Subgroups — #44553228

Order 32128.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548672	C₂	1	1	yes	maximal, characteristic
2	#30631571	C₂	1	2	yes	maximal, complemented
3	#30631568	C₂	1	2	yes	maximal, complemented
4	#30631558	C₂	1	2	yes	maximal
5	#30631555	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631551	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631549	C₂	1	1	yes	maximal, characteristic
8	C₄×(C₂×D₂₅₁)⋊C₄	C₂	1	2	yes	maximal
9	#30631532	C₂	1	1	yes	maximal, characteristic, complemented
10	#95×D₂₅₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
14	#22548672	C₄	1	8	yes	complemented
15	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
19	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
23	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
24	C₂₅₁⋊#95	C₂²	1	2	yes	—
25	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
26	C₂₅₁⋊#95	C₂²	1	2	yes	—
27	C₂₅₁⋊#95	C₂²	1	2	yes	—
28	C₂₅₁⋊#95	C₂²	1	2	yes	—
29	C₂₅₁⋊#95	C₂²	1	2	yes	—
30	C₂₅₁⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
34	#95×C₂₅₁	C₂²	1	1	yes	characteristic
35	#95×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
38	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
39	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₂₅₁	—	2	4	no	—
43	C₂×C₄×D₂₅₁	—	2	4	no	—
44	C₂×C₄×D₂₅₁	—	2	4	no	—
45	C₂×C₄×D₂₅₁	—	4	2	no	—
46	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
47	Dic₂₅₁⋊C₄	—	4	2	no	—
48	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	complemented
49	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	complemented
50	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
51	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
52	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
53	C₄×Dic₂₅₁	—	2	4	no	—
54	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	complemented
55	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₅₁	—	2	4	no	—
61	C₄²×C₂₅₁	—	4	2	no	—
62	C₄²×C₂₅₁	C₂×C₄	1	4	yes	complemented
63	C₂²×D₂₅₁	—	2	2	no	—
64	C₂²×D₂₅₁	—	2	1	no	—
65	C₂²×D₂₅₁	—	2	1	no	—
66	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
67	C₂²×D₂₅₁	#41	1	4	yes	—
68	C₂×Dic₂₅₁	—	2	2	no	—
69	C₂×Dic₂₅₁	—	2	2	no	—
70	C₂×Dic₂₅₁	—	2	1	no	—
71	C₂×Dic₂₅₁	—	2	1	no	—
72	C₂×Dic₂₅₁	—	2	2	no	—
73	C₂×Dic₂₅₁	—	2	2	no	—
74	C₂×Dic₂₅₁	—	2	2	no	—
75	C₂×Dic₂₅₁	—	2	2	no	—
76	C₂×Dic₂₅₁	—	2	2	no	—
77	C₂×Dic₂₅₁	—	2	2	no	—
78	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
81	C₄×D₂₅₁	—	2	8	no	—
82	C₄×D₂₅₁	—	4	8	no	—
83	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
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	1	no	—
88	C₂×C₄×C₂₅₁	—	2	2	no	—
89	C₂×C₄×C₂₅₁	—	2	1	no	—
90	C₂×C₄×C₂₅₁	—	2	2	no	—
91	C₂×C₄×C₂₅₁	—	2	2	no	—
92	C₂×C₄×C₂₅₁	—	2	2	no	—
93	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₅₁	#41	1	4	yes	—
97	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
99	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
100	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×D₂₅₁	—	2	2	no	—
103	C₂×D₂₅₁	—	2	2	no	—
104	C₂×D₂₅₁	—	2	4	no	—
105	C₂×D₂₅₁	—	2	4	no	—
106	C₂×D₂₅₁	#117	1	4	yes	—
107	C₄×C₂₅₁	—	2	2	no	—
108	C₄×C₂₅₁	—	2	2	no	—
109	C₄×C₂₅₁	—	2	2	no	—
110	C₄×C₂₅₁	—	2	4	no	—
111	C₄×C₂₅₁	—	4	2	no	—
112	Dic₂₅₁	—	2	2	no	—
113	Dic₂₅₁	—	2	2	no	—
114	Dic₂₅₁	—	2	2	no	—
115	Dic₂₅₁	—	4	2	no	—
116	Dic₂₅₁	—	4	2	no	—
117	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
118	C₂×C₂₅₁	C₄×#41	1	1	yes	characteristic
119	C₂×C₂₅₁	#520	1	1	yes	characteristic
120	C₂×C₂₅₁	#551	1	2	yes	—
121	C₂×C₂₅₁	#552	1	2	yes	—
122	D₂₅₁	—	2	4	no	—
123	C₂₅₁	#2343	1	1	yes	characteristic, complemented, Sylow
124	#2343	—	251	1	no	maximal, Sylow
125	C₂×#117	—	251	1	no	—
126	#387	—	251	2	no	—
127	#387	—	251	2	no	—
128	#385	—	251	2	no	—
129	#383	—	251	1	no	—
130	#382	—	251	1	no	—
131	#381	—	251	1	no	—
132	C₄×C₄⋊C₄	D₂₅₁	1	1	yes	characteristic, complemented
133	C₄×C₂²⋊C₄	—	251	2	no	—
134	#376	—	251	1	no	—
135	C₂×#95	—	251	1	no	—
136	C₂³×C₄	—	251	2	no	—
137	C₂³×C₄	—	251	1	no	—
138	#117	—	251	8	no	—
139	C₂×C₄⋊C₄	—	251	2	no	—
140	C₂×C₄⋊C₄	—	251	1	no	—
141	C₂×C₄⋊C₄	—	251	1	no	—
142	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
143	C₂×C₂²⋊C₄	—	251	2	no	—
144	C₂×C₂²⋊C₄	—	251	1	no	—
145	C₂×C₂²⋊C₄	—	251	1	no	—
146	C₂×C₄²	—	251	2	no	—
147	C₂×C₄²	—	251	1	no	—
148	C₂×C₄²	C₂×D₂₅₁	1	2	yes	—
149	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
150	#95	—	251	1	no	—
151	#95	—	251	2	no	—
152	#95	—	251	1	no	—
153	#95	—	251	2	no	—
154	#95	—	251	2	no	—
155	#95	—	251	2	no	—
156	#95	—	251	2	no	—
157	#95	—	251	2	no	—
158	#95	C₂×D₂₅₁	1	1	yes	characteristic
159	#95	C₂×D₂₅₁	1	1	yes	characteristic
160	C₂⁴	—	251	1	no	—
161	C₂²×C₄	—	1004	2	no	—
162	C₂²×C₄	—	502	4	no	—
163	C₂²×C₄	—	502	4	no	—
164	C₂²×C₄	—	502	4	no	—
165	C₂²×C₄	—	251	2	no	—
166	C₂²×C₄	—	251	2	no	—
167	C₂²×C₄	—	251	1	no	—
168	C₂²×C₄	—	251	1	no	—
169	C₂²×C₄	—	251	1	no	—
170	C₂²×C₄	—	251	4	no	—
171	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
172	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	1004	2	no	—
178	C₄⋊C₄	—	251	4	no	—
179	C₄⋊C₄	—	251	4	no	—
180	C₂²⋊C₄	—	502	4	no	—
181	C₂²⋊C₄	—	251	4	no	—
182	C₂²⋊C₄	—	251	4	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	502	4	no	—
185	C₄²	—	251	4	no	—
186	C₄²	C₄×D₂₅₁	1	4	yes	complemented
187	C₂³	—	502	2	no	—
188	C₂³	—	502	1	no	—
189	C₂³	—	502	1	no	—
190	C₂³	—	251	2	no	—
191	C₂³	—	251	4	no	—
192	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	1004	8	no	—
203	C₂×C₄	—	502	8	no	—
204	C₂×C₄	—	502	2	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	2	no	—
210	C₂×C₄	—	502	2	no	—
211	C₂×C₄	—	502	2	no	—
212	C₂×C₄	—	502	1	no	—
213	C₂×C₄	—	502	1	no	—
214	C₂×C₄	—	251	2	no	—
215	C₂×C₄	—	251	2	no	—
216	C₂×C₄	—	251	2	no	—
217	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
220	C₂×C₄	#16860810	1	4	yes	—
221	C₂²	—	502	4	no	—
222	C₂²	—	502	4	no	—
223	C₂²	—	502	2	no	—
224	C₂²	—	502	2	no	—
225	C₂²	—	251	4	no	—
226	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
227	C₂²	#41×D₂₅₁	1	2	yes	—
228	C₂²	C₂×#16860810	1	2	yes	—
229	C₂²	#22548760	1	1	yes	characteristic
230	C₂²	#22548763	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	1004	2	no	—
237	C₄	—	1004	2	no	—
238	C₄	—	502	2	no	—
239	C₄	—	502	2	no	—
240	C₄	—	502	2	no	—
241	C₂	—	502	4	no	—
242	C₂	#117×D₂₅₁	1	1	yes	characteristic
243	C₂	C₄×#16860810	1	1	yes	characteristic
244	C₂	#30632168	1	1	yes	characteristic
245	C₂	#30632366	1	2	yes	—
246	C₂	#30632378	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂³×D₂₅₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₅₁
hypocenter	C₂₅₁	#2343
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #30631571	C₂	semidirect
3. #30631568	C₂	semidirect
5. #30631555	C₂	semidirect
6. #30631551	C₂	semidirect
9. #30631532	C₂	semidirect
11. C₄×C₄⋊C₄×C₂₅₁	C₂	semidirect
14. #22548672	C₄	semidirect
48. Dic₂₅₁⋊C₄	C₂×C₄	semidirect
49. Dic₂₅₁⋊C₄	C₂×C₄	semidirect
54. C₄×Dic₂₅₁	C₂×C₄	semidirect
62. C₄²×C₂₅₁	C₂×C₄	semidirect
123. C₂₅₁	#2343	semidirect
132. C₄×C₄⋊C₄	D₂₅₁	semidirect
186. C₄²	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548672	2
#30631571	2
#30631568	2
#30631558	2
#30631555	2
#30631551	2
#30631549	2
C₄×(C₂×D₂₅₁)⋊C₄	2
#30631532	2
#95×D₂₅₁	2
C₄×C₄⋊C₄×C₂₅₁	2
#2343	251