Subgroups — #44559161

Order 32128.

492 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	C₂×#22548735	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632155	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631796	C₂	1	2	yes	maximal, complemented
5	#30631791	C₂	1	2	yes	maximal, complemented
6	#30631790	C₂	1	2	yes	maximal, complemented
7	#30631784	C₂	1	2	yes	maximal, complemented
8	#30631723	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631684	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22548625	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	#22548735	C₄	1	8	yes	complemented
14	C₂×#16860800	C₂²	1	1	yes	characteristic
15	C₂×#16860798	C₂²	1	1	yes	characteristic
16	C₂×#16860797	C₂²	1	1	yes	characteristic
17	C₂×#16860795	C₂²	1	1	yes	characteristic
18	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₂₅₁	C₂²	1	2	yes	—
20	#22548680	C₂²	1	2	yes	—
21	D₄×Dic₂₅₁	—	2	2	no	—
22	D₄×Dic₂₅₁	C₂²	1	2	yes	—
23	#22548677	C₂²	1	2	yes	—
24	(C₂²⋊D₂₅₁)⋊C₄	—	2	2	no	—
25	#22548674	—	2	1	no	—
26	#22548674	C₂²	1	1	yes	characteristic
27	#22548672	—	2	1	no	—
28	C₂×C₄×Dic₂₅₁	—	2	1	no	—
29	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
30	C₄²×D₂₅₁	—	2	1	no	—
31	#22548648	C₂²	1	2	yes	—
32	#22548646	C₂²	1	2	yes	—
33	#22548640	C₂²	1	2	yes	—
34	#22548637	C₂²	1	2	yes	—
35	#22548625	C₂²	1	2	yes	—
36	#22548625	C₂²	1	2	yes	—
37	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
39	#117×C₂₅₁	C₂²	1	1	yes	characteristic
40	#103×C₂₅₁	C₂²	1	2	yes	—
41	#102×C₂₅₁	C₂²	1	2	yes	—
42	#16860810	—	2	2	no	—
43	#16860810	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
45	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
47	C₂²×Dic₂₅₁	—	2	2	no	—
48	C₂²×Dic₂₅₁	D₄	1	2	yes	—
49	C₂×C₄×D₂₅₁	—	2	2	no	—
50	C₂×C₄×D₂₅₁	—	2	1	no	—
51	#16860800	C₂×C₄	1	4	yes	complemented
52	#16860798	C₂×C₄	1	4	yes	complemented
53	#16860797	C₂×C₄	1	4	yes	complemented
54	#16860795	C₂×C₄	1	4	yes	complemented
55	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₂₅₁	—	2	2	no	—
59	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
60	Dic₂₅₁⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₂₅₁	—	2	2	no	—
62	C₂²⋊Dic₂₅₁	—	2	1	no	—
63	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
64	C₄×Dic₂₅₁	—	2	2	no	—
65	C₄×Dic₂₅₁	—	2	2	no	—
66	C₄×Dic₂₅₁	—	2	1	no	—
67	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
68	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
69	C₄×Dic₂₅₁	C₂³	1	2	yes	—
70	#41×C₂₅₁	—	2	2	no	—
71	#41×C₂₅₁	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₂₅₁	—	2	1	no	—
74	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₂₅₁	—	2	1	no	—
76	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₅₁	—	2	2	no	—
78	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
79	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
81	C₄²×C₂₅₁	—	2	1	no	—
82	C₂²×D₂₅₁	—	2	1	no	—
83	Dic₅₀₂	—	2	1	no	—
84	Dic₅₀₂	C₂²×C₄	1	2	yes	—
85	D₁₀₀₄	—	2	1	no	—
86	D₁₀₀₄	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₂₅₁	—	2	4	no	—
88	C₂×Dic₂₅₁	—	2	1	no	—
89	C₂×Dic₂₅₁	—	2	2	no	—
90	C₂×Dic₂₅₁	—	4	2	no	—
91	C₂×Dic₂₅₁	—	4	4	no	—
92	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
93	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
94	C₄×D₂₅₁	—	2	4	no	—
95	C₄×D₂₅₁	—	4	4	no	—
96	C₂₅₁⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂₅₁⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂³×C₂₅₁	—	2	1	no	—
99	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₂₅₁	—	2	1	no	—
101	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
102	D₄×C₂₅₁	—	2	4	no	—
103	D₄×C₂₅₁	—	2	1	no	—
104	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₂₅₁	—	2	1	no	—
106	C₂×C₄×C₂₅₁	—	2	4	no	—
107	C₂×C₄×C₂₅₁	—	2	2	no	—
108	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₅₁	#41	1	2	yes	—
111	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
112	C₈×C₂₅₁	—	2	2	no	—
113	C₂²×C₂₅₁	—	2	2	no	—
114	C₂²×C₂₅₁	—	2	2	no	—
115	C₂²×C₂₅₁	—	4	2	no	—
116	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
117	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₂₅₁	—	2	2	no	—
119	C₂×D₂₅₁	—	4	2	no	—
120	C₄×C₂₅₁	—	2	2	no	—
121	C₄×C₂₅₁	—	4	2	no	—
122	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
123	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
125	Dic₂₅₁	—	2	2	no	—
126	Dic₂₅₁	—	2	2	no	—
127	Dic₂₅₁	—	2	2	no	—
128	Dic₂₅₁	—	4	2	no	—
129	C₂×C₂₅₁	—	2	2	no	—
130	C₂×C₂₅₁	—	4	2	no	—
131	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₂₅₁	#575	1	2	yes	—
133	D₂₅₁	—	4	2	no	—
134	C₂₅₁	#2943	1	1	yes	characteristic, complemented, Sylow
135	#2943	—	251	1	no	maximal, Sylow
136	C₂×#135	—	251	1	no	—
137	C₄×#41	—	251	1	no	—
138	C₄×#41	—	251	1	no	—
139	#442	—	251	2	no	—
140	#441	—	251	2	no	—
141	#440	—	251	2	no	—
142	#440	—	251	2	no	—
143	#427	—	251	1	no	—
144	#416	—	251	1	no	—
145	#416	D₂₅₁	1	1	yes	characteristic, complemented
146	C₂×#97	—	251	1	no	—
147	C₂×#41	—	251	1	no	—
148	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
149	#135	—	251	8	no	—
150	C₂×Q₁₆	—	251	1	no	—
151	C₂×SD₁₆	—	251	1	no	—
152	C₂×SD₁₆	—	251	1	no	—
153	C₂×D₈	—	251	1	no	—
154	C₂²×C₈	—	251	1	no	—
155	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
156	C₄×Q₈	—	251	2	no	—
157	C₄×Q₈	—	251	2	no	—
158	C₄×D₄	—	502	2	no	—
159	C₄×D₄	—	502	2	no	—
160	C₄×D₄	—	251	2	no	—
161	C₄×D₄	—	251	2	no	—
162	#117	—	502	1	no	—
163	#117	—	502	1	no	—
164	#117	—	251	1	no	—
165	#117	C₂×D₂₅₁	1	1	yes	characteristic
166	C₂×C₄²	—	502	1	no	—
167	C₂×C₄²	—	502	1	no	—
168	C₂×C₄²	—	251	1	no	—
169	#107	—	251	2	no	—
170	#106	—	251	2	no	—
171	#103	—	251	2	no	—
172	#103	C₂×D₂₅₁	1	2	yes	—
173	#102	—	251	2	no	—
174	#102	C₂×D₂₅₁	1	2	yes	—
175	#97	—	251	2	no	—
176	#97	—	251	2	no	—
177	#41	—	2	2	no	—
178	#41	—	502	2	no	—
179	#41	—	251	4	no	—
180	#41	C₄×D₂₅₁	1	4	yes	complemented
181	C₂×Q₈	—	251	1	no	—
182	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	502	1	no	—
185	C₂×D₄	—	251	1	no	—
186	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	502	2	no	—
189	C₂²×C₄	—	502	2	no	—
190	C₂²×C₄	—	502	1	no	—
191	C₂²×C₄	—	251	2	no	—
192	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
193	Q₁₆	—	251	4	no	—
194	SD₁₆	—	251	4	no	—
195	SD₁₆	—	251	4	no	—
196	D₈	—	251	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	251	4	no	—
199	C₂×C₈	—	251	1	no	—
200	C₂×C₈	—	251	1	no	—
201	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
202	C₄⋊C₄	—	502	2	no	—
203	C₄⋊C₄	—	502	2	no	—
204	C₄⋊C₄	—	251	2	no	—
205	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	502	2	no	—
208	C₂²⋊C₄	—	502	1	no	—
209	C₂²⋊C₄	—	502	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	502	2	no	—
212	C₄²	—	502	2	no	—
213	C₄²	—	502	1	no	—
214	C₄²	—	251	1	no	—
215	C₄²	—	251	1	no	—
216	C₄²	—	251	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	502	1	no	—
219	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	502	1	no	—
222	Q₈	—	251	2	no	—
223	Q₈	C₂×C₄×D₂₅₁	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	502	4	no	—
227	D₄	—	502	1	no	—
228	D₄	—	251	2	no	—
229	D₄	C₂×C₄×D₂₅₁	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	1004	4	no	—
234	C₂×C₄	—	1004	4	no	—
235	C₂×C₄	—	1004	2	no	—
236	C₂×C₄	—	502	2	no	—
237	C₂×C₄	—	502	1	no	—
238	C₂×C₄	—	502	4	no	—
239	C₂×C₄	—	251	2	no	—
240	C₂×C₄	—	251	2	no	—
241	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
242	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
243	C₂×C₄	#16860809	1	2	yes	—
244	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	251	2	no	—
247	C₈	—	251	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	1004	2	no	—
252	C₂²	—	502	2	no	—
253	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
254	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	1004	2	no	—
258	C₄	—	502	2	no	—
259	C₄	—	502	2	no	—
260	C₄	—	502	2	no	—
261	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
262	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
263	C₄	C₂×#16860809	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	1004	2	no	—
267	C₂	C₂×(C₂²⋊D₂₅₁)⋊C₄	1	1	yes	characteristic
268	C₂	#30632484	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#416×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548735	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#416	D₂₅₁
hypocenter	C₂₅₁	#2943
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₂₅₁	C₂	semidirect
3. #30632155	C₂	semidirect
4. #30631796	C₂	semidirect
5. #30631791	C₂	semidirect
6. #30631790	C₂	semidirect
7. #30631784	C₂	semidirect
8. #30631723	C₂	semidirect
9. #30631684	C₂	semidirect
10. C₂×#22548625	C₂	semidirect
11. #416×C₂₅₁	C₂	semidirect
13. #22548735	C₄	semidirect
43. #16860810	C₂×C₄	semidirect
51. #16860800	C₂×C₄	semidirect
52. #16860798	C₂×C₄	semidirect
53. #16860797	C₂×C₄	semidirect
54. #16860795	C₂×C₄	semidirect
55. C₂×C₂₅₁⋊C₈	C₂×C₄	semidirect
71. #41×C₂₅₁	C₂×C₄	semidirect
134. C₂₅₁	#2943	semidirect
145. #416	D₂₅₁	semidirect
180. #41	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×#22548735	2
#41×Dic₂₅₁	2
#30632155	2
#30631796	2
#30631791	2
#30631790	2
#30631784	2
#30631723	2
#30631684	2
C₂×#22548625	2
#416×C₂₅₁	2
#2943	251