Subgroups — C₅₃×#22235 #27229747

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3135	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1963	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1929	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1866	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#533	C₂²	1	1	yes	characteristic
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#491	C₄	1	2	yes	complemented
14	C₅₃×#490	C₄	1	2	yes	complemented
15	C₅₃×#489	C₄	1	2	yes	complemented
16	C₅₃×#489	C₄	1	2	yes	complemented
17	C₅₃×#446	—	2	1	no	—
18	C₅₃×#446	—	2	1	no	—
19	C₈×Q₈×C₅₃	—	2	1	no	—
20	C₈×Q₈×C₅₃	—	2	1	no	—
21	C₅₃×#436	—	2	1	no	—
22	C₅₃×#436	—	2	1	no	—
23	C₈×D₄×C₅₃	—	2	1	no	—
24	C₈×D₄×C₅₃	—	2	1	no	—
25	C₂×#108×C₅₃	C₂²	1	4	yes	—
26	C₅₃×#424	C₂²	1	2	yes	—
27	C₅₃×#423	C₂²	1	1	yes	characteristic
28	C₅₃×#423	C₂²	1	1	yes	characteristic
29	C₂×#104×C₅₃	C₂²	1	4	yes	—
30	C₅₃×#416	C₂²	1	4	yes	—
31	C₅₃×#408	C₂²	1	2	yes	—
32	C₅₃×#405	C₂²	1	2	yes	—
33	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₅₃×#339	C₂²	1	4	yes	—
37	C₅₃×#336	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₂×#41×C₅₃	C₂³	1	2	yes	—
40	#131×C₅₃	—	4	4	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
43	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	—	2	2	no	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#125×C₅₃	C₂×C₄	1	2	yes	—
53	#124×C₅₃	C₂×C₄	1	2	yes	—
54	C₂²⋊Q₈×C₅₃	—	2	1	no	—
55	C₂²⋊Q₈×C₅₃	—	2	1	no	—
56	#121×C₅₃	—	2	1	no	—
57	#121×C₅₃	—	2	1	no	—
58	C₄×Q₈×C₅₃	—	2	1	no	—
59	C₄×Q₈×C₅₃	—	2	1	no	—
60	C₄×D₄×C₅₃	—	2	1	no	—
61	C₄×D₄×C₅₃	—	2	1	no	—
62	#117×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#108×C₅₃	—	4	4	no	—
66	C₄⋊C₈×C₅₃	—	2	2	no	—
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	—	2	1	no	—
69	C₄⋊C₈×C₅₃	—	2	1	no	—
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	#104×C₅₃	—	4	4	no	—
73	#103×C₅₃	C₂×C₄	1	4	yes	—
74	#103×C₅₃	C₂×C₄	1	4	yes	—
75	#102×C₅₃	C₂×C₄	1	4	yes	—
76	#102×C₅₃	C₂×C₄	1	4	yes	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	#97×C₅₃	—	2	2	no	—
80	#97×C₅₃	C₂×C₄	1	2	yes	—
81	#97×C₅₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
87	#95×C₅₃	—	2	1	no	—
88	#95×C₅₃	—	2	1	no	—
89	#41×C₅₃	—	2	4	no	—
90	#41×C₅₃	—	2	4	no	—
91	C₂×Q₈×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	1	no	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₅₃	—	2	4	no	—
104	M₁₆×C₅₃	—	2	4	no	—
105	M₁₆×C₅₃	—	4	4	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	4	4	no	—
113	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
125	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
138	Q₈×C₅₃	—	2	4	no	—
139	Q₈×C₅₃	—	4	2	no	—
140	Q₈×C₅₃	—	4	2	no	—
141	D₄×C₅₃	—	2	4	no	—
142	D₄×C₅₃	—	4	2	no	—
143	D₄×C₅₃	—	4	2	no	—
144	D₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	4	1	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	—	4	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
162	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
163	C₈×C₅₃	—	2	4	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	1	no	—
166	C₈×C₅₃	—	2	1	no	—
167	C₈×C₅₃	—	2	1	no	—
168	C₈×C₅₃	—	4	2	no	—
169	C₈×C₅₃	—	4	2	no	—
170	#22235	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	—	4	2	no	—
175	C₂²×C₅₃	—	4	2	no	—
176	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂²×C₅₃	#542	1	2	yes	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	4	2	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	—	4	2	no	—
185	C₄×C₅₃	—	8	2	no	—
186	C₄×C₅₃	#520	1	2	yes	—
187	C₄×C₅₃	#535	1	1	yes	characteristic
188	C₄×C₅₃	#536	1	1	yes	characteristic
189	#3135	C₂×C₅₃	1	1	yes	characteristic
190	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#2931	C₂×C₅₃	1	1	yes	characteristic
192	#1995	C₂×C₅₃	1	1	yes	characteristic, complemented
193	#1963	C₂×C₅₃	1	1	yes	characteristic
194	#1929	C₂×C₅₃	1	4	yes	complemented
195	#1866	C₂×C₅₃	1	4	yes	complemented
196	#1849	C₂×C₅₃	1	1	yes	characteristic, complemented
197	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₂×C₅₃	—	2	2	no	—
199	C₂×C₅₃	—	8	2	no	—
200	C₂×C₅₃	#2326	1	1	yes	characteristic
201	C₂×C₅₃	#2955	1	1	yes	characteristic
202	C₂×C₅₃	#2955	1	1	yes	characteristic
203	C₂×#131	C₂²×C₅₃	1	2	yes	—
204	#533	C₂²×C₅₃	1	1	yes	characteristic
205	#532	C₂²×C₅₃	1	1	yes	characteristic
206	#491	C₄×C₅₃	1	2	yes	complemented
207	#490	C₄×C₅₃	1	2	yes	complemented
208	#489	C₄×C₅₃	1	2	yes	complemented
209	#489	C₄×C₅₃	1	2	yes	complemented
210	#446	—	2	1	no	—
211	#446	—	2	1	no	—
212	C₈×Q₈	—	2	1	no	—
213	C₈×Q₈	—	2	1	no	—
214	#436	—	2	1	no	—
215	#436	—	2	1	no	—
216	C₈×D₄	—	2	1	no	—
217	C₈×D₄	—	2	1	no	—
218	C₂×#108	C₂²×C₅₃	1	4	yes	—
219	#424	C₂²×C₅₃	1	2	yes	—
220	#423	C₂²×C₅₃	1	1	yes	characteristic
221	#423	C₂²×C₅₃	1	1	yes	characteristic
222	C₂×#104	C₂²×C₅₃	1	4	yes	—
223	#416	C₂²×C₅₃	1	4	yes	—
224	#408	C₂²×C₅₃	1	2	yes	—
225	#405	C₂²×C₅₃	1	2	yes	—
226	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
227	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
228	#376	C₂²×C₅₃	1	1	yes	characteristic
229	#339	C₂²×C₅₃	1	4	yes	—
230	#336	C₂²×C₅₃	1	2	yes	—
231	#336	C₂²×C₅₃	1	2	yes	—
232	C₅₃	#22235	1	1	yes	characteristic, direct product factor, Sylow
233	C₂×#41	C₂³×C₅₃	1	2	yes	—
234	#131	—	4	4	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
237	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
240	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
242	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
243	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
244	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
246	#125	C₂×C₄×C₅₃	1	2	yes	—
247	#124	C₂×C₄×C₅₃	1	2	yes	—
248	C₂²⋊Q₈	—	2	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	#121	—	2	1	no	—
251	#121	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	1	no	—
256	#117	C₂³×C₅₃	1	2	yes	—
257	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
258	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
259	#108	—	4	4	no	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	#104	—	4	4	no	—
267	#103	C₂×C₄×C₅₃	1	4	yes	—
268	#103	C₂×C₄×C₅₃	1	4	yes	—
269	#102	C₂×C₄×C₅₃	1	4	yes	—
270	#102	C₂×C₄×C₅₃	1	4	yes	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₂²⋊C₈	—	2	2	no	—
273	#97	—	2	2	no	—
274	#97	C₂×C₄×C₅₃	1	2	yes	—
275	#97	C₂×C₄×C₅₃	1	2	yes	—
276	C₄×C₈	—	2	2	no	—
277	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
278	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
279	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
280	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
281	#95	—	2	1	no	—
282	#95	—	2	1	no	—
283	#41	—	2	4	no	—
284	#41	—	2	4	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
295	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
297	M₁₆	—	2	4	no	—
298	M₁₆	—	2	4	no	—
299	M₁₆	—	4	4	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	4	4	no	—
307	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
308	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
319	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
329	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
330	C₂³	—	2	2	no	—
331	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
332	Q₈	—	2	4	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	D₄	—	2	4	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
350	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
356	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
357	C₈	—	2	4	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
370	C₂²	C₅₃×#542	1	2	yes	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	C₅₃×#520	1	2	yes	—
380	C₄	C₅₃×#535	1	1	yes	characteristic
381	C₄	C₅₃×#536	1	1	yes	characteristic
382	C₂	—	2	2	no	—
383	C₂	—	8	2	no	—
384	C₂	C₅₃×#2326	1	1	yes	characteristic
385	C₂	C₅₃×#2955	1	1	yes	characteristic
386	C₂	C₅₃×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2960	C₂	semidirect
4. C₅₃×#1995	C₂	semidirect
6. C₅₃×#1929	C₂	semidirect
7. C₅₃×#1866	C₂	semidirect
8. C₅₃×#1849	C₂	semidirect
9. C₅₃×#1746	C₂	semidirect
13. C₅₃×#491	C₄	semidirect
14. C₅₃×#490	C₄	semidirect
15. C₅₃×#489	C₄	semidirect
16. C₅₃×#489	C₄	semidirect
170. #22235	C₅₃	direct (N × H)
190. #2960	C₂×C₅₃	semidirect
192. #1995	C₂×C₅₃	semidirect
194. #1929	C₂×C₅₃	semidirect
195. #1866	C₂×C₅₃	semidirect
196. #1849	C₂×C₅₃	semidirect
197. #1746	C₂×C₅₃	semidirect
206. #491	C₄×C₅₃	semidirect
207. #490	C₄×C₅₃	semidirect
208. #489	C₄×C₅₃	semidirect
209. #489	C₄×C₅₃	semidirect
232. C₅₃	#22235	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3135	2
C₅₃×#2960	2
C₅₃×#2931	2
C₅₃×#1995	2
C₅₃×#1963	2
C₅₃×#1929	2
C₅₃×#1866	2
C₅₃×#1849	2
C₅₃×#1746	2
#22235	53

Subgroups — C53×#22235 #27229747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22235 #27229747