Subgroups — #27273570

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810679	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810675	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
4	#20272550	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272549	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	#16810679	—	2	2	no	—
10	#16810679	C₄	1	4	yes	—
11	#16810675	—	2	2	no	—
12	#16810675	C₄	1	4	yes	—
13	#16810674	—	2	2	no	—
14	#16810674	—	2	2	no	—
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×#13133146	C₄	1	2	yes	—
18	C₂×#13133129	—	2	1	no	—
19	C₂×#13133129	—	2	1	no	—
20	C₂×#13133129	C₄	1	2	yes	—
21	C₂×#13133127	—	2	1	no	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	—	2	1	no	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	—	2	1	no	—
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
36	C₄²×D₅₃	C₂×C₄	1	2	yes	—
37	C₄²×D₅₃	C₂×C₄	1	2	yes	—
38	C₄²×D₅₃	D₄	1	2	yes	—
39	C₄²×D₅₃	D₄	1	2	yes	—
40	C₄⋊#88322	—	2	1	no	—
41	C₄⋊#88322	—	2	1	no	—
42	C₄⋊#88322	C₂×C₄	1	2	yes	—
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	#13133146	—	2	4	no	—
45	#13133129	—	2	4	no	—
46	#13133129	—	2	4	no	—
47	#13133129	—	2	4	no	—
48	#13133127	—	2	4	no	—
49	C₄×#88322	—	2	1	no	—
50	C₄×#88322	—	2	1	no	—
51	C₈×Dic₅₃	—	2	4	no	—
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	1	no	—
57	C₄⋊C₈×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	—	2	2	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×C₄×D₅₃	—	2	1	no	—
74	C₂×C₄×D₅₃	—	2	1	no	—
75	C₂×C₄×D₅₃	—	4	2	no	—
76	C₂×C₄×D₅₃	C₄²	1	2	yes	—
77	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄×Dic₅₃	C₄²	1	2	yes	—
96	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
97	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
98	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
99	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
100	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	—	2	2	no	—
106	C₂×#88322	—	4	1	no	—
107	C₂×#88322	—	4	1	no	—
108	C₂×#88322	—	4	1	no	—
109	C₂×#88322	—	4	1	no	—
110	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×D₅₃	—	2	2	no	—
112	C₂²×D₅₃	—	2	1	no	—
113	C₂²×D₅₃	—	2	1	no	—
114	C₂²×D₅₃	—	2	2	no	—
115	C₂²×D₅₃	#95	1	2	yes	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#131	1	2	yes	—
124	C₂×C₄×C₅₃	#131	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	1	no	—
131	C₂×Dic₅₃	—	2	1	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	#95	1	2	yes	—
134	C₂×Dic₅₃	#95	1	2	yes	—
135	C₂×Dic₅₃	#95	1	2	yes	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	4	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	2	2	no	—
143	C₄×D₅₃	—	4	2	no	—
144	C₄×D₅₃	—	4	2	no	—
145	C₈×C₅₃	—	4	2	no	—
146	C₈×C₅₃	—	4	2	no	—
147	C₈×C₅₃	—	4	2	no	—
148	#88322	—	4	2	no	—
149	#88322	—	4	2	no	—
150	#88322	—	4	2	no	—
151	#7878	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
153	C₂²×C₅₃	#424	1	2	yes	—
154	C₂²×C₅₃	#427	1	1	yes	characteristic
155	C₂²×C₅₃	#427	1	1	yes	characteristic
156	C₂²×C₅₃	#430	1	2	yes	—
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	2	2	no	—
159	C₂×D₅₃	—	2	2	no	—
160	C₂×D₅₃	—	4	2	no	—
161	C₂×D₅₃	—	4	2	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	#405	1	1	yes	characteristic
165	C₄×C₅₃	#405	1	1	yes	characteristic
166	C₄×C₅₃	#408	1	2	yes	—
167	C₄×C₅₃	#416	1	1	yes	characteristic
168	C₄×C₅₃	#416	1	1	yes	characteristic
169	C₄×C₅₃	#416	1	1	yes	characteristic
170	C₄×C₅₃	#416	1	1	yes	characteristic
171	Dic₅₃	—	2	2	no	—
172	Dic₅₃	—	2	2	no	—
173	Dic₅₃	—	2	2	no	—
174	C₂×#432	—	53	1	no	—
175	C₂×#431	—	53	1	no	—
176	C₂×#431	—	53	1	no	—
177	#1277	—	53	1	no	—
178	#1277	—	53	1	no	—
179	#1277	—	53	1	no	—
180	#1277	D₅₃	1	1	yes	characteristic, complemented
181	C₂×C₅₃	#1572	1	2	yes	—
182	C₂×C₅₃	#1572	1	2	yes	—
183	C₂×C₅₃	#1729	1	1	yes	characteristic
184	C₂×C₅₃	#1735	1	1	yes	characteristic
185	C₂×C₅₃	#1738	1	1	yes	characteristic
186	D₅₃	—	4	2	no	—
187	C₂²×C₄²	—	53	1	no	—
188	#432	—	53	4	no	—
189	#432	—	106	2	no	—
190	#431	—	53	4	no	—
191	#431	—	106	2	no	—
192	#431	—	106	2	no	—
193	#431	—	106	2	no	—
194	C₂×C₄⋊C₈	—	53	1	no	—
195	C₂×C₄⋊C₈	—	53	1	no	—
196	C₂×C₄⋊C₈	—	53	2	no	—
197	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
199	C₂×C₂²⋊C₈	—	53	2	no	—
200	C₂×C₂²⋊C₈	—	106	1	no	—
201	C₂×C₂²⋊C₈	—	106	1	no	—
202	C₂×#97	—	106	1	no	—
203	C₂×C₄×C₈	—	53	1	no	—
204	C₂×C₄×C₈	—	106	1	no	—
205	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
206	C₅₃	#7878	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₄	—	53	1	no	—
208	C₂³×C₄	—	53	1	no	—
209	C₂³×C₄	—	53	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	53	2	no	—
213	C₂²×C₈	—	106	1	no	—
214	C₂²×C₈	—	106	1	no	—
215	C₂²×C₈	C₄×D₅₃	1	2	yes	—
216	C₂×C₄²	—	53	1	no	—
217	C₂×C₄²	—	53	1	no	—
218	C₂×C₄²	—	53	1	no	—
219	C₂×C₄²	—	53	2	no	—
220	C₂×C₄²	—	53	2	no	—
221	C₂×C₄²	—	53	2	no	—
222	C₂×C₄²	—	53	2	no	—
223	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	53	2	no	—
227	C₄⋊C₈	—	53	2	no	—
228	C₄⋊C₈	—	106	4	no	—
229	C₄⋊C₈	—	106	1	no	—
230	C₄⋊C₈	—	106	1	no	—
231	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
232	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
233	C₂²⋊C₈	—	106	4	no	—
234	C₂²⋊C₈	—	106	4	no	—
235	C₂²⋊C₈	—	106	4	no	—
236	#97	—	106	4	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	106	4	no	—
240	C₄×C₈	—	106	1	no	—
241	C₄×C₈	—	106	1	no	—
242	C₂⁴	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	53	2	no	—
249	C₂²×C₄	—	53	2	no	—
250	C₂²×C₄	—	53	2	no	—
251	C₂²×C₄	—	53	2	no	—
252	C₂²×C₄	—	106	2	no	—
253	C₂²×C₄	—	106	2	no	—
254	C₂²×C₄	—	106	1	no	—
255	C₂²×C₄	—	106	1	no	—
256	C₂²×C₄	—	106	1	no	—
257	C₂²×C₄	—	106	1	no	—
258	C₂²×C₄	—	212	2	no	—
259	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
261	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	—	106	2	no	—
274	C₂×C₈	—	106	2	no	—
275	C₂×C₈	—	106	2	no	—
276	C₂×C₈	—	212	1	no	—
277	C₂×C₈	—	212	1	no	—
278	C₂×C₈	—	212	1	no	—
279	C₂×C₈	—	212	1	no	—
280	C₄²	—	53	2	no	—
281	C₄²	—	53	2	no	—
282	C₄²	—	53	2	no	—
283	C₄²	—	53	2	no	—
284	C₄²	—	53	2	no	—
285	C₄²	—	53	2	no	—
286	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₄²	D₄×D₅₃	1	1	yes	characteristic
289	C₄²	D₄×D₅₃	1	1	yes	characteristic
290	C₂³	—	53	2	no	—
291	C₂³	—	106	2	no	—
292	C₂³	—	106	1	no	—
293	C₂³	—	106	1	no	—
294	C₂³	—	106	2	no	—
295	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	106	2	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	4	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	106	2	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	—	212	2	no	—
313	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
316	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
317	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
318	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
319	C₂×C₄	#13133223	1	2	yes	—
320	C₂×C₄	#13133225	1	2	yes	—
321	C₂×C₄	#13133236	1	1	yes	characteristic
322	C₂×C₄	#13133238	1	1	yes	characteristic
323	C₂×C₄	#13133238	1	1	yes	characteristic
324	C₂×C₄	#13133240	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₈	—	212	2	no	—
329	C₈	—	212	2	no	—
330	C₈	—	212	2	no	—
331	C₂²	—	106	2	no	—
332	C₂²	—	106	2	no	—
333	C₂²	—	106	2	no	—
334	C₂²	—	212	2	no	—
335	C₂²	—	212	2	no	—
336	C₂²	#95×D₅₃	1	1	yes	characteristic
337	C₂²	#16810648	1	2	yes	—
338	C₂²	#16810658	1	1	yes	characteristic
339	C₂²	#16810660	1	1	yes	characteristic
340	C₂²	#16810670	1	2	yes	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	106	2	no	—
344	C₄	—	106	2	no	—
345	C₄	—	106	2	no	—
346	C₄	#16810577	1	1	yes	characteristic
347	C₄	#16810580	1	1	yes	characteristic
348	C₄	#16810588	1	2	yes	—
349	C₄	#16810616	1	1	yes	characteristic
350	C₄	#16810616	1	1	yes	characteristic
351	C₄	#16810618	1	1	yes	characteristic
352	C₄	#16810618	1	1	yes	characteristic
353	C₂	—	212	2	no	—
354	C₂	#20273492	1	2	yes	—
355	C₂	#20273494	1	2	yes	—
356	C₂	#20274410	1	1	yes	characteristic
357	C₂	#20274430	1	1	yes	characteristic
358	C₂	#20274440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1277	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1277	D₅₃
hypocenter	C₅₃	#7878
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #20272550	C₂	semidirect
5. #20272549	C₂	semidirect
6. #20272549	C₂	semidirect
7. C₅₃×#1277	C₂	semidirect
180. #1277	D₅₃	semidirect
206. C₅₃	#7878	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810679	2
C₂×#16810675	2
C₂×#16810674	2
#20272550	2
#20272549	2
C₅₃×#1277	2
#7878	53