Subgroups — #44560327

Order 32128.

386 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	#30632260	C₂	1	1	yes	maximal, characteristic
2	#30632168	C₂	1	1	yes	maximal, characteristic
3	#30632031	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632029	C₂	1	1	yes	maximal, characteristic, complemented
5	#30632026	C₂	1	1	yes	maximal, characteristic, complemented
6	#30632025	C₂	1	1	yes	maximal, characteristic, complemented
7	#30632022	C₂	1	1	yes	maximal, characteristic, complemented
8	#30632021	C₂	1	1	yes	maximal, characteristic, complemented
9	#30632019	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631726	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631722	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631694	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631652	C₂	1	1	yes	maximal, characteristic
15	#483×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#16860810	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×#16860793	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
20	#22548714	—	2	1	no	—
21	#22548701	—	2	1	no	—
22	#22548700	—	2	1	no	—
23	#22548697	—	2	1	no	—
24	#22548695	C₂²	1	1	yes	characteristic
25	#22548694	C₂²	1	1	yes	characteristic
26	#22548691	C₂²	1	1	yes	characteristic
27	#22548680	—	2	1	no	—
28	#22548677	—	2	1	no	—
29	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
30	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
31	#22548674	C₂²	1	1	yes	characteristic
32	#22548673	C₂²	1	1	yes	characteristic
33	#22548673	C₂²	1	1	yes	characteristic
34	#22548673	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
37	#22548649	C₂²	1	1	yes	characteristic
38	#22548648	C₂²	1	1	yes	characteristic
39	#22548648	C₂²	1	1	yes	characteristic
40	#22548648	C₂²	1	1	yes	characteristic
41	#22548647	C₂²	1	1	yes	characteristic
42	#22548646	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	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
54	#106×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	D₄	1	1	yes	characteristic
63	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
64	Q₈×D₂₅₁	—	2	2	no	—
65	Q₈×D₂₅₁	—	2	2	no	—
66	C₂×D₁₀₀₄	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₂₅₁	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₂₅₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₂₅₁	—	2	1	no	—
70	C₂×C₄×D₂₅₁	—	2	1	no	—
71	C₂×C₄×D₂₅₁	—	2	1	no	—
72	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
74	#16860793	—	2	2	no	—
75	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×D₂₅₁	—	2	2	no	—
78	C₄⋊Dic₂₅₁	—	2	1	no	—
79	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₂₅₁	—	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	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
92	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
93	C₄×Dic₂₅₁	—	2	1	no	—
94	C₄×Dic₂₅₁	—	2	1	no	—
95	C₄×Dic₂₅₁	—	2	1	no	—
96	C₄×Dic₂₅₁	—	2	1	no	—
97	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₂₅₁	—	2	1	no	—
99	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₂₅₁	—	2	1	no	—
103	C₄⋊C₄×C₂₅₁	—	2	1	no	—
104	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
108	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
110	Dic₅₀₂	—	2	2	no	—
111	Dic₅₀₂	—	2	1	no	—
112	Dic₅₀₂	—	2	1	no	—
113	Dic₅₀₂	—	4	2	no	—
114	D₁₀₀₄	—	2	2	no	—
115	C₂²⋊D₂₅₁	—	2	2	no	—
116	C₂²⋊D₂₅₁	—	2	2	no	—
117	C₂×Dic₂₅₁	—	2	1	no	—
118	C₂×Dic₂₅₁	—	2	1	no	—
119	C₂×Dic₂₅₁	—	2	1	no	—
120	C₂×Dic₂₅₁	—	2	1	no	—
121	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₂₅₁	—	2	1	no	—
124	C₄×D₂₅₁	—	2	1	no	—
125	C₄×D₂₅₁	—	4	2	no	—
126	C₄×D₂₅₁	—	8	1	no	—
127	C₄×D₂₅₁	—	8	1	no	—
128	C₄×D₂₅₁	#41	1	2	yes	—
129	C₄×D₂₅₁	#41	1	2	yes	—
130	C₂₅₁⋊C₈	—	2	1	no	—
131	C₂₅₁⋊C₈	—	2	1	no	—
132	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
133	Q₈×C₂₅₁	—	2	1	no	—
134	Q₈×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	—	2	1	no	—
141	C₂×C₄×C₂₅₁	—	8	1	no	—
142	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
144	C₈×C₂₅₁	—	2	1	no	—
145	C₈×C₂₅₁	—	2	1	no	—
146	C₂²×C₂₅₁	—	2	1	no	—
147	C₂²×C₂₅₁	—	2	1	no	—
148	C₂²×C₂₅₁	—	2	1	no	—
149	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₂₅₁	—	2	1	no	—
151	C₂×D₂₅₁	—	2	1	no	—
152	C₂×D₂₅₁	—	2	1	no	—
153	C₂×D₂₅₁	—	2	1	no	—
154	C₂×D₂₅₁	—	2	1	no	—
155	C₂×D₂₅₁	#123	1	2	yes	—
156	C₄×C₂₅₁	—	2	1	no	—
157	C₄×C₂₅₁	—	4	1	no	—
158	C₄×C₂₅₁	—	4	1	no	—
159	C₄×C₂₅₁	—	4	1	no	—
160	C₄×C₂₅₁	—	4	1	no	—
161	C₄×C₂₅₁	C₂×#41	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	—	4	1	no	—
168	Dic₂₅₁	#123	1	2	yes	—
169	C₂×C₂₅₁	—	4	1	no	—
170	C₂×C₂₅₁	C₂×#123	1	1	yes	characteristic
171	C₂×C₂₅₁	#577	1	1	yes	characteristic
172	C₂×C₂₅₁	#578	1	1	yes	characteristic
173	D₂₅₁	—	2	2	no	—
174	D₂₅₁	—	4	1	no	—
175	C₂₅₁	#3097	1	1	yes	characteristic, complemented, Sylow
176	#3097	—	251	1	no	maximal, Sylow
177	#536	—	251	1	no	—
178	#520	—	251	1	no	—
179	#485	—	251	1	no	—
180	#485	—	251	1	no	—
181	#485	—	251	1	no	—
182	#485	—	251	1	no	—
183	#484	—	251	1	no	—
184	#484	—	251	1	no	—
185	#483	—	251	1	no	—
186	#483	D₂₅₁	1	1	yes	characteristic, complemented
187	#428	—	251	1	no	—
188	#427	—	251	1	no	—
189	#419	—	251	1	no	—
190	C₂×#103	—	251	1	no	—
191	#408	—	251	1	no	—
192	C₂×#41	—	251	1	no	—
193	C₂²×Q₈	—	251	1	no	—
194	C₂×M₁₆	—	251	1	no	—
195	C₂²×C₈	—	251	1	no	—
196	C₄⋊Q₈	—	502	1	no	—
197	#124	—	502	1	no	—
198	#123	—	502	1	no	—
199	#123	—	502	1	no	—
200	C₂²⋊Q₈	—	251	1	no	—
201	C₂²⋊Q₈	—	251	1	no	—
202	C₂²⋊Q₈	—	251	1	no	—
203	C₂²⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
204	C₄×Q₈	—	502	1	no	—
205	C₄×D₄	—	502	1	no	—
206	C₄×D₄	—	502	1	no	—
207	C₄×D₄	—	502	1	no	—
208	#117	—	251	1	no	—
209	#117	—	251	1	no	—
210	#117	—	251	1	no	—
211	#117	—	251	1	no	—
212	C₂×C₄⋊C₄	—	251	1	no	—
213	C₂×C₄⋊C₄	—	251	1	no	—
214	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
215	#107	—	251	1	no	—
216	#107	—	251	1	no	—
217	#107	—	251	1	no	—
218	#107	—	251	1	no	—
219	#106	—	251	1	no	—
220	#106	—	251	1	no	—
221	#106	C₂×D₂₅₁	1	1	yes	characteristic
222	#106	C₂×D₂₅₁	1	1	yes	characteristic
223	#103	—	251	1	no	—
224	#103	—	251	1	no	—
225	#103	—	251	1	no	—
226	#103	—	251	1	no	—
227	#103	—	251	1	no	—
228	#103	—	251	1	no	—
229	#103	C₂×D₂₅₁	1	1	yes	characteristic
230	#103	C₂×D₂₅₁	1	1	yes	characteristic
231	C₂²⋊C₈	—	251	1	no	—
232	C₂²⋊C₈	—	251	1	no	—
233	C₂²⋊C₈	—	251	1	no	—
234	C₂²⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
235	#41	—	502	2	no	—
236	#41	—	502	2	no	—
237	C₂×Q₈	—	502	1	no	—
238	C₂×Q₈	—	502	2	no	—
239	C₂×Q₈	—	502	2	no	—
240	C₂×Q₈	—	251	1	no	—
241	C₂×Q₈	—	251	1	no	—
242	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
243	C₂×D₄	—	251	1	no	—
244	C₂×D₄	—	251	1	no	—
245	C₂×D₄	—	251	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	502	1	no	—
248	C₂²×C₄	—	502	1	no	—
249	C₂²×C₄	—	502	1	no	—
250	C₂²×C₄	—	251	1	no	—
251	C₂²×C₄	—	251	1	no	—
252	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
253	M₁₆	—	502	2	no	—
254	C₂×C₈	—	502	2	no	—
255	C₂×C₈	—	251	1	no	—
256	C₂×C₈	—	251	1	no	—
257	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	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	—	502	1	no	—
266	C₄⋊C₄	—	251	1	no	—
267	C₄⋊C₄	—	251	1	no	—
268	C₄⋊C₄	—	251	1	no	—
269	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	502	1	no	—
278	C₂²⋊C₄	—	502	1	no	—
279	C₂²⋊C₄	—	502	1	no	—
280	C₄²	—	502	1	no	—
281	C₄²	—	502	1	no	—
282	C₄²	—	502	1	no	—
283	C₄²	—	502	1	no	—
284	C₂³	—	251	1	no	—
285	C₂³	—	251	1	no	—
286	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	1004	2	no	—
290	Q₈	—	502	2	no	—
291	Q₈	—	502	1	no	—
292	Q₈	—	502	1	no	—
293	D₄	—	502	2	no	—
294	D₄	—	502	2	no	—
295	D₄	—	502	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	2008	1	no	—
304	C₂×C₄	—	2008	1	no	—
305	C₂×C₄	—	1004	2	no	—
306	C₂×C₄	—	502	1	no	—
307	C₂×C₄	—	502	1	no	—
308	C₂×C₄	—	502	1	no	—
309	C₂×C₄	—	502	1	no	—
310	C₂×C₄	—	502	1	no	—
311	C₂×C₄	—	502	1	no	—
312	C₂×C₄	—	251	1	no	—
313	C₂×C₄	—	251	1	no	—
314	C₂×C₄	—	251	2	no	—
315	C₂×C₄	—	251	2	no	—
316	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	502	1	no	—
321	C₈	—	502	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	502	1	no	—
326	C₂²	—	502	1	no	—
327	C₂²	—	502	1	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	—	4	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	—	1004	1	no	—
341	C₄	—	502	1	no	—
342	C₄	—	251	2	no	—
343	C₄	#41×D₂₅₁	1	1	yes	characteristic
344	C₄	#41×D₂₅₁	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	1004	1	no	—
347	C₂	—	502	2	no	—
348	C₂	#123×D₂₅₁	1	1	yes	characteristic
349	C₂	#30632493	1	1	yes	characteristic
350	C₂	#30632503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#483×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860810	C₂²
hypercenter	#483	D₂₅₁
hypocenter	C₂₅₁	#3097
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
3. #30632031	C₂	semidirect
4. #30632029	C₂	semidirect
5. #30632026	C₂	semidirect
6. #30632025	C₂	semidirect
7. #30632022	C₂	semidirect
8. #30632021	C₂	semidirect
9. #30632019	C₂	semidirect
10. #30631726	C₂	semidirect
11. #30631722	C₂	semidirect
12. #30631694	C₂	semidirect
13. #103×D₂₅₁	C₂	semidirect
15. #483×C₂₅₁	C₂	semidirect
175. C₂₅₁	#3097	semidirect
186. #483	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632260	2
#30632168	2
#30632031	2
#30632029	2
#30632026	2
#30632025	2
#30632022	2
#30632021	2
#30632019	2
#30631726	2
#30631722	2
#30631694	2
#103×D₂₅₁	2
#30631652	2
#483×C₂₅₁	2
#3097	251