Subgroups — #27367885

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284508	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20274163	C₂	1	4	yes	maximal, complemented
4	#20274111	C₂	1	2	yes	maximal, complemented
5	#20274077	C₂	1	2	yes	maximal, complemented
6	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273405	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₃	—	2	1	no	—
10	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#16810920	—	2	4	no	—
12	#16810910	C₂²	1	4	yes	—
13	#16810900	C₂²	1	4	yes	—
14	C₈×D₂₁₂	—	2	1	no	—
15	C₈×D₂₁₂	—	2	1	no	—
16	C₈×D₂₁₂	C₂²	1	2	yes	—
17	C₈×D₂₁₂	C₂²	1	2	yes	—
18	#16810666	—	2	2	no	—
19	#16810647	C₂²	1	2	yes	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	#16810627	—	2	2	no	—
22	C₂×#13133129	—	2	1	no	—
23	#16810376	C₂²	1	2	yes	complemented
24	#16810376	C₂²	1	2	yes	complemented
25	#16810372	C₂²	1	4	yes	complemented
26	#16810369	C₂²	1	4	yes	complemented
27	C₅₃×#533	C₂²	1	2	yes	—
28	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
29	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
30	C₅₃×#487	C₂²	1	2	yes	complemented
31	C₅₃×#467	—	2	1	no	—
32	C₅₃×#467	—	2	1	no	—
33	C₄×Q₁₆×C₅₃	—	2	1	no	—
34	C₄×Q₁₆×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₅₃	—	2	1	no	—
38	C₂×#11719917	—	2	2	no	—
39	C₂²×#88322	—	2	1	no	—
40	C₂×C₈×D₅₃	—	2	2	no	—
41	C₂×C₈×D₅₃	—	2	2	no	—
42	C₂×C₈×D₅₃	—	4	4	no	—
43	C₄×D₂₁₂	—	2	1	no	—
44	C₄×D₂₁₂	—	2	1	no	—
45	C₄×D₂₁₂	D₄	1	2	yes	—
46	C₄×D₂₁₂	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
48	C₂×#11719921	—	2	1	no	—
49	#13133150	—	2	4	no	—
50	#13133148	—	2	4	no	—
51	C₄⋊#88322	—	2	1	no	—
52	C₄⋊#88322	—	2	2	no	—
53	C₄⋊#88322	C₂³	1	2	yes	—
54	C₄⋊#88322	C₂³	1	4	yes	—
55	#13133137	—	2	4	no	—
56	#13133137	—	2	4	no	—
57	#13133129	—	2	2	no	—
58	#13133129	—	2	2	no	—
59	C₂×Q₁₆×C₅₃	—	2	1	no	—
60	C₂×Q₁₆×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
65	#125×C₅₃	—	2	2	no	—
66	C₂²⋊Q₈×C₅₃	—	2	2	no	—
67	C₂²⋊Q₈×C₅₃	—	2	2	no	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	C₄×Q₈×C₅₃	—	2	2	no	—
70	#117×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	—	2	1	no	—
74	#103×C₅₃	—	2	2	no	—
75	#103×C₅₃	—	2	2	no	—
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂³×D₅₃	—	2	1	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	C₂²×Dic₅₃	—	2	1	no	—
87	C₂×D₂₁₂	—	2	1	no	—
88	C₂×D₂₁₂	—	2	2	no	—
89	C₂×D₂₁₂	Q₁₆	1	4	yes	complemented
90	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₅₃	—	2	2	no	—
92	C₂×C₄×D₅₃	—	2	2	no	—
93	C₂×C₄×D₅₃	—	4	2	no	—
94	C₂×C₄×D₅₃	—	4	2	no	—
95	Q₁₆×C₅₃	—	4	2	no	—
96	Q₁₆×C₅₃	—	4	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	4	1	no	—
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₄⋊C₄×C₅₃	—	4	2	no	—
109	C₂²⋊C₄×C₅₃	—	4	2	no	—
110	C₄²×C₅₃	—	4	2	no	—
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	#41	1	1	yes	characteristic
114	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
115	#11719921	—	2	2	no	—
116	#11719921	—	2	2	no	—
117	C₄⋊Dic₅₃	—	2	1	no	—
118	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
119	#11719917	—	4	4	no	—
120	C₂×#88322	—	2	2	no	—
121	C₂×#88322	—	2	4	no	—
122	C₂×#88322	—	4	2	no	—
123	C₈×D₅₃	—	4	4	no	—
124	C₈×D₅₃	—	4	4	no	—
125	C₂³×C₅₃	#121	1	1	yes	characteristic
126	C₂²×D₅₃	—	2	2	no	—
127	C₂²×D₅₃	—	2	4	no	—
128	C₂²×D₅₃	—	4	2	no	—
129	C₂²×D₅₃	—	4	2	no	—
130	Q₈×C₅₃	—	4	2	no	—
131	Q₈×C₅₃	—	4	2	no	—
132	Q₈×C₅₃	—	8	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	2	no	—
137	C₂×C₄×C₅₃	—	4	2	no	—
138	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₅₃	#135	1	2	yes	—
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	4	2	no	—
148	D₂₁₂	—	2	2	no	—
149	D₂₁₂	—	2	4	no	—
150	D₂₁₂	C₂×Q₁₆	1	4	yes	—
151	C₄×D₅₃	—	4	4	no	—
152	C₄×D₅₃	—	4	2	no	—
153	C₄×D₅₃	—	4	2	no	—
154	C₈×C₅₃	—	2	1	no	—
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
158	#88322	—	4	4	no	—
159	#88322	—	4	2	no	—
160	#19256	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	#467	1	2	yes	—
164	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
165	C₂×D₅₃	—	2	4	no	—
166	C₂×D₅₃	—	4	8	no	—
167	C₂×D₅₃	—	4	2	no	—
168	C₂×D₅₃	—	4	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	8	2	no	—
173	C₄×C₅₃	—	8	2	no	—
174	C₄×C₅₃	#467	1	2	yes	—
175	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
177	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₄×C₅₃	#575	1	1	yes	characteristic
179	Dic₅₃	—	4	2	no	—
180	#3170	D₅₃	1	1	yes	characteristic, complemented
181	#3077	—	53	2	no	—
182	C₂×C₈×D₄	—	53	1	no	—
183	#1683	—	53	4	no	—
184	#1675	—	53	2	no	—
185	#1669	—	53	2	no	—
186	#1567	—	53	1	no	—
187	#1550	—	53	2	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	#3036	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
191	C₂×C₅₃	#3067	1	1	yes	characteristic
192	D₅₃	—	4	4	no	—
193	C₂³×C₈	—	106	1	no	—
194	#533	C₂×D₅₃	1	2	yes	—
195	C₂×C₄×D₄	—	53	1	no	—
196	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
197	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
198	#487	C₂×D₅₃	1	2	yes	complemented
199	#478	—	106	4	no	—
200	#476	—	53	4	no	—
201	#474	—	53	4	no	—
202	#467	—	2	1	no	—
203	#467	—	2	1	no	—
204	C₄×Q₁₆	—	2	1	no	—
205	C₄×Q₁₆	—	2	1	no	—
206	C₈×D₄	—	53	2	no	—
207	C₈×D₄	—	53	2	no	—
208	C₈×D₄	—	106	1	no	—
209	C₈×D₄	—	106	1	no	—
210	#428	—	106	2	no	—
211	#423	—	53	2	no	—
212	C₂×C₄⋊C₈	—	53	1	no	—
213	#418	—	106	2	no	—
214	C₂×C₂²⋊C₈	—	106	1	no	—
215	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
216	#335	—	53	2	no	—
217	#335	—	53	2	no	—
218	#333	—	53	4	no	—
219	#332	—	53	4	no	—
220	C₅₃	#19256	1	1	yes	characteristic, complemented, Sylow
221	C₂²×D₄	—	53	1	no	—
222	C₂³×C₄	—	106	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×M₁₆	—	106	2	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂²×C₈	—	106	2	no	—
228	C₂²×C₈	—	106	2	no	—
229	C₂²×C₈	—	212	4	no	—
230	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
232	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
233	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
234	#125	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×D₄	—	53	2	no	—
240	C₄×D₄	—	53	2	no	—
241	C₄×D₄	—	106	1	no	—
242	C₄×D₄	—	106	1	no	—
243	#117	—	2	2	no	—
244	C₂×C₄⋊C₄	—	53	1	no	—
245	C₂×C₂²⋊C₄	—	106	1	no	—
246	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
247	#107	—	2	1	no	—
248	#107	—	2	1	no	—
249	#107	—	106	4	no	—
250	#106	—	106	4	no	—
251	C₄⋊C₈	—	53	2	no	—
252	C₄⋊C₈	—	53	4	no	—
253	C₄⋊C₈	—	106	2	no	—
254	C₄⋊C₈	—	106	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#102	—	106	4	no	—
258	#102	—	106	4	no	—
259	C₂²⋊C₈	—	106	2	no	—
260	C₂²⋊C₈	—	106	2	no	—
261	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
265	C₂⁴	—	106	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×D₄	—	53	4	no	—
269	C₂×D₄	—	53	2	no	—
270	C₂×D₄	—	106	1	no	—
271	C₂×D₄	—	106	2	no	—
272	C₂²×C₄	—	106	1	no	—
273	C₂²×C₄	—	106	2	no	—
274	C₂²×C₄	—	106	2	no	—
275	C₂²×C₄	—	212	2	no	—
276	C₂²×C₄	—	212	2	no	—
277	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
278	C₂²×C₄	#11719947	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
280	Q₁₆	—	4	2	no	—
281	Q₁₆	—	4	2	no	—
282	M₁₆	—	212	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	106	4	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	212	4	no	—
288	C₂×C₈	—	212	2	no	—
289	C₂×C₈	—	212	4	no	—
290	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	53	2	no	—
301	C₄⋊C₄	—	106	1	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	106	2	no	—
304	C₂²⋊C₄	—	106	2	no	—
305	C₄²	—	4	2	no	—
306	C₄²	D₄×D₅₃	1	1	yes	characteristic
307	C₄²	#11719947	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
309	C₄²	C₂³×D₅₃	1	1	yes	characteristic
310	C₂³	—	106	2	no	—
311	C₂³	—	106	4	no	—
312	C₂³	—	212	2	no	—
313	C₂³	—	212	2	no	—
314	C₂³	#13133188	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	8	2	no	—
318	D₄	—	53	4	no	—
319	D₄	—	106	4	no	—
320	D₄	—	106	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	212	4	no	—
328	C₂×C₄	—	212	2	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	#13133188	1	1	yes	characteristic
332	C₂×C₄	#13133188	1	1	yes	characteristic
333	C₂×C₄	#13133188	1	1	yes	characteristic
334	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
335	C₂×C₄	#13133240	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	212	4	no	—
343	C₈	—	212	2	no	—
344	C₈	#13133242	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	106	4	no	—
348	C₂²	—	212	8	no	—
349	C₂²	—	212	2	no	—
350	C₂²	—	212	2	no	—
351	C₂²	#16810861	1	2	yes	—
352	C₂²	C₂×#13133188	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	—	212	2	no	—
359	C₄	#16810861	1	2	yes	—
360	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
361	C₄	C₂×#13133240	1	1	yes	characteristic
362	C₄	C₂×#13133242	1	1	yes	characteristic
363	C₄	#16811421	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	212	4	no	—
366	C₂	#20284175	1	1	yes	characteristic
367	C₂	C₂×#16810861	1	1	yes	characteristic
368	C₂	#20284431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₂₁₂	D₄
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#19256
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #20274163	C₂	semidirect
4. #20274111	C₂	semidirect
5. #20274077	C₂	semidirect
6. #20273474	C₂	semidirect
7. #20273405	C₂	semidirect
8. C₅₃×#3170	C₂	semidirect
23. #16810376	C₂²	semidirect
24. #16810376	C₂²	semidirect
25. #16810372	C₂²	semidirect
26. #16810369	C₂²	semidirect
28. C₅₃×#500	C₂²	semidirect
29. C₅₃×#494	C₂²	semidirect
30. C₅₃×#487	C₂²	semidirect
89. C₂×D₂₁₂	Q₁₆	semidirect
180. #3170	D₅₃	semidirect
196. #500	C₂×D₅₃	semidirect
197. #494	C₂×D₅₃	semidirect
198. #487	C₂×D₅₃	semidirect
220. C₅₃	#19256	semidirect

Maximal subgroups

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

subgroup	index
#20284508	2
C₂×C₈×D₂₁₂	2
#20274163	2
#20274111	2
#20274077	2
#20273474	2
#20273405	2
C₅₃×#3170	2
#19256	53