Subgroups — #27349817

Order 13568.

814 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	#20283681	C₂	1	1	yes	maximal, characteristic
2	#20283676	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133225	C₂	1	1	yes	maximal, characteristic
4	#20273019	C₂	1	2	yes	maximal, complemented
5	#20273011	C₂	1	2	yes	maximal, complemented
6	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
7	#20272999	C₂	1	1	yes	maximal, characteristic, complemented
8	#20272994	C₂	1	2	yes	maximal, complemented
9	#20272994	C₂	1	2	yes	maximal, complemented
10	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	C₄	1	2	yes	—
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	#16810753	C₄	1	2	yes	—
15	#16810750	C₄	1	2	yes	—
16	#16810702	C₄	1	2	yes	—
17	#16810700	C₄	1	2	yes	—
18	#16810694	C₄	1	4	yes	—
19	#16810691	C₄	1	4	yes	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810674	C₂²	1	2	yes	—
22	#16810674	C₂²	1	2	yes	—
23	#16810581	C₄	1	2	yes	—
24	#16810580	C₄	1	4	yes	—
25	C₄×#11719912	C₂²	1	2	yes	—
26	C₄×#11719912	C₂²	1	2	yes	—
27	C₄×#11719912	C₂²	1	2	yes	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅₃	C₂²	1	2	yes	—
31	#16810345	C₂²	1	2	yes	—
32	#16810345	C₂²	1	2	yes	—
33	#16810345	C₂²	1	2	yes	—
34	C₈×#88322	C₂²	1	1	yes	characteristic
35	C₈×#88322	C₂²	1	1	yes	characteristic
36	C₈×#88322	C₂²	1	1	yes	characteristic
37	C₈×#88322	C₂²	1	1	yes	characteristic
38	C₈×#88322	C₂²	1	2	yes	—
39	C₅₃×#431	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	2	yes	—
43	C₅₃×#322	C₂²	1	2	yes	—
44	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
45	#13133225	—	2	8	no	—
46	C₂×#11719912	C₂×C₄	1	4	yes	—
47	C₂×#11719912	C₂×C₄	1	4	yes	—
48	C₂×#11719912	C₂×C₄	1	4	yes	—
49	C₂²×#88322	C₂×C₄	1	2	yes	—
50	C₂²×#88322	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
54	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
56	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
57	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
58	#13133176	C₂×C₄	1	1	yes	characteristic
59	#13133174	C₂×C₄	1	2	yes	—
60	C₄²×D₅₃	C₂³	1	2	yes	—
61	C₄⋊#88322	C₂×C₄	1	2	yes	—
62	#13133146	C₂×C₄	1	4	yes	—
63	#13133146	C₂×C₄	1	4	yes	—
64	C₂²⋊#88322	C₂×C₄	1	2	yes	—
65	#13133129	C₂×C₄	1	4	yes	—
66	#13133129	C₂×C₄	1	4	yes	—
67	#13133127	C₂×C₄	1	4	yes	—
68	#13133127	C₂×C₄	1	4	yes	—
69	#13133127	C₂×C₄	1	4	yes	—
70	C₄×#88322	C₂×C₄	1	2	yes	—
71	C₄×#88322	C₂×C₄	1	2	yes	—
72	C₄×#88322	C₂³	1	1	yes	characteristic
73	C₄×#88322	C₂³	1	1	yes	characteristic
74	C₄×#88322	C₂³	1	1	yes	characteristic
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₄×#88322	C₂³	1	2	yes	—
77	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
79	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
82	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
83	#97×C₅₃	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	2	yes	—
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₃	C₄²	1	2	yes	—
93	#11719945	—	2	4	no	—
94	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
99	M₁₆×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	—	2	4	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₅₃	C₄²	1	2	yes	—
110	#11719921	C₄²	1	2	yes	—
111	#11719921	C₄²	1	2	yes	—
112	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
113	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
114	#11719919	C₄²	1	2	yes	—
115	#11719919	C₄²	1	2	yes	—
116	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
119	C₂×#88322	—	2	4	no	—
120	C₂×#88322	M₁₆	1	8	yes	complemented
121	C₂×#88322	C₂²×C₄	1	2	yes	—
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂×#88322	C₂²×C₄	1	2	yes	—
124	C₂×#88322	C₂²×C₄	1	2	yes	—
125	C₂×#88322	C₂²×C₄	1	2	yes	—
126	#11719912	—	2	8	no	—
127	#11719912	—	2	8	no	—
128	#11719912	—	2	8	no	—
129	C₈×D₅₃	—	2	8	no	—
130	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
131	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	C₂²⋊D₅₃	—	2	4	no	—
142	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
143	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
144	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
145	D₂₁₂	—	2	2	no	—
146	C₄×D₅₃	—	2	4	no	—
147	C₄×D₅₃	—	2	4	no	—
148	C₄×D₅₃	—	2	4	no	—
149	Dic₁₀₆	—	2	2	no	—
150	C₈×C₅₃	—	2	4	no	—
151	C₈×C₅₃	—	2	4	no	—
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	#131	1	4	yes	complemented
154	#88322	—	2	4	no	—
155	#88322	—	2	2	no	—
156	#88322	C₂×M₁₆	1	4	yes	—
157	#88322	C₂×M₁₆	1	4	yes	—
158	#88322	#131	1	4	yes	complemented
159	#17335	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
163	C₂²×C₅₃	C₂²×C₄²	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	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
170	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
171	C₄×C₅₃	C₂×#131	1	2	yes	—
172	C₄×C₅₃	C₂×#131	1	2	yes	—
173	Dic₅₃	—	2	2	no	—
174	Dic₅₃	—	2	2	no	—
175	Dic₅₃	—	2	2	no	—
176	#2965	—	53	1	no	—
177	#2965	—	53	1	no	—
178	C₄×#131	—	53	1	no	—
179	#1455	—	53	2	no	—
180	#1453	—	53	2	no	—
181	#1451	—	53	1	no	—
182	#1451	D₅₃	1	1	yes	characteristic, complemented
183	#1450	—	53	1	no	—
184	C₈×M₁₆	—	53	2	no	—
185	C₈×M₁₆	—	53	2	no	—
186	C₂×C₈²	—	53	1	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
189	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
190	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
191	D₅₃	—	4	2	no	—
192	C₂×#131	—	53	2	no	—
193	C₄×#41	—	53	1	no	—
194	#446	—	53	2	no	—
195	#446	—	53	2	no	—
196	#436	—	53	2	no	—
197	#436	—	53	2	no	—
198	#435	—	53	4	no	—
199	#435	—	53	4	no	—
200	#431	—	53	1	no	—
201	#431	—	53	2	no	—
202	#431	—	53	2	no	—
203	#431	C₂×D₅₃	1	1	yes	characteristic
204	#405	—	53	2	no	—
205	#405	—	53	4	no	—
206	C₄×M₁₆	—	53	2	no	—
207	C₄×M₁₆	—	53	2	no	—
208	C₄×M₁₆	—	53	2	no	—
209	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₂×C₄×C₈	—	53	1	no	—
211	C₂×C₄×C₈	—	53	1	no	—
212	C₂×C₄×C₈	—	53	2	no	—
213	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
214	#322	—	53	2	no	—
215	#322	—	53	2	no	—
216	#322	—	53	2	no	—
217	#322	C₂×D₅₃	1	2	yes	—
218	#322	C₂×D₅₃	1	2	yes	—
219	C₈²	—	53	1	no	—
220	C₈²	—	53	1	no	—
221	C₈²	—	53	1	no	—
222	C₈²	—	53	1	no	—
223	C₈²	—	53	2	no	—
224	C₅₃	#17335	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	—	53	1	no	—
226	#131	—	106	8	no	—
227	C₂×M₁₆	—	53	4	no	—
228	C₂×M₁₆	—	53	4	no	—
229	C₂×M₁₆	—	53	4	no	—
230	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
231	C₂²×C₈	—	53	2	no	—
232	C₂²×C₈	—	53	2	no	—
233	C₂²×C₈	—	53	4	no	—
234	C₂²×C₈	C₄×D₅₃	1	2	yes	—
235	C₄×Q₈	—	53	1	no	—
236	C₄×Q₈	—	53	1	no	—
237	C₄×D₄	—	53	2	no	—
238	C₄×D₄	—	53	2	no	—
239	C₄×D₄	—	53	1	no	—
240	C₄×D₄	—	53	1	no	—
241	#117	—	53	1	no	—
242	#117	—	53	2	no	—
243	C₂×C₄²	—	53	2	no	—
244	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	C₄⋊C₈	—	53	4	no	—
246	C₄⋊C₈	—	53	4	no	—
247	C₄⋊C₈	—	53	2	no	—
248	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
249	C₂²⋊C₈	—	53	2	no	—
250	C₂²⋊C₈	—	53	4	no	—
251	C₂²⋊C₈	—	53	4	no	—
252	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
253	#97	—	53	4	no	—
254	#97	—	53	4	no	—
255	#97	—	53	4	no	—
256	#97	C₄×D₅₃	1	2	yes	—
257	C₄×C₈	—	53	4	no	—
258	C₄×C₈	—	53	2	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	1	no	—
264	C₄×C₈	—	53	2	no	—
265	C₄×C₈	C₄×D₅₃	1	2	yes	—
266	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₅₃	1	2	yes	—
269	C₄×C₈	C₂²×D₅₃	1	2	yes	—
270	#41	—	106	4	no	—
271	C₂×Q₈	—	53	1	no	—
272	C₂×D₄	—	53	2	no	—
273	C₂×D₄	—	53	1	no	—
274	C₂²×C₄	—	53	2	no	—
275	C₂²×C₄	—	53	2	no	—
276	C₂²×C₄	—	53	2	no	—
277	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
280	M₁₆	—	2	4	no	—
281	M₁₆	—	106	8	no	—
282	M₁₆	—	106	8	no	—
283	M₁₆	—	106	8	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	53	8	no	—
286	C₂×C₈	—	53	2	no	—
287	C₂×C₈	—	53	2	no	—
288	C₂×C₈	—	53	2	no	—
289	C₂×C₈	—	53	2	no	—
290	C₂×C₈	—	53	4	no	—
291	C₂×C₈	—	106	8	no	—
292	C₂×C₈	—	106	4	no	—
293	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
294	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
296	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
297	C₄⋊C₄	—	53	2	no	—
298	C₄⋊C₄	—	53	2	no	—
299	C₄⋊C₄	—	53	1	no	—
300	C₄⋊C₄	—	53	1	no	—
301	C₂²⋊C₄	—	53	2	no	—
302	C₂²⋊C₄	—	53	2	no	—
303	C₂²⋊C₄	—	53	2	no	—
304	C₄²	—	53	2	no	—
305	C₄²	—	53	2	no	—
306	C₄²	—	53	2	no	—
307	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₄²	C₂³×D₅₃	1	1	yes	characteristic
311	C₂³	—	53	2	no	—
312	C₂³	C₄²×D₅₃	1	1	yes	characteristic
313	Q₈	—	106	2	no	—
314	D₄	—	106	4	no	—
315	D₄	—	106	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	53	2	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	106	4	no	—
322	C₂×C₄	—	106	4	no	—
323	C₂×C₄	—	106	4	no	—
324	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
326	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	2	4	no	—
333	C₈	—	2	2	no	—
334	C₈	—	53	4	no	—
335	C₈	—	53	4	no	—
336	C₈	—	53	4	no	—
337	C₈	—	106	2	no	—
338	C₈	—	106	4	no	—
339	C₈	#13133225	1	4	yes	complemented
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	106	2	no	—
343	C₂²	—	106	2	no	—
344	C₂²	—	106	2	no	—
345	C₂²	#16810575	1	2	yes	—
346	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	106	2	no	—
350	C₄	—	106	2	no	—
351	C₄	—	106	2	no	—
352	C₄	#16810575	1	2	yes	—
353	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
354	C₄	#131×D₅₃	1	2	yes	—
355	C₄	C₂×#13133225	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	212	2	no	—
358	C₂	C₂×#16810575	1	1	yes	characteristic
359	C₂	#20283137	1	1	yes	characteristic
360	C₂	C₄×#13133225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1451	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#17335
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #20273019	C₂	semidirect
5. #20273011	C₂	semidirect
6. #20273002	C₂	semidirect
7. #20272999	C₂	semidirect
8. #20272994	C₂	semidirect
9. #20272994	C₂	semidirect
10. C₂×C₈×#88322	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
120. C₂×#88322	M₁₆	semidirect
153. C₈×C₅₃	#131	semidirect
158. #88322	#131	semidirect
182. #1451	D₅₃	semidirect
224. C₅₃	#17335	semidirect
339. C₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20283681	2
#20283676	2
C₄×#13133225	2
#20273019	2
#20273011	2
#20273002	2
#20272999	2
#20272994	2
C₂×C₈×#88322	2
C₅₃×#1451	2
#17335	53