Subgroups — C₅₃×#24695 #27232207

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#440	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2563	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2356	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2074	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2055	C₂	1	2	yes	maximal
7	C₅₃×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1949	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1926	C₂	1	2	yes	maximal
11	C₅₃×#1804	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#475	—	2	4	no	—
17	C₅₃×#440	—	2	4	no	—
18	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#106×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₅₃	C₂²	1	2	yes	—
24	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#102×C₅₃	C₂²	1	2	yes	—
26	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#393	—	2	1	no	—
28	C₅₃×#393	—	2	1	no	—
29	C₅₃×#391	C₂²	1	2	yes	—
30	C₅₃×#390	C₂²	1	2	yes	—
31	C₅₃×#387	—	2	2	no	—
32	C₅₃×#383	—	2	1	no	—
33	C₅₃×#382	C₂²	1	2	yes	—
34	C₅₃×#382	C₂²	1	1	yes	characteristic
35	C₅₃×#380	—	2	2	no	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#376	—	2	1	no	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	—	2	1	no	—
47	C₂×SD₁₆×C₅₃	—	2	2	no	—
48	C₂×SD₁₆×C₅₃	—	2	2	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	—	2	2	no	—
54	C₄⋊Q₈×C₅₃	—	2	2	no	—
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	D₄	1	4	yes	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
69	C₂×C₄²×C₅₃	—	2	1	no	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	#107×C₅₃	—	2	2	no	—
74	#106×C₅₃	—	2	4	no	—
75	C₄⋊C₈×C₅₃	—	2	4	no	—
76	C₄⋊C₈×C₅₃	—	2	2	no	—
77	#103×C₅₃	—	2	2	no	—
78	#103×C₅₃	—	2	4	no	—
79	#102×C₅₃	—	2	2	no	—
80	#102×C₅₃	—	2	4	no	—
81	#97×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	1	no	—
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	—	2	2	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	—	2	1	no	—
88	#95×C₅₃	—	2	2	no	—
89	#95×C₅₃	D₄	1	1	yes	characteristic
90	#95×C₅₃	D₄	1	1	yes	characteristic
91	C₂⁴×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	—	2	1	no	—
97	C₂×Q₈×C₅₃	—	4	2	no	—
98	C₂×Q₈×C₅₃	—	8	2	no	—
99	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	—	2	1	no	—
102	C₂×D₄×C₅₃	—	2	1	no	—
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	—	2	1	no	—
105	C₂×D₄×C₅₃	#41	1	2	yes	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	4	4	no	—
112	C₂²×C₄×C₅₃	—	4	2	no	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	SD₁₆×C₅₃	—	2	8	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	#41	1	2	yes	—
128	C₄⋊C₄×C₅₃	—	2	4	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	4	2	no	—
137	C₄⋊C₄×C₅₃	—	4	2	no	—
138	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₅₃	—	4	2	no	—
140	C₂²⋊C₄×C₅₃	—	8	2	no	—
141	C₂²⋊C₄×C₅₃	—	8	2	no	—
142	C₄²×C₅₃	—	2	4	no	—
143	C₄²×C₅₃	—	2	2	no	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	4	4	no	—
146	C₄²×C₅₃	C₂×D₄	1	2	yes	—
147	C₄²×C₅₃	#41	1	2	yes	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	4	1	no	—
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	—	4	2	no	—
152	C₂³×C₅₃	—	4	2	no	—
153	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₅₃	—	2	2	no	—
155	Q₈×C₅₃	—	2	2	no	—
156	Q₈×C₅₃	—	4	1	no	—
157	Q₈×C₅₃	—	4	1	no	—
158	Q₈×C₅₃	—	8	4	no	—
159	D₄×C₅₃	—	2	2	no	—
160	D₄×C₅₃	—	2	2	no	—
161	D₄×C₅₃	—	4	1	no	—
162	D₄×C₅₃	—	4	1	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	4	1	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	8	4	no	—
185	C₂×C₄×C₅₃	#121	1	2	yes	—
186	C₂×C₄×C₅₃	#121	1	2	yes	—
187	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
192	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	4	2	no	—
195	C₈×C₅₃	—	4	1	no	—
196	#24695	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂²×C₅₃	—	4	4	no	—
198	C₂²×C₅₃	—	4	2	no	—
199	C₂²×C₅₃	—	8	2	no	—
200	C₂²×C₅₃	—	8	1	no	—
201	C₂²×C₅₃	—	8	1	no	—
202	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
203	C₂²×C₅₃	#547	1	1	yes	characteristic
204	C₂²×C₅₃	#548	1	1	yes	characteristic
205	C₂²×C₅₃	#575	1	2	yes	—
206	C₂²×C₅₃	#577	1	2	yes	—
207	C₄×C₅₃	—	2	2	no	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	4	2	no	—
210	C₄×C₅₃	—	4	2	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	8	2	no	—
214	C₄×C₅₃	—	8	1	no	—
215	C₄×C₅₃	—	8	2	no	—
216	C₄×C₅₃	#525	1	1	yes	characteristic
217	C₄×C₅₃	#536	1	1	yes	characteristic
218	C₄×C₅₃	#552	1	1	yes	characteristic
219	C₄×C₅₃	#556	1	1	yes	characteristic
220	C₂×#475	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₂×#440	C₂×C₅₃	1	1	yes	characteristic
222	#2563	C₂×C₅₃	1	1	yes	characteristic
223	#2356	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#2074	C₂×C₅₃	1	2	yes	complemented
225	#2055	C₂×C₅₃	1	2	yes	—
226	#1993	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#1949	C₂×C₅₃	1	2	yes	complemented
228	#1940	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#1926	C₂×C₅₃	1	2	yes	—
230	#1804	C₂×C₅₃	1	1	yes	characteristic
231	C₂×C₅₃	—	8	2	no	—
232	C₂×C₅₃	#2415	1	1	yes	characteristic
233	C₂×C₅₃	#3040	1	1	yes	characteristic
234	C₂×C₅₃	#3043	1	1	yes	characteristic
235	C₂×C₅₃	#3348	1	2	yes	—
236	C₂×C₅₃	#3376	1	2	yes	—
237	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
238	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
241	#475	—	2	4	no	—
242	#440	—	2	4	no	—
243	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#106	C₂²×C₅₃	1	2	yes	—
245	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
247	C₂×#103	C₂²×C₅₃	1	2	yes	—
248	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×#102	C₂²×C₅₃	1	2	yes	—
250	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
252	#393	—	2	1	no	—
253	#393	—	2	1	no	—
254	#391	C₂²×C₅₃	1	2	yes	—
255	#390	C₂²×C₅₃	1	2	yes	—
256	#387	—	2	2	no	—
257	#383	—	2	1	no	—
258	#382	C₂²×C₅₃	1	2	yes	—
259	#382	C₂²×C₅₃	1	1	yes	characteristic
260	#380	—	2	2	no	—
261	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
262	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	#376	—	2	1	no	—
264	#340	C₂²×C₅₃	1	2	yes	—
265	#340	C₂²×C₅₃	1	2	yes	—
266	#340	C₂²×C₅₃	1	2	yes	—
267	#336	C₂²×C₅₃	1	2	yes	—
268	C₅₃	#24695	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
272	C₂³×C₄	—	2	1	no	—
273	C₂×SD₁₆	—	2	2	no	—
274	C₂×SD₁₆	—	2	2	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
278	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₄⋊Q₈	—	2	2	no	—
280	C₄⋊Q₈	—	2	2	no	—
281	C₄×Q₈	—	2	2	no	—
282	C₄×Q₈	D₄×C₅₃	1	4	yes	—
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	—	2	2	no	—
297	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
298	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
299	#107	—	2	2	no	—
300	#106	—	2	4	no	—
301	C₄⋊C₈	—	2	4	no	—
302	C₄⋊C₈	—	2	2	no	—
303	#103	—	2	2	no	—
304	#103	—	2	4	no	—
305	#102	—	2	4	no	—
306	#102	—	2	2	no	—
307	#97	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	1	no	—
314	#95	—	2	2	no	—
315	#95	D₄×C₅₃	1	1	yes	characteristic
316	#95	D₄×C₅₃	1	1	yes	characteristic
317	C₂⁴	—	2	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	4	2	no	—
324	C₂×Q₈	—	8	2	no	—
325	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	#41×C₅₃	1	2	yes	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	—	4	2	no	—
339	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
344	SD₁₆	—	2	8	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	#41×C₅₃	1	2	yes	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	8	2	no	—
367	C₂²⋊C₄	—	8	2	no	—
368	C₄²	—	2	4	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	4	4	no	—
372	C₄²	C₂×D₄×C₅₃	1	2	yes	—
373	C₄²	#41×C₅₃	1	2	yes	—
374	C₂³	—	2	2	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	2	no	—
379	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
380	Q₈	—	2	2	no	—
381	Q₈	—	2	2	no	—
382	Q₈	—	4	1	no	—
383	Q₈	—	4	1	no	—
384	Q₈	—	8	4	no	—
385	D₄	—	2	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	4	1	no	—
388	D₄	—	4	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	8	4	no	—
411	C₂×C₄	#121×C₅₃	1	2	yes	—
412	C₂×C₄	#121×C₅₃	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
417	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
418	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
419	C₈	—	2	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	1	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	2	no	—
427	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
428	C₂²	C₅₃×#547	1	1	yes	characteristic
429	C₂²	C₅₃×#548	1	1	yes	characteristic
430	C₂²	C₅₃×#575	1	2	yes	—
431	C₂²	C₅₃×#577	1	2	yes	—
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	8	1	no	—
439	C₄	—	8	2	no	—
440	C₄	—	8	2	no	—
441	C₄	C₅₃×#525	1	1	yes	characteristic
442	C₄	C₅₃×#536	1	1	yes	characteristic
443	C₄	C₅₃×#552	1	1	yes	characteristic
444	C₄	C₅₃×#556	1	1	yes	characteristic
445	C₂	—	8	2	no	—
446	C₂	C₅₃×#2415	1	1	yes	characteristic
447	C₂	C₅₃×#3040	1	1	yes	characteristic
448	C₂	C₅₃×#3043	1	1	yes	characteristic
449	C₂	C₅₃×#3348	1	2	yes	—
450	C₂	C₅₃×#3376	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#475	C₂	semidirect
4. C₅₃×#2356	C₂	semidirect
5. C₅₃×#2074	C₂	semidirect
7. C₅₃×#1993	C₂	semidirect
8. C₅₃×#1949	C₂	semidirect
9. C₅₃×#1940	C₂	semidirect
196. #24695	C₅₃	direct (N × H)
220. C₂×#475	C₂×C₅₃	semidirect
223. #2356	C₂×C₅₃	semidirect
224. #2074	C₂×C₅₃	semidirect
226. #1993	C₂×C₅₃	semidirect
227. #1949	C₂×C₅₃	semidirect
228. #1940	C₂×C₅₃	semidirect
268. C₅₃	#24695	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×#475	2
C₂×C₅₃×#440	2
C₅₃×#2563	2
C₅₃×#2356	2
C₅₃×#2074	2
C₅₃×#2055	2
C₅₃×#1993	2
C₅₃×#1949	2
C₅₃×#1940	2
C₅₃×#1926	2
C₅₃×#1804	2
#24695	53

Subgroups — C53×#24695 #27232207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24695 #27232207