Subgroups — #44560857

Order 32128.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548747	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548744	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22548736	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#30632262	C₂	1	1	yes	maximal, characteristic, complemented
6	#30632257	C₂	1	1	yes	maximal, characteristic, complemented
7	#30632081	C₂	1	1	yes	maximal, characteristic, complemented
8	#30632080	C₂	1	1	yes	maximal, characteristic, complemented
9	#30632078	C₂	1	1	yes	maximal, characteristic, complemented
10	#30632069	C₂	1	1	yes	maximal, characteristic, complemented
11	#30632068	C₂	1	1	yes	maximal, characteristic, complemented
12	#30632066	C₂	1	1	yes	maximal, characteristic, complemented
13	#30632065	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631644	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#16860811	C₂²	1	1	yes	characteristic
17	C₂×#16860809	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
20	#22548747	—	2	4	no	—
21	#22548744	—	2	4	no	—
22	#22548736	—	2	4	no	—
23	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
25	C₂×#16860800	C₂²	1	1	yes	characteristic
26	C₂×#16860800	C₂²	1	1	yes	characteristic
27	C₂×#16860800	C₂²	1	1	yes	characteristic
28	C₂×#16860800	C₂²	1	1	yes	characteristic
29	C₂×#16860800	C₂²	1	1	yes	characteristic
30	Q₁₆×D₂₅₁	—	2	4	no	—
31	C₂×#16860799	C₂²	1	1	yes	characteristic
32	C₂×#16860799	C₂²	1	1	yes	characteristic
33	C₂×#16860798	C₂²	1	1	yes	characteristic
34	C₂×#16860797	C₂²	1	1	yes	characteristic
35	C₂×#16860795	C₂²	1	1	yes	characteristic
36	C₂×#16860793	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
38	#22548715	C₂²	1	1	yes	characteristic
39	#22548713	C₂²	1	1	yes	characteristic
40	#22548713	C₂²	1	1	yes	characteristic
41	#22548704	C₂²	1	1	yes	characteristic
42	#22548703	C₂²	1	1	yes	characteristic
43	#22548702	C₂²	1	1	yes	characteristic
44	#22548700	—	2	2	no	—
45	#22548697	—	2	2	no	—
46	#22548693	—	2	2	no	—
47	#22548692	—	2	2	no	—
48	#22548672	C₂²	1	1	yes	characteristic
49	#22548625	C₂²	1	1	yes	characteristic
50	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
57	#124×C₂₅₁	C₂²	1	1	yes	characteristic
58	#97×C₂₅₁	C₂²	1	1	yes	characteristic
59	#16860811	—	2	2	no	—
60	#16860811	—	2	2	no	—
61	#16860809	—	2	2	no	—
62	#16860809	—	2	2	no	—
63	C₂×Dic₅₀₂	—	2	1	no	—
64	C₂×Dic₅₀₂	—	2	1	no	—
65	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
66	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
67	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
68	Q₈×D₂₅₁	—	2	2	no	—
69	Q₈×D₂₅₁	—	2	2	no	—
70	Q₈×D₂₅₁	—	2	2	no	—
71	Q₈×D₂₅₁	—	2	2	no	—
72	C₂×D₁₀₀₄	—	2	1	no	—
73	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
75	C₂²×Dic₂₅₁	—	2	1	no	—
76	C₂×C₄×D₂₅₁	—	2	1	no	—
77	C₂×C₄×D₂₅₁	—	2	1	no	—
78	C₂×C₄×D₂₅₁	—	2	1	no	—
79	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
80	Dic₁₀₀₄	—	2	2	no	—
81	Dic₁₀₀₄	—	2	2	no	—
82	#16860800	—	2	2	no	—
83	#16860800	—	2	2	no	—
84	#16860800	—	2	2	no	—
85	#16860800	—	2	2	no	—
86	#16860800	—	2	2	no	—
87	#16860799	—	2	2	no	—
88	#16860799	—	2	2	no	—
89	#16860798	—	2	2	no	—
90	#16860797	—	2	2	no	—
91	#16860795	—	2	2	no	—
92	#16860793	D₄	1	4	yes	—
93	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
94	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
95	C₈×D₂₅₁	D₄	1	4	yes	—
96	C₄⋊Dic₂₅₁	—	2	2	no	—
97	Dic₂₅₁⋊C₄	—	2	2	no	—
98	Dic₂₅₁⋊C₄	—	2	2	no	—
99	Dic₂₅₁⋊C₄	D₄	1	2	yes	—
100	C₂²⋊Dic₂₅₁	—	2	2	no	—
101	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
102	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
103	(C₂×D₂₅₁)⋊C₄	D₄	1	2	yes	—
104	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
106	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
109	Q₁₆×C₂₅₁	—	2	2	no	—
110	Q₁₆×C₂₅₁	—	2	2	no	—
111	SD₁₆×C₂₅₁	—	2	2	no	—
112	SD₁₆×C₂₅₁	—	2	2	no	—
113	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₂₅₁	—	2	2	no	—
116	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
117	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
118	C₂²×D₂₅₁	—	2	1	no	—
119	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
120	Dic₅₀₂	—	2	1	no	—
121	Dic₅₀₂	—	2	1	no	—
122	Dic₅₀₂	—	2	1	no	—
123	Dic₅₀₂	—	2	1	no	—
124	Dic₅₀₂	—	2	1	no	—
125	Dic₅₀₂	—	2	1	no	—
126	Dic₅₀₂	—	4	2	no	—
127	Dic₅₀₂	—	4	2	no	—
128	D₁₀₀₄	—	2	1	no	—
129	D₁₀₀₄	—	2	1	no	—
130	D₁₀₀₄	—	4	2	no	—
131	C₂²⋊D₂₅₁	—	4	2	no	—
132	C₂×Dic₂₅₁	—	2	1	no	—
133	C₂×Dic₂₅₁	—	2	1	no	—
134	C₂×Dic₂₅₁	—	2	1	no	—
135	C₂×Dic₂₅₁	—	4	2	no	—
136	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
137	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
138	C₄×D₂₅₁	—	4	2	no	—
139	C₄×D₂₅₁	—	4	2	no	—
140	C₄×D₂₅₁	—	4	2	no	—
141	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
142	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
143	C₂₅₁⋊C₈	C₂×D₄	1	2	yes	—
144	C₂₅₁⋊C₈	C₂×D₄	1	2	yes	—
145	C₂³×C₂₅₁	—	2	1	no	—
146	Q₈×C₂₅₁	—	2	1	no	—
147	Q₈×C₂₅₁	—	2	1	no	—
148	Q₈×C₂₅₁	—	2	1	no	—
149	Q₈×C₂₅₁	—	2	1	no	—
150	Q₈×C₂₅₁	—	2	1	no	—
151	Q₈×C₂₅₁	—	2	1	no	—
152	D₄×C₂₅₁	—	2	1	no	—
153	D₄×C₂₅₁	—	2	1	no	—
154	C₂×C₄×C₂₅₁	—	2	1	no	—
155	C₂×C₄×C₂₅₁	—	2	1	no	—
156	C₂×C₄×C₂₅₁	—	2	1	no	—
157	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
158	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
159	C₈×C₂₅₁	C₂×D₄	1	2	yes	—
160	C₈×C₂₅₁	C₂×D₄	1	2	yes	—
161	C₂²×C₂₅₁	—	4	1	no	—
162	C₂²×C₂₅₁	—	4	2	no	—
163	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₂₅₁	—	2	2	no	—
165	C₂×D₂₅₁	—	4	1	no	—
166	C₂×D₂₅₁	—	4	2	no	—
167	C₂×D₂₅₁	#127	1	2	yes	—
168	C₄×C₂₅₁	—	2	2	no	—
169	C₄×C₂₅₁	—	4	1	no	—
170	C₄×C₂₅₁	—	4	1	no	—
171	C₄×C₂₅₁	—	4	1	no	—
172	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
174	Dic₂₅₁	—	2	2	no	—
175	Dic₂₅₁	—	4	1	no	—
176	Dic₂₅₁	—	4	1	no	—
177	Dic₂₅₁	—	4	1	no	—
178	Dic₂₅₁	#127	1	2	yes	—
179	C₂×C₂₅₁	—	8	1	no	—
180	C₂×C₂₅₁	C₂×#127	1	1	yes	characteristic
181	C₂×C₂₅₁	#578	1	2	yes	—
182	D₂₅₁	—	2	2	no	—
183	D₂₅₁	—	8	1	no	—
184	C₂₅₁	#3156	1	1	yes	characteristic, complemented, Sylow
185	#3156	—	251	1	no	maximal, Sylow
186	C₂×#137	—	251	1	no	—
187	C₂×#137	—	251	1	no	—
188	C₂×#135	—	251	1	no	—
189	C₂²×Q₁₆	—	251	1	no	—
190	#536	—	251	1	no	—
191	#536	—	251	1	no	—
192	#497	—	251	1	no	—
193	#497	—	251	1	no	—
194	#497	—	251	1	no	—
195	#497	D₂₅₁	1	1	yes	characteristic, complemented
196	#495	—	251	1	no	—
197	#495	—	251	1	no	—
198	#494	—	251	1	no	—
199	#494	—	251	1	no	—
200	#405	—	251	1	no	—
201	C₂×#41	—	251	1	no	—
202	C₂×#41	—	251	1	no	—
203	C₂²×Q₈	—	251	1	no	—
204	C₂²×Q₈	—	251	1	no	—
205	#137	—	502	4	no	—
206	#137	—	502	4	no	—
207	#135	—	502	4	no	—
208	C₂×Q₁₆	—	502	4	no	—
209	C₂×Q₁₆	—	251	1	no	—
210	C₂×Q₁₆	—	251	1	no	—
211	C₂×Q₁₆	—	251	1	no	—
212	C₂×Q₁₆	—	251	1	no	—
213	C₂×Q₁₆	—	251	1	no	—
214	C₂×Q₁₆	—	251	1	no	—
215	C₂×Q₁₆	—	251	1	no	—
216	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
217	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
218	C₂×SD₁₆	—	251	1	no	—
219	C₂×SD₁₆	—	251	1	no	—
220	C₂×SD₁₆	—	251	1	no	—
221	C₂×SD₁₆	—	251	1	no	—
222	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
223	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
224	C₂×D₈	—	251	1	no	—
225	C₂×M₁₆	—	251	1	no	—
226	C₂²×C₈	—	251	1	no	—
227	C₄⋊Q₈	—	251	1	no	—
228	C₄⋊Q₈	—	251	1	no	—
229	C₄⋊Q₈	—	251	1	no	—
230	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
231	#124	—	251	1	no	—
232	#124	—	251	1	no	—
233	#124	—	251	1	no	—
234	#124	C₂×D₂₅₁	1	1	yes	characteristic
235	#123	—	502	2	no	—
236	#123	—	502	2	no	—
237	C₂²⋊Q₈	—	502	2	no	—
238	C₂²⋊Q₈	—	502	2	no	—
239	#117	—	251	1	no	—
240	#97	—	251	1	no	—
241	#97	C₂×D₂₅₁	1	1	yes	characteristic
242	C₄×C₈	—	251	1	no	—
243	C₄×C₈	—	251	1	no	—
244	#41	—	502	2	no	—
245	#41	—	502	2	no	—
246	#41	—	502	2	no	—
247	#41	—	502	2	no	—
248	C₂×Q₈	—	502	1	no	—
249	C₂×Q₈	—	502	1	no	—
250	C₂×Q₈	—	502	2	no	—
251	C₂×Q₈	—	502	2	no	—
252	C₂×Q₈	—	502	2	no	—
253	C₂×Q₈	—	502	2	no	—
254	C₂×Q₈	—	251	1	no	—
255	C₂×Q₈	—	251	1	no	—
256	C₂×Q₈	—	251	1	no	—
257	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
260	C₂×D₄	—	502	1	no	—
261	C₂×D₄	—	502	1	no	—
262	C₂×D₄	—	251	1	no	—
263	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
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	—	251	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	502	2	no	—
272	Q₁₆	—	502	2	no	—
273	Q₁₆	—	502	2	no	—
274	Q₁₆	—	502	2	no	—
275	Q₁₆	—	502	2	no	—
276	Q₁₆	—	502	2	no	—
277	Q₁₆	—	502	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	502	2	no	—
281	SD₁₆	—	502	2	no	—
282	SD₁₆	—	502	2	no	—
283	SD₁₆	—	502	2	no	—
284	D₈	—	502	2	no	—
285	M₁₆	—	251	4	no	—
286	C₂×C₈	—	251	1	no	—
287	C₂×C₈	—	251	1	no	—
288	C₂×C₈	—	251	4	no	—
289	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	502	2	no	—
293	C₄⋊C₄	—	502	2	no	—
294	C₄⋊C₄	—	502	2	no	—
295	C₄⋊C₄	—	251	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	502	2	no	—
298	C₂²⋊C₄	—	502	2	no	—
299	C₂²⋊C₄	—	502	2	no	—
300	C₂²⋊C₄	—	251	2	no	—
301	C₄²	—	251	1	no	—
302	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	502	1	no	—
305	C₂³	—	251	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	1004	2	no	—
313	Q₈	—	1004	2	no	—
314	Q₈	—	502	1	no	—
315	Q₈	—	502	1	no	—
316	Q₈	—	502	1	no	—
317	Q₈	—	502	1	no	—
318	Q₈	—	502	1	no	—
319	Q₈	—	502	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	1004	2	no	—
323	D₄	—	1004	2	no	—
324	D₄	—	502	1	no	—
325	D₄	—	502	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	1004	2	no	—
330	C₂×C₄	—	1004	2	no	—
331	C₂×C₄	—	1004	2	no	—
332	C₂×C₄	—	1004	2	no	—
333	C₂×C₄	—	502	1	no	—
334	C₂×C₄	—	502	1	no	—
335	C₂×C₄	—	502	1	no	—
336	C₂×C₄	—	251	2	no	—
337	C₂×C₄	—	251	1	no	—
338	C₂×C₄	—	251	2	no	—
339	C₂×C₄	—	251	2	no	—
340	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
341	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
342	C₈	—	251	2	no	—
343	C₈	—	251	2	no	—
344	C₈	D₄×D₂₅₁	1	2	yes	—
345	C₈	D₄×D₂₅₁	1	2	yes	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	1004	1	no	—
349	C₂²	—	1004	2	no	—
350	C₂²	—	502	2	no	—
351	C₂²	—	251	2	no	—
352	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	1004	1	no	—
358	C₄	—	1004	1	no	—
359	C₄	—	1004	1	no	—
360	C₄	—	502	2	no	—
361	C₄	—	251	2	no	—
362	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
363	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	2008	1	no	—
366	C₂	—	502	2	no	—
367	C₂	#127×D₂₅₁	1	1	yes	characteristic
368	C₂	#30632503	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#497×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548736	C₂
hypercenter	#497	D₂₅₁
hypocenter	C₂₅₁	#3156
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#22548747	C₂	semidirect
2. C₂×#22548744	C₂	semidirect
4. C₂×Q₁₆×D₂₅₁	C₂	semidirect
5. #30632262	C₂	semidirect
6. #30632257	C₂	semidirect
7. #30632081	C₂	semidirect
8. #30632080	C₂	semidirect
9. #30632078	C₂	semidirect
10. #30632069	C₂	semidirect
11. #30632068	C₂	semidirect
12. #30632066	C₂	semidirect
13. #30632065	C₂	semidirect
14. #30631644	C₂	semidirect
15. #497×C₂₅₁	C₂	semidirect
184. C₂₅₁	#3156	semidirect
195. #497	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548747	2
C₂×#22548744	2
C₂×#22548736	2
C₂×Q₁₆×D₂₅₁	2
#30632262	2
#30632257	2
#30632081	2
#30632080	2
#30632078	2
#30632069	2
#30632068	2
#30632066	2
#30632065	2
#30631644	2
#497×C₂₅₁	2
#3156	251