Subgroups — #44563080

Order 32128.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632406	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632352	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632045	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632036	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631980	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631979	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631965	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631901	C₂	1	1	yes	maximal, characteristic
9	#30631899	C₂	1	1	yes	maximal, characteristic
10	#30631891	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631815	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631790	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631789	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631767	C₂	1	1	yes	maximal, characteristic
15	#489×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₅₁	—	2	1	no	—
17	C₂×#16860799	C₂²	1	1	yes	characteristic
18	C₂×#16860797	C₂²	1	1	yes	characteristic
19	C₂×#16860795	C₂²	1	1	yes	characteristic
20	#22548714	C₂²	1	1	yes	characteristic
21	#22548714	C₂²	1	1	yes	characteristic
22	#22548713	C₂²	1	1	yes	characteristic
23	#22548708	—	2	1	no	—
24	#22548708	—	2	1	no	—
25	#22548702	C₂²	1	1	yes	characteristic
26	#22548693	—	2	1	no	—
27	#22548693	—	2	1	no	—
28	#22548692	—	2	1	no	—
29	#22548691	—	2	1	no	—
30	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
31	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
33	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
34	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
35	#22548677	C₂²	1	1	yes	characteristic
36	#22548677	C₂²	1	1	yes	characteristic
37	#22548671	—	2	1	no	—
38	#22548649	C₂²	1	1	yes	characteristic
39	#22548648	C₂²	1	1	yes	characteristic
40	#22548648	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
44	#22548640	C₂²	1	1	yes	characteristic
45	#22548639	C₂²	1	1	yes	characteristic
46	#22548638	C₂²	1	1	yes	characteristic
47	#22548638	C₂²	1	1	yes	characteristic
48	#22548637	C₂²	1	1	yes	characteristic
49	#22548636	C₂²	1	1	yes	characteristic
50	#22548625	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
53	#124×C₂₅₁	C₂²	1	1	yes	characteristic
54	#103×C₂₅₁	C₂²	1	1	yes	characteristic
55	#103×C₂₅₁	C₂²	1	1	yes	characteristic
56	#102×C₂₅₁	C₂²	1	1	yes	characteristic
57	#102×C₂₅₁	C₂²	1	1	yes	characteristic
58	#97×C₂₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₅₀₂	—	2	1	no	—
60	C₂×Dic₅₀₂	—	2	1	no	—
61	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
62	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
63	Q₈×D₂₅₁	—	4	4	no	—
64	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
65	C₂²×Dic₂₅₁	—	2	1	no	—
66	C₂×C₄×D₂₅₁	—	2	1	no	—
67	C₂×C₄×D₂₅₁	—	2	1	no	—
68	C₂×C₄×D₂₅₁	—	2	1	no	—
69	#16860799	—	2	2	no	—
70	#16860797	—	2	2	no	—
71	#16860795	—	2	2	no	—
72	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₂₅₁	—	2	1	no	—
75	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
78	Dic₂₅₁⋊C₄	—	2	1	no	—
79	Dic₂₅₁⋊C₄	—	2	1	no	—
80	Dic₂₅₁⋊C₄	—	2	1	no	—
81	Dic₂₅₁⋊C₄	—	2	1	no	—
82	Dic₂₅₁⋊C₄	—	2	1	no	—
83	Dic₂₅₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₂₅₁⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₂₅₁	—	2	1	no	—
86	C₂²⋊Dic₂₅₁	—	2	1	no	—
87	C₂²⋊Dic₂₅₁	—	2	1	no	—
88	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
89	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
90	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
91	C₄×Dic₂₅₁	—	2	1	no	—
92	C₄×Dic₂₅₁	—	2	1	no	—
93	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
94	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₂₅₁	—	2	1	no	—
96	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₂₅₁	—	2	1	no	—
101	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
104	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
105	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
106	C₂²×D₂₅₁	—	2	1	no	—
107	Dic₅₀₂	—	2	2	no	—
108	Dic₅₀₂	—	2	1	no	—
109	Dic₅₀₂	—	2	1	no	—
110	Dic₅₀₂	—	4	2	no	—
111	Dic₅₀₂	—	4	2	no	—
112	D₁₀₀₄	—	2	1	no	—
113	D₁₀₀₄	#41	1	2	yes	—
114	C₂×Dic₂₅₁	—	2	1	no	—
115	C₂×Dic₂₅₁	—	2	1	no	—
116	C₂×Dic₂₅₁	—	2	1	no	—
117	C₂×Dic₂₅₁	—	4	2	no	—
118	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₂₅₁	—	4	2	no	—
123	C₄×D₂₅₁	—	4	2	no	—
124	C₄×D₂₅₁	—	4	2	no	—
125	C₂₅₁⋊C₈	—	2	1	no	—
126	C₂₅₁⋊C₈	—	2	1	no	—
127	C₂³×C₂₅₁	—	2	1	no	—
128	Q₈×C₂₅₁	—	2	1	no	—
129	Q₈×C₂₅₁	—	2	1	no	—
130	Q₈×C₂₅₁	—	4	2	no	—
131	D₄×C₂₅₁	—	2	1	no	—
132	D₄×C₂₅₁	—	2	1	no	—
133	C₂×C₄×C₂₅₁	—	2	1	no	—
134	C₂×C₄×C₂₅₁	—	2	1	no	—
135	C₂×C₄×C₂₅₁	—	2	1	no	—
136	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
139	C₈×C₂₅₁	—	2	1	no	—
140	C₈×C₂₅₁	—	2	1	no	—
141	C₂²×C₂₅₁	—	4	1	no	—
142	C₂²×C₂₅₁	—	4	1	no	—
143	C₂²×C₂₅₁	—	4	1	no	—
144	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₂₅₁	—	2	2	no	—
146	C₂×D₂₅₁	—	4	1	no	—
147	C₂×D₂₅₁	—	4	1	no	—
148	C₄×C₂₅₁	—	2	1	no	—
149	C₄×C₂₅₁	—	2	1	no	—
150	C₄×C₂₅₁	—	4	1	no	—
151	C₄×C₂₅₁	—	4	1	no	—
152	C₄×C₂₅₁	—	4	1	no	—
153	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
154	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₂₅₁	—	2	1	no	—
156	Dic₂₅₁	—	2	1	no	—
157	Dic₂₅₁	—	2	1	no	—
158	Dic₂₅₁	—	4	1	no	—
159	Dic₂₅₁	—	4	1	no	—
160	Dic₂₅₁	—	4	1	no	—
161	Dic₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₂₅₁	—	8	1	no	—
163	C₂×C₂₅₁	#548	1	1	yes	characteristic
164	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₂₅₁	#578	1	1	yes	characteristic
166	D₂₅₁	—	4	2	no	—
167	C₂₅₁	#3367	1	1	yes	characteristic, complemented, Sylow
168	#3367	—	251	1	no	maximal, Sylow
169	#556	—	251	1	no	—
170	D₄×Q₈	—	251	1	no	—
171	#489	—	251	1	no	—
172	#489	D₂₅₁	1	1	yes	characteristic, complemented
173	#487	—	251	1	no	—
174	#477	—	251	1	no	—
175	#477	—	251	1	no	—
176	#474	—	251	1	no	—
177	#463	—	251	1	no	—
178	#463	—	251	1	no	—
179	#461	—	251	1	no	—
180	#446	—	251	1	no	—
181	#440	—	251	1	no	—
182	#440	—	251	1	no	—
183	C₄×D₈	—	251	1	no	—
184	C₂²×Q₈	—	502	1	no	—
185	C₂×SD₁₆	—	251	1	no	—
186	C₂×SD₁₆	—	251	1	no	—
187	C₂×D₈	—	251	1	no	—
188	C₄⋊Q₈	—	251	1	no	—
189	C₄⋊Q₈	—	251	1	no	—
190	C₄⋊Q₈	—	251	1	no	—
191	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
192	#126	—	502	1	no	—
193	#126	—	502	1	no	—
194	#124	—	251	1	no	—
195	#124	C₂×D₂₅₁	1	1	yes	characteristic
196	C₂²⋊Q₈	—	502	1	no	—
197	C₂²⋊Q₈	—	502	1	no	—
198	C₂²⋊Q₈	—	502	1	no	—
199	C₂²⋊Q₈	—	502	1	no	—
200	C₄×Q₈	—	251	1	no	—
201	C₄×Q₈	—	251	1	no	—
202	C₄×Q₈	—	251	1	no	—
203	C₄×D₄	—	251	1	no	—
204	C₄×D₄	—	251	1	no	—
205	C₄×D₄	—	251	1	no	—
206	C₄×D₄	—	251	1	no	—
207	#117	—	502	1	no	—
208	#107	—	251	1	no	—
209	#107	—	251	1	no	—
210	#107	—	251	1	no	—
211	C₄⋊C₈	—	251	1	no	—
212	C₄⋊C₈	—	251	1	no	—
213	C₄⋊C₈	—	251	1	no	—
214	#103	—	251	1	no	—
215	#103	—	251	1	no	—
216	#103	—	251	1	no	—
217	#103	—	251	1	no	—
218	#103	C₂×D₂₅₁	1	1	yes	characteristic
219	#103	C₂×D₂₅₁	1	1	yes	characteristic
220	#102	—	251	1	no	—
221	#102	—	251	1	no	—
222	#102	C₂×D₂₅₁	1	1	yes	characteristic
223	#102	C₂×D₂₅₁	1	1	yes	characteristic
224	#97	—	251	1	no	—
225	#97	C₂×D₂₅₁	1	1	yes	characteristic
226	C₄×C₈	—	251	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	1004	4	no	—
229	C₂×Q₈	—	502	1	no	—
230	C₂×Q₈	—	502	1	no	—
231	C₂×Q₈	—	251	1	no	—
232	C₂×Q₈	—	251	1	no	—
233	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
234	C₂×D₄	—	251	1	no	—
235	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
236	C₂²×C₄	—	502	1	no	—
237	C₂²×C₄	—	502	1	no	—
238	C₂²×C₄	—	502	1	no	—
239	C₂²×C₄	—	502	1	no	—
240	SD₁₆	—	502	2	no	—
241	SD₁₆	—	502	2	no	—
242	D₈	—	502	2	no	—
243	C₂×C₈	—	251	1	no	—
244	C₂×C₈	—	251	1	no	—
245	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	502	1	no	—
249	C₄⋊C₄	—	502	1	no	—
250	C₄⋊C₄	—	502	1	no	—
251	C₄⋊C₄	—	502	1	no	—
252	C₄⋊C₄	—	502	1	no	—
253	C₄⋊C₄	—	502	1	no	—
254	C₄⋊C₄	—	251	1	no	—
255	C₄⋊C₄	—	251	1	no	—
256	C₄⋊C₄	—	251	1	no	—
257	C₄⋊C₄	—	251	1	no	—
258	C₄⋊C₄	—	251	1	no	—
259	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	502	1	no	—
264	C₂²⋊C₄	—	502	1	no	—
265	C₂²⋊C₄	—	502	1	no	—
266	C₂²⋊C₄	—	502	1	no	—
267	C₂²⋊C₄	—	502	1	no	—
268	C₂²⋊C₄	—	502	1	no	—
269	C₄²	—	502	1	no	—
270	C₄²	—	502	1	no	—
271	C₄²	—	251	1	no	—
272	C₄²	—	251	1	no	—
273	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	502	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	1004	2	no	—
280	Q₈	—	1004	2	no	—
281	Q₈	—	502	2	no	—
282	Q₈	—	502	1	no	—
283	Q₈	—	502	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	502	1	no	—
287	D₄	—	251	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	1004	2	no	—
292	C₂×C₄	—	1004	2	no	—
293	C₂×C₄	—	1004	2	no	—
294	C₂×C₄	—	1004	2	no	—
295	C₂×C₄	—	502	1	no	—
296	C₂×C₄	—	502	1	no	—
297	C₂×C₄	—	502	1	no	—
298	C₂×C₄	—	251	1	no	—
299	C₂×C₄	—	251	1	no	—
300	C₂×C₄	—	251	1	no	—
301	C₂×C₄	—	251	1	no	—
302	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
303	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	502	1	no	—
308	C₈	—	502	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	1004	1	no	—
313	C₂²	—	1004	1	no	—
314	C₂²	—	502	2	no	—
315	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	1004	1	no	—
322	C₄	—	1004	1	no	—
323	C₄	—	1004	1	no	—
324	C₄	—	502	1	no	—
325	C₄	—	502	1	no	—
326	C₄	—	502	1	no	—
327	C₄	—	251	2	no	—
328	C₄	#41×D₂₅₁	1	1	yes	characteristic
329	C₄	#22548762	1	1	yes	characteristic
330	C₂	—	8	1	no	—
331	C₂	—	1004	2	no	—
332	C₂	#30632345	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₂₅₁	1	1	yes	characteristic
334	C₂	#30632503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#489×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860795	C₂²
hypercenter	#489	D₂₅₁
hypocenter	C₂₅₁	#3367
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #30632406	C₂	semidirect
2. #30632352	C₂	semidirect
3. #30632045	C₂	semidirect
4. #30632036	C₂	semidirect
5. #30631980	C₂	semidirect
6. #30631979	C₂	semidirect
7. #30631965	C₂	semidirect
10. #30631891	C₂	semidirect
11. #30631815	C₂	semidirect
12. #30631790	C₂	semidirect
13. #30631789	C₂	semidirect
15. #489×C₂₅₁	C₂	semidirect
167. C₂₅₁	#3367	semidirect
172. #489	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632406	2
#30632352	2
#30632045	2
#30632036	2
#30631980	2
#30631979	2
#30631965	2
#30631901	2
#30631899	2
#30631891	2
#30631815	2
#30631790	2
#30631789	2
#30631767	2
#489×C₂₅₁	2
#3367	251