Subgroups — #44560522

Order 32128.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632239	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632155	C₂	1	1	yes	maximal, characteristic
3	#30632028	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631981	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631976	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631896	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631891	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631851	C₂	1	1	yes	maximal, characteristic
9	#30631847	C₂	1	1	yes	maximal, characteristic
10	#30631793	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631791	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631790	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631786	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631741	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#16860810	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
18	C₂×#16860800	C₂²	1	1	yes	characteristic
19	C₂×#16860799	C₂²	1	1	yes	characteristic
20	C₂×#16860797	C₂²	1	1	yes	characteristic
21	#22548714	C₂²	1	1	yes	characteristic
22	#22548713	C₂²	1	1	yes	characteristic
23	#22548705	—	2	1	no	—
24	#22548694	C₂²	1	1	yes	characteristic
25	#22548692	—	2	1	no	—
26	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	#22548680	—	2	1	no	—
28	#22548680	C₂²	1	1	yes	characteristic
29	#22548680	C₂²	1	1	yes	characteristic
30	#22548680	C₂²	1	1	yes	characteristic
31	#22548677	C₂²	1	1	yes	characteristic
32	#22548677	C₂²	1	1	yes	characteristic
33	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
34	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
35	#22548672	—	2	1	no	—
36	#22548671	—	2	1	no	—
37	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
38	C₄²×D₂₅₁	—	2	1	no	—
39	#22548648	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	#22548640	C₂²	1	1	yes	characteristic
44	#22548638	C₂²	1	1	yes	characteristic
45	#22548638	C₂²	1	1	yes	characteristic
46	#22548638	C₂²	1	1	yes	characteristic
47	#22548637	C₂²	1	1	yes	characteristic
48	#22548635	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	C₂²⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
53	#117×C₂₅₁	C₂²	1	1	yes	characteristic
54	#107×C₂₅₁	C₂²	1	1	yes	characteristic
55	#106×C₂₅₁	C₂²	1	1	yes	characteristic
56	#103×C₂₅₁	C₂²	1	1	yes	characteristic
57	#103×C₂₅₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
59	#16860810	—	2	2	no	—
60	#16860810	—	2	2	no	—
61	C₂×Dic₅₀₂	—	2	1	no	—
62	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
64	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
66	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
67	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₂₅₁	—	2	1	no	—
69	C₂×C₄×D₂₅₁	—	2	1	no	—
70	C₂×C₄×D₂₅₁	—	2	1	no	—
71	Dic₁₀₀₄	—	2	2	no	—
72	#16860800	—	2	2	no	—
73	#16860799	—	2	2	no	—
74	#16860797	—	2	2	no	—
75	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₂₅₁	—	2	1	no	—
78	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
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	—	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	C₄×Dic₂₅₁	—	2	1	no	—
90	C₄×Dic₂₅₁	—	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	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
107	C₄²×C₂₅₁	—	2	1	no	—
108	C₂²×D₂₅₁	—	2	1	no	—
109	Dic₅₀₂	—	2	1	no	—
110	Dic₅₀₂	—	2	1	no	—
111	Dic₅₀₂	—	2	1	no	—
112	Dic₅₀₂	—	4	2	no	—
113	Dic₅₀₂	#41	1	2	yes	—
114	D₁₀₀₄	—	2	1	no	—
115	D₁₀₀₄	#41	1	2	yes	—
116	C₂²⋊D₂₅₁	—	2	4	no	—
117	C₂×Dic₂₅₁	—	2	1	no	—
118	C₂×Dic₂₅₁	—	2	2	no	—
119	C₂×Dic₂₅₁	—	2	1	no	—
120	C₂×Dic₂₅₁	—	2	1	no	—
121	C₂×Dic₂₅₁	—	4	1	no	—
122	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
126	C₄×D₂₅₁	—	2	4	no	—
127	C₄×D₂₅₁	—	4	2	no	—
128	C₄×D₂₅₁	—	4	2	no	—
129	C₂₅₁⋊C₈	—	2	1	no	—
130	C₂₅₁⋊C₈	—	2	1	no	—
131	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
132	Q₈×C₂₅₁	—	2	1	no	—
133	Q₈×C₂₅₁	—	2	1	no	—
134	C₂×C₄×C₂₅₁	—	2	1	no	—
135	C₂×C₄×C₂₅₁	—	2	1	no	—
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	—	2	1	no	—
140	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₂₅₁	—	2	1	no	—
143	C₈×C₂₅₁	—	2	1	no	—
144	C₂²×C₂₅₁	—	2	1	no	—
145	C₂²×C₂₅₁	—	2	1	no	—
146	C₂²×C₂₅₁	—	2	1	no	—
147	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₂₅₁	—	2	2	no	—
149	C₂×D₂₅₁	—	4	1	no	—
150	C₂×D₂₅₁	—	4	1	no	—
151	C₄×C₂₅₁	—	2	1	no	—
152	C₄×C₂₅₁	—	4	1	no	—
153	C₄×C₂₅₁	—	4	1	no	—
154	C₄×C₂₅₁	—	4	1	no	—
155	C₄×C₂₅₁	—	4	1	no	—
156	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
158	Dic₂₅₁	—	2	2	no	—
159	Dic₂₅₁	—	2	1	no	—
160	Dic₂₅₁	—	2	1	no	—
161	Dic₂₅₁	—	2	1	no	—
162	Dic₂₅₁	—	4	1	no	—
163	Dic₂₅₁	—	4	1	no	—
164	Dic₂₅₁	#137	1	2	yes	—
165	C₂×C₂₅₁	—	4	1	no	—
166	C₂×C₂₅₁	#525	1	1	yes	characteristic
167	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
168	C₂×C₂₅₁	#575	1	1	yes	characteristic
169	D₂₅₁	—	4	2	no	—
170	C₂₅₁	#3116	1	1	yes	characteristic, complemented, Sylow
171	#3116	—	251	1	no	maximal, Sylow
172	#533	—	251	1	no	—
173	C₄×#41	—	251	1	no	—
174	#485	—	251	1	no	—
175	#485	D₂₅₁	1	1	yes	characteristic, complemented
176	#477	—	251	1	no	—
177	#476	—	251	1	no	—
178	#462	—	251	1	no	—
179	#461	—	251	1	no	—
180	#452	—	251	1	no	—
181	#452	—	251	1	no	—
182	#441	—	251	1	no	—
183	#441	—	251	1	no	—
184	#440	—	251	1	no	—
185	#440	—	251	1	no	—
186	#432	—	251	1	no	—
187	C₂×#41	—	251	1	no	—
188	C₂×Q₁₆	—	251	1	no	—
189	C₂×Q₁₆	—	251	1	no	—
190	C₂×SD₁₆	—	251	1	no	—
191	C₂×SD₁₆	—	251	1	no	—
192	C₄⋊Q₈	—	251	1	no	—
193	C₄⋊Q₈	—	251	1	no	—
194	#125	—	502	1	no	—
195	C₂²⋊Q₈	—	502	1	no	—
196	C₂²⋊Q₈	—	251	1	no	—
197	C₂²⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
198	C₄×Q₈	—	502	1	no	—
199	C₄×Q₈	—	251	1	no	—
200	C₄×Q₈	—	251	1	no	—
201	C₄×Q₈	—	251	1	no	—
202	C₄×Q₈	—	251	1	no	—
203	C₄×D₄	—	502	1	no	—
204	C₄×D₄	—	502	1	no	—
205	C₄×D₄	—	251	1	no	—
206	C₄×D₄	—	251	1	no	—
207	#117	—	502	1	no	—
208	#117	—	502	1	no	—
209	#117	C₂×D₂₅₁	1	1	yes	characteristic
210	C₂×C₄²	—	502	1	no	—
211	C₂×C₄²	—	251	1	no	—
212	#107	—	251	1	no	—
213	#107	C₂×D₂₅₁	1	1	yes	characteristic
214	#106	—	251	1	no	—
215	#106	C₂×D₂₅₁	1	1	yes	characteristic
216	C₄⋊C₈	—	251	1	no	—
217	C₄⋊C₈	—	251	1	no	—
218	#103	—	251	1	no	—
219	#103	—	251	1	no	—
220	#103	—	251	1	no	—
221	#103	—	251	1	no	—
222	#103	C₂×D₂₅₁	1	1	yes	characteristic
223	#103	C₂×D₂₅₁	1	1	yes	characteristic
224	#102	—	251	1	no	—
225	#102	—	251	1	no	—
226	C₂²⋊C₈	—	251	1	no	—
227	C₂²⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
228	#97	—	251	1	no	—
229	#97	—	251	1	no	—
230	#41	—	502	2	no	—
231	#41	—	502	2	no	—
232	C₂×Q₈	—	502	1	no	—
233	C₂×Q₈	—	251	1	no	—
234	C₂×Q₈	—	251	1	no	—
235	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
236	C₂×D₄	—	502	1	no	—
237	C₂×D₄	—	251	1	no	—
238	C₂²×C₄	—	502	1	no	—
239	C₂²×C₄	—	502	1	no	—
240	C₂²×C₄	—	502	1	no	—
241	C₂²×C₄	—	251	1	no	—
242	C₂²×C₄	—	251	1	no	—
243	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
244	Q₁₆	—	502	2	no	—
245	Q₁₆	—	502	2	no	—
246	SD₁₆	—	502	2	no	—
247	SD₁₆	—	502	2	no	—
248	C₂×C₈	—	251	1	no	—
249	C₂×C₈	—	251	1	no	—
250	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	502	1	no	—
254	C₄⋊C₄	—	502	1	no	—
255	C₄⋊C₄	—	502	1	no	—
256	C₄⋊C₄	—	502	1	no	—
257	C₄⋊C₄	—	502	1	no	—
258	C₄⋊C₄	—	502	1	no	—
259	C₄⋊C₄	—	502	1	no	—
260	C₄⋊C₄	—	502	1	no	—
261	C₄⋊C₄	—	251	1	no	—
262	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	502	1	no	—
268	C₂²⋊C₄	—	502	1	no	—
269	C₂²⋊C₄	—	502	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	502	1	no	—
272	C₄²	—	502	1	no	—
273	C₄²	—	502	1	no	—
274	C₄²	—	502	1	no	—
275	C₄²	—	251	1	no	—
276	C₄²	—	251	1	no	—
277	C₄²	—	251	1	no	—
278	C₄²	—	251	1	no	—
279	C₂³	—	502	1	no	—
280	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	1004	2	no	—
284	Q₈	—	502	1	no	—
285	Q₈	—	502	1	no	—
286	Q₈	—	502	1	no	—
287	Q₈	—	251	2	no	—
288	D₄	—	502	4	no	—
289	D₄	—	502	1	no	—
290	D₄	—	251	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	1004	2	no	—
298	C₂×C₄	—	1004	2	no	—
299	C₂×C₄	—	1004	1	no	—
300	C₂×C₄	—	502	4	no	—
301	C₂×C₄	—	502	1	no	—
302	C₂×C₄	—	502	1	no	—
303	C₂×C₄	—	502	1	no	—
304	C₂×C₄	—	502	2	no	—
305	C₂×C₄	—	251	1	no	—
306	C₂×C₄	—	251	1	no	—
307	C₂×C₄	—	251	1	no	—
308	C₂×C₄	—	251	1	no	—
309	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	502	1	no	—
314	C₈	—	502	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	1004	1	no	—
319	C₂²	—	1004	1	no	—
320	C₂²	—	502	2	no	—
321	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	1004	1	no	—
328	C₄	—	1004	1	no	—
329	C₄	—	502	2	no	—
330	C₄	—	502	1	no	—
331	C₄	—	502	1	no	—
332	C₄	—	502	1	no	—
333	C₄	—	251	2	no	—
334	C₄	#41×D₂₅₁	1	1	yes	characteristic
335	C₄	#41×D₂₅₁	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	1004	2	no	—
338	C₂	#30632195	1	1	yes	characteristic
339	C₂	#137×D₂₅₁	1	1	yes	characteristic
340	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	#485×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860810	C₂²
hypercenter	#485	D₂₅₁
hypocenter	C₂₅₁	#3116
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #30632239	C₂	semidirect
3. #30632028	C₂	semidirect
4. #30631981	C₂	semidirect
5. #30631976	C₂	semidirect
6. #30631896	C₂	semidirect
7. #30631891	C₂	semidirect
10. #30631793	C₂	semidirect
11. #30631791	C₂	semidirect
12. #30631790	C₂	semidirect
13. #30631786	C₂	semidirect
14. #30631741	C₂	semidirect
15. #485×C₂₅₁	C₂	semidirect
170. C₂₅₁	#3116	semidirect
175. #485	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632239	2
#30632155	2
#30632028	2
#30631981	2
#30631976	2
#30631896	2
#30631891	2
#30631851	2
#30631847	2
#30631793	2
#30631791	2
#30631790	2
#30631786	2
#30631741	2
#485×C₂₅₁	2
#3116	251