Subgroups — #27368139

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284810	C₂	1	2	yes	maximal
2	C₂×#16810702	C₂	1	1	yes	maximal, characteristic
3	#20274259	C₂	1	4	yes	maximal, complemented
4	#20274225	C₂	1	2	yes	maximal, complemented
5	#20274201	C₂	1	2	yes	maximal, complemented
6	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273415	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719912	—	2	1	no	—
10	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#16810956	—	2	2	no	—
12	#16810900	C₂²	1	4	yes	—
13	#16810825	C₂²	1	4	yes	—
14	#16810773	—	2	2	no	—
15	#16810721	—	2	4	no	—
16	#16810702	—	2	1	no	—
17	#16810702	—	2	1	no	—
18	#16810702	C₂²	1	2	yes	—
19	#16810702	C₂²	1	2	yes	—
20	#16810680	C₂²	1	2	yes	—
21	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×#13133129	—	2	1	no	—
23	#16810376	C₂²	1	2	yes	complemented
24	#16810374	C₂²	1	2	yes	complemented
25	#16810369	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×#11719924	—	2	4	no	—
39	C₂×#11719912	—	2	2	no	—
40	C₂×#11719912	—	2	2	no	—
41	C₂×#11719912	—	4	4	no	—
42	C₂²×#88322	—	2	1	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	C₄⋊#88322	—	2	1	no	—
51	C₄⋊#88322	C₂³	1	4	yes	—
52	C₄⋊#88322	C₂³	1	2	yes	—
53	#13133141	—	2	4	no	—
54	#13133137	—	2	4	no	—
55	C₂²⋊#88322	—	2	2	no	—
56	#13133129	—	2	2	no	—
57	#13133129	—	2	2	no	—
58	#13133128	—	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	—	4	2	no	—
89	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₅₃	—	2	2	no	—
91	C₂×C₄×D₅₃	—	2	2	no	—
92	C₂×C₄×D₅₃	—	4	2	no	—
93	C₂×C₄×D₅₃	—	4	2	no	—
94	Q₁₆×C₅₃	—	4	2	no	—
95	Q₁₆×C₅₃	—	4	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	4	1	no	—
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	—	2	2	no	—
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	—	4	2	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₂²⋊C₄×C₅₃	—	4	2	no	—
109	C₄²×C₅₃	—	4	2	no	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	#41	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	#11719924	—	4	8	no	—
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	C₂×#88322	—	2	4	no	—
120	C₂×#88322	—	2	2	no	—
121	C₂×#88322	—	4	2	no	—
122	#11719912	—	4	4	no	—
123	#11719912	—	4	4	no	—
124	C₂³×C₅₃	#121	1	1	yes	characteristic
125	C₂²×D₅₃	—	2	2	no	—
126	C₂²×D₅₃	—	4	2	no	—
127	C₂²×D₅₃	—	4	2	no	—
128	C₂²×D₅₃	—	4	2	no	—
129	Q₈×C₅₃	—	4	2	no	—
130	Q₈×C₅₃	—	4	4	no	—
131	Q₈×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	2	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	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₅₃	—	2	2	no	—
145	C₂×Dic₅₃	—	4	2	no	—
146	D₂₁₂	—	2	4	no	—
147	D₂₁₂	—	4	2	no	—
148	C₄×D₅₃	—	4	4	no	—
149	C₄×D₅₃	—	4	2	no	—
150	C₄×D₅₃	—	4	2	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
155	#88322	—	4	2	no	—
156	#88322	—	4	4	no	—
157	#19284	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	#473	1	2	yes	—
161	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
162	C₂×D₅₃	—	4	2	no	—
163	C₂×D₅₃	—	4	2	no	—
164	C₂×D₅₃	—	4	2	no	—
165	C₂×D₅₃	—	8	4	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	8	2	no	—
171	C₄×C₅₃	—	8	2	no	—
172	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₄×C₅₃	#575	1	2	yes	—
174	C₄×C₅₃	#575	1	1	yes	characteristic
175	Dic₅₃	—	4	2	no	—
176	#3170	D₅₃	1	1	yes	characteristic, complemented
177	#3110	—	53	2	no	—
178	C₂×#436	—	53	1	no	—
179	#1699	—	53	4	no	—
180	#1693	—	53	2	no	—
181	#1689	—	53	2	no	—
182	#1568	—	53	1	no	—
183	#1552	—	53	2	no	—
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	#3036	1	1	yes	characteristic
186	C₂×C₅₃	#3054	1	1	yes	characteristic
187	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
188	D₅₃	—	8	2	no	—
189	C₂²×M₁₆	—	106	1	no	—
190	#533	C₂×D₅₃	1	2	yes	—
191	C₂×C₄×D₄	—	53	1	no	—
192	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
193	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
194	#487	C₂×D₅₃	1	2	yes	complemented
195	#484	—	106	2	no	—
196	#474	—	53	4	no	—
197	#467	—	2	1	no	—
198	#467	—	2	1	no	—
199	#461	—	53	4	no	—
200	#450	—	106	2	no	—
201	#440	—	106	4	no	—
202	C₄×Q₁₆	—	2	1	no	—
203	C₄×Q₁₆	—	2	1	no	—
204	#436	—	53	2	no	—
205	#436	—	53	2	no	—
206	#436	—	106	1	no	—
207	#436	—	106	1	no	—
208	#432	—	53	2	no	—
209	C₂×C₄⋊C₈	—	53	1	no	—
210	C₂×C₂²⋊C₈	—	106	1	no	—
211	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
212	#335	—	53	2	no	—
213	#334	—	53	2	no	—
214	#332	—	53	4	no	—
215	#332	—	53	4	no	—
216	C₅₃	#19284	1	1	yes	characteristic, complemented, Sylow
217	C₂²×D₄	—	53	1	no	—
218	C₂³×C₄	—	106	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×SD₁₆	—	106	4	no	—
222	C₂×M₁₆	—	106	2	no	—
223	C₂×M₁₆	—	106	2	no	—
224	C₂×M₁₆	—	212	4	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
228	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
229	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
230	#125	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	53	2	no	—
236	C₄×D₄	—	53	2	no	—
237	C₄×D₄	—	106	1	no	—
238	C₄×D₄	—	106	1	no	—
239	#117	—	2	2	no	—
240	C₂×C₄⋊C₄	—	53	1	no	—
241	C₂×C₂²⋊C₄	—	106	1	no	—
242	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
243	#107	—	2	1	no	—
244	#107	—	2	1	no	—
245	#107	—	106	4	no	—
246	C₄⋊C₈	—	53	2	no	—
247	C₄⋊C₈	—	53	4	no	—
248	C₄⋊C₈	—	106	1	no	—
249	#103	—	2	2	no	—
250	#103	—	2	2	no	—
251	#103	—	106	4	no	—
252	#102	—	106	4	no	—
253	C₂²⋊C₈	—	106	2	no	—
254	C₂²⋊C₈	—	106	2	no	—
255	C₂²⋊C₈	—	106	2	no	—
256	#97	—	106	2	no	—
257	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₂⁴	—	106	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	53	2	no	—
265	C₂×D₄	—	106	1	no	—
266	C₂×D₄	—	212	2	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	—	106	2	no	—
269	C₂²×C₄	—	106	2	no	—
270	C₂²×C₄	—	212	2	no	—
271	C₂²×C₄	—	212	2	no	—
272	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	#11719947	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
275	Q₁₆	—	4	2	no	—
276	Q₁₆	—	4	2	no	—
277	SD₁₆	—	212	8	no	—
278	M₁₆	—	212	4	no	—
279	M₁₆	—	212	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	—	212	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	53	2	no	—
296	C₄⋊C₄	—	106	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	106	2	no	—
299	C₂²⋊C₄	—	106	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	D₄×D₅₃	1	1	yes	characteristic
302	C₄²	#11719947	1	1	yes	characteristic
303	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
304	C₄²	C₂³×D₅₃	1	1	yes	characteristic
305	C₂³	—	106	2	no	—
306	C₂³	—	212	2	no	—
307	C₂³	—	212	2	no	—
308	C₂³	—	212	2	no	—
309	C₂³	#13133188	1	1	yes	characteristic
310	Q₈	—	4	2	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	4	2	no	—
313	D₄	—	106	4	no	—
314	D₄	—	212	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	212	4	no	—
323	C₂×C₄	—	212	2	no	—
324	C₂×C₄	—	212	2	no	—
325	C₂×C₄	—	212	2	no	—
326	C₂×C₄	#13133188	1	1	yes	characteristic
327	C₂×C₄	#13133188	1	1	yes	characteristic
328	C₂×C₄	#13133188	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	212	2	no	—
336	C₈	—	212	4	no	—
337	C₈	#13133242	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	424	4	no	—
341	C₂²	—	212	2	no	—
342	C₂²	—	212	2	no	—
343	C₂²	—	212	2	no	—
344	C₂²	#16810895	1	2	yes	—
345	C₂²	C₂×#13133188	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	212	2	no	—
353	C₄	C₂×#13133242	1	1	yes	characteristic
354	C₄	#16811420	1	2	yes	—
355	C₄	#16811421	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	424	2	no	—
358	C₂	#20284175	1	1	yes	characteristic
359	C₂	#20284320	1	1	yes	characteristic
360	C₂	C₂×#16810895	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	#19284
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. #20274259	C₂	semidirect
4. #20274225	C₂	semidirect
5. #20274201	C₂	semidirect
6. #20273478	C₂	semidirect
7. #20273415	C₂	semidirect
8. C₅₃×#3170	C₂	semidirect
23. #16810376	C₂²	semidirect
24. #16810374	C₂²	semidirect
25. #16810369	C₂²	semidirect
26. #16810369	C₂²	semidirect
28. C₅₃×#500	C₂²	semidirect
29. C₅₃×#494	C₂²	semidirect
30. C₅₃×#487	C₂²	semidirect
176. #3170	D₅₃	semidirect
192. #500	C₂×D₅₃	semidirect
193. #494	C₂×D₅₃	semidirect
194. #487	C₂×D₅₃	semidirect
216. C₅₃	#19284	semidirect

Maximal subgroups

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

subgroup	index
#20284810	2
C₂×#16810702	2
#20274259	2
#20274225	2
#20274201	2
#20273478	2
#20273415	2
C₅₃×#3170	2
#19284	53