Subgroups — #44559280

Order 32128.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#30632453	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631807	C₂	1	4	yes	maximal, complemented
4	#30631802	C₂	1	4	yes	maximal, complemented
5	C₂×#22548650	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631704	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631703	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631645	C₂	1	1	yes	maximal, characteristic, complemented
9	#568×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16860810	C₂²	1	1	yes	characteristic
11	#22548735	C₄	1	2	yes	complemented
12	#22548735	C₄	1	2	yes	complemented
13	#22548735	C₂²	1	2	yes	—
14	#22548735	C₂²	1	2	yes	—
15	C₂×#16860800	C₄	1	1	yes	characteristic, complemented
16	C₂×#16860798	C₄	1	1	yes	characteristic, complemented
17	C₂×#16860797	C₄	1	1	yes	characteristic, complemented
18	C₂×#16860795	C₄	1	1	yes	characteristic, complemented
19	#22548723	—	2	2	no	—
20	#22548721	C₂²	1	4	yes	complemented
21	C₂×#16860794	C₂²	1	1	yes	characteristic
22	M₁₆×D₂₅₁	—	2	2	no	—
23	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
24	#22548674	C₂²	1	1	yes	characteristic
25	#22548650	C₂²	1	2	yes	complemented
26	#22548650	C₂²	1	2	yes	complemented
27	#22548642	C₂²	1	4	yes	complemented
28	#22548641	C₂²	1	4	yes	complemented
29	#22548625	C₂²	1	2	yes	complemented
30	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
31	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
32	#131×C₂₅₁	—	2	2	no	—
33	#131×C₂₅₁	C₂²	1	4	yes	complemented
34	C₂×M₁₆×C₂₅₁	—	2	2	no	—
35	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
37	#16860810	—	2	1	no	—
38	#16860810	—	2	1	no	—
39	#16860810	C₂×C₄	1	2	yes	—
40	#16860810	C₂³	1	2	yes	—
41	C₂×Dic₅₀₂	C₂×C₄	1	1	yes	characteristic
42	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
44	C₂×C₄×D₂₅₁	—	2	1	no	—
45	#16860800	C₂×C₄	1	2	yes	—
46	#16860800	C₂×C₄	1	2	yes	—
47	#16860798	C₂×C₄	1	2	yes	—
48	#16860798	C₂×C₄	1	2	yes	—
49	#16860797	C₂×C₄	1	2	yes	—
50	#16860797	C₂×C₄	1	2	yes	—
51	#16860795	C₂×C₄	1	2	yes	—
52	#16860795	C₂×C₄	1	2	yes	—
53	#16860794	—	2	1	no	—
54	#16860794	C₂³	1	2	yes	—
55	#16860793	—	2	4	no	—
56	C₂×C₂₅₁⋊C₈	—	2	1	no	—
57	C₂×C₂₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂₅₁⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
63	C₈×D₂₅₁	—	2	4	no	—
64	C₄⋊Dic₂₅₁	—	2	1	no	—
65	C₂²⋊Dic₂₅₁	—	2	1	no	—
66	C₄×Dic₂₅₁	C₂³	1	2	yes	—
67	#41×C₂₅₁	—	2	1	no	—
68	#41×C₂₅₁	—	2	1	no	—
69	#41×C₂₅₁	C₂×C₄	1	2	yes	—
70	#41×C₂₅₁	C₂³	1	2	yes	—
71	C₂×Q₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
72	C₂×D₄×C₂₅₁	—	2	1	no	—
73	C₂×D₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₅₁	—	2	1	no	—
75	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
76	M₁₆×C₂₅₁	—	2	1	no	—
77	M₁₆×C₂₅₁	—	2	4	no	—
78	M₁₆×C₂₅₁	D₄	1	2	yes	—
79	M₁₆×C₂₅₁	D₄	1	2	yes	—
80	M₁₆×C₂₅₁	C₂³	1	2	yes	—
81	C₂×C₈×C₂₅₁	—	2	1	no	—
82	C₂×C₈×C₂₅₁	—	2	4	no	—
83	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
84	C₂²×D₂₅₁	—	2	1	no	—
85	Dic₅₀₂	—	2	1	no	—
86	Dic₅₀₂	C₂²×C₄	1	2	yes	—
87	D₁₀₀₄	—	2	1	no	—
88	D₁₀₀₄	C₂²×C₄	1	2	yes	—
89	C₂²⋊D₂₅₁	—	2	2	no	—
90	C₂²⋊D₂₅₁	—	2	2	no	—
91	C₂×Dic₂₅₁	—	2	1	no	—
92	C₂×Dic₂₅₁	—	2	2	no	—
93	C₄×D₂₅₁	—	2	2	no	—
94	C₄×D₂₅₁	—	2	2	no	—
95	C₂₅₁⋊C₈	—	2	2	no	—
96	C₂₅₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂₅₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂₅₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂₅₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂³×C₂₅₁	—	2	1	no	—
101	C₂³×C₂₅₁	#41	1	1	yes	characteristic
102	Q₈×C₂₅₁	—	2	1	no	—
103	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
104	D₄×C₂₅₁	—	2	2	no	—
105	D₄×C₂₅₁	—	2	2	no	—
106	D₄×C₂₅₁	—	2	1	no	—
107	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₂₅₁	—	2	1	no	—
109	C₂×C₄×C₂₅₁	—	2	2	no	—
110	C₂×C₄×C₂₅₁	—	2	2	no	—
111	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
116	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
117	C₈×C₂₅₁	—	2	2	no	—
118	C₈×C₂₅₁	C₂×D₄	1	4	yes	—
119	C₂²×C₂₅₁	—	2	2	no	—
120	C₂²×C₂₅₁	—	4	1	no	—
121	C₂²×C₂₅₁	—	8	1	no	—
122	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
125	C₂×D₂₅₁	—	2	2	no	—
126	C₂×D₂₅₁	—	8	1	no	—
127	C₄×C₂₅₁	—	2	2	no	—
128	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
129	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
130	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
132	Dic₂₅₁	—	2	2	no	—
133	Dic₂₅₁	—	4	2	no	—
134	C₂×C₂₅₁	—	2	1	no	—
135	C₂×C₂₅₁	—	2	1	no	—
136	C₂×C₂₅₁	—	2	1	no	—
137	C₂×C₂₅₁	—	4	2	no	—
138	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
139	D₂₅₁	—	4	2	no	—
140	C₂₅₁	#2957	1	1	yes	characteristic, complemented, Sylow
141	#2957	—	251	1	no	maximal, Sylow
142	C₂×#135	—	251	1	no	—
143	#568	—	251	1	no	—
144	#568	D₂₅₁	1	1	yes	characteristic, complemented
145	#444	—	251	4	no	—
146	#443	—	251	4	no	—
147	C₂×#108	—	251	1	no	—
148	#421	—	251	1	no	—
149	#421	—	251	1	no	—
150	#405	—	251	1	no	—
151	C₂×#41	—	251	1	no	—
152	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
153	#135	—	251	2	no	—
154	#135	—	251	2	no	—
155	#135	—	251	2	no	—
156	#135	—	251	2	no	—
157	C₂×Q₁₆	—	251	1	no	—
158	C₂×SD₁₆	—	251	1	no	—
159	C₂×SD₁₆	—	251	1	no	—
160	C₂×D₈	—	251	1	no	—
161	#131	—	2	2	no	—
162	#131	—	502	2	no	—
163	#131	—	251	4	no	—
164	#131	C₂×D₂₅₁	1	4	yes	complemented
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	502	2	no	—
167	C₂×M₁₆	—	251	1	no	—
168	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
169	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
170	C₂²×C₈	—	251	1	no	—
171	#117	—	251	1	no	—
172	#108	—	251	2	no	—
173	#108	—	251	2	no	—
174	#104	—	251	4	no	—
175	#104	—	251	4	no	—
176	#97	—	251	2	no	—
177	C₄×C₈	—	251	2	no	—
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	502	1	no	—
181	#41	—	502	1	no	—
182	#41	—	251	2	no	—
183	#41	—	251	2	no	—
184	#41	C₄×D₂₅₁	1	2	yes	—
185	#41	C₂²×D₂₅₁	1	2	yes	—
186	C₂×Q₈	—	251	1	no	—
187	C₂×Q₈	C₄×D₂₅₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	502	1	no	—
190	C₂×D₄	—	251	1	no	—
191	C₂×D₄	C₄×D₂₅₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	502	1	no	—
194	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
195	Q₁₆	—	251	2	no	—
196	Q₁₆	—	251	2	no	—
197	SD₁₆	—	251	2	no	—
198	SD₁₆	—	251	2	no	—
199	SD₁₆	—	251	2	no	—
200	SD₁₆	—	251	2	no	—
201	D₈	—	251	2	no	—
202	D₈	—	251	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	502	1	no	—
206	M₁₆	—	502	4	no	—
207	M₁₆	—	251	2	no	—
208	M₁₆	C₂²⋊D₂₅₁	1	2	yes	—
209	M₁₆	C₂²⋊D₂₅₁	1	2	yes	—
210	M₁₆	C₂²×D₂₅₁	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	502	1	no	—
214	C₂×C₈	—	502	4	no	—
215	C₂×C₈	—	251	1	no	—
216	C₂×C₈	—	251	1	no	—
217	C₂×C₈	—	251	1	no	—
218	C₂×C₈	—	251	1	no	—
219	C₂×C₈	—	251	1	no	—
220	C₂×C₈	—	251	1	no	—
221	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
222	C₄⋊C₄	—	502	1	no	—
223	C₂²⋊C₄	—	502	1	no	—
224	C₄²	—	251	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	502	1	no	—
227	C₂³	#16860810	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	502	1	no	—
230	Q₈	—	251	2	no	—
231	Q₈	C₂×C₄×D₂₅₁	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	502	2	no	—
236	D₄	—	502	2	no	—
237	D₄	—	502	1	no	—
238	D₄	—	251	2	no	—
239	D₄	C₂×C₄×D₂₅₁	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	502	2	no	—
244	C₂×C₄	—	502	1	no	—
245	C₂×C₄	—	502	2	no	—
246	C₂×C₄	—	502	2	no	—
247	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
251	C₂×C₄	#16860810	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	502	2	no	—
255	C₈	—	251	1	no	—
256	C₈	—	251	1	no	—
257	C₈	—	251	1	no	—
258	C₈	—	251	1	no	—
259	C₈	C₂×C₂²⋊D₂₅₁	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	2008	1	no	—
264	C₂²	—	502	2	no	—
265	C₂²	C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
266	C₂²	C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
267	C₂²	C₂×#16860810	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	1004	2	no	—
270	C₄	—	502	2	no	—
271	C₄	C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
272	C₄	C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
274	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	1004	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#568×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂×C₄×D₄
prime radical	C₂×#22548735	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#568	D₂₅₁
hypocenter	C₂₅₁	#2957
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #30632453	C₂	semidirect
3. #30631807	C₂	semidirect
4. #30631802	C₂	semidirect
5. C₂×#22548650	C₂	semidirect
6. #30631704	C₂	semidirect
7. #30631703	C₂	semidirect
8. #30631645	C₂	semidirect
9. #568×C₂₅₁	C₂	semidirect
11. #22548735	C₄	semidirect
12. #22548735	C₄	semidirect
15. C₂×#16860800	C₄	semidirect
16. C₂×#16860798	C₄	semidirect
17. C₂×#16860797	C₄	semidirect
18. C₂×#16860795	C₄	semidirect
20. #22548721	C₂²	semidirect
25. #22548650	C₂²	semidirect
26. #22548650	C₂²	semidirect
27. #22548642	C₂²	semidirect
28. #22548641	C₂²	semidirect
29. #22548625	C₂²	semidirect
30. C₈×Dic₂₅₁	C₂²	semidirect
33. #131×C₂₅₁	C₂²	semidirect
140. C₂₅₁	#2957	semidirect
144. #568	D₂₅₁	semidirect
164. #131	C₂×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
#30632453	2
#30631807	2
#30631802	2
C₂×#22548650	2
#30631704	2
#30631703	2
#30631645	2
#568×C₂₅₁	2
#2957	251