Subgroups — #44559890

Order 32128.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 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
2	C₂×Q₁₆×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632262	C₂	1	1	yes	maximal, characteristic
4	#30632163	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631912	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631911	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631910	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631909	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631896	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631895	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631894	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631712	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631694	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631682	C₂	1	1	yes	maximal, characteristic
15	#462×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#16860811	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
19	#22548747	—	2	4	no	—
20	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
21	C₂×#16860800	C₂²	1	1	yes	characteristic
22	C₂×#16860800	C₂²	1	1	yes	characteristic
23	C₂×#16860800	C₂²	1	1	yes	characteristic
24	Q₁₆×D₂₅₁	—	2	4	no	—
25	C₂×#16860799	C₂²	1	1	yes	characteristic
26	C₂×#16860797	C₂²	1	1	yes	characteristic
27	C₂×#16860793	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
29	#22548713	C₂²	1	1	yes	characteristic
30	#22548704	C₂²	1	1	yes	characteristic
31	#22548703	C₂²	1	1	yes	characteristic
32	#22548700	—	2	1	no	—
33	#22548700	—	2	1	no	—
34	#22548693	—	2	1	no	—
35	#22548693	—	2	1	no	—
36	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
37	C₄×Dic₅₀₂	—	2	1	no	—
38	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
39	#22548680	—	2	1	no	—
40	#22548680	C₂²	1	1	yes	characteristic
41	#22548673	—	2	1	no	—
42	#22548673	C₂²	1	1	yes	characteristic
43	#22548672	—	2	1	no	—
44	#22548672	C₂²	1	1	yes	characteristic
45	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
46	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
48	#22548640	C₂²	1	1	yes	characteristic
49	#22548639	C₂²	1	1	yes	characteristic
50	#22548639	C₂²	1	1	yes	characteristic
51	#22548638	C₂²	1	1	yes	characteristic
52	#22548637	C₂²	1	1	yes	characteristic
53	#22548635	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
55	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
59	#103×C₂₅₁	C₂²	1	1	yes	characteristic
60	#103×C₂₅₁	C₂²	1	1	yes	characteristic
61	#16860811	—	2	2	no	—
62	#16860811	—	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	Q₈×D₂₅₁	—	2	2	no	—
68	Q₈×D₂₅₁	—	2	2	no	—
69	Q₈×D₂₅₁	—	2	2	no	—
70	Q₈×D₂₅₁	D₄	1	4	yes	—
71	C₂×D₁₀₀₄	—	2	1	no	—
72	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₂₅₁	—	2	1	no	—
74	C₂×C₄×D₂₅₁	—	2	1	no	—
75	C₂×C₄×D₂₅₁	—	2	1	no	—
76	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
77	Dic₁₀₀₄	—	2	2	no	—
78	#16860800	—	2	2	no	—
79	#16860800	—	2	2	no	—
80	#16860800	—	2	2	no	—
81	#16860799	—	2	2	no	—
82	#16860797	—	2	2	no	—
83	#16860793	—	2	2	no	—
84	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
85	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
86	C₈×D₂₅₁	—	2	2	no	—
87	C₄⋊Dic₂₅₁	—	2	1	no	—
88	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
89	Dic₂₅₁⋊C₄	—	2	1	no	—
90	Dic₂₅₁⋊C₄	—	2	1	no	—
91	Dic₂₅₁⋊C₄	—	2	1	no	—
92	Dic₂₅₁⋊C₄	—	2	1	no	—
93	Dic₂₅₁⋊C₄	D₄	1	1	yes	characteristic
94	Dic₂₅₁⋊C₄	D₄	1	1	yes	characteristic
95	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
96	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
97	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
98	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
99	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
100	(C₂×D₂₅₁)⋊C₄	D₄	1	1	yes	characteristic
101	(C₂×D₂₅₁)⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₂₅₁	—	2	1	no	—
103	C₄×Dic₂₅₁	—	2	1	no	—
104	C₄×Dic₂₅₁	—	2	1	no	—
105	C₄×Dic₂₅₁	—	2	1	no	—
106	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
110	Q₁₆×C₂₅₁	—	2	2	no	—
111	Q₁₆×C₂₅₁	—	2	2	no	—
112	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₂₅₁	—	2	1	no	—
115	C₄⋊C₄×C₂₅₁	—	2	1	no	—
116	C₄⋊C₄×C₂₅₁	—	2	1	no	—
117	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
118	C₄²×C₂₅₁	—	2	1	no	—
119	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
120	C₂²×D₂₅₁	—	2	1	no	—
121	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
122	Dic₅₀₂	—	2	1	no	—
123	Dic₅₀₂	—	2	4	no	—
124	Dic₅₀₂	—	2	1	no	—
125	Dic₅₀₂	—	2	1	no	—
126	Dic₅₀₂	—	4	2	no	—
127	Dic₅₀₂	C₂×D₄	1	2	yes	—
128	D₁₀₀₄	—	2	1	no	—
129	D₁₀₀₄	—	2	1	no	—
130	D₁₀₀₄	—	4	2	no	—
131	C₂×Dic₂₅₁	—	2	1	no	—
132	C₂×Dic₂₅₁	—	2	1	no	—
133	C₂×Dic₂₅₁	—	2	1	no	—
134	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
137	C₄×D₂₅₁	—	2	4	no	—
138	C₄×D₂₅₁	—	4	2	no	—
139	C₄×D₂₅₁	—	4	2	no	—
140	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
141	C₄×D₂₅₁	#41	1	2	yes	—
142	C₂₅₁⋊C₈	—	2	1	no	—
143	C₂₅₁⋊C₈	—	2	1	no	—
144	Q₈×C₂₅₁	—	2	1	no	—
145	Q₈×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	C₂×D₄	1	2	yes	—
150	C₂×C₄×C₂₅₁	—	2	1	no	—
151	C₂×C₄×C₂₅₁	—	2	1	no	—
152	C₂×C₄×C₂₅₁	—	2	1	no	—
153	C₂×C₄×C₂₅₁	—	2	1	no	—
154	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
155	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
156	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
157	C₈×C₂₅₁	—	2	1	no	—
158	C₈×C₂₅₁	—	2	1	no	—
159	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₂₅₁	—	2	1	no	—
161	C₂×D₂₅₁	—	2	1	no	—
162	C₂×D₂₅₁	—	4	1	no	—
163	C₂×D₂₅₁	—	4	1	no	—
164	C₂×D₂₅₁	—	4	1	no	—
165	C₂×D₂₅₁	#121	1	2	yes	—
166	C₄×C₂₅₁	—	2	2	no	—
167	C₄×C₂₅₁	—	2	1	no	—
168	C₄×C₂₅₁	—	2	1	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	C₂×#41	1	1	yes	characteristic
174	Dic₂₅₁	—	2	2	no	—
175	Dic₂₅₁	—	2	1	no	—
176	Dic₂₅₁	—	2	1	no	—
177	Dic₂₅₁	—	4	1	no	—
178	Dic₂₅₁	—	4	1	no	—
179	Dic₂₅₁	#121	1	2	yes	—
180	C₂×C₂₅₁	C₂×#121	1	1	yes	characteristic
181	C₂×C₂₅₁	#578	1	1	yes	characteristic
182	C₂×C₂₅₁	#578	1	1	yes	characteristic
183	D₂₅₁	—	2	2	no	—
184	D₂₅₁	—	8	1	no	—
185	C₂₅₁	#3044	1	1	yes	characteristic, complemented, Sylow
186	#3044	—	251	1	no	maximal, Sylow
187	C₂×#137	—	251	1	no	—
188	C₂²×Q₁₆	—	251	1	no	—
189	#536	—	251	1	no	—
190	#519	—	251	1	no	—
191	#464	—	251	1	no	—
192	#464	—	251	1	no	—
193	#464	—	251	1	no	—
194	#464	—	251	1	no	—
195	#462	—	251	1	no	—
196	#462	—	251	1	no	—
197	#462	—	251	1	no	—
198	#462	D₂₅₁	1	1	yes	characteristic, complemented
199	#424	—	251	1	no	—
200	#419	—	251	1	no	—
201	#416	—	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	C₂×Q₁₆	—	502	4	no	—
207	C₂×Q₁₆	—	251	1	no	—
208	C₂×Q₁₆	—	251	1	no	—
209	C₂×Q₁₆	—	251	1	no	—
210	C₂×Q₁₆	—	251	1	no	—
211	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
212	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
213	C₂×SD₁₆	—	251	1	no	—
214	C₂×SD₁₆	—	251	1	no	—
215	C₂×M₁₆	—	251	1	no	—
216	C₂²×C₈	—	251	1	no	—
217	C₄⋊Q₈	—	251	1	no	—
218	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
219	#124	—	251	1	no	—
220	#124	—	251	1	no	—
221	#123	—	502	1	no	—
222	#123	—	502	1	no	—
223	C₂²⋊Q₈	—	502	1	no	—
224	C₂²⋊Q₈	—	502	1	no	—
225	C₄×Q₈	—	502	1	no	—
226	C₄×Q₈	—	502	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	#117	—	502	1	no	—
232	#117	—	502	1	no	—
233	#117	—	251	1	no	—
234	#117	—	251	1	no	—
235	C₄⋊C₈	—	251	1	no	—
236	C₄⋊C₈	—	251	1	no	—
237	C₄⋊C₈	—	251	1	no	—
238	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
239	#103	—	251	1	no	—
240	#103	—	251	1	no	—
241	#103	—	251	1	no	—
242	#103	—	251	1	no	—
243	#103	C₂×D₂₅₁	1	1	yes	characteristic
244	#103	C₂×D₂₅₁	1	1	yes	characteristic
245	#102	—	251	1	no	—
246	#102	—	251	1	no	—
247	#41	—	502	2	no	—
248	#41	—	502	2	no	—
249	C₂×Q₈	—	502	1	no	—
250	C₂×Q₈	—	502	1	no	—
251	C₂×Q₈	—	502	2	no	—
252	C₂×Q₈	—	502	2	no	—
253	C₂×Q₈	—	502	2	no	—
254	C₂×Q₈	—	251	4	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	—	251	1	no	—
262	C₂²×C₄	—	502	1	no	—
263	C₂²×C₄	—	502	1	no	—
264	C₂²×C₄	—	502	1	no	—
265	C₂²×C₄	—	251	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	502	2	no	—
269	Q₁₆	—	502	2	no	—
270	Q₁₆	—	502	2	no	—
271	Q₁₆	—	502	2	no	—
272	SD₁₆	—	502	2	no	—
273	SD₁₆	—	502	2	no	—
274	M₁₆	—	502	2	no	—
275	C₂×C₈	—	502	2	no	—
276	C₂×C₈	—	251	1	no	—
277	C₂×C₈	—	251	1	no	—
278	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
279	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	502	1	no	—
284	C₄⋊C₄	—	502	1	no	—
285	C₄⋊C₄	—	502	1	no	—
286	C₄⋊C₄	—	502	1	no	—
287	C₄⋊C₄	—	502	1	no	—
288	C₄⋊C₄	—	251	1	no	—
289	C₄⋊C₄	—	251	1	no	—
290	C₄⋊C₄	—	251	1	no	—
291	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
292	C₂²⋊C₄	—	502	1	no	—
293	C₂²⋊C₄	—	502	1	no	—
294	C₂²⋊C₄	—	502	1	no	—
295	C₂²⋊C₄	—	502	1	no	—
296	C₂²⋊C₄	—	502	1	no	—
297	C₂²⋊C₄	—	251	1	no	—
298	C₂²⋊C₄	—	251	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	502	1	no	—
301	C₄²	—	502	1	no	—
302	C₄²	—	502	1	no	—
303	C₄²	—	502	1	no	—
304	C₄²	—	251	1	no	—
305	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
306	C₂³	—	502	1	no	—
307	C₂³	—	251	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	1004	2	no	—
314	Q₈	—	502	4	no	—
315	Q₈	—	502	1	no	—
316	Q₈	—	502	1	no	—
317	Q₈	—	502	1	no	—
318	Q₈	—	251	2	no	—
319	Q₈	D₄×D₂₅₁	1	2	yes	—
320	D₄	—	1004	2	no	—
321	D₄	—	502	1	no	—
322	D₄	—	502	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	1004	2	no	—
328	C₂×C₄	—	1004	2	no	—
329	C₂×C₄	—	502	4	no	—
330	C₂×C₄	—	502	1	no	—
331	C₂×C₄	—	502	1	no	—
332	C₂×C₄	—	502	1	no	—
333	C₂×C₄	—	251	1	no	—
334	C₂×C₄	—	251	1	no	—
335	C₂×C₄	—	251	1	no	—
336	C₂×C₄	—	251	2	no	—
337	C₂×C₄	—	251	2	no	—
338	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
339	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	502	1	no	—
344	C₈	—	502	1	no	—
345	C₂²	—	1004	1	no	—
346	C₂²	—	1004	1	no	—
347	C₂²	—	1004	1	no	—
348	C₂²	—	502	1	no	—
349	C₂²	—	502	1	no	—
350	C₂²	—	251	2	no	—
351	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	1004	1	no	—
359	C₄	—	1004	1	no	—
360	C₄	—	502	1	no	—
361	C₄	—	502	1	no	—
362	C₄	—	502	2	no	—
363	C₄	—	251	2	no	—
364	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
365	C₄	#41×D₂₅₁	1	1	yes	characteristic
366	C₂	—	2008	1	no	—
367	C₂	—	502	2	no	—
368	C₂	#121×D₂₅₁	1	1	yes	characteristic
369	C₂	#30632500	1	1	yes	characteristic
370	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	#462×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860811	C₂²
hypercenter	#462	D₂₅₁
hypocenter	C₂₅₁	#3044
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₂₅₁	C₂	semidirect
4. #30632163	C₂	semidirect
5. #30631912	C₂	semidirect
6. #30631911	C₂	semidirect
7. #30631910	C₂	semidirect
8. #30631909	C₂	semidirect
9. #30631896	C₂	semidirect
10. #30631895	C₂	semidirect
11. #30631894	C₂	semidirect
12. #30631712	C₂	semidirect
13. #30631694	C₂	semidirect
15. #462×C₂₅₁	C₂	semidirect
185. C₂₅₁	#3044	semidirect
198. #462	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×Q₁₆×D₂₅₁	2
#30632262	2
#30632163	2
#30631912	2
#30631911	2
#30631910	2
#30631909	2
#30631896	2
#30631895	2
#30631894	2
#30631712	2
#30631694	2
#30631682	2
#462×C₂₅₁	2
#3044	251