Subgroups — #44548618

Order 32128.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₂²×C₄⋊C₄×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	2	no	—
12	C₂₅₁⋊#95	—	2	4	no	—
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	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₅₁	C₄	1	4	yes	complemented
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	—	2	2	no	—
30	C₂²×Dic₂₅₁	—	2	2	no	—
31	C₂²×Dic₂₅₁	D₄	1	4	yes	—
32	C₂²×Dic₂₅₁	Q₈	1	4	yes	—
33	C₄⋊Dic₂₅₁	—	2	8	no	—
34	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
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	1	no	—
40	C₂²×C₄×C₂₅₁	—	2	1	no	—
41	C₂²×C₄×C₂₅₁	—	2	1	no	—
42	C₂²×C₄×C₂₅₁	—	2	1	no	—
43	C₂²×C₄×C₂₅₁	—	2	2	no	—
44	C₂²×C₄×C₂₅₁	—	2	2	no	—
45	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
49	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
50	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
51	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
52	C₄⋊C₄×C₂₅₁	—	2	8	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	—	4	2	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	—	4	2	no	—
65	C₂×Dic₂₅₁	—	4	2	no	—
66	C₂×Dic₂₅₁	—	4	2	no	—
67	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
68	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
69	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₂₅₁	C₂×Q₈	1	2	yes	—
73	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₂₅₁	#41	1	2	yes	—
75	C₂³×C₂₅₁	#41	1	1	yes	characteristic
76	C₂³×C₂₅₁	#41	1	1	yes	characteristic
77	C₂³×C₂₅₁	#41	1	2	yes	—
78	C₂×C₄×C₂₅₁	—	2	2	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	—	2	2	no	—
85	C₂×C₄×C₂₅₁	—	2	2	no	—
86	C₂×C₄×C₂₅₁	—	2	2	no	—
87	C₂×C₄×C₂₅₁	—	2	2	no	—
88	C₂×C₄×C₂₅₁	—	2	2	no	—
89	C₂×C₄×C₂₅₁	—	2	4	no	—
90	C₂×C₄×C₂₅₁	—	4	1	no	—
91	C₂×C₄×C₂₅₁	—	4	2	no	—
92	C₂×C₄×C₂₅₁	—	4	1	no	—
93	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	complemented
95	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₂₅₁	#117	1	1	yes	characteristic
98	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
99	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
100	C₂²×C₂₅₁	#120	1	1	yes	characteristic
101	C₂²×C₂₅₁	#121	1	2	yes	—
102	C₂²×C₂₅₁	#121	1	1	yes	characteristic
103	C₂²×C₂₅₁	#121	1	1	yes	characteristic
104	C₂²×C₂₅₁	#121	1	2	yes	—
105	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
106	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂²×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
110	C₂²×C₂₅₁	#123	1	1	yes	characteristic
111	C₂²×C₂₅₁	#123	1	2	yes	—
112	C₂²×C₂₅₁	#124	1	1	yes	characteristic
113	C₂²×C₂₅₁	#125	1	2	yes	—
114	C₂²×C₂₅₁	#125	1	2	yes	—
115	C₂²×C₂₅₁	#126	1	2	yes	—
116	C₂²×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
118	C₄×C₂₅₁	—	2	4	no	—
119	C₄×C₂₅₁	—	2	4	no	—
120	C₄×C₂₅₁	—	4	2	no	—
121	Dic₂₅₁	—	4	4	no	—
122	Dic₂₅₁	—	4	4	no	—
123	C₂×C₂₅₁	#380	1	1	yes	characteristic
124	C₂×C₂₅₁	#386	1	1	yes	characteristic
125	C₂×C₂₅₁	#387	1	2	yes	—
126	C₂×C₂₅₁	#389	1	2	yes	—
127	C₂×C₂₅₁	#391	1	1	yes	characteristic
128	C₂×C₂₅₁	#394	1	1	yes	characteristic
129	C₂×C₂₅₁	#395	1	1	yes	characteristic
130	C₂×C₂₅₁	#396	1	2	yes	—
131	C₂×C₂₅₁	#398	1	1	yes	characteristic
132	C₂×C₂₅₁	#398	1	2	yes	—
133	C₂×C₂₅₁	#400	1	1	yes	characteristic
134	C₂₅₁	#1441	1	1	yes	characteristic, complemented, Sylow
135	#1441	—	251	1	no	maximal, Sylow
136	C₂²×C₄⋊C₄	—	251	1	no	—
137	C₂²×C₄⋊C₄	D₂₅₁	1	1	yes	characteristic
138	C₂×#95	—	251	2	no	—
139	C₂×#95	—	251	1	no	—
140	C₂×#95	—	251	2	no	—
141	C₂³×C₄	—	251	2	no	—
142	C₂³×C₄	—	251	2	no	—
143	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
144	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
145	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	502	4	no	—
149	C₂×C₄⋊C₄	—	502	2	no	—
150	C₂×C₄⋊C₄	Dic₂₅₁	1	4	yes	complemented
151	#95	—	502	4	no	—
152	#95	—	502	2	no	—
153	#95	—	502	4	no	—
154	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
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	—	2	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	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	—	502	2	no	—
172	C₂²×C₄	—	502	2	no	—
173	C₂²×C₄	—	502	2	no	—
174	C₂²×C₄	—	502	2	no	—
175	C₂²×C₄	—	502	2	no	—
176	C₂²×C₄	—	502	2	no	—
177	C₂²×C₄	—	251	4	no	—
178	C₂²×C₄	—	251	4	no	—
179	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
180	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
181	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
182	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
183	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
184	C₂²×C₄	D₁₀₀₄	1	2	yes	—
185	C₂²×C₄	Dic₅₀₂	1	2	yes	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	502	8	no	—
188	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
189	C₂³	D₄×D₂₅₁	1	2	yes	—
190	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
191	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
192	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
193	C₂³	Q₈×D₂₅₁	1	2	yes	—
194	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
195	C₂³	#16860809	1	2	yes	—
196	C₂³	#16860810	1	1	yes	characteristic
197	C₂³	#16860810	1	1	yes	characteristic
198	C₂³	#16860811	1	2	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	1004	2	no	—
215	C₂×C₄	—	1004	2	no	—
216	C₂×C₄	—	1004	2	no	—
217	C₂×C₄	—	1004	2	no	—
218	C₂×C₄	—	1004	2	no	—
219	C₂×C₄	—	1004	2	no	—
220	C₂×C₄	—	1004	2	no	—
221	C₂×C₄	—	1004	2	no	—
222	C₂×C₄	—	502	4	no	—
223	C₂×C₄	—	502	4	no	—
224	C₂×C₄	—	502	4	no	—
225	C₂×C₄	—	502	4	no	—
226	C₂×C₄	—	502	4	no	—
227	C₂×C₄	—	502	4	no	—
228	C₂×C₄	C₂²⋊Dic₂₅₁	1	4	yes	—
229	C₂×C₄	C₄⋊Dic₂₅₁	1	4	yes	complemented
230	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
231	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
232	C₂²	#22548674	1	1	yes	characteristic
233	C₂²	D₄×Dic₂₅₁	1	2	yes	—
234	C₂²	Q₈×Dic₂₅₁	1	2	yes	—
235	C₂²	#22548683	1	1	yes	characteristic
236	C₂²	#22548687	1	2	yes	—
237	C₂²	#22548689	1	1	yes	characteristic
238	C₂²	#22548689	1	1	yes	characteristic
239	C₂²	#22548690	1	2	yes	—
240	C₂²	#22548691	1	2	yes	—
241	C₂²	#22548693	1	2	yes	—
242	C₂²	#22548694	1	1	yes	characteristic
243	C₂²	#22548694	1	1	yes	characteristic
244	C₂²	#22548695	1	2	yes	—
245	C₂²	#22548696	1	1	yes	characteristic
246	C₂²	#22548700	1	2	yes	—
247	C₂²	#22548704	1	1	yes	characteristic
248	C₂²	#22548705	1	2	yes	—
249	C₂²	#22548706	1	2	yes	—
250	C₂²	#22548710	1	2	yes	—
251	C₂²	#22548713	1	1	yes	characteristic
252	C₂²	#22548714	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	—	1004	4	no	—
257	C₄	—	1004	4	no	—
258	C₂	#30631547	1	1	yes	characteristic
259	C₂	#30631567	1	1	yes	characteristic
260	C₂	#30631573	1	2	yes	—
261	C₂	#30631586	1	2	yes	—
262	C₂	#30631594	1	1	yes	characteristic
263	C₂	#30631604	1	1	yes	characteristic
264	C₂	#30631607	1	1	yes	characteristic
265	C₂	#30631609	1	2	yes	—
266	C₂	#30631617	1	1	yes	characteristic
267	C₂	#30631620	1	2	yes	—
268	C₂	#30631626	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₅₁
commutator	C₂³×C₂₅₁	C₂²×C₄
Fitting	C₂²×C₄⋊C₄×C₂₅₁	C₂
socle, prime radical, cosocle	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₂₅₁
hypocenter	C₂₅₁	#1441
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
18. C₂×C₄⋊C₄×C₂₅₁	C₄	semidirect
94. C₂×C₄×C₂₅₁	C₄⋊C₄	semidirect
134. C₂₅₁	#1441	semidirect
150. C₂×C₄⋊C₄	Dic₂₅₁	semidirect
229. C₂×C₄	C₄⋊Dic₂₅₁	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
C₂²×C₄⋊C₄×C₂₅₁	2
#1441	251