C₄	—	251	2	no	—
454	C₄⋊C₄	—	251	2	no	—
455	C₄⋊C₄	—	251	2	no	—
456	C₄⋊C₄	—	251	2	no	—
457	C₄⋊C₄	—	251	2	no	—
458	C₄⋊C₄	—	251	2	no	—
459	C₄⋊C₄	—	251	2	no	—
460	C₄⋊C₄	—	251	2	no	—
461	C₄⋊C₄	—	251	2	no	—
462	C₄⋊C₄	—	251	1	no	—
463	C₄⋊C₄	—	251	2	no	—
464	C₄⋊C₄	—	251	2	no	—
465	C₄⋊C₄	—	251	1	no	—
466	C₄⋊C₄	—	251	2	no	—
467	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
468	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
469	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
470	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
471	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
472	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
473	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
474	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
475	C₂²⋊C₄	—	251	2	no	—
476	C₂²⋊C₄	—	251	2	no	—
477	C₂²⋊C₄	—	251	2	no	—
478	C₂²⋊C₄	—	251	2	no	—
479	C₂²⋊C₄	—	251	2	no	—
480	C₂²⋊C₄	—	251	2	no	—
481	C₂²⋊C₄	—	251	2	no	—
482	C₂²⋊C₄	—	251	2	no	—
483	C₂²⋊C₄	—	251	2	no	—
484	C₂²⋊C₄	—	251	2	no	—
485	C₂²⋊C₄	—	251	2	no	—
486	C₂²⋊C₄	—	251	2	no	—
487	C₂²⋊C₄	—	251	2	no	—
488	C₂²⋊C₄	—	251	2	no	—
489	C₂²⋊C₄	—	251	2	no	—
490	C₂²⋊C₄	C₂²×D₂₅₁	1	2	yes	—
491	C₂²⋊C₄	C₂²×D₂₅₁	1	2	yes	—
492	C₂²⋊C₄	C₂²×D₂₅₁	1	2	yes	—
493	C₂²⋊C₄	C₂²×D₂₅₁	1	2	yes	—
494	C₂²⋊C₄	C₂²×D₂₅₁	1	2	yes	—
495	C₄²	—	251	2	no	—
496	C₄²	—	251	2	no	—
497	C₄²	—	251	2	no	—
498	C₄²	C₂²×D₂₅₁	1	2	yes	—
499	C₂³	—	502	4	no	—
500	C₂³	—	502	4	no	—
501	C₂³	—	502	4	no	—
502	C₂³	—	502	2	no	—
503	C₂³	—	502	2	no	—
504	C₂³	—	251	2	no	—
505	C₂³	—	251	4	no	—
506	C₂³	—	251	1	no	—
507	C₂³	—	251	1	no	—
508	C₂³	C₂³×D₂₅₁	1	2	yes	—
509	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
510	Q₈	—	4	1	no	—
511	Q₈	—	1004	2	no	—
512	Q₈	—	1004	1	no	—
513	D₄	—	2	4	no	—
514	D₄	—	4	1	no	—
515	D₄	—	1004	1	no	—
516	D₄	—	1004	2	no	—
517	D₄	—	502	4	no	—
518	D₄	—	502	4	no	—
519	D₄	—	502	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	1004	2	no	—
525	C₂×C₄	—	1004	2	no	—
526	C₂×C₄	—	1004	1	no	—
527	C₂×C₄	—	1004	2	no	—
528	C₂×C₄	—	1004	1	no	—
529	C₂×C₄	—	1004	1	no	—
530	C₂×C₄	—	1004	2	no	—
531	C₂×C₄	—	1004	1	no	—
532	C₂×C₄	—	502	4	no	—
533	C₂×C₄	—	502	2	no	—
534	C₂×C₄	—	502	4	no	—
535	C₂×C₄	—	502	2	no	—
536	C₂×C₄	—	502	4	no	—
537	C₂×C₄	—	502	4	no	—
538	C₂×C₄	—	502	4	no	—
539	C₂×C₄	—	502	4	no	—
540	C₂×C₄	—	502	2	no	—
541	C₂×C₄	—	251	2	no	—
542	C₂×C₄	—	251	2	no	—
543	C₂×C₄	—	251	2	no	—
544	C₂×C₄	—	251	2	no	—
545	C₂×C₄	—	251	2	no	—
546	C₂×C₄	—	251	1	no	—
547	C₂×C₄	—	251	1	no	—
548	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
549	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
550	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
551	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
552	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
553	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
554	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	1004	8	no	—
561	C₂²	—	502	8	no	—
562	C₂²	—	502	2	no	—
563	C₂²	—	502	2	no	—
564	C₂²	—	502	2	no	—
565	C₂²	—	502	1	no	—
566	C₂²	—	502	1	no	—
567	C₂²	—	251	2	no	—
568	C₂²	—	251	2	no	—
569	C₂²	—	251	2	no	—
570	C₂²	—	251	2	no	—
571	C₂²	#41×D₂₅₁	1	2	yes	—
572	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	2	no	—
580	C₄	—	502	2	no	—
581	C₄	—	502	2	no	—
582	C₄	—	502	2	no	—
583	C₄	—	502	1	no	—
584	C₄	—	502	2	no	—
585	C₄	—	502	1	no	—
586	C₄	—	502	2	no	—
587	C₂	—	2	2	no	—
588	C₂	—	4	2	no	—
589	C₂	—	1004	2	no	—
590	C₂	—	502	2	no	—
591	C₂	—	251	4	no	—
592	C₂	C₂×#41×D₂₅₁	1	1	yes	characteristic
593	C₂	H(5,𝔽₂)×D₂₅₁	1	1	yes	characteristic
594	C₂	2₋¹⁺⁴×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁵
Fitting	#539×C₂₅₁	C₂
prime radical	#121×D₂₅₁	C₂
hypercenter	#539	D₂₅₁
hypocenter	C₂₅₁	C₂×#539
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #539

Complements

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

N	H ≅ G/N	type
1. #30632286	C₂	semidirect
2. #30632285	C₂	semidirect
3. #30632284	C₂	semidirect
4. #30632283	C₂	semidirect
5. #30632282	C₂	semidirect
6. #30632281	C₂	semidirect
7. #30632280	C₂	semidirect
8. #30632279	C₂	semidirect
9. #30632278	C₂	semidirect
10. #126×D₂₅₁	C₂	semidirect
11. #125×D₂₅₁	C₂	semidirect
12. #123×D₂₅₁	C₂	semidirect
13. #123×D₂₅₁	C₂	semidirect
14. C₂²⋊Q₈×D₂₅₁	C₂	semidirect
15. C₂²⋊Q₈×D₂₅₁	C₂	semidirect
17. C₄×D₄×D₂₅₁	C₂	semidirect
18. C₂×C₄⋊C₄×D₂₅₁	C₂	semidirect
19. #539×C₂₅₁	C₂	semidirect
27. #22548710	C₂²	semidirect
28. #22548710	C₂²	semidirect
29. #22548710	C₂²	semidirect
30. #22548709	C₂²	semidirect
31. #22548708	C₂²	semidirect
32. #22548708	C₂²	semidirect
33. #22548708	C₂²	semidirect
34. #22548707	C₂²	semidirect
35. #22548706	C₂²	semidirect
36. #22548706	C₂²	semidirect
37. #22548705	C₂²	semidirect
38. #22548705	C₂²	semidirect
39. #22548705	C₂²	semidirect
40. #22548705	C₂²	semidirect
41. #22548700	C₂²	semidirect
42. #22548700	C₂²	semidirect
43. #22548700	C₂²	semidirect
44. #22548700	C₂²	semidirect
45. #22548699	C₂²	semidirect
46. #22548699	C₂²	semidirect
47. #22548698	C₂²	semidirect
48. #22548698	C₂²	semidirect
49. #22548697	C₂²	semidirect
50. #22548697	C₂²	semidirect
51. #22548697	C₂²	semidirect
52. #22548697	C₂²	semidirect
53. #22548696	C₂²	semidirect
54. #22548696	C₂²	semidirect
55. #22548695	C₂²	semidirect
56. #22548695	C₂²	semidirect
57. #22548694	C₂²	semidirect
58. #22548694	C₂²	semidirect
59. #22548693	C₂²	semidirect
60. #22548693	C₂²	semidirect
61. #22548693	C₂²	semidirect
62. #22548692	C₂²	semidirect
63. #22548692	C₂²	semidirect
64. #22548692	C₂²	semidirect
65. #22548691	C₂²	semidirect
66. #22548691	C₂²	semidirect
72. C₄×D₁₀₀₄	C₂²	semidirect
73. D₄×Dic₂₅₁	C₂²	semidirect
74. #22548677	C₂²	semidirect
75. C₄×C₂²⋊D₂₅₁	C₂²	semidirect
76. C₄×C₂²⋊D₂₅₁	C₂²	semidirect
77. (C₂²⋊D₂₅₁)⋊C₄	C₂²	semidirect
78. (C₂²⋊D₂₅₁)⋊C₄	C₂²	semidirect
79. C₂×C₄⋊Dic₂₅₁	C₂²	semidirect
80. C₂×Dic₂₅₁⋊C₄	C₂²	semidirect
81. C₂×Dic₂₅₁⋊C₄	C₂²	semidirect
90. C₄⋊C₄×D₂₅₁	C₂²	semidirect
91. C₄⋊C₄×D₂₅₁	C₂²	semidirect
98. #126×C₂₅₁	C₂²	semidirect
99. #125×C₂₅₁	C₂²	semidirect
100. #123×C₂₅₁	C₂²	semidirect
101. #123×C₂₅₁	C₂²	semidirect
102. C₂²⋊Q₈×C₂₅₁	C₂²	semidirect
103. C₂²⋊Q₈×C₂₅₁	C₂²	semidirect
105. C₄×D₄×C₂₅₁	C₂²	semidirect
106. C₂×C₄⋊C₄×C₂₅₁	C₂²	semidirect
296. D₂₅₁	#539	direct (N × H)
297. C₂₅₁	C₂×#539	semidirect
308. #539	D₂₅₁	direct (N × H)
332. #126	C₂×D₂₅₁	semidirect
340. #125	C₂×D₂₅₁	semidirect
355. #123	C₂×D₂₅₁	semidirect
356. #123	C₂×D₂₅₁	semidirect
369. C₂²⋊Q₈	C₂×D₂₅₁	semidirect
370. C₂²⋊Q₈	C₂×D₂₅₁	semidirect
384. C₄×D₄	C₂×D₂₅₁	semidirect
398. C₂×C₄⋊C₄	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632286	2
#30632285	2
#30632284	2
#30632283	2
#30632282	2
#30632281	2
#30632280	2
#30632279	2
#30632278	2
#126×D₂₅₁	2
#125×D₂₅₁	2
#123×D₂₅₁	2
C₂²⋊Q₈×D₂₅₁	2
#121×D₂₅₁	2
C₄×D₄×D₂₅₁	2
C₂×C₄⋊C₄×D₂₅₁	2
#539×C₂₅₁	2
C₂×#539	251

Subgroups — #539×D251 #44563709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #539×D₂₅₁ #44563709