Subgroups — #44548591

Order 32128.

568 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	1	yes	maximal, characteristic
3	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal
4	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal
5	C₂×#95×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
7	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₂₅₁	—	2	4	no	—
10	C₂×C₄×Dic₂₅₁	C₄	1	4	yes	complemented
11	C₂₅₁⋊#95	—	2	4	no	—
12	C₂₅₁⋊#95	—	2	2	no	—
13	C₂₅₁⋊#95	C₄	1	8	yes	complemented
14	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	—
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	1	no	—
21	C₂²×Dic₂₅₁	—	2	1	no	—
22	C₂²×Dic₂₅₁	—	2	2	no	—
23	C₂²×Dic₂₅₁	—	2	1	no	—
24	C₂²×Dic₂₅₁	—	2	1	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	C₂×C₄	1	4	yes	—
31	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
32	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
33	C₂²×Dic₂₅₁	D₄	1	4	yes	—
34	C₄×Dic₂₅₁	—	2	8	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	2	no	—
39	C₂²×C₄×C₂₅₁	—	2	2	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	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
47	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
48	C₂×Dic₂₅₁	—	2	2	no	—
49	C₂×Dic₂₅₁	—	2	2	no	—
50	C₂×Dic₂₅₁	—	2	2	no	—
51	C₂×Dic₂₅₁	—	2	2	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	—	4	2	no	—
60	C₂×Dic₂₅₁	—	4	2	no	—
61	C₂×Dic₂₅₁	—	4	2	no	—
62	C₂×Dic₂₅₁	—	4	2	no	—
63	C₂×Dic₂₅₁	C₄²	1	4	yes	—
64	C₂×Dic₂₅₁	C₂²⋊C₄	1	4	yes	—
65	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
66	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
67	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
68	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₂₅₁	#41	1	2	yes	—
71	C₂³×C₂₅₁	#41	1	1	yes	characteristic
72	C₂³×C₂₅₁	#41	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	2	no	—
77	C₂×C₄×C₂₅₁	—	2	2	no	—
78	C₂×C₄×C₂₅₁	—	2	4	no	—
79	C₂×C₄×C₂₅₁	—	2	4	no	—
80	C₂×C₄×C₂₅₁	—	2	4	no	—
81	C₂×C₄×C₂₅₁	—	2	4	no	—
82	C₂×C₄×C₂₅₁	—	2	4	no	—
83	C₂×C₄×C₂₅₁	—	4	2	no	—
84	C₂×C₄×C₂₅₁	—	4	2	no	—
85	C₂×C₄×C₂₅₁	—	4	2	no	—
86	C₂×C₄×C₂₅₁	—	4	2	no	—
87	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
88	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
89	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₅₁	#117	1	1	yes	characteristic
92	C₂²×C₂₅₁	#117	1	2	yes	—
93	C₂²×C₂₅₁	#117	1	1	yes	characteristic
94	C₂²×C₂₅₁	#117	1	1	yes	characteristic
95	C₂²×C₂₅₁	#117	1	2	yes	—
96	C₂²×C₂₅₁	#117	1	1	yes	characteristic
97	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
98	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
99	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
100	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
101	C₂²×C₂₅₁	#121	1	2	yes	—
102	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
103	C₂²×C₂₅₁	#123	1	2	yes	—
104	C₂²×C₂₅₁	#123	1	2	yes	—
105	C₂²×C₂₅₁	#124	1	1	yes	characteristic
106	C₂²×C₂₅₁	#124	1	1	yes	characteristic
107	C₂²×C₂₅₁	#125	1	1	yes	characteristic
108	C₂²×C₂₅₁	#125	1	1	yes	characteristic
109	C₂²×C₂₅₁	#126	1	2	yes	—
110	C₂²×C₂₅₁	#126	1	2	yes	—
111	C₄×C₂₅₁	—	2	4	no	—
112	C₄×C₂₅₁	—	4	4	no	—
113	Dic₂₅₁	—	2	4	no	—
114	Dic₂₅₁	—	2	4	no	—
115	Dic₂₅₁	—	4	4	no	—
116	C₂×C₂₅₁	#376	1	1	yes	characteristic
117	C₂×C₂₅₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₂₅₁	#380	1	1	yes	characteristic
120	C₂×C₂₅₁	#381	1	1	yes	characteristic
121	C₂×C₂₅₁	#382	1	1	yes	characteristic
122	C₂×C₂₅₁	#383	1	1	yes	characteristic
123	C₂×C₂₅₁	#387	1	2	yes	—
124	C₂×C₂₅₁	#387	1	2	yes	—
125	C₂×C₂₅₁	#388	1	2	yes	—
126	C₂×C₂₅₁	#388	1	2	yes	—
127	C₂₅₁	#1434	1	1	yes	characteristic, complemented, Sylow
128	#1434	—	251	1	no	maximal, Sylow
129	C₂²×C₄²	—	251	1	no	—
130	C₂×#95	—	251	2	no	—
131	C₂×#95	—	251	1	no	—
132	C₂×#95	—	251	2	no	—
133	C₂×#95	D₂₅₁	1	1	yes	characteristic
134	C₂³×C₄	—	251	2	no	—
135	C₂³×C₄	—	251	1	no	—
136	C₂³×C₄	—	251	1	no	—
137	C₂³×C₄	C₂×D₂₅₁	1	2	yes	—
138	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
139	C₂×C₄²	—	502	4	no	—
140	C₂×C₄²	—	251	4	no	—
141	#95	—	2	2	no	—
142	#95	—	502	4	no	—
143	#95	—	502	2	no	—
144	#95	—	251	8	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	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	502	2	no	—
155	C₂²×C₄	—	502	2	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	1	no	—
162	C₂²×C₄	—	502	1	no	—
163	C₂²×C₄	—	502	2	no	—
164	C₂²×C₄	—	502	1	no	—
165	C₂²×C₄	—	502	1	no	—
166	C₂²×C₄	—	251	4	no	—
167	C₂²×C₄	—	251	4	no	—
168	C₂²×C₄	—	251	2	no	—
169	C₂²×C₄	—	251	2	no	—
170	C₂²×C₄	—	251	4	no	—
171	C₂²×C₄	C₄×D₂₅₁	1	4	yes	—
172	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
173	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
174	C₂²×C₄	Dic₅₀₂	1	2	yes	—
175	C₄²	—	502	8	no	—
176	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
177	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
178	C₂³	D₄×D₂₅₁	1	2	yes	—
179	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
180	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
181	C₂³	#16860809	1	2	yes	—
182	C₂³	#16860809	1	1	yes	characteristic
183	C₂³	#16860809	1	1	yes	characteristic
184	C₂³	#16860810	1	2	yes	—
185	C₂³	#16860811	1	1	yes	characteristic
186	C₂³	#16860811	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	1004	2	no	—
199	C₂×C₄	—	1004	2	no	—
200	C₂×C₄	—	1004	2	no	—
201	C₂×C₄	—	1004	2	no	—
202	C₂×C₄	—	502	4	no	—
203	C₂×C₄	—	502	4	no	—
204	C₂×C₄	—	502	4	no	—
205	C₂×C₄	—	502	2	no	—
206	C₂×C₄	—	502	2	no	—
207	C₂×C₄	—	502	2	no	—
208	C₂×C₄	—	502	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	—	251	4	no	—
214	C₂×C₄	—	251	4	no	—
215	C₂×C₄	Dic₂₅₁⋊C₄	1	4	yes	—
216	C₂²	C₄²×D₂₅₁	1	1	yes	characteristic
217	C₂²	C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
218	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
219	C₂²	#22548671	1	1	yes	characteristic
220	C₂²	#22548671	1	2	yes	—
221	C₂²	#22548672	1	1	yes	characteristic
222	C₂²	#22548673	1	1	yes	characteristic
223	C₂²	#22548673	1	2	yes	—
224	C₂²	#22548673	1	1	yes	characteristic
225	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
226	C₂²	C₄×C₂²⋊D₂₅₁	1	2	yes	—
227	C₂²	#22548677	1	2	yes	—
228	C₂²	C₄×Dic₅₀₂	1	2	yes	—
229	C₂²	#22548687	1	2	yes	—
230	C₂²	#22548692	1	2	yes	—
231	C₂²	#22548697	1	2	yes	—
232	C₂²	#22548700	1	2	yes	—
233	C₂²	#22548702	1	1	yes	characteristic
234	C₂²	#22548704	1	1	yes	characteristic
235	C₂²	#22548706	1	1	yes	characteristic
236	C₂²	#22548706	1	1	yes	characteristic
237	C₂²	#22548708	1	2	yes	—
238	C₂²	#22548710	1	2	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	4	4	no	—
241	C₄	—	1004	4	no	—
242	C₄	—	502	4	no	—
243	C₄	—	502	4	no	—
244	C₂	#30631532	1	1	yes	characteristic
245	C₂	#30631534	1	1	yes	characteristic
246	C₂	C₄×Dic₂₅₁⋊C₄	1	1	yes	characteristic
247	C₂	#30631545	1	1	yes	characteristic
248	C₂	#30631549	1	1	yes	characteristic
249	C₂	#30631552	1	1	yes	characteristic
250	C₂	#30631554	1	1	yes	characteristic
251	C₂	#30631570	1	2	yes	—
252	C₂	#30631574	1	2	yes	—
253	C₂	#30631577	1	2	yes	—
254	C₂	#30631581	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂×C₄²
Fitting	C₂×#95×C₂₅₁	C₂
socle, prime radical, cosocle	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂×#95	D₂₅₁
hypocenter	C₂₅₁	#1434
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×Dic₂₅₁	C₄	semidirect
13. C₂₅₁⋊#95	C₄	semidirect
127. C₂₅₁	#1434	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
#1434	251