Subgroups — #44560580

Order 32128.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632297	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632170	C₂	1	1	yes	maximal, characteristic
3	#30632023	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632015	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631996	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631985	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631895	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631890	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631849	C₂	1	1	yes	maximal, characteristic
10	#30631791	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631788	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631778	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631745	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₂₅₁	C₂	1	1	yes	maximal, characteristic
16	C₂×#16860800	C₂²	1	1	yes	characteristic
17	C₂×#16860798	C₂²	1	1	yes	characteristic
18	#22548713	C₂²	1	1	yes	characteristic
19	#22548708	—	2	1	no	—
20	#22548708	—	2	1	no	—
21	#22548706	—	2	1	no	—
22	#22548706	C₂²	1	1	yes	characteristic
23	#22548705	—	2	1	no	—
24	#22548694	C₂²	1	1	yes	characteristic
25	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
26	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
28	#22548680	C₂²	1	1	yes	characteristic
29	D₄×Dic₂₅₁	—	2	1	no	—
30	D₄×Dic₂₅₁	—	2	1	no	—
31	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
32	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
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	#22548649	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	#22548639	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	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
52	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×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	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
60	C₂²×Dic₂₅₁	—	2	1	no	—
61	C₂²×Dic₂₅₁	—	2	1	no	—
62	C₂²×Dic₂₅₁	—	2	1	no	—
63	#16860800	—	2	2	no	—
64	#16860798	—	2	2	no	—
65	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₂₅₁	—	2	1	no	—
68	C₄⋊Dic₂₅₁	—	2	1	no	—
69	C₄⋊Dic₂₅₁	—	2	1	no	—
70	C₄⋊Dic₂₅₁	—	2	1	no	—
71	C₄⋊Dic₂₅₁	—	2	1	no	—
72	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
73	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
74	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
77	Dic₂₅₁⋊C₄	—	2	1	no	—
78	Dic₂₅₁⋊C₄	—	2	1	no	—
79	Dic₂₅₁⋊C₄	—	2	1	no	—
80	Dic₂₅₁⋊C₄	—	2	1	no	—
81	Dic₂₅₁⋊C₄	—	2	1	no	—
82	C₂²⋊Dic₂₅₁	—	2	1	no	—
83	C₂²⋊Dic₂₅₁	—	2	1	no	—
84	C₂²⋊Dic₂₅₁	—	2	1	no	—
85	C₂²⋊Dic₂₅₁	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₂₅₁	D₄	1	1	yes	characteristic
87	C₄×Dic₂₅₁	—	2	1	no	—
88	C₄×Dic₂₅₁	—	2	1	no	—
89	C₄×Dic₂₅₁	—	2	1	no	—
90	C₄×Dic₂₅₁	—	2	1	no	—
91	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
92	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
93	#41×C₂₅₁	—	2	2	no	—
94	#41×C₂₅₁	—	2	2	no	—
95	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₂₅₁	—	2	1	no	—
98	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₂₅₁	—	2	1	no	—
100	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
101	Q₁₆×C₂₅₁	—	2	2	no	—
102	SD₁₆×C₂₅₁	—	2	2	no	—
103	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₂₅₁	—	2	1	no	—
106	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
108	Dic₅₀₂	—	2	1	no	—
109	Dic₅₀₂	—	2	1	no	—
110	C₂×Dic₂₅₁	—	2	1	no	—
111	C₂×Dic₂₅₁	—	2	1	no	—
112	C₂×Dic₂₅₁	—	2	1	no	—
113	C₂×Dic₂₅₁	—	2	1	no	—
114	C₂×Dic₂₅₁	—	8	1	no	—
115	C₂×Dic₂₅₁	—	8	1	no	—
116	C₂×Dic₂₅₁	—	8	1	no	—
117	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂₅₁⋊C₈	—	2	1	no	—
122	C₂₅₁⋊C₈	—	2	1	no	—
123	C₂³×C₂₅₁	—	2	1	no	—
124	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₂₅₁	—	2	1	no	—
126	Q₈×C₂₅₁	—	2	1	no	—
127	Q₈×C₂₅₁	—	2	1	no	—
128	Q₈×C₂₅₁	#41	1	2	yes	—
129	D₄×C₂₅₁	—	2	4	no	—
130	D₄×C₂₅₁	—	2	1	no	—
131	D₄×C₂₅₁	#41	1	2	yes	—
132	C₂×C₄×C₂₅₁	—	2	4	no	—
133	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
140	C₈×C₂₅₁	—	2	1	no	—
141	C₈×C₂₅₁	—	2	1	no	—
142	C₂²×C₂₅₁	—	2	1	no	—
143	C₂²×C₂₅₁	—	2	2	no	—
144	C₂²×C₂₅₁	—	2	1	no	—
145	C₂²×C₂₅₁	—	2	1	no	—
146	C₂²×C₂₅₁	—	4	1	no	—
147	C₂²×C₂₅₁	—	4	1	no	—
148	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₂₅₁	—	2	1	no	—
150	C₄×C₂₅₁	—	2	2	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	1	no	—
156	Dic₂₅₁	—	2	1	no	—
157	Dic₂₅₁	—	2	1	no	—
158	Dic₂₅₁	—	2	1	no	—
159	Dic₂₅₁	—	4	1	no	—
160	Dic₂₅₁	—	4	1	no	—
161	Dic₂₅₁	—	4	1	no	—
162	Dic₂₅₁	—	4	1	no	—
163	C₂×C₂₅₁	—	4	1	no	—
164	C₂×C₂₅₁	—	4	2	no	—
165	C₂×C₂₅₁	#525	1	1	yes	characteristic
166	C₂×C₂₅₁	#577	1	1	yes	characteristic
167	C₂×C₂₅₁	#578	1	1	yes	characteristic
168	C₂₅₁	#3120	1	1	yes	characteristic, complemented, Sylow
169	#3120	—	251	1	no	maximal, Sylow
170	#541	—	251	1	no	—
171	#520	—	251	1	no	—
172	#484	—	251	1	no	—
173	#483	—	251	1	no	—
174	#479	—	251	1	no	—
175	#478	—	251	1	no	—
176	#462	—	251	1	no	—
177	#461	—	251	1	no	—
178	#452	—	251	1	no	—
179	#452	D₂₅₁	1	1	yes	characteristic
180	#441	—	251	1	no	—
181	#440	—	251	1	no	—
182	C₄×Q₁₆	—	251	1	no	—
183	C₄×SD₁₆	—	251	1	no	—
184	#433	—	251	1	no	—
185	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
186	C₂×Q₁₆	—	251	1	no	—
187	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
188	C₂×SD₁₆	—	251	1	no	—
189	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
190	C₄⋊Q₈	—	251	1	no	—
191	#126	—	502	1	no	—
192	#126	—	502	1	no	—
193	#125	—	502	1	no	—
194	#125	—	502	1	no	—
195	#125	—	251	1	no	—
196	C₂²⋊Q₈	—	251	1	no	—
197	C₂²⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
198	C₄×Q₈	—	251	1	no	—
199	C₄×Q₈	—	251	1	no	—
200	C₄×Q₈	—	251	1	no	—
201	C₄×Q₈	—	251	1	no	—
202	C₄×D₄	—	502	1	no	—
203	C₄×D₄	—	502	1	no	—
204	C₄×D₄	—	251	1	no	—
205	C₄×D₄	—	251	1	no	—
206	#117	—	502	1	no	—
207	#117	—	251	1	no	—
208	C₂×C₄⋊C₄	—	502	1	no	—
209	C₂×C₄⋊C₄	—	251	1	no	—
210	#107	—	251	1	no	—
211	#107	—	251	1	no	—
212	#106	—	251	1	no	—
213	#106	—	251	1	no	—
214	C₄⋊C₈	—	251	1	no	—
215	C₄⋊C₈	—	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	—	251	1	no	—
221	#103	C₂×D₂₅₁	1	1	yes	characteristic
222	#102	—	251	1	no	—
223	#102	C₂×D₂₅₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	251	1	no	—
225	C₂²⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
226	#97	—	251	1	no	—
227	C₄×C₈	—	251	1	no	—
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	C₂×Q₈	—	251	1	no	—
231	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	502	1	no	—
237	C₂²×C₄	—	502	1	no	—
238	C₂²×C₄	—	502	1	no	—
239	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	502	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	502	2	no	—
244	C₂×C₈	—	251	1	no	—
245	C₂×C₈	—	251	1	no	—
246	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
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	—	251	1	no	—
262	C₄⋊C₄	—	251	1	no	—
263	C₄⋊C₄	—	251	1	no	—
264	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	502	1	no	—
267	C₂²⋊C₄	—	502	1	no	—
268	C₂²⋊C₄	—	502	1	no	—
269	C₂²⋊C₄	—	251	1	no	—
270	C₂²⋊C₄	—	251	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	C₂³	—	2	1	no	—
278	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	502	1	no	—
283	Q₈	—	502	1	no	—
284	Q₈	#16860809	1	2	yes	—
285	D₄	—	2	4	no	—
286	D₄	—	2	1	no	—
287	D₄	#16860809	1	2	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2008	1	no	—
295	C₂×C₄	—	2008	1	no	—
296	C₂×C₄	—	2008	1	no	—
297	C₂×C₄	—	502	1	no	—
298	C₂×C₄	—	502	1	no	—
299	C₂×C₄	—	502	1	no	—
300	C₂×C₄	—	502	1	no	—
301	C₂×C₄	—	251	1	no	—
302	C₂×C₄	—	251	1	no	—
303	C₂×C₄	—	251	1	no	—
304	C₂×C₄	—	251	1	no	—
305	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	502	1	no	—
310	C₈	—	502	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	1004	1	no	—
323	C₄	—	1004	1	no	—
324	C₄	—	1004	1	no	—
325	C₄	—	1004	1	no	—
326	C₄	—	502	1	no	—
327	C₄	—	502	1	no	—
328	C₄	—	502	1	no	—
329	C₄	—	502	1	no	—
330	C₄	C₂×#16860809	1	1	yes	characteristic
331	C₄	C₂×#16860809	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	2	no	—
334	C₂	#30632193	1	1	yes	characteristic
335	C₂	#30632496	1	1	yes	characteristic
336	C₂	#30632501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#452×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#41×C₂₅₁	C₂²
hypercenter	#452	D₂₅₁
hypocenter	C₂₅₁	#3120
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #30632297	C₂	semidirect
3. #30632023	C₂	semidirect
4. #30632015	C₂	semidirect
5. #30631996	C₂	semidirect
6. #30631985	C₂	semidirect
7. #30631895	C₂	semidirect
8. #30631890	C₂	semidirect
10. #30631791	C₂	semidirect
11. #30631788	C₂	semidirect
12. Q₁₆×Dic₂₅₁	C₂	semidirect
13. #30631778	C₂	semidirect
14. #30631745	C₂	semidirect
168. C₂₅₁	#3120	semidirect

Maximal subgroups

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

subgroup	index
#30632297	2
#30632170	2
#30632023	2
#30632015	2
#30631996	2
#30631985	2
#30631895	2
#30631890	2
#30631849	2
#30631791	2
#30631788	2
Q₁₆×Dic₂₅₁	2
#30631778	2
#30631745	2
#452×C₂₅₁	2
#3120	251