Subgroups — C₅₃×#20784 #27228296

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1839	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1831	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1818	C₂	1	2	yes	maximal
5	C₅₃×#1798	C₂	1	2	yes	maximal
6	C₅₃×#1768	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1746	C₂	1	2	yes	maximal, complemented
10	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
20	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
22	C₅₃×#376	C₂²	1	1	yes	characteristic
23	C₅₃×#376	C₂²	1	1	yes	characteristic
24	C₄³×C₅₃	C₂²	1	1	yes	characteristic
25	C₄³×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#336	C₂²	1	4	yes	—
27	C₅₃×#336	C₂²	1	4	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
35	#117×C₅₃	—	2	4	no	—
36	#117×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	—	2	4	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	8	no	—
56	C₂²⋊C₈×C₅₃	—	2	4	no	—
57	C₂²⋊C₈×C₅₃	—	2	4	no	—
58	#97×C₅₃	—	2	4	no	—
59	C₄×C₈×C₅₃	—	2	4	no	—
60	#95×C₅₃	C₂×C₄	1	2	yes	—
61	#95×C₅₃	C₂×C₄	1	2	yes	—
62	#95×C₅₃	C₂×C₄	1	2	yes	—
63	#95×C₅₃	C₂×C₄	1	2	yes	—
64	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	#41	1	4	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	4	2	no	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	#41	1	4	yes	—
103	C₄²×C₅₃	#41	1	4	yes	—
104	C₂³×C₅₃	—	2	1	no	—
105	C₂³×C₅₃	—	2	1	no	—
106	C₂³×C₅₃	—	2	2	no	—
107	C₂³×C₅₃	—	2	2	no	—
108	C₂³×C₅₃	#117	1	2	yes	—
109	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	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	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	8	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	8	no	—
131	C₂×C₄×C₅₃	#117	1	4	yes	—
132	C₂×C₄×C₅₃	#117	1	2	yes	—
133	C₂×C₄×C₅₃	#117	1	4	yes	—
134	C₂×C₄×C₅₃	#131	1	4	yes	—
135	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
138	C₈×C₅₃	—	4	4	no	—
139	C₈×C₅₃	—	4	4	no	—
140	#20784	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	4	2	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×#131	1	2	yes	—
150	C₂²×C₅₃	#568	1	2	yes	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	—	2	4	no	—
153	C₄×C₅₃	—	2	4	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	4	1	no	—
156	C₄×C₅₃	—	4	1	no	—
157	C₄×C₅₃	—	4	1	no	—
158	C₄×C₅₃	—	4	1	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	#529	1	4	yes	—
161	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
162	#1839	C₂×C₅₃	1	2	yes	complemented
163	#1831	C₂×C₅₃	1	2	yes	complemented
164	#1818	C₂×C₅₃	1	2	yes	—
165	#1798	C₂×C₅₃	1	2	yes	—
166	#1768	C₂×C₅₃	1	2	yes	complemented
167	#1749	C₂×C₅₃	1	1	yes	characteristic
168	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic
169	#1746	C₂×C₅₃	1	2	yes	complemented
170	C₂×C₅₃	—	4	2	no	—
171	C₂×C₅₃	#2283	1	1	yes	characteristic
172	C₂×C₅₃	#2922	1	1	yes	characteristic
173	C₂×C₅₃	#2923	1	1	yes	characteristic
174	C₂×C₅₃	#2924	1	2	yes	—
175	C₂×C₅₃	#2925	1	2	yes	—
176	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
180	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
181	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
182	C₂×#97	C₂²×C₅₃	1	2	yes	—
183	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
184	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
185	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
186	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
187	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
188	#376	C₂²×C₅₃	1	1	yes	characteristic
189	#376	C₂²×C₅₃	1	1	yes	characteristic
190	C₄³	C₂²×C₅₃	1	1	yes	characteristic
191	C₄³	C₂²×C₅₃	1	1	yes	characteristic
192	#336	C₂²×C₅₃	1	4	yes	—
193	#336	C₂²×C₅₃	1	4	yes	—
194	#336	C₂²×C₅₃	1	4	yes	—
195	#336	C₂²×C₅₃	1	2	yes	—
196	#336	C₂²×C₅₃	1	2	yes	—
197	C₅₃	#20784	1	1	yes	characteristic, direct product factor, Sylow
198	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
199	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
201	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
202	#117	—	2	4	no	—
203	#117	—	2	4	no	—
204	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
212	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
213	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
214	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
215	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
216	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	8	no	—
223	C₂²⋊C₈	—	2	4	no	—
224	C₂²⋊C₈	—	2	4	no	—
225	#97	—	2	4	no	—
226	C₄×C₈	—	2	4	no	—
227	#95	C₂×C₄×C₅₃	1	2	yes	—
228	#95	C₂×C₄×C₅₃	1	2	yes	—
229	#95	C₂×C₄×C₅₃	1	2	yes	—
230	#95	C₂×C₄×C₅₃	1	2	yes	—
231	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
246	C₂²×C₄	#41×C₅₃	1	4	yes	—
247	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	#41×C₅₃	1	4	yes	—
270	C₄²	#41×C₅₃	1	4	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	#117×C₅₃	1	2	yes	—
276	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	#117×C₅₃	1	4	yes	—
299	C₂×C₄	#117×C₅₃	1	4	yes	—
300	C₂×C₄	#117×C₅₃	1	2	yes	—
301	C₂×C₄	#131×C₅₃	1	4	yes	—
302	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
303	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
304	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
305	C₈	—	4	4	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
313	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
314	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
315	C₂²	C₂×#131×C₅₃	1	2	yes	—
316	C₂²	C₅₃×#568	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	4	no	—
320	C₄	—	2	4	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	2	no	—
326	C₄	C₅₃×#529	1	4	yes	—
327	C₂	—	4	2	no	—
328	C₂	C₅₃×#2283	1	1	yes	characteristic
329	C₂	C₅₃×#2922	1	1	yes	characteristic
330	C₂	C₅₃×#2923	1	1	yes	characteristic
331	C₂	C₅₃×#2924	1	2	yes	—
332	C₂	C₅₃×#2925	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1839	C₂	semidirect
3. C₅₃×#1831	C₂	semidirect
6. C₅₃×#1768	C₂	semidirect
9. C₅₃×#1746	C₂	semidirect
140. #20784	C₅₃	direct (N × H)
162. #1839	C₂×C₅₃	semidirect
163. #1831	C₂×C₅₃	semidirect
166. #1768	C₂×C₅₃	semidirect
169. #1746	C₂×C₅₃	semidirect
197. C₅₃	#20784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#117×C₅₃	2
C₅₃×#1839	2
C₅₃×#1831	2
C₅₃×#1818	2
C₅₃×#1798	2
C₅₃×#1768	2
C₅₃×#1749	2
C₄×C₂²⋊C₈×C₅₃	2
C₅₃×#1746	2
#20784	53

Subgroups — C53×#20784 #27228296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20784 #27228296