Subgroups — C₅₃×#22745 #27230257

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2355	C₂	1	2	yes	maximal
3	C₅₃×#1949	C₂	1	2	yes	maximal
4	C₅₃×#1949	C₂	1	2	yes	maximal
5	C₅₃×#1948	C₂	1	2	yes	maximal
6	C₅₃×#1836	C₂	1	2	yes	maximal
7	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1776	C₂	1	2	yes	maximal
9	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#446	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#106×C₅₃	C₂²	1	2	yes	—
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#391	—	2	2	no	—
20	C₅₃×#387	—	2	2	no	—
21	C₅₃×#387	—	2	2	no	—
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₅₃×#382	C₂²	1	2	yes	—
24	C₅₃×#382	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#340	C₂²	1	1	yes	characteristic
30	C₅₃×#340	C₂²	1	2	yes	—
31	C₅₃×#340	C₂²	1	1	yes	characteristic
32	C₅₃×#340	C₂²	1	2	yes	—
33	C₅₃×#340	C₂²	1	1	yes	characteristic
34	C₅₃×#340	C₂²	1	1	yes	characteristic
35	C₅₃×#340	C₂²	1	2	yes	—
36	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
55	#107×C₅₃	—	2	4	no	—
56	#106×C₅₃	—	2	4	no	—
57	#106×C₅₃	—	2	4	no	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
60	#97×C₅₃	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
62	#95×C₅₃	—	2	2	no	—
63	#95×C₅₃	—	2	2	no	—
64	#95×C₅₃	—	2	2	no	—
65	#95×C₅₃	—	2	2	no	—
66	#95×C₅₃	—	2	2	no	—
67	#95×C₅₃	—	2	2	no	—
68	#95×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	—	4	1	no	—
70	C₂×Q₈×C₅₃	—	4	1	no	—
71	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	4	4	no	—
96	C₄⋊C₄×C₅₃	—	8	2	no	—
97	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
98	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₅₃	#41	1	4	yes	—
100	C₄²×C₅₃	—	4	4	no	—
101	C₄²×C₅₃	—	8	2	no	—
102	C₄²×C₅₃	—	8	2	no	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
105	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
106	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
107	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
135	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
136	C₈×C₅₃	—	2	2	no	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	2	2	no	—
139	#22745	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₅₃	#548	1	2	yes	—
142	C₂²×C₅₃	#575	1	1	yes	characteristic
143	C₂²×C₅₃	#575	1	1	yes	characteristic
144	C₂²×C₅₃	#577	1	2	yes	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	4	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	4	no	—
149	C₄×C₅₃	—	4	4	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	—	8	2	no	—
153	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
154	C₄×C₅₃	#519	1	1	yes	characteristic
155	C₄×C₅₃	#557	1	2	yes	—
156	C₂×#446	C₂×C₅₃	1	1	yes	characteristic
157	#2355	C₂×C₅₃	1	2	yes	—
158	#1949	C₂×C₅₃	1	2	yes	—
159	#1949	C₂×C₅₃	1	2	yes	—
160	#1948	C₂×C₅₃	1	2	yes	—
161	#1836	C₂×C₅₃	1	2	yes	—
162	#1835	C₂×C₅₃	1	1	yes	characteristic
163	#1776	C₂×C₅₃	1	2	yes	—
164	C₄×#106	C₂×C₅₃	1	1	yes	characteristic
165	C₂×C₅₃	#2351	1	1	yes	characteristic
166	C₂×C₅₃	#2911	1	1	yes	characteristic
167	C₂×C₅₃	#2912	1	1	yes	characteristic
168	C₂×C₅₃	#3371	1	2	yes	—
169	C₂×C₅₃	#3376	1	2	yes	—
170	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
171	#446	C₄×C₅₃	1	8	yes	complemented
172	C₂×#107	C₂²×C₅₃	1	2	yes	—
173	C₂×#106	C₂²×C₅₃	1	2	yes	—
174	C₂×#106	C₂²×C₅₃	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×#97	C₂²×C₅₃	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	#391	—	2	2	no	—
180	#387	—	2	2	no	—
181	#387	—	2	2	no	—
182	#384	C₂²×C₅₃	1	2	yes	—
183	#382	C₂²×C₅₃	1	2	yes	—
184	#382	C₂²×C₅₃	1	2	yes	—
185	C₄×C₄⋊C₄	—	2	2	no	—
186	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
189	#340	C₂²×C₅₃	1	2	yes	—
190	#340	C₂²×C₅₃	1	1	yes	characteristic
191	#340	C₂²×C₅₃	1	1	yes	characteristic
192	#340	C₂²×C₅₃	1	1	yes	characteristic
193	#340	C₂²×C₅₃	1	2	yes	—
194	#340	C₂²×C₅₃	1	2	yes	—
195	#340	C₂²×C₅₃	1	1	yes	characteristic
196	C₅₃	#22745	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
201	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
202	C₄×Q₈	Q₈×C₅₃	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
206	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	#106	—	2	4	no	—
219	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
220	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
221	#97	C₂×C₄×C₅₃	1	8	yes	—
222	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
259	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
260	C₄⋊C₄	#41×C₅₃	1	4	yes	—
261	C₄²	—	4	4	no	—
262	C₄²	—	8	2	no	—
263	C₄²	—	8	2	no	—
264	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
265	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
266	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
267	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
268	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
296	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
297	C₈	—	2	2	no	—
298	C₈	—	2	4	no	—
299	C₈	—	2	2	no	—
300	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
301	C₂²	C₅₃×#548	1	2	yes	—
302	C₂²	C₅₃×#575	1	1	yes	characteristic
303	C₂²	C₅₃×#575	1	1	yes	characteristic
304	C₂²	C₅₃×#577	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
314	C₄	C₅₃×#519	1	1	yes	characteristic
315	C₄	C₅₃×#557	1	2	yes	—
316	C₂	C₅₃×#2351	1	1	yes	characteristic
317	C₂	C₅₃×#2911	1	1	yes	characteristic
318	C₂	C₅₃×#2912	1	1	yes	characteristic
319	C₂	C₅₃×#3371	1	2	yes	—
320	C₂	C₅₃×#3376	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#446	C₄	semidirect
139. #22745	C₅₃	direct (N × H)
171. #446	C₄×C₅₃	semidirect
196. C₅₃	#22745	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#446	2
C₅₃×#2355	2
C₅₃×#1949	2
C₅₃×#1948	2
C₅₃×#1836	2
C₅₃×#1835	2
C₅₃×#1776	2
C₄×#106×C₅₃	2
#22745	53

Subgroups — C53×#22745 #27230257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22745 #27230257