Subgroups — #44564698

Order 32128.

1260 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632424	C₂	1	4	yes	maximal, complemented
2	#30632423	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632357	C₂	1	2	yes	maximal, complemented
4	#30632331	C₂	1	4	yes	maximal, complemented
5	#30632322	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₂²⋊D₂₅₁	C₂	1	4	yes	maximal, complemented
7	#30632291	C₂	1	4	yes	maximal, complemented
8	#30632250	C₂	1	4	yes	maximal, complemented
9	#30632246	C₂	1	4	yes	maximal, complemented
10	#30632221	C₂	1	2	yes	maximal, complemented
11	C₂×#127×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860809	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₂₅₁	C₂²	1	4	yes	complemented
14	C₂×D₄×D₂₅₁	C₂²	1	4	yes	complemented
15	#22548711	C₂²	1	4	yes	complemented
16	#22548709	C₂²	1	4	yes	complemented
17	#22548707	C₂²	1	2	yes	complemented
18	#22548703	C₂²	1	2	yes	complemented
19	#22548702	C₂²	1	4	yes	complemented
20	#22548698	C₂²	1	4	yes	complemented
21	#22548696	C₂²	1	4	yes	complemented
22	#22548696	C₂²	1	4	yes	complemented
23	#22548696	C₂²	1	8	yes	complemented
24	#22548694	C₂²	1	2	yes	complemented
25	#22548689	C₂²	1	2	yes	—
26	#22548687	C₂²	1	8	yes	complemented
27	#22548687	C₂²	1	8	yes	complemented
28	#22548687	C₂²	1	8	yes	complemented
29	#22548687	C₂²	1	8	yes	complemented
30	#22548686	C₂²	1	4	yes	complemented
31	#22548686	C₂²	1	4	yes	complemented
32	#22548684	C₂²	1	8	yes	complemented
33	#22548683	C₂²	1	4	yes	complemented
34	#22548683	C₂²	1	4	yes	complemented
35	#22548683	C₂²	1	8	yes	complemented
36	C₄×Dic₅₀₂	C₂²	1	2	yes	complemented
37	C₄×D₁₀₀₄	C₂²	1	2	yes	complemented
38	D₄×Dic₂₅₁	C₂²	1	4	yes	complemented
39	D₄×Dic₂₅₁	C₂²	1	4	yes	complemented
40	C₄×C₂²⋊D₂₅₁	C₂²	1	4	yes	complemented
41	#22548672	C₂²	1	4	yes	complemented
42	C₂×C₂²⋊Dic₂₅₁	C₂²	1	4	yes	complemented
43	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	complemented
44	C₂²×D₄×C₂₅₁	C₂²	1	4	yes	complemented
45	#127×C₂₅₁	C₂²	1	1	yes	characteristic
46	#127×C₂₅₁	C₂²	1	4	yes	complemented
47	#127×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
48	#127×C₂₅₁	C₂²	1	2	yes	complemented
49	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₂₅₁	C₂³	1	4	yes	—
51	#16860809	—	4	8	no	—
52	C₂×Dic₅₀₂	C₂³	1	2	yes	—
53	C₂×D₁₀₀₄	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₂₅₁	—	4	8	no	—
55	C₂×C₂²⋊D₂₅₁	C₂³	1	4	yes	—
56	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
59	D₄×D₂₅₁	—	4	8	no	—
60	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
61	C₂²×Dic₂₅₁	C₂³	1	8	yes	—
62	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
63	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
64	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
65	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
66	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
67	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
68	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
69	C₂²⋊Dic₂₅₁	C₂³	1	4	yes	—
70	C₂²⋊Dic₂₅₁	C₂³	1	4	yes	—
71	C₂²⋊Dic₂₅₁	C₂³	1	4	yes	—
72	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
73	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
74	(C₂×D₂₅₁)⋊C₄	C₂³	1	4	yes	—
75	(C₂×D₂₅₁)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₂₅₁)⋊C₄	C₂³	1	4	yes	—
77	(C₂×D₂₅₁)⋊C₄	C₂³	1	4	yes	—
78	C₄×Dic₂₅₁	C₂³	1	4	yes	—
79	C₂⁴×C₂₅₁	C₂³	1	4	yes	—
80	C₂×D₄×C₂₅₁	—	4	8	no	—
81	C₂×D₄×C₂₅₁	—	4	4	no	—
82	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
83	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
84	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
85	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
86	C₂×D₄×C₂₅₁	C₂³	1	8	yes	—
87	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
88	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
89	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
90	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
91	C₄²×C₂₅₁	C₂³	1	2	yes	—
92	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
93	C₂²×D₂₅₁	—	4	8	no	—
94	C₂²×D₂₅₁	—	4	4	no	—
95	C₂²×D₂₅₁	C₂⁴	1	4	yes	—
96	Dic₅₀₂	—	2	4	no	—
97	D₁₀₀₄	—	2	4	no	—
98	C₂²⋊D₂₅₁	—	2	8	no	—
99	C₂²⋊D₂₅₁	—	4	8	no	—
100	C₂²⋊D₂₅₁	—	4	8	no	—
101	C₂²⋊D₂₅₁	—	4	8	no	—
102	C₂×Dic₂₅₁	—	4	4	no	—
103	C₂×Dic₂₅₁	—	4	8	no	—
104	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
105	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
106	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
107	C₄×D₂₅₁	—	2	8	no	—
108	C₂³×C₂₅₁	—	2	8	no	—
109	C₂³×C₂₅₁	—	4	8	no	—
110	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
111	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
112	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
113	D₄×C₂₅₁	—	2	8	no	—
114	D₄×C₂₅₁	—	2	8	no	—
115	D₄×C₂₅₁	—	4	8	no	—
116	D₄×C₂₅₁	—	4	2	no	—
117	D₄×C₂₅₁	—	4	4	no	—
118	D₄×C₂₅₁	—	4	2	no	—
119	C₂×C₄×C₂₅₁	—	2	2	no	—
120	C₂×C₄×C₂₅₁	—	2	4	no	—
121	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
124	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
125	C₂²×C₂₅₁	—	2	2	no	—
126	C₂²×C₂₅₁	—	2	8	no	—
127	C₂²×C₂₅₁	—	2	8	no	—
128	C₂²×C₂₅₁	—	2	4	no	—
129	C₂²×C₂₅₁	—	2	4	no	—
130	C₂²×C₂₅₁	—	8	8	no	—
131	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
133	C₂×D₂₅₁	—	2	4	no	—
134	C₂×D₂₅₁	—	2	4	no	—
135	C₂×D₂₅₁	—	2	4	no	—
136	C₂×D₂₅₁	—	8	8	no	—
137	C₄×C₂₅₁	—	2	1	no	—
138	C₄×C₂₅₁	—	2	1	no	—
139	C₄×C₂₅₁	—	2	2	no	—
140	C₄×C₂₅₁	H(5,𝔽₂)	1	4	yes	—
141	Dic₂₅₁	—	2	4	no	—
142	Dic₂₅₁	—	2	4	no	—
143	Dic₂₅₁	—	2	4	no	—
144	C₂×C₂₅₁	—	2	2	no	—
145	C₂×C₂₅₁	—	4	4	no	—
146	C₂×C₂₅₁	—	4	4	no	—
147	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	2	yes	—
149	D₂₅₁	—	4	4	no	—
150	C₂₅₁	#3543	1	1	yes	characteristic, complemented, Sylow
151	#3543	—	251	1	no	maximal, Sylow
152	#560	—	251	4	no	—
153	#559	—	251	1	no	—
154	#550	—	251	2	no	—
155	#546	—	251	4	no	—
156	D₄²	—	251	4	no	—
157	D₄²	—	251	1	no	—
158	#540	—	251	4	no	—
159	#535	—	251	4	no	—
160	#534	—	251	4	no	—
161	C₂×#127	D₂₅₁	1	1	yes	characteristic, complemented
162	#529	—	251	2	no	—
163	C₂×#41	—	251	4	no	—
164	C₂²×D₄	—	251	4	no	—
165	C₂²×D₄	—	251	4	no	—
166	C₂²×D₄	C₂×D₂₅₁	1	2	yes	complemented
167	C₂²×D₄	C₂×D₂₅₁	1	4	yes	complemented
168	#127	—	251	4	no	—
169	#127	C₂×D₂₅₁	1	1	yes	characteristic
170	#127	C₂×D₂₅₁	1	4	yes	complemented
171	#127	C₂×D₂₅₁	1	1	yes	characteristic, complemented
172	#127	C₂×D₂₅₁	1	2	yes	complemented
173	#126	—	251	4	no	—
174	#125	—	251	2	no	—
175	#124	—	251	4	no	—
176	#124	—	251	2	no	—
177	#123	—	251	4	no	—
178	#123	—	251	4	no	—
179	#123	—	251	8	no	—
180	#123	—	251	4	no	—
181	C₂²⋊Q₈	—	251	2	no	—
182	#121	—	251	8	no	—
183	#121	—	251	8	no	—
184	#121	—	251	8	no	—
185	#121	—	251	4	no	—
186	#121	—	251	4	no	—
187	#121	—	251	8	no	—
188	#121	—	251	2	no	—
189	#120	—	251	8	no	—
190	#120	—	251	4	no	—
191	#120	—	251	4	no	—
192	#120	—	251	8	no	—
193	C₄×Q₈	—	251	2	no	—
194	C₄×D₄	—	251	4	no	—
195	C₄×D₄	—	251	4	no	—
196	C₄×D₄	—	251	4	no	—
197	C₄×D₄	—	251	2	no	—
198	#117	—	251	4	no	—
199	C₂×C₂²⋊C₄	—	251	4	no	—
200	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic, complemented
201	C₂⁴	—	251	4	no	—
202	C₂⁴	C₂²×D₂₅₁	1	4	yes	—
203	#41	—	1004	8	no	—
204	C₂×Q₈	—	251	2	no	—
205	C₂×D₄	—	4	8	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	1004	8	no	—
208	C₂×D₄	—	1004	8	no	—
209	C₂×D₄	—	251	8	no	—
210	C₂×D₄	—	251	8	no	—
211	C₂×D₄	—	251	8	no	—
212	C₂×D₄	—	251	4	no	—
213	C₂×D₄	—	251	2	no	—
214	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
215	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
216	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
217	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
218	C₂×D₄	C₂²×D₂₅₁	1	8	yes	—
219	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
220	C₂²×C₄	—	251	8	no	—
221	C₂²×C₄	—	251	4	no	—
222	C₂²×C₄	—	251	4	no	—
223	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
224	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
225	C₄⋊C₄	—	251	4	no	—
226	C₄⋊C₄	—	251	4	no	—
227	C₄⋊C₄	—	251	4	no	—
228	C₄⋊C₄	—	251	4	no	—
229	C₄⋊C₄	—	251	2	no	—
230	C₄⋊C₄	—	251	2	no	—
231	C₂²⋊C₄	—	251	4	no	—
232	C₂²⋊C₄	—	251	4	no	—
233	C₂²⋊C₄	—	251	4	no	—
234	C₂²⋊C₄	—	251	8	no	—
235	C₂²⋊C₄	—	251	8	no	—
236	C₂²⋊C₄	—	251	4	no	—
237	C₂²⋊C₄	—	251	4	no	—
238	C₂²⋊C₄	—	251	4	no	—
239	C₂²⋊C₄	—	251	4	no	—
240	C₄²	—	251	4	no	—
241	C₄²	C₂²×D₂₅₁	1	2	yes	—
242	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
243	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
244	C₂³	—	2	8	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	1004	4	no	—
247	C₂³	—	1004	8	no	—
248	C₂³	—	251	4	no	—
249	C₂³	C₂³×D₂₅₁	1	4	yes	—
250	C₂³	C₂³×D₂₅₁	1	4	yes	—
251	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
252	Q₈	—	502	4	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	1004	8	no	—
260	D₄	—	1004	8	no	—
261	D₄	—	1004	8	no	—
262	D₄	—	502	8	no	—
263	D₄	—	502	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	1004	4	no	—
267	C₂×C₄	—	1004	8	no	—
268	C₂×C₄	—	502	8	no	—
269	C₂×C₄	—	251	4	no	—
270	C₂×C₄	—	251	4	no	—
271	C₂×C₄	—	251	4	no	—
272	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
274	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	8	8	no	—
282	C₂²	—	2008	8	no	—
283	C₂²	—	502	4	no	—
284	C₂²	—	502	4	no	—
285	C₂²	—	502	4	no	—
286	C₂²	#22548760	1	2	yes	—
287	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	502	4	no	—
292	C₄	—	502	4	no	—
293	C₄	—	502	4	no	—
294	C₄	#22548760	1	4	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	4	4	no	—
298	C₂	—	1004	4	no	—
299	C₂	C₂×#22548760	1	1	yes	characteristic
300	C₂	C₂×#22548760	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁵
Fitting	C₂×#127×C₂₅₁	C₂
hypercenter	C₂×#127	D₂₅₁
hypocenter	C₂₅₁	#3543
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #30632424	C₂	semidirect
2. #30632423	C₂	semidirect
3. #30632357	C₂	semidirect
4. #30632331	C₂	semidirect
5. #30632322	C₂	semidirect
6. D₄×C₂²⋊D₂₅₁	C₂	semidirect
7. #30632291	C₂	semidirect
8. #30632250	C₂	semidirect
9. #30632246	C₂	semidirect
10. #30632221	C₂	semidirect
11. C₂×#127×C₂₅₁	C₂	semidirect
12. C₂×#16860809	C₂²	semidirect
13. C₂²×C₂²⋊D₂₅₁	C₂²	semidirect
14. C₂×D₄×D₂₅₁	C₂²	semidirect
15. #22548711	C₂²	semidirect
16. #22548709	C₂²	semidirect
17. #22548707	C₂²	semidirect
18. #22548703	C₂²	semidirect
19. #22548702	C₂²	semidirect
20. #22548698	C₂²	semidirect
21. #22548696	C₂²	semidirect
22. #22548696	C₂²	semidirect
23. #22548696	C₂²	semidirect
24. #22548694	C₂²	semidirect
26. #22548687	C₂²	semidirect
27. #22548687	C₂²	semidirect
28. #22548687	C₂²	semidirect
29. #22548687	C₂²	semidirect
30. #22548686	C₂²	semidirect
31. #22548686	C₂²	semidirect
32. #22548684	C₂²	semidirect
33. #22548683	C₂²	semidirect
34. #22548683	C₂²	semidirect
35. #22548683	C₂²	semidirect
36. C₄×Dic₅₀₂	C₂²	semidirect
37. C₄×D₁₀₀₄	C₂²	semidirect
38. D₄×Dic₂₅₁	C₂²	semidirect
39. D₄×Dic₂₅₁	C₂²	semidirect
40. C₄×C₂²⋊D₂₅₁	C₂²	semidirect
41. #22548672	C₂²	semidirect
42. C₂×C₂²⋊Dic₂₅₁	C₂²	semidirect
43. C₂²×D₄×C₂₅₁	C₂²	semidirect
44. C₂²×D₄×C₂₅₁	C₂²	semidirect
46. #127×C₂₅₁	C₂²	semidirect
47. #127×C₂₅₁	C₂²	semidirect
48. #127×C₂₅₁	C₂²	semidirect
49. C₂×C₄²×C₂₅₁	C₂²	semidirect
150. C₂₅₁	#3543	semidirect
161. C₂×#127	D₂₅₁	semidirect
166. C₂²×D₄	C₂×D₂₅₁	semidirect
167. C₂²×D₄	C₂×D₂₅₁	semidirect
170. #127	C₂×D₂₅₁	semidirect
171. #127	C₂×D₂₅₁	semidirect
172. #127	C₂×D₂₅₁	semidirect
200. C₂×C₄²	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632424	2
#30632423	2
#30632357	2
#30632331	2
#30632322	2
D₄×C₂²⋊D₂₅₁	2
#30632291	2
#30632250	2
#30632246	2
#30632221	2
C₂×#127×C₂₅₁	2
#3543	251