Subgroups — #27537205

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283137	C₂	1	1	yes	maximal, characteristic
2	#20276523	C₂	1	2	yes	maximal, complemented
3	#20276520	C₂	1	2	yes	maximal, complemented
4	#20276516	C₂	1	2	yes	maximal, complemented
5	#20276512	C₂	1	2	yes	maximal, complemented
6	#20276470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810441	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276330	C₂	1	1	yes	maximal, characteristic
9	#20276328	C₂	1	1	yes	maximal, characteristic
10	#20276312	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	C₄	1	1	yes	characteristic
13	C₂×#13133223	C₂²	1	1	yes	characteristic
14	#16811091	C₄	1	1	yes	characteristic
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	C₂²×#11719908	C₂²	1	1	yes	characteristic
17	#16810580	C₄	1	2	yes	—
18	#16810579	C₄	1	1	yes	characteristic
19	#16810577	C₂²	1	2	yes	—
20	#16810575	C₄	1	2	yes	—
21	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₅₃	C₂²	1	2	yes	—
24	#16810441	C₂²	1	1	yes	characteristic
25	#16810441	C₂²	1	2	yes	—
26	#16810441	C₂²	1	1	yes	characteristic
27	#16810441	C₂²	1	1	yes	characteristic
28	#16810441	C₂²	1	2	yes	—
29	#16810441	C₂²	1	1	yes	characteristic
30	#16810402	C₂²	1	2	yes	—
31	#16810402	C₂²	1	2	yes	—
32	#16810402	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	#16810399	C₂²	1	2	yes	—
35	C₄×#11719908	C₂²	1	2	yes	—
36	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#405	C₂²	1	1	yes	characteristic
39	C₅₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
42	#13133225	—	2	4	no	—
43	#13133223	—	2	2	no	—
44	#13133223	—	2	2	no	—
45	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719912	C₂×C₄	1	2	yes	—
47	C₂×#11719912	C₂×C₄	1	2	yes	—
48	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₅₃	C₂³	1	2	yes	—
51	#13133182	C₈	1	1	yes	characteristic
52	#13133182	C₈	1	1	yes	characteristic
53	#13133179	C₈	1	1	yes	characteristic
54	#13133179	C₈	1	1	yes	characteristic
55	#13133177	C₈	1	2	yes	—
56	#13133177	C₈	1	2	yes	—
57	#13133174	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₅₃	C₂×C₄	1	2	yes	—
60	#13133157	—	2	2	no	—
61	C₂×#11719908	—	2	2	no	—
62	C₂×#11719908	C₂³	1	1	yes	characteristic
63	C₂×#11719908	C₂³	1	1	yes	characteristic
64	C₂×#11719908	C₂³	1	2	yes	—
65	#13133128	C₂×C₄	1	2	yes	—
66	#13133127	C₂×C₄	1	2	yes	—
67	C₄×#88322	C₂×C₄	1	2	yes	—
68	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
76	M₃₂×C₅₃	—	2	2	no	—
77	C₂×C₁₆×C₅₃	—	2	2	no	—
78	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
81	#97×C₅₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₅₃	C₂³	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
85	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
86	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
87	#11719945	—	2	2	no	—
88	#11719945	—	2	2	no	—
89	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
91	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
94	M₁₆×C₅₃	M₁₆	1	4	yes	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	#41	1	2	yes	—
98	C₂×C₈×C₅₃	#41	1	2	yes	—
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	#11719921	C₂×C₈	1	2	yes	—
105	#11719921	C₂×C₈	1	2	yes	—
106	#11719919	C₂×C₈	1	2	yes	—
107	#11719919	C₂×C₈	1	2	yes	—
108	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
109	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
110	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
114	#11719917	—	2	2	no	—
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
117	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
118	C₂×#88322	C₂²×C₄	1	2	yes	—
119	#11719912	—	2	4	no	—
120	#11719912	—	2	4	no	—
121	C₈×D₅₃	—	2	4	no	—
122	C₈×D₅₃	—	2	2	no	—
123	C₈×D₅₃	—	2	2	no	—
124	C₁₆×C₅₃	—	2	1	no	—
125	C₁₆×C₅₃	—	2	1	no	—
126	C₁₆×C₅₃	—	2	2	no	—
127	#11719908	—	2	1	no	—
128	#11719908	—	2	1	no	—
129	#11719908	—	2	2	no	—
130	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅₃	#117	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₅₃	—	2	4	no	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	4	1	no	—
141	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
142	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
146	D₂₁₂	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	2	2	no	—
149	C₄×D₅₃	—	2	4	no	—
150	C₄×D₅₃	—	4	2	no	—
151	Dic₁₀₆	—	2	2	no	—
152	C₈×C₅₃	#117	1	2	yes	—
153	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
154	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
156	#88322	—	2	2	no	—
157	#88322	—	2	1	no	—
158	#88322	—	2	1	no	—
159	#33423	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	#431	1	2	yes	—
163	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	2	2	no	—
166	C₂×D₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	#431	1	2	yes	—
170	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
172	Dic₅₃	—	2	1	no	—
173	Dic₅₃	—	2	1	no	—
174	Dic₅₃	—	2	1	no	—
175	Dic₅₃	—	2	2	no	—
176	Dic₅₃	#504	1	2	yes	—
177	C₄×#131	—	53	1	no	—
178	#2164	—	53	2	no	—
179	#2164	—	53	2	no	—
180	#2163	—	53	2	no	—
181	#2162	—	53	2	no	—
182	#2150	—	53	1	no	—
183	C₂×C₄⋊C₁₆	—	53	1	no	—
184	#2113	—	53	1	no	—
185	#2113	—	53	1	no	—
186	#2108	—	53	1	no	—
187	#2108	D₅₃	1	1	yes	characteristic, complemented
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
191	C₂×C₅₃	#3407	1	1	yes	characteristic
192	D₅₃	—	4	2	no	—
193	C₂×#131	—	53	1	no	—
194	C₂×#131	—	53	1	no	—
195	C₄×#41	—	53	1	no	—
196	C₂×M₃₂	—	53	1	no	—
197	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
198	C₂²×C₁₆	—	53	1	no	—
199	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
200	#405	—	53	1	no	—
201	#405	—	53	2	no	—
202	#405	—	53	2	no	—
203	#405	C₂×D₅₃	1	1	yes	characteristic
204	C₄×M₁₆	—	53	1	no	—
205	C₄×M₁₆	—	53	2	no	—
206	C₂×C₄×C₈	—	53	1	no	—
207	C₂×C₄×C₈	—	53	2	no	—
208	C₄⋊C₁₆	—	53	1	no	—
209	C₄⋊C₁₆	—	53	1	no	—
210	C₄⋊C₁₆	—	53	1	no	—
211	C₄⋊C₁₆	—	53	1	no	—
212	C₄⋊C₁₆	—	53	2	no	—
213	C₄⋊C₁₆	—	53	2	no	—
214	C₂²⋊C₁₆	—	53	2	no	—
215	C₂²⋊C₁₆	—	53	2	no	—
216	C₂²⋊C₁₆	—	53	2	no	—
217	C₂²⋊C₁₆	—	53	2	no	—
218	#346	—	53	2	no	—
219	#346	C₂×D₅₃	1	2	yes	—
220	C₄×C₁₆	—	53	2	no	—
221	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
222	C₅₃	#33423	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	53	1	no	—
224	#131	—	106	4	no	—
225	#131	—	106	2	no	—
226	#131	—	106	2	no	—
227	C₂×M₁₆	—	53	1	no	—
228	C₂×M₁₆	—	53	2	no	—
229	C₂×M₁₆	—	53	2	no	—
230	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
231	C₂²×C₈	—	53	1	no	—
232	C₂²×C₈	—	53	2	no	—
233	C₂²×C₈	—	53	2	no	—
234	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄×Q₈	—	53	1	no	—
236	C₄×Q₈	—	53	1	no	—
237	C₄×D₄	—	53	1	no	—
238	C₄×D₄	—	53	1	no	—
239	C₄×D₄	—	53	2	no	—
240	C₄×D₄	—	53	2	no	—
241	#117	—	53	2	no	—
242	#117	C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	—	53	1	no	—
244	C₂×C₄²	—	53	2	no	—
245	M₃₂	—	2	2	no	—
246	M₃₂	—	106	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	53	1	no	—
249	C₂×C₁₆	—	53	2	no	—
250	C₂×C₁₆	—	53	1	no	—
251	C₂×C₁₆	—	106	2	no	—
252	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
254	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
255	#97	—	53	2	no	—
256	#97	—	53	2	no	—
257	#97	C₄×D₅₃	1	2	yes	—
258	C₄×C₈	—	53	1	no	—
259	C₄×C₈	—	53	2	no	—
260	C₄×C₈	—	53	1	no	—
261	C₄×C₈	—	53	1	no	—
262	C₄×C₈	—	53	1	no	—
263	C₄×C₈	—	53	2	no	—
264	C₄×C₈	C₂²×D₅₃	1	2	yes	—
265	#41	—	106	2	no	—
266	#41	—	106	2	no	—
267	C₂×Q₈	—	53	1	no	—
268	C₂×D₄	—	53	2	no	—
269	C₂×D₄	—	53	1	no	—
270	C₂²×C₄	—	53	1	no	—
271	C₂²×C₄	—	53	1	no	—
272	C₂²×C₄	—	53	2	no	—
273	C₂²×C₄	—	53	2	no	—
274	C₂²×C₄	—	53	2	no	—
275	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
276	M₁₆	—	106	2	no	—
277	M₁₆	—	106	4	no	—
278	M₁₆	—	106	4	no	—
279	M₁₆	#11719912	1	4	yes	—
280	C₂×C₈	—	53	1	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	53	1	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	—	106	4	no	—
285	C₂×C₈	—	106	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
289	C₂×C₈	#11719945	1	2	yes	—
290	C₂×C₈	#11719945	1	2	yes	—
291	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
292	C₄⋊C₄	—	53	2	no	—
293	C₄⋊C₄	—	53	2	no	—
294	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
295	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
296	C₂²⋊C₄	—	53	2	no	—
297	C₂²⋊C₄	—	53	2	no	—
298	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
299	C₄²	—	53	1	no	—
300	C₄²	—	53	2	no	—
301	C₄²	—	53	1	no	—
302	C₄²	—	53	2	no	—
303	C₄²	—	53	1	no	—
304	C₄²	—	53	1	no	—
305	C₄²	C₂×C₄×D₅₃	1	2	yes	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	106	1	no	—
310	C₁₆	—	106	1	no	—
311	C₁₆	—	106	2	no	—
312	C₂³	—	53	2	no	—
313	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
314	Q₈	—	106	2	no	—
315	D₄	—	106	4	no	—
316	D₄	—	106	2	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	106	4	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	212	2	no	—
327	C₂×C₄	—	212	1	no	—
328	C₂×C₄	#13133174	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×#11719912	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
334	C₈	—	106	2	no	—
335	C₈	—	106	1	no	—
336	C₈	—	106	1	no	—
337	C₈	#13133174	1	2	yes	—
338	C₈	C₂×#11719912	1	4	yes	—
339	C₈	C₂×#11719945	1	1	yes	characteristic
340	C₈	C₂×#11719945	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	106	2	no	—
344	C₂²	—	106	2	no	—
345	C₂²	—	106	2	no	—
346	C₂²	#16810674	1	2	yes	—
347	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	53	2	no	—
351	C₄	—	106	1	no	—
352	C₄	—	106	1	no	—
353	C₄	—	106	1	no	—
354	C₄	—	106	2	no	—
355	C₄	#16810674	1	2	yes	—
356	C₄	C₂×#13133174	1	1	yes	characteristic
357	C₄	C₂²×#11719912	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	212	2	no	—
360	C₂	C₂×#16810674	1	1	yes	characteristic
361	C₂	#504×D₅₃	1	1	yes	characteristic
362	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2108	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33423
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20276523	C₂	semidirect
3. #20276520	C₂	semidirect
4. #20276516	C₂	semidirect
5. #20276512	C₂	semidirect
6. #20276470	C₂	semidirect
7. C₂×#16810441	C₂	semidirect
10. #20276312	C₂	semidirect
11. C₅₃×#2108	C₂	semidirect
187. #2108	D₅₃	semidirect
222. C₅₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#20283137	2
#20276523	2
#20276520	2
#20276516	2
#20276512	2
#20276470	2
C₂×#16810441	2
#20276330	2
#20276328	2
#20276312	2
C₅₃×#2108	2
#33423	53