Subgroups — C₅₃×#22606 #27230118

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2924	C₂	1	2	yes	maximal
2	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1945	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1922	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1826	C₂	1	2	yes	maximal
6	C₅₃×#1810	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1772	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#433	—	2	4	no	—
12	C₅₃×#432	—	2	2	no	—
13	C₅₃×#431	—	2	2	no	—
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#427	C₄	1	1	yes	characteristic, complemented
16	C₅₃×#427	C₄	1	1	yes	characteristic, complemented
17	C₅₃×#427	C₄	1	2	yes	complemented
18	C₅₃×#427	C₂²	1	2	yes	—
19	C₅₃×#427	C₂²	1	1	yes	characteristic
20	C₅₃×#427	C₂²	1	1	yes	characteristic
21	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
22	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂×#106×C₅₃	C₄	1	2	yes	complemented
24	C₅₃×#424	C₂²	1	4	yes	—
25	C₅₃×#407	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#405	C₂²	1	4	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#341	C₂²	1	2	yes	—
31	C₅₃×#339	C₂²	1	4	yes	—
32	C₅₃×#339	C₂²	1	4	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	#117×C₅₃	C₂×C₄	1	2	yes	—
47	#117×C₅₃	C₂×C₄	1	2	yes	—
48	#117×C₅₃	C₂³	1	2	yes	—
49	#117×C₅₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	—	2	2	no	—
55	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
56	#108×C₅₃	—	2	4	no	—
57	#107×C₅₃	C₂×C₄	1	2	yes	—
58	#107×C₅₃	C₂×C₄	1	2	yes	—
59	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
60	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
61	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
63	#106×C₅₃	C₂×C₄	1	2	yes	—
64	#106×C₅₃	C₂×C₄	1	2	yes	—
65	#106×C₅₃	C₂×C₄	1	2	yes	—
66	#106×C₅₃	C₂×C₄	1	2	yes	—
67	C₄⋊C₈×C₅₃	—	2	4	no	—
68	C₄⋊C₈×C₅₃	—	2	4	no	—
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	C₂²⋊C₈×C₅₃	—	2	2	no	—
71	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
72	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
73	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
74	#97×C₅₃	—	2	4	no	—
75	C₄×C₈×C₅₃	—	2	4	no	—
76	C₄×C₈×C₅₃	—	2	4	no	—
77	#95×C₅₃	—	2	1	no	—
78	C₂⁴×C₅₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₅₃	—	2	4	no	—
91	M₁₆×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₅₃	#41	1	4	yes	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂²⋊C₄×C₅₃	—	2	4	no	—
114	C₂²⋊C₄×C₅₃	—	2	2	no	—
115	C₂²⋊C₄×C₅₃	—	2	2	no	—
116	C₂²⋊C₄×C₅₃	—	2	2	no	—
117	C₂²⋊C₄×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	—	2	2	no	—
119	C₄²×C₅₃	—	2	2	no	—
120	C₄²×C₅₃	—	2	2	no	—
121	C₄²×C₅₃	—	2	2	no	—
122	C₄²×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	4	1	no	—
127	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	—	8	2	no	—
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
151	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	4	4	no	—
157	C₈×C₅₃	#117	1	2	yes	—
158	C₈×C₅₃	#117	1	2	yes	—
159	#22606	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	4	4	no	—
168	C₂²×C₅₃	#443	1	2	yes	—
169	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₅₃	#552	1	2	yes	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	4	4	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	—	4	2	no	—
176	C₄×C₅₃	—	4	2	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
179	C₄×C₅₃	#519	1	1	yes	characteristic
180	C₄×C₅₃	#547	1	2	yes	—
181	#2924	C₂×C₅₃	1	2	yes	—
182	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
183	#1945	C₂×C₅₃	1	2	yes	complemented
184	#1922	C₂×C₅₃	1	4	yes	complemented
185	#1826	C₂×C₅₃	1	2	yes	—
186	#1810	C₂×C₅₃	1	1	yes	characteristic
187	#1772	C₂×C₅₃	1	2	yes	complemented
188	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
189	C₂×C₅₃	—	2	2	no	—
190	C₂×C₅₃	—	2	2	no	—
191	C₂×C₅₃	—	4	1	no	—
192	C₂×C₅₃	#2344	1	1	yes	characteristic
193	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
194	C₂×C₅₃	#2957	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×#117	C₂²×C₅₃	1	2	yes	—
197	#433	—	2	4	no	—
198	#432	—	2	2	no	—
199	#431	—	2	2	no	—
200	C₂×#108	C₂²×C₅₃	1	2	yes	—
201	#427	C₄×C₅₃	1	1	yes	characteristic, complemented
202	#427	C₄×C₅₃	1	1	yes	characteristic, complemented
203	#427	C₄×C₅₃	1	2	yes	complemented
204	#427	C₂²×C₅₃	1	2	yes	—
205	#427	C₂²×C₅₃	1	1	yes	characteristic
206	#427	C₂²×C₅₃	1	1	yes	characteristic
207	C₂×#107	C₄×C₅₃	1	1	yes	characteristic, complemented
208	C₂×#107	C₄×C₅₃	1	1	yes	characteristic, complemented
209	C₂×#106	C₄×C₅₃	1	2	yes	complemented
210	#424	C₂²×C₅₃	1	4	yes	—
211	#407	C₂²×C₅₃	1	2	yes	—
212	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
213	#405	C₂²×C₅₃	1	4	yes	—
214	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
215	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
216	#341	C₂²×C₅₃	1	2	yes	—
217	#339	C₂²×C₅₃	1	4	yes	—
218	#339	C₂²×C₅₃	1	4	yes	—
219	#336	C₂²×C₅₃	1	2	yes	—
220	C₅₃	#22606	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
222	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
223	C₂²×C₈	—	2	4	no	—
224	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
230	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
231	#117	—	2	2	no	—
232	#117	—	2	2	no	—
233	#117	C₂×C₄×C₅₃	1	2	yes	—
234	#117	C₂×C₄×C₅₃	1	2	yes	—
235	#117	C₂³×C₅₃	1	2	yes	—
236	#117	C₂³×C₅₃	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
238	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
239	C₂×C₂²⋊C₄	—	2	2	no	—
240	C₂×C₂²⋊C₄	—	2	1	no	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
243	#108	—	2	4	no	—
244	#107	C₂×C₄×C₅₃	1	2	yes	—
245	#107	C₂×C₄×C₅₃	1	2	yes	—
246	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
247	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
248	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
249	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
250	#106	C₂×C₄×C₅₃	1	2	yes	—
251	#106	C₂×C₄×C₅₃	1	2	yes	—
252	#106	C₂×C₄×C₅₃	1	2	yes	—
253	#106	C₂×C₄×C₅₃	1	2	yes	—
254	C₄⋊C₈	—	2	4	no	—
255	C₄⋊C₈	—	2	4	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
259	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
260	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
261	#97	—	2	4	no	—
262	C₄×C₈	—	2	4	no	—
263	C₄×C₈	—	2	4	no	—
264	#95	—	2	1	no	—
265	C₂⁴	#41×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
293	C₂×C₈	#41×C₅₃	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
297	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
298	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
299	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
315	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
316	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
338	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	2	no	—
342	C₈	—	2	2	no	—
343	C₈	—	4	4	no	—
344	C₈	#117×C₅₃	1	2	yes	—
345	C₈	#117×C₅₃	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	4	no	—
354	C₂²	C₅₃×#443	1	2	yes	—
355	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
356	C₂²	C₅₃×#552	1	2	yes	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
365	C₄	C₅₃×#519	1	1	yes	characteristic
366	C₄	C₅₃×#547	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	C₅₃×#2344	1	1	yes	characteristic
371	C₂	C₂×C₅₃×#443	1	1	yes	characteristic
372	C₂	C₅₃×#2957	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	C₂⁴×C₅₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₂²⋊C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1945	C₂	semidirect
4. C₅₃×#1922	C₂	semidirect
7. C₅₃×#1772	C₂	semidirect
15. C₅₃×#427	C₄	semidirect
16. C₅₃×#427	C₄	semidirect
17. C₅₃×#427	C₄	semidirect
21. C₂×#107×C₅₃	C₄	semidirect
22. C₂×#107×C₅₃	C₄	semidirect
23. C₂×#106×C₅₃	C₄	semidirect
159. #22606	C₅₃	direct (N × H)
183. #1945	C₂×C₅₃	semidirect
184. #1922	C₂×C₅₃	semidirect
187. #1772	C₂×C₅₃	semidirect
201. #427	C₄×C₅₃	semidirect
202. #427	C₄×C₅₃	semidirect
203. #427	C₄×C₅₃	semidirect
207. C₂×#107	C₄×C₅₃	semidirect
208. C₂×#107	C₄×C₅₃	semidirect
209. C₂×#106	C₄×C₅₃	semidirect
220. C₅₃	#22606	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×#2924	2
C₂×C₅₃×#427	2
C₅₃×#1945	2
C₅₃×#1922	2
C₅₃×#1826	2
C₅₃×#1810	2
C₅₃×#1772	2
C₈×C₂²⋊C₄×C₅₃	2
#22606	53

Subgroups — C53×#22606 #27230118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22606 #27230118