Subgroups — C₅₃×#22373 #27229885

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1939	C₂	1	2	yes	maximal
3	C₅₃×#1916	C₂	1	2	yes	maximal
4	C₅₃×#1916	C₂	1	2	yes	maximal
5	C₅₃×#1916	C₂	1	2	yes	maximal
6	C₅₃×#1916	C₂	1	2	yes	maximal
7	C₅₃×#1839	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#389	C₄	1	1	yes	characteristic
19	C₅₃×#389	C₄	1	1	yes	characteristic
20	C₅₃×#387	C₄	1	2	yes	—
21	C₅₃×#387	C₄	1	2	yes	—
22	C₅₃×#382	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₄³×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	1	yes	characteristic
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₅₃×#336	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	#125×C₅₃	C₈	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₄⋊C₈×C₅₃	—	2	4	no	—
61	C₄⋊C₈×C₅₃	—	4	2	no	—
62	C₄×C₈×C₅₃	—	2	4	no	—
63	C₄×C₈×C₅₃	—	4	1	no	—
64	C₄×C₈×C₅₃	—	4	1	no	—
65	C₄×C₈×C₅₃	—	4	2	no	—
66	#95×C₅₃	C₂×C₄	1	2	yes	—
67	#95×C₅₃	C₂×C₄	1	2	yes	—
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	2	yes	—
71	#95×C₅₃	C₂×C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	#41	1	4	yes	—
95	C₄⋊C₄×C₅₃	—	2	4	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
100	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
101	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
102	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	—	2	1	no	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	—	4	1	no	—
113	C₄²×C₅₃	—	4	1	no	—
114	C₄²×C₅₃	C₂×C₈	1	4	yes	—
115	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	2	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	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
144	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
145	C₂×C₄×C₅₃	#131	1	2	yes	—
146	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
147	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
149	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
150	C₈×C₅₃	—	2	4	no	—
151	C₈×C₅₃	—	4	2	no	—
152	C₈×C₅₃	—	4	2	no	—
153	C₈×C₅₃	—	4	2	no	—
154	#22373	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
156	C₂²×C₅₃	#519	1	1	yes	characteristic
157	C₂²×C₅₃	#520	1	1	yes	characteristic
158	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
159	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
160	C₂²×C₅₃	#568	1	1	yes	characteristic
161	C₂²×C₅₃	#568	1	1	yes	characteristic
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	—	2	4	no	—
164	C₄×C₅₃	—	2	2	no	—
165	C₄×C₅₃	—	2	2	no	—
166	C₄×C₅₃	—	2	4	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	4	1	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	—	4	1	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	#539	1	2	yes	—
173	C₄×C₅₃	#541	1	2	yes	—
174	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
175	#1939	C₂×C₅₃	1	2	yes	—
176	#1916	C₂×C₅₃	1	2	yes	—
177	#1916	C₂×C₅₃	1	2	yes	—
178	#1916	C₂×C₅₃	1	2	yes	—
179	#1916	C₂×C₅₃	1	2	yes	—
180	#1839	C₂×C₅₃	1	1	yes	characteristic
181	C₈×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
182	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
183	C₂×C₅₃	#2336	1	1	yes	characteristic
184	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
185	C₂×C₅₃	#2967	1	1	yes	characteristic
186	C₂×C₅₃	#2982	1	1	yes	characteristic
187	C₂×C₅₃	#2985	1	1	yes	characteristic
188	C₂×C₅₃	#2988	1	1	yes	characteristic
189	C₂×C₅₃	#2992	1	1	yes	characteristic
190	C₂×#125	C₄×C₅₃	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
192	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
197	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
198	#389	C₄×C₅₃	1	1	yes	characteristic
199	#389	C₄×C₅₃	1	1	yes	characteristic
200	#387	C₄×C₅₃	1	2	yes	—
201	#387	C₄×C₅₃	1	2	yes	—
202	#382	C₄×C₅₃	1	1	yes	characteristic
203	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
204	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
205	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
206	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
207	C₄³	C₂²×C₅₃	1	1	yes	characteristic
208	#336	C₂²×C₅₃	1	2	yes	—
209	#336	C₂²×C₅₃	1	1	yes	characteristic
210	#336	C₂²×C₅₃	1	2	yes	—
211	#336	C₂²×C₅₃	1	1	yes	characteristic
212	#336	C₂²×C₅₃	1	2	yes	—
213	#336	C₂²×C₅₃	1	2	yes	—
214	#336	C₂²×C₅₃	1	2	yes	—
215	#336	C₂²×C₅₃	1	2	yes	—
216	#336	C₂²×C₅₃	1	2	yes	—
217	C₅₃	#22373	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
219	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
220	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
221	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
222	#125	C₈×C₅₃	1	8	yes	complemented
223	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
239	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
240	C₄⋊C₈	—	2	4	no	—
241	C₄⋊C₈	—	2	4	no	—
242	C₄⋊C₈	—	4	2	no	—
243	C₄×C₈	—	2	4	no	—
244	C₄×C₈	—	4	1	no	—
245	C₄×C₈	—	4	1	no	—
246	C₄×C₈	—	4	2	no	—
247	#95	C₂×C₄×C₅₃	1	2	yes	—
248	#95	C₂×C₄×C₅₃	1	2	yes	—
249	#95	C₂×C₄×C₅₃	1	2	yes	—
250	#95	C₂×C₄×C₅₃	1	2	yes	—
251	#95	C₂×C₄×C₅₃	1	2	yes	—
252	#95	C₂×C₄×C₅₃	1	2	yes	—
253	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
260	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
261	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	#41×C₅₃	1	4	yes	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	C₂×C₈×C₅₃	1	8	yes	—
281	C₄⋊C₄	C₂×C₈×C₅₃	1	8	yes	—
282	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
283	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	C₂×C₈×C₅₃	1	4	yes	—
296	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
322	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
323	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
324	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
325	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
326	C₂×C₄	#131×C₅₃	1	2	yes	—
327	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
328	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
329	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
330	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
336	C₂²	C₅₃×#519	1	1	yes	characteristic
337	C₂²	C₅₃×#520	1	1	yes	characteristic
338	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
339	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
340	C₂²	C₅₃×#568	1	1	yes	characteristic
341	C₂²	C₅₃×#568	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	4	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	4	no	—
347	C₄	—	2	2	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	C₅₃×#539	1	2	yes	—
353	C₄	C₅₃×#541	1	2	yes	—
354	C₂	C₅₃×#2336	1	1	yes	characteristic
355	C₂	C₈×#41×C₅₃	1	1	yes	characteristic
356	C₂	C₅₃×#2967	1	1	yes	characteristic
357	C₂	C₅₃×#2982	1	1	yes	characteristic
358	C₂	C₅₃×#2985	1	1	yes	characteristic
359	C₂	C₅₃×#2988	1	1	yes	characteristic
360	C₂	C₅₃×#2992	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
41. #125×C₅₃	C₈	semidirect
154. #22373	C₅₃	direct (N × H)
217. C₅₃	#22373	direct (N × H)
222. #125	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₅₃	2
C₅₃×#1939	2
C₅₃×#1916	2
C₅₃×#1839	2
C₈×C₄⋊C₄×C₅₃	2
C₄×C₄⋊C₈×C₅₃	2
#22373	53

Subgroups — C53×#22373 #27229885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22373 #27229885