Subgroups — #44560493

Order 32128.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632241	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#30632019	C₂	1	1	yes	maximal, characteristic
4	#30632016	C₂	1	1	yes	maximal, characteristic
5	#30631971	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631970	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631896	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631895	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631844	C₂	1	1	yes	maximal, characteristic
10	#30631841	C₂	1	1	yes	maximal, characteristic
11	#30631795	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631792	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631792	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631742	C₂	1	1	yes	maximal, characteristic
15	#441×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
18	C₂×#16860800	C₂²	1	1	yes	characteristic
19	C₂×#16860800	C₂²	1	1	yes	characteristic
20	#22548714	C₂²	1	1	yes	characteristic
21	#22548713	C₂²	1	1	yes	characteristic
22	#22548705	—	2	1	no	—
23	#22548695	C₂²	1	1	yes	characteristic
24	#22548693	—	2	1	no	—
25	#22548691	C₂²	1	1	yes	characteristic
26	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₀₂	—	2	1	no	—
28	C₄×Dic₅₀₂	—	2	1	no	—
29	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
31	#22548680	—	2	1	no	—
32	#22548680	C₂²	1	1	yes	characteristic
33	#22548672	—	2	1	no	—
34	C₄⋊C₄×D₂₅₁	—	2	1	no	—
35	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
36	C₄²×D₂₅₁	—	2	1	no	—
37	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
38	#22548647	C₂²	1	1	yes	characteristic
39	#22548646	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	#22548640	C₂²	1	1	yes	characteristic
45	#22548639	C₂²	1	1	yes	characteristic
46	#22548639	C₂²	1	1	yes	characteristic
47	#22548638	C₂²	1	1	yes	characteristic
48	#22548638	C₂²	1	1	yes	characteristic
49	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#22548626	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×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	#97×C₂₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₅₀₂	—	2	1	no	—
60	C₂×Dic₅₀₂	—	2	1	no	—
61	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
63	Q₈×D₂₅₁	—	2	2	no	—
64	Q₈×D₂₅₁	—	2	2	no	—
65	C₂×C₄×D₂₅₁	—	2	1	no	—
66	C₂×C₄×D₂₅₁	—	2	1	no	—
67	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
70	Dic₁₀₀₄	—	2	2	no	—
71	#16860800	—	2	2	no	—
72	#16860800	—	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	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	Dic₂₅₁⋊C₄	—	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	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₂₅₁	—	2	2	no	—
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	—	2	1	no	—
103	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
105	C₄²×C₂₅₁	—	2	1	no	—
106	C₄²×C₂₅₁	—	2	1	no	—
107	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
108	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
109	Dic₅₀₂	—	2	4	no	—
110	Dic₅₀₂	—	2	1	no	—
111	Dic₅₀₂	—	2	1	no	—
112	Dic₅₀₂	—	2	1	no	—
113	Dic₅₀₂	—	4	2	no	—
114	Dic₅₀₂	#41	1	2	yes	—
115	C₂×Dic₂₅₁	—	2	1	no	—
116	C₂×Dic₂₅₁	—	2	1	no	—
117	C₂×Dic₂₅₁	—	2	1	no	—
118	C₂×Dic₂₅₁	—	2	1	no	—
119	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₂₅₁	—	2	4	no	—
123	C₄×D₂₅₁	—	2	2	no	—
124	C₄×D₂₅₁	—	2	1	no	—
125	C₄×D₂₅₁	—	2	1	no	—
126	C₄×D₂₅₁	—	2	2	no	—
127	C₄×D₂₅₁	—	4	2	no	—
128	C₂₅₁⋊C₈	—	2	1	no	—
129	C₂₅₁⋊C₈	—	2	1	no	—
130	Q₈×C₂₅₁	—	2	1	no	—
131	Q₈×C₂₅₁	—	2	1	no	—
132	Q₈×C₂₅₁	—	2	1	no	—
133	Q₈×C₂₅₁	#41	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
141	C₈×C₂₅₁	—	2	1	no	—
142	C₈×C₂₅₁	—	2	1	no	—
143	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₂₅₁	—	2	1	no	—
145	C₂×D₂₅₁	—	2	1	no	—
146	C₂×D₂₅₁	#137	1	2	yes	—
147	C₄×C₂₅₁	—	2	2	no	—
148	C₄×C₂₅₁	—	2	1	no	—
149	C₄×C₂₅₁	—	2	1	no	—
150	C₄×C₂₅₁	—	4	1	no	—
151	C₄×C₂₅₁	—	4	1	no	—
152	C₄×C₂₅₁	—	4	1	no	—
153	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
154	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
155	Dic₂₅₁	—	2	2	no	—
156	Dic₂₅₁	—	2	1	no	—
157	Dic₂₅₁	—	2	1	no	—
158	Dic₂₅₁	—	4	1	no	—
159	Dic₂₅₁	—	4	1	no	—
160	Dic₂₅₁	—	4	1	no	—
161	Dic₂₅₁	#137	1	2	yes	—
162	C₂×C₂₅₁	#525	1	1	yes	characteristic
163	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
164	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
165	D₂₅₁	—	2	2	no	—
166	C₂₅₁	#3113	1	1	yes	characteristic, complemented, Sylow
167	#3113	—	251	1	no	maximal, Sylow
168	#533	—	251	1	no	—
169	C₂×C₄×Q₈	—	251	1	no	—
170	#483	—	251	1	no	—
171	#483	—	251	1	no	—
172	#475	—	251	1	no	—
173	#475	—	251	1	no	—
174	#462	—	251	1	no	—
175	#462	—	251	1	no	—
176	#451	—	251	1	no	—
177	#451	—	251	1	no	—
178	#441	—	251	1	no	—
179	#441	—	251	1	no	—
180	#441	—	251	1	no	—
181	#441	D₂₅₁	1	1	yes	characteristic, complemented
182	#432	—	251	1	no	—
183	C₂²×Q₈	—	251	1	no	—
184	C₂×Q₁₆	—	251	1	no	—
185	C₂×Q₁₆	—	251	1	no	—
186	C₂×Q₁₆	—	251	1	no	—
187	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	251	1	no	—
189	C₄⋊Q₈	—	251	1	no	—
190	#125	—	502	1	no	—
191	C₂²⋊Q₈	—	502	1	no	—
192	C₂²⋊Q₈	—	251	1	no	—
193	C₂²⋊Q₈	—	251	1	no	—
194	C₄×Q₈	—	502	1	no	—
195	C₄×Q₈	—	502	1	no	—
196	C₄×Q₈	—	502	1	no	—
197	C₄×Q₈	—	251	1	no	—
198	C₄×Q₈	—	251	1	no	—
199	C₄×Q₈	—	251	1	no	—
200	C₄×Q₈	—	251	1	no	—
201	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
202	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
203	#117	—	502	1	no	—
204	C₂×C₄⋊C₄	—	502	1	no	—
205	C₂×C₄⋊C₄	—	251	1	no	—
206	C₂×C₄²	—	502	1	no	—
207	C₂×C₄²	—	251	1	no	—
208	#106	—	251	1	no	—
209	#106	—	251	1	no	—
210	#106	—	251	1	no	—
211	#106	C₂×D₂₅₁	1	1	yes	characteristic
212	C₄⋊C₈	—	251	1	no	—
213	C₄⋊C₈	—	251	1	no	—
214	#103	—	251	1	no	—
215	#103	—	251	1	no	—
216	#103	—	251	1	no	—
217	#103	—	251	1	no	—
218	#103	—	251	1	no	—
219	#103	—	251	1	no	—
220	#103	C₂×D₂₅₁	1	1	yes	characteristic
221	#103	C₂×D₂₅₁	1	1	yes	characteristic
222	C₂²⋊C₈	—	251	1	no	—
223	C₂²⋊C₈	—	251	1	no	—
224	#97	—	251	1	no	—
225	#97	C₂×D₂₅₁	1	1	yes	characteristic
226	C₂×Q₈	—	502	1	no	—
227	C₂×Q₈	—	502	1	no	—
228	C₂×Q₈	—	502	2	no	—
229	C₂×Q₈	—	502	2	no	—
230	C₂×Q₈	—	251	1	no	—
231	C₂×Q₈	—	251	1	no	—
232	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
234	C₂²×C₄	—	502	1	no	—
235	C₂²×C₄	—	502	1	no	—
236	C₂²×C₄	—	251	1	no	—
237	C₂²×C₄	—	251	1	no	—
238	C₂²×C₄	—	251	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	502	2	no	—
241	Q₁₆	—	502	2	no	—
242	Q₁₆	—	502	2	no	—
243	C₂×C₈	—	251	1	no	—
244	C₂×C₈	—	251	1	no	—
245	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	502	1	no	—
250	C₄⋊C₄	—	502	1	no	—
251	C₄⋊C₄	—	502	1	no	—
252	C₄⋊C₄	—	502	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	—	251	1	no	—
260	C₄⋊C₄	—	251	1	no	—
261	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	502	1	no	—
264	C₂²⋊C₄	—	502	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	502	1	no	—
268	C₄²	—	502	1	no	—
269	C₄²	—	502	1	no	—
270	C₄²	—	502	1	no	—
271	C₄²	—	251	1	no	—
272	C₄²	—	251	1	no	—
273	C₄²	—	251	1	no	—
274	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
275	C₂³	—	251	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	1004	2	no	—
280	Q₈	—	502	4	no	—
281	Q₈	—	502	1	no	—
282	Q₈	—	502	1	no	—
283	Q₈	—	502	1	no	—
284	Q₈	—	251	2	no	—
285	Q₈	#16860810	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	1004	2	no	—
291	C₂×C₄	—	502	4	no	—
292	C₂×C₄	—	502	2	no	—
293	C₂×C₄	—	502	2	no	—
294	C₂×C₄	—	502	1	no	—
295	C₂×C₄	—	502	1	no	—
296	C₂×C₄	—	502	1	no	—
297	C₂×C₄	—	502	1	no	—
298	C₂×C₄	—	502	1	no	—
299	C₂×C₄	—	502	1	no	—
300	C₂×C₄	—	251	1	no	—
301	C₂×C₄	—	251	1	no	—
302	C₂×C₄	—	251	1	no	—
303	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
304	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	502	1	no	—
309	C₈	—	502	1	no	—
310	C₂²	—	502	1	no	—
311	C₂²	—	502	1	no	—
312	C₂²	—	251	2	no	—
313	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	1004	1	no	—
321	C₄	—	1004	1	no	—
322	C₄	—	1004	1	no	—
323	C₄	—	502	2	no	—
324	C₄	—	502	1	no	—
325	C₄	—	502	1	no	—
326	C₄	—	251	2	no	—
327	C₄	#41×D₂₅₁	1	1	yes	characteristic
328	C₄	C₂×#16860810	1	1	yes	characteristic
329	C₂	—	502	2	no	—
330	C₂	#30632196	1	1	yes	characteristic
331	C₂	#137×D₂₅₁	1	1	yes	characteristic
332	C₂	#137×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#441×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂²×D₂₅₁	C₂×D₄
hypercenter	#441	D₂₅₁
hypocenter	C₂₅₁	#3113
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
5. #30631971	C₂	semidirect
6. #30631970	C₂	semidirect
7. #30631896	C₂	semidirect
8. #30631895	C₂	semidirect
11. #30631795	C₂	semidirect
12. #30631792	C₂	semidirect
13. #30631792	C₂	semidirect
15. #441×C₂₅₁	C₂	semidirect
166. C₂₅₁	#3113	semidirect
181. #441	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632241	2
C₄×Q₈×D₂₅₁	2
#30632019	2
#30632016	2
#30631971	2
#30631970	2
#30631896	2
#30631895	2
#30631844	2
#30631841	2
#30631795	2
#30631792	2
#30631742	2
#441×C₂₅₁	2
#3113	251