Subgroups — #44560160

Order 32128.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632243	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#30631988	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631987	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631984	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631976	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631975	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631966	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631965	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631728	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631726	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631709	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631690	C₂	1	1	yes	maximal, characteristic
14	#30631690	C₂	1	1	yes	maximal, characteristic
15	#478×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₂₅₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₅₁	—	2	1	no	—
18	C₂×#16860793	C₂²	1	1	yes	characteristic
19	C₂×#16860793	C₂²	1	1	yes	characteristic
20	#22548714	C₂²	1	1	yes	characteristic, complemented
21	#22548714	C₂²	1	1	yes	characteristic, complemented
22	#22548707	C₂²	1	1	yes	characteristic, complemented
23	#22548693	—	2	1	no	—
24	#22548693	—	2	1	no	—
25	#22548680	—	2	1	no	—
26	#22548680	—	2	1	no	—
27	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
28	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
29	#22548677	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
31	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
32	#22548673	C₂²	1	1	yes	characteristic
33	#22548673	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₂₅₁	—	2	1	no	—
36	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
37	#22548649	C₂²	1	1	yes	characteristic, complemented
38	#22548648	C₂²	1	1	yes	characteristic, complemented
39	#22548648	C₂²	1	1	yes	characteristic, complemented
40	#22548647	C₂²	1	1	yes	characteristic, complemented
41	#22548646	C₂²	1	1	yes	characteristic, complemented
42	#22548646	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	#22548637	C₂²	1	1	yes	characteristic
47	#22548637	C₂²	1	1	yes	characteristic
48	#22548636	C₂²	1	1	yes	characteristic
49	#22548636	C₂²	1	1	yes	characteristic
50	#22548635	C₂²	1	1	yes	characteristic
51	#22548635	C₂²	1	1	yes	characteristic
52	#125×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
54	#107×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
55	#106×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
57	#102×C₂₅₁	C₂²	1	1	yes	characteristic
58	#102×C₂₅₁	C₂²	1	1	yes	characteristic
59	C₂³×D₂₅₁	—	2	1	no	—
60	C₂×Dic₅₀₂	—	2	1	no	—
61	C₂×Dic₅₀₂	—	2	1	no	—
62	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
64	D₄×D₂₅₁	—	2	2	no	—
65	D₄×D₂₅₁	Q₈	1	4	yes	complemented
66	C₂²×Dic₂₅₁	—	2	1	no	—
67	C₂×C₄×D₂₅₁	—	2	1	no	—
68	C₂×C₄×D₂₅₁	—	2	1	no	—
69	C₂×C₄×D₂₅₁	—	2	1	no	—
70	C₂×C₄×D₂₅₁	—	4	4	no	—
71	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
72	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
73	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
74	#16860793	—	2	2	no	—
75	#16860793	—	2	2	no	—
76	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
81	Dic₂₅₁⋊C₄	—	2	1	no	—
82	Dic₂₅₁⋊C₄	—	2	1	no	—
83	Dic₂₅₁⋊C₄	—	2	1	no	—
84	Dic₂₅₁⋊C₄	—	2	1	no	—
85	Dic₂₅₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₂₅₁	—	2	1	no	—
87	(C₂×D₂₅₁)⋊C₄	—	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	D₄	1	1	yes	characteristic
94	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
95	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₂₅₁	—	2	1	no	—
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	—	2	1	no	—
102	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
106	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
107	C₂²×D₂₅₁	—	2	1	no	—
108	C₂²×D₂₅₁	—	2	4	no	—
109	C₂²×D₂₅₁	—	4	2	no	—
110	C₂²×D₂₅₁	—	4	2	no	—
111	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
112	Dic₅₀₂	—	4	2	no	—
113	Dic₅₀₂	—	4	2	no	—
114	D₁₀₀₄	—	2	1	no	—
115	D₁₀₀₄	C₂×Q₈	1	2	yes	—
116	C₂²⋊D₂₅₁	—	2	4	no	—
117	C₂×Dic₂₅₁	—	2	1	no	—
118	C₂×Dic₂₅₁	—	2	1	no	—
119	C₂×Dic₂₅₁	—	2	1	no	—
120	C₂×Dic₂₅₁	—	4	2	no	—
121	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
124	C₄×D₂₅₁	—	2	2	no	—
125	C₄×D₂₅₁	—	4	1	no	—
126	C₄×D₂₅₁	—	4	2	no	—
127	C₄×D₂₅₁	—	4	2	no	—
128	C₄×D₂₅₁	—	4	2	no	—
129	C₄×D₂₅₁	C₂×Q₈	1	2	yes	—
130	C₄×D₂₅₁	#41	1	2	yes	—
131	C₂₅₁⋊C₈	—	2	1	no	—
132	C₂₅₁⋊C₈	—	2	1	no	—
133	C₂³×C₂₅₁	—	2	1	no	—
134	D₄×C₂₅₁	—	2	1	no	—
135	D₄×C₂₅₁	C₂×Q₈	1	2	yes	—
136	C₂×C₄×C₂₅₁	—	2	1	no	—
137	C₂×C₄×C₂₅₁	—	2	1	no	—
138	C₂×C₄×C₂₅₁	—	2	1	no	—
139	C₂×C₄×C₂₅₁	—	4	2	no	—
140	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
143	C₈×C₂₅₁	—	2	1	no	—
144	C₈×C₂₅₁	—	2	1	no	—
145	C₂²×C₂₅₁	—	2	2	no	—
146	C₂²×C₂₅₁	—	4	1	no	—
147	C₂²×C₂₅₁	—	4	1	no	—
148	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₂₅₁	—	2	1	no	—
150	C₂×D₂₅₁	—	2	1	no	—
151	C₂×D₂₅₁	—	2	2	no	—
152	C₂×D₂₅₁	—	4	8	no	—
153	C₂×D₂₅₁	—	4	1	no	—
154	C₂×D₂₅₁	—	4	1	no	—
155	C₂×D₂₅₁	C₂²⋊Q₈	1	2	yes	—
156	C₄×C₂₅₁	—	2	1	no	—
157	C₄×C₂₅₁	—	2	1	no	—
158	C₄×C₂₅₁	—	4	1	no	—
159	C₄×C₂₅₁	—	4	1	no	—
160	C₄×C₂₅₁	—	4	1	no	—
161	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
162	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
163	Dic₂₅₁	—	2	1	no	—
164	Dic₂₅₁	—	4	1	no	—
165	Dic₂₅₁	—	4	1	no	—
166	Dic₂₅₁	—	4	1	no	—
167	Dic₂₅₁	C₂²⋊Q₈	1	2	yes	—
168	Dic₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₂₅₁	—	4	2	no	—
170	C₂×C₂₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₂₅₁	#575	1	1	yes	characteristic
173	D₂₅₁	—	2	2	no	—
174	D₂₅₁	—	4	2	no	—
175	C₂₅₁	#3077	1	1	yes	characteristic, complemented, Sylow
176	#3077	—	251	1	no	maximal, Sylow
177	#533	—	251	1	no	—
178	C₂×C₄×D₄	—	251	1	no	—
179	#478	—	251	1	no	—
180	#478	—	251	1	no	—
181	#478	—	251	1	no	—
182	#478	D₂₅₁	1	1	yes	characteristic, complemented
183	#476	—	251	1	no	—
184	#476	—	251	1	no	—
185	#474	—	251	1	no	—
186	#474	—	251	1	no	—
187	#428	—	251	1	no	—
188	#428	—	251	1	no	—
189	#423	—	251	1	no	—
190	#418	—	251	1	no	—
191	#418	—	251	1	no	—
192	C₂²×D₄	—	251	1	no	—
193	C₂³×C₄	—	502	1	no	—
194	C₂×M₁₆	—	251	1	no	—
195	C₂×M₁₆	—	251	1	no	—
196	C₄⋊Q₈	—	251	1	no	—
197	C₄⋊Q₈	—	251	1	no	—
198	#125	—	251	1	no	—
199	#125	C₂×D₂₅₁	1	1	yes	characteristic, complemented
200	C₂²⋊Q₈	—	502	1	no	—
201	C₂²⋊Q₈	—	502	1	no	—
202	C₄×Q₈	—	502	1	no	—
203	C₄×Q₈	—	502	1	no	—
204	C₄×D₄	—	502	1	no	—
205	C₄×D₄	—	502	1	no	—
206	C₄×D₄	—	251	1	no	—
207	C₄×D₄	—	251	1	no	—
208	C₄×D₄	—	251	1	no	—
209	C₄×D₄	C₂×D₂₅₁	1	1	yes	characteristic
210	#117	—	251	1	no	—
211	#117	—	251	1	no	—
212	C₂×C₄⋊C₄	—	251	1	no	—
213	C₂×C₂²⋊C₄	—	502	1	no	—
214	C₂×C₄²	—	251	1	no	—
215	#107	—	251	1	no	—
216	#107	—	251	1	no	—
217	#107	—	251	1	no	—
218	#107	C₂×D₂₅₁	1	1	yes	characteristic, complemented
219	#106	—	251	1	no	—
220	#106	—	251	1	no	—
221	#106	—	251	1	no	—
222	#106	C₂×D₂₅₁	1	1	yes	characteristic, complemented
223	C₄⋊C₈	—	251	1	no	—
224	C₄⋊C₈	—	251	1	no	—
225	C₄⋊C₈	—	251	1	no	—
226	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic, complemented
227	#102	—	251	1	no	—
228	#102	—	251	1	no	—
229	#102	—	251	1	no	—
230	#102	—	251	1	no	—
231	#102	—	251	1	no	—
232	#102	—	251	1	no	—
233	#102	C₂×D₂₅₁	1	1	yes	characteristic
234	#102	C₂×D₂₅₁	1	1	yes	characteristic
235	C₂⁴	—	502	1	no	—
236	C₂×Q₈	—	502	1	no	—
237	C₂×Q₈	—	502	1	no	—
238	C₂×D₄	—	502	2	no	—
239	C₂×D₄	—	502	1	no	—
240	C₂×D₄	—	251	4	no	—
241	C₂×D₄	—	251	1	no	—
242	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	1004	4	no	—
245	C₂²×C₄	—	502	1	no	—
246	C₂²×C₄	—	502	1	no	—
247	C₂²×C₄	—	502	1	no	—
248	C₂²×C₄	—	502	1	no	—
249	C₂²×C₄	—	251	1	no	—
250	C₂²×C₄	—	251	1	no	—
251	C₂²×C₄	—	251	1	no	—
252	M₁₆	—	502	2	no	—
253	M₁₆	—	502	2	no	—
254	C₂×C₈	—	251	1	no	—
255	C₂×C₈	—	251	1	no	—
256	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	502	1	no	—
261	C₄⋊C₄	—	502	1	no	—
262	C₄⋊C₄	—	502	1	no	—
263	C₄⋊C₄	—	502	1	no	—
264	C₄⋊C₄	—	502	1	no	—
265	C₄⋊C₄	—	251	1	no	—
266	C₄⋊C₄	—	251	1	no	—
267	C₄⋊C₄	—	251	1	no	—
268	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	502	1	no	—
273	C₂²⋊C₄	—	502	1	no	—
274	C₂²⋊C₄	—	502	1	no	—
275	C₂²⋊C₄	—	502	1	no	—
276	C₂²⋊C₄	—	502	1	no	—
277	C₄²	—	502	1	no	—
278	C₄²	—	502	1	no	—
279	C₄²	—	251	1	no	—
280	C₄²	—	251	1	no	—
281	C₄²	—	251	1	no	—
282	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	1004	2	no	—
285	C₂³	—	1004	2	no	—
286	C₂³	—	502	4	no	—
287	C₂³	—	502	1	no	—
288	C₂³	—	251	1	no	—
289	Q₈	—	1004	2	no	—
290	Q₈	—	1004	2	no	—
291	D₄	—	2	1	no	—
292	D₄	—	502	4	no	—
293	D₄	—	502	1	no	—
294	D₄	—	251	2	no	—
295	D₄	Q₈×D₂₅₁	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	1004	2	no	—
301	C₂×C₄	—	1004	1	no	—
302	C₂×C₄	—	1004	2	no	—
303	C₂×C₄	—	1004	2	no	—
304	C₂×C₄	—	1004	2	no	—
305	C₂×C₄	—	502	2	no	—
306	C₂×C₄	—	502	1	no	—
307	C₂×C₄	—	502	1	no	—
308	C₂×C₄	—	502	1	no	—
309	C₂×C₄	—	251	1	no	—
310	C₂×C₄	—	251	1	no	—
311	C₂×C₄	—	251	1	no	—
312	C₂×C₄	—	251	2	no	—
313	C₂×C₄	—	251	2	no	—
314	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
315	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	502	1	no	—
320	C₈	—	502	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	1004	8	no	—
325	C₂²	—	1004	1	no	—
326	C₂²	—	1004	1	no	—
327	C₂²	—	502	2	no	—
328	C₂²	—	502	1	no	—
329	C₂²	—	502	1	no	—
330	C₂²	—	251	2	no	—
331	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	1004	1	no	—
338	C₄	—	1004	1	no	—
339	C₄	—	1004	1	no	—
340	C₄	—	502	1	no	—
341	C₄	—	251	2	no	—
342	C₄	—	251	2	no	—
343	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
344	C₄	#41×D₂₅₁	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	1004	2	no	—
347	C₂	—	502	2	no	—
348	C₂	C₂²⋊Q₈×D₂₅₁	1	1	yes	characteristic
349	C₂	AGL(1,ℤ/8)×D₂₅₁	1	1	yes	characteristic
350	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	#478×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×D₄×D₂₅₁	C₂²
hypercenter	#478	D₂₅₁
hypocenter	C₂₅₁	#3077
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #30632243	C₂	semidirect
3. #30631988	C₂	semidirect
4. #30631987	C₂	semidirect
5. #30631984	C₂	semidirect
6. #30631976	C₂	semidirect
7. #30631975	C₂	semidirect
8. #30631966	C₂	semidirect
9. #30631965	C₂	semidirect
10. #30631728	C₂	semidirect
11. #30631726	C₂	semidirect
12. #30631709	C₂	semidirect
15. #478×C₂₅₁	C₂	semidirect
20. #22548714	C₂²	semidirect
21. #22548714	C₂²	semidirect
22. #22548707	C₂²	semidirect
37. #22548649	C₂²	semidirect
38. #22548648	C₂²	semidirect
39. #22548648	C₂²	semidirect
40. #22548647	C₂²	semidirect
41. #22548646	C₂²	semidirect
42. #22548646	C₂²	semidirect
43. C₄⋊(C₂₅₁⋊C₈)	C₂²	semidirect
44. Dic₂₅₁⋊C₈	C₂²	semidirect
45. Dic₂₅₁⋊C₈	C₂²	semidirect
52. #125×C₂₅₁	C₂²	semidirect
54. #107×C₂₅₁	C₂²	semidirect
55. #106×C₂₅₁	C₂²	semidirect
56. C₄⋊C₈×C₂₅₁	C₂²	semidirect
65. D₄×D₂₅₁	Q₈	semidirect
175. C₂₅₁	#3077	semidirect
182. #478	D₂₅₁	semidirect
199. #125	C₂×D₂₅₁	semidirect
218. #107	C₂×D₂₅₁	semidirect
222. #106	C₂×D₂₅₁	semidirect
226. C₄⋊C₈	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632243	2
C₄×D₄×D₂₅₁	2
#30631988	2
#30631987	2
#30631984	2
#30631976	2
#30631975	2
#30631966	2
#30631965	2
#30631728	2
#30631726	2
#30631709	2
#30631690	2
#478×C₂₅₁	2
#3077	251