Subgroups — #44560509

Order 32128.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632232	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632012	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632008	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631970	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631965	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631885	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631880	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631832	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631799	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631796	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631790	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631788	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631741	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#16860809	—	2	1	no	—
17	C₂×#16860797	C₂²	1	1	yes	characteristic
18	C₂×#16860795	C₂²	1	1	yes	characteristic
19	#22548714	C₂²	1	1	yes	characteristic
20	#22548711	C₂²	1	1	yes	characteristic
21	#22548701	—	2	1	no	—
22	#22548689	C₂²	1	1	yes	characteristic
23	#22548687	—	2	1	no	—
24	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
25	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
26	D₄×Dic₂₅₁	—	2	1	no	—
27	D₄×Dic₂₅₁	—	2	1	no	—
28	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
29	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
30	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
31	#22548677	C₂²	1	1	yes	characteristic
32	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
33	#22548674	—	2	1	no	—
34	#22548674	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₂₅₁	—	2	1	no	—
36	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
37	#22548649	C₂²	1	1	yes	characteristic
38	#22548648	C₂²	1	1	yes	characteristic
39	#22548647	C₂²	1	1	yes	characteristic
40	#22548646	C₂²	1	1	yes	characteristic
41	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
43	#22548639	C₂²	1	1	yes	characteristic
44	#22548637	C₂²	1	1	yes	characteristic
45	#22548637	C₂²	1	1	yes	characteristic
46	#22548636	C₂²	1	1	yes	characteristic
47	#22548636	C₂²	1	1	yes	characteristic
48	#22548636	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
50	#22548625	C₂²	1	1	yes	characteristic
51	#22548625	C₂²	1	1	yes	characteristic
52	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₂₅₁	C₂²	1	1	yes	characteristic
55	#121×C₂₅₁	C₂²	1	1	yes	characteristic
56	#103×C₂₅₁	C₂²	1	1	yes	characteristic
57	#102×C₂₅₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
59	#16860809	—	4	4	no	—
60	C₂×Dic₅₀₂	—	2	1	no	—
61	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
63	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
64	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
65	C₂²×Dic₂₅₁	—	2	1	no	—
66	C₂²×Dic₂₅₁	—	2	1	no	—
67	C₂²×Dic₂₅₁	—	2	1	no	—
68	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
69	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₂₅₁	—	2	1	no	—
71	#16860797	—	2	2	no	—
72	#16860795	—	2	2	no	—
73	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₂₅₁	—	2	1	no	—
76	C₄⋊Dic₂₅₁	—	2	1	no	—
77	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
80	Dic₂₅₁⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₂₅₁	—	2	1	no	—
82	C₂²⋊Dic₂₅₁	—	2	1	no	—
83	C₂²⋊Dic₂₅₁	—	2	1	no	—
84	C₂²⋊Dic₂₅₁	—	2	1	no	—
85	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
86	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
87	C₄×Dic₂₅₁	—	2	1	no	—
88	C₄×Dic₂₅₁	—	2	1	no	—
89	C₄×Dic₂₅₁	—	2	1	no	—
90	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
91	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
92	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
94	#41×C₂₅₁	—	2	2	no	—
95	#41×C₂₅₁	—	2	2	no	—
96	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₂₅₁	—	2	1	no	—
98	C₂×D₄×C₂₅₁	—	2	1	no	—
99	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₂₅₁	—	2	1	no	—
102	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
103	SD₁₆×C₂₅₁	—	2	2	no	—
104	D₈×C₂₅₁	—	2	2	no	—
105	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
109	C₂²×D₂₅₁	—	2	1	no	—
110	Dic₅₀₂	—	4	2	no	—
111	D₁₀₀₄	—	2	1	no	—
112	D₁₀₀₄	—	2	1	no	—
113	C₂²⋊D₂₅₁	—	4	2	no	—
114	C₂²⋊D₂₅₁	—	4	2	no	—
115	C₂²⋊D₂₅₁	—	4	2	no	—
116	C₂×Dic₂₅₁	—	2	2	no	—
117	C₂×Dic₂₅₁	—	2	1	no	—
118	C₂×Dic₂₅₁	—	2	1	no	—
119	C₂×Dic₂₅₁	—	2	1	no	—
120	C₂×Dic₂₅₁	—	4	1	no	—
121	C₂×Dic₂₅₁	—	4	2	no	—
122	C₂×Dic₂₅₁	—	4	2	no	—
123	C₂×Dic₂₅₁	—	4	2	no	—
124	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
128	C₄×D₂₅₁	—	4	2	no	—
129	C₂₅₁⋊C₈	—	2	1	no	—
130	C₂₅₁⋊C₈	—	2	1	no	—
131	C₂³×C₂₅₁	—	2	1	no	—
132	C₂³×C₂₅₁	—	2	1	no	—
133	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
134	Q₈×C₂₅₁	—	2	1	no	—
135	Q₈×C₂₅₁	#41	1	2	yes	complemented
136	D₄×C₂₅₁	—	2	4	no	—
137	D₄×C₂₅₁	—	2	1	no	—
138	D₄×C₂₅₁	—	2	1	no	—
139	D₄×C₂₅₁	—	2	1	no	—
140	D₄×C₂₅₁	—	4	2	no	—
141	D₄×C₂₅₁	#41	1	2	yes	complemented
142	C₂×C₄×C₂₅₁	—	2	4	no	—
143	C₂×C₄×C₂₅₁	—	2	1	no	—
144	C₂×C₄×C₂₅₁	—	2	1	no	—
145	C₂×C₄×C₂₅₁	—	2	1	no	—
146	C₂×C₄×C₂₅₁	—	2	1	no	—
147	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
149	C₈×C₂₅₁	—	2	1	no	—
150	C₈×C₂₅₁	—	2	1	no	—
151	C₂²×C₂₅₁	—	2	1	no	—
152	C₂²×C₂₅₁	—	2	2	no	—
153	C₂²×C₂₅₁	—	2	1	no	—
154	C₂²×C₂₅₁	—	2	1	no	—
155	C₂²×C₂₅₁	—	4	1	no	—
156	C₂²×C₂₅₁	—	4	1	no	—
157	C₂²×C₂₅₁	—	4	1	no	—
158	C₂²×C₂₅₁	—	4	1	no	—
159	C₂²×C₂₅₁	—	4	1	no	—
160	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₂₅₁	—	4	1	no	—
162	C₂×D₂₅₁	—	4	1	no	—
163	C₂×D₂₅₁	—	4	1	no	—
164	C₄×C₂₅₁	—	2	1	no	—
165	C₄×C₂₅₁	—	2	2	no	—
166	C₄×C₂₅₁	—	4	1	no	—
167	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
168	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
169	Dic₂₅₁	—	2	1	no	—
170	Dic₂₅₁	—	2	1	no	—
171	Dic₂₅₁	—	2	1	no	—
172	Dic₂₅₁	—	4	1	no	—
173	Dic₂₅₁	—	4	1	no	—
174	Dic₂₅₁	—	4	1	no	—
175	Dic₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₂₅₁	—	4	1	no	—
177	C₂×C₂₅₁	—	4	2	no	—
178	C₂×C₂₅₁	—	8	1	no	—
179	C₂×C₂₅₁	#525	1	1	yes	characteristic
180	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₂₅₁	#575	1	1	yes	characteristic
182	D₂₅₁	—	8	1	no	—
183	C₂₅₁	#3115	1	1	yes	characteristic, complemented, Sylow
184	#3115	—	251	1	no	maximal, Sylow
185	#532	—	251	1	no	—
186	C₄×#41	—	251	1	no	—
187	#482	—	251	1	no	—
188	#482	—	251	1	no	—
189	#475	—	251	1	no	—
190	#474	—	251	1	no	—
191	#460	—	251	1	no	—
192	#459	—	251	1	no	—
193	#449	—	251	1	no	—
194	#449	D₂₅₁	1	1	yes	characteristic, complemented
195	#442	—	251	1	no	—
196	#442	—	251	1	no	—
197	#440	—	251	1	no	—
198	#440	—	251	1	no	—
199	#432	—	251	1	no	—
200	C₂×#41	—	502	1	no	—
201	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
202	C₂×SD₁₆	—	251	1	no	—
203	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
204	C₂×D₈	—	251	1	no	—
205	C₂×D₈	C₂×D₂₅₁	1	1	yes	characteristic
206	C₄⋊Q₈	—	251	1	no	—
207	#127	—	251	1	no	—
208	#124	—	502	1	no	—
209	#121	—	502	1	no	—
210	#121	—	251	1	no	—
211	#121	C₂×D₂₅₁	1	1	yes	characteristic
212	C₄×Q₈	—	251	1	no	—
213	C₄×Q₈	—	251	1	no	—
214	C₄×D₄	—	502	1	no	—
215	C₄×D₄	—	502	1	no	—
216	C₄×D₄	—	502	1	no	—
217	C₄×D₄	—	251	1	no	—
218	C₄×D₄	—	251	1	no	—
219	C₄×D₄	—	251	1	no	—
220	C₄×D₄	—	251	1	no	—
221	#117	—	502	1	no	—
222	#117	—	251	1	no	—
223	C₂×C₄²	—	502	1	no	—
224	C₂×C₄²	—	251	1	no	—
225	#107	—	251	1	no	—
226	#107	—	251	1	no	—
227	#106	—	251	1	no	—
228	#106	—	251	1	no	—
229	C₄⋊C₈	—	251	1	no	—
230	C₄⋊C₈	—	251	1	no	—
231	#103	—	251	1	no	—
232	#103	C₂×D₂₅₁	1	1	yes	characteristic
233	#102	—	251	1	no	—
234	#102	—	251	1	no	—
235	#102	—	251	1	no	—
236	#102	—	251	1	no	—
237	#102	—	251	1	no	—
238	#102	C₂×D₂₅₁	1	1	yes	characteristic
239	C₂²⋊C₈	—	251	1	no	—
240	C₂²⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
241	#97	—	251	1	no	—
242	#97	—	251	1	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	1004	4	no	—
246	C₂×Q₈	—	502	1	no	—
247	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	502	1	no	—
251	C₂×D₄	—	502	1	no	—
252	C₂×D₄	—	502	1	no	—
253	C₂×D₄	—	251	1	no	—
254	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	502	1	no	—
258	C₂²×C₄	—	502	1	no	—
259	C₂²×C₄	—	502	1	no	—
260	C₂²×C₄	—	502	1	no	—
261	C₂²×C₄	—	251	1	no	—
262	C₂²×C₄	—	251	1	no	—
263	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	502	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	502	2	no	—
268	C₂×C₈	—	251	1	no	—
269	C₂×C₈	—	251	1	no	—
270	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
272	C₄⋊C₄	—	502	1	no	—
273	C₄⋊C₄	—	502	1	no	—
274	C₄⋊C₄	—	502	1	no	—
275	C₄⋊C₄	—	251	1	no	—
276	C₄⋊C₄	—	251	1	no	—
277	C₄⋊C₄	—	251	1	no	—
278	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	502	1	no	—
281	C₂²⋊C₄	—	502	1	no	—
282	C₂²⋊C₄	—	502	1	no	—
283	C₂²⋊C₄	—	502	1	no	—
284	C₂²⋊C₄	—	502	1	no	—
285	C₂²⋊C₄	—	502	1	no	—
286	C₄²	—	502	1	no	—
287	C₄²	—	502	1	no	—
288	C₄²	—	502	1	no	—
289	C₄²	—	251	1	no	—
290	C₄²	—	251	1	no	—
291	C₄²	—	251	1	no	—
292	C₄²	—	251	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	502	1	no	—
296	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	1004	2	no	—
299	Q₈	#16860809	1	2	yes	complemented
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	1004	2	no	—
306	D₄	—	1004	2	no	—
307	D₄	—	1004	2	no	—
308	D₄	—	502	1	no	—
309	D₄	—	502	1	no	—
310	D₄	#16860809	1	2	yes	complemented
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	1004	2	no	—
317	C₂×C₄	—	1004	2	no	—
318	C₂×C₄	—	1004	2	no	—
319	C₂×C₄	—	1004	2	no	—
320	C₂×C₄	—	1004	1	no	—
321	C₂×C₄	—	502	1	no	—
322	C₂×C₄	—	502	1	no	—
323	C₂×C₄	—	502	1	no	—
324	C₂×C₄	—	502	2	no	—
325	C₂×C₄	—	251	1	no	—
326	C₂×C₄	—	251	1	no	—
327	C₂×C₄	—	251	1	no	—
328	C₂×C₄	—	251	1	no	—
329	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	502	1	no	—
334	C₈	—	502	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	1004	1	no	—
345	C₂²	—	1004	1	no	—
346	C₂²	—	1004	1	no	—
347	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	1004	1	no	—
352	C₄	—	1004	1	no	—
353	C₄	—	1004	1	no	—
354	C₄	—	502	1	no	—
355	C₄	—	502	1	no	—
356	C₄	—	502	1	no	—
357	C₄	—	251	2	no	—
358	C₄	C₂×#16860809	1	1	yes	characteristic
359	C₄	C₂×#16860809	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	2008	1	no	—
364	C₂	#30632193	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₂₅₁	1	1	yes	characteristic
366	C₂	#30632484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#449×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	#449	D₂₅₁
hypocenter	C₂₅₁	#3115
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #30632232	C₂	semidirect
2. #41×Dic₂₅₁	C₂	semidirect
3. #30632012	C₂	semidirect
4. #30632008	C₂	semidirect
5. #30631970	C₂	semidirect
6. #30631965	C₂	semidirect
7. #30631885	C₂	semidirect
8. #30631880	C₂	semidirect
9. #30631832	C₂	semidirect
10. #30631799	C₂	semidirect
11. #30631796	C₂	semidirect
12. #30631790	C₂	semidirect
13. #30631788	C₂	semidirect
14. #30631741	C₂	semidirect
15. #449×C₂₅₁	C₂	semidirect
135. Q₈×C₂₅₁	#41	semidirect
141. D₄×C₂₅₁	#41	semidirect
183. C₂₅₁	#3115	semidirect
194. #449	D₂₅₁	semidirect
299. Q₈	#16860809	semidirect
310. D₄	#16860809	semidirect

Maximal subgroups

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

subgroup	index
#30632232	2
#41×Dic₂₅₁	2
#30632012	2
#30632008	2
#30631970	2
#30631965	2
#30631885	2
#30631880	2
#30631832	2
#30631799	2
#30631796	2
#30631790	2
#30631788	2
#30631741	2
#449×C₂₅₁	2
#3115	251