Subgroups — C₅₃×#20468 #27227980

Order 13568.

950 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#433	C₂	1	2	yes	maximal
2	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1768	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1750	C₂	1	2	yes	maximal
7	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1726	C₂	1	2	yes	maximal, complemented
12	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#433	C₄	1	16	yes	complemented
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×#97×C₅₃	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
30	C₅₃×#376	C₂²	1	1	yes	characteristic
31	C₅₃×#376	C₂²	1	1	yes	characteristic
32	C₄³×C₅₃	C₂²	1	1	yes	characteristic
33	C₄³×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#336	C₂²	1	1	yes	characteristic
35	C₅₃×#336	C₂²	1	1	yes	characteristic
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₅₃×#336	C₂²	1	1	yes	characteristic
38	C₅₃×#336	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	#117×C₅₃	—	2	2	no	—
47	#117×C₅₃	—	2	2	no	—
48	#117×C₅₃	C₂×C₄	1	8	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	—	2	4	no	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	complemented
68	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	complemented
69	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
71	#97×C₅₃	—	2	4	no	—
72	#97×C₅₃	—	2	4	no	—
73	#97×C₅₃	C₂×C₄	1	8	yes	complemented
74	C₄×C₈×C₅₃	—	2	4	no	—
75	C₄×C₈×C₅₃	—	2	4	no	—
76	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
77	#95×C₅₃	C₂×C₄	1	2	yes	—
78	#95×C₅₃	C₂×C₄	1	2	yes	—
79	#95×C₅₃	C₂×C₄	1	2	yes	—
80	#95×C₅₃	C₂×C₄	1	2	yes	—
81	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	1	no	—
84	C₂²×C₄×C₅₃	—	2	1	no	—
85	C₂²×C₄×C₅₃	—	2	4	no	—
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
108	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
111	C₄⋊C₄×C₅₃	—	2	4	no	—
112	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
113	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
114	C₂²⋊C₄×C₅₃	—	2	4	no	—
115	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
116	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
117	C₄²×C₅₃	—	2	4	no	—
118	C₄²×C₅₃	—	2	4	no	—
119	C₄²×C₅₃	—	2	4	no	—
120	C₄²×C₅₃	—	2	2	no	—
121	C₄²×C₅₃	—	2	1	no	—
122	C₄²×C₅₃	—	2	1	no	—
123	C₄²×C₅₃	—	4	2	no	—
124	C₄²×C₅₃	—	4	2	no	—
125	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
126	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
127	C₄²×C₅₃	#41	1	2	yes	—
128	C₄²×C₅₃	#41	1	4	yes	—
129	C₄²×C₅₃	#41	1	2	yes	—
130	C₂³×C₅₃	—	2	1	no	—
131	C₂³×C₅₃	—	2	1	no	—
132	C₂³×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
135	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	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	—	2	4	no	—
149	C₂×C₄×C₅₃	—	2	4	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	4	8	no	—
154	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
155	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
157	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
158	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
159	C₂×C₄×C₅₃	#117	1	4	yes	—
160	C₂×C₄×C₅₃	#117	1	4	yes	—
161	C₂×C₄×C₅₃	#131	1	4	yes	—
162	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
163	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₈×C₅₃	—	2	4	no	—
167	C₈×C₅₃	—	2	4	no	—
168	C₈×C₅₃	—	2	4	no	—
169	C₈×C₅₃	—	2	4	no	—
170	#20468	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	—	2	2	no	—
174	C₂²×C₅₃	—	4	2	no	—
175	C₂²×C₅₃	—	4	2	no	—
176	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
177	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
178	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
179	C₂²×C₅₃	C₂×#131	1	2	yes	—
180	C₂²×C₅₃	#568	1	2	yes	—
181	C₄×C₅₃	—	2	2	no	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	2	2	no	—
184	C₄×C₅₃	—	2	2	no	—
185	C₄×C₅₃	—	2	4	no	—
186	C₄×C₅₃	—	2	4	no	—
187	C₄×C₅₃	—	2	4	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	4	2	no	—
190	C₄×C₅₃	C₄×#41	1	2	yes	—
191	C₄×C₅₃	C₄×#41	1	2	yes	—
192	C₂×#433	C₂×C₅₃	1	2	yes	—
193	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
194	#1768	C₂×C₅₃	1	2	yes	complemented
195	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
196	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
197	#1750	C₂×C₅₃	1	2	yes	—
198	#1749	C₂×C₅₃	1	1	yes	characteristic
199	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic
200	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
201	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#1726	C₂×C₅₃	1	2	yes	complemented
203	C₂×C₅₃	—	4	2	no	—
204	C₂×C₅₃	C₄×#117	1	1	yes	characteristic
205	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
206	C₂×C₅₃	#2875	1	1	yes	characteristic
207	C₂×C₅₃	#2924	1	2	yes	—
208	C₂×C₅₃	#2925	1	2	yes	—
209	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
211	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
212	#433	C₄×C₅₃	1	16	yes	complemented
213	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
214	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
215	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
216	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
217	C₂×#97	C₂²×C₅₃	1	2	yes	—
218	C₂×#97	C₂²×C₅₃	1	2	yes	—
219	C₂×#97	C₂²×C₅₃	1	2	yes	—
220	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
221	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
222	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
223	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
224	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
225	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
226	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
227	#376	C₂²×C₅₃	1	1	yes	characteristic
228	#376	C₂²×C₅₃	1	1	yes	characteristic
229	C₄³	C₂²×C₅₃	1	1	yes	characteristic
230	C₄³	C₂²×C₅₃	1	1	yes	characteristic
231	#336	C₂²×C₅₃	1	1	yes	characteristic
232	#336	C₂²×C₅₃	1	1	yes	characteristic
233	#336	C₂²×C₅₃	1	2	yes	—
234	#336	C₂²×C₅₃	1	1	yes	characteristic
235	#336	C₂²×C₅₃	1	1	yes	characteristic
236	#336	C₂²×C₅₃	1	2	yes	—
237	C₅₃	#20468	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
239	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
240	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
241	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
242	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
243	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
244	#117	—	2	2	no	—
245	#117	—	2	2	no	—
246	#117	C₂×C₄×C₅₃	1	8	yes	—
247	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
248	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
251	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
252	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄²	—	2	4	no	—
254	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
255	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
256	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
257	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
258	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
259	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
260	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
265	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	complemented
266	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	complemented
267	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
268	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
269	#97	—	2	4	no	—
270	#97	—	2	4	no	—
271	#97	C₂×C₄×C₅₃	1	8	yes	complemented
272	C₄×C₈	—	2	4	no	—
273	C₄×C₈	—	2	4	no	—
274	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	complemented
275	#95	C₂×C₄×C₅₃	1	2	yes	—
276	#95	C₂×C₄×C₅₃	1	2	yes	—
277	#95	C₂×C₄×C₅₃	1	2	yes	—
278	#95	C₂×C₄×C₅₃	1	2	yes	—
279	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
287	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
293	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
294	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
296	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	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	C₂²×C₄×C₅₃	1	4	yes	—
306	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
307	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
308	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	C₄²×C₅₃	1	4	yes	—
311	C₄⋊C₄	C₄²×C₅₃	1	4	yes	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	C₄²×C₅₃	1	4	yes	—
314	C₂²⋊C₄	C₄²×C₅₃	1	4	yes	—
315	C₄²	—	2	4	no	—
316	C₄²	—	2	4	no	—
317	C₄²	—	2	4	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	1	no	—
320	C₄²	—	2	1	no	—
321	C₄²	—	4	2	no	—
322	C₄²	—	4	2	no	—
323	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
324	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
325	C₄²	#41×C₅₃	1	2	yes	—
326	C₄²	#41×C₅₃	1	4	yes	—
327	C₄²	#41×C₅₃	1	2	yes	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	2	no	—
332	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
333	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
353	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
354	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
355	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
356	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
357	C₂×C₄	#117×C₅₃	1	4	yes	—
358	C₂×C₄	#117×C₅₃	1	4	yes	—
359	C₂×C₄	#131×C₅₃	1	4	yes	—
360	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
361	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
362	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
364	C₈	—	2	4	no	—
365	C₈	—	2	4	no	—
366	C₈	—	2	4	no	—
367	C₈	—	2	4	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
374	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
375	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
376	C₂²	C₂×#131×C₅₃	1	2	yes	—
377	C₂²	C₅₃×#568	1	2	yes	—
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	4	no	—
384	C₄	—	2	4	no	—
385	C₄	—	2	4	no	—
386	C₄	—	4	2	no	—
387	C₄	C₄×#41×C₅₃	1	2	yes	—
388	C₄	C₄×#41×C₅₃	1	2	yes	—
389	C₂	—	4	2	no	—
390	C₂	C₄×#117×C₅₃	1	1	yes	characteristic
391	C₂	C₄×#131×C₅₃	1	1	yes	characteristic
392	C₂	C₅₃×#2875	1	1	yes	characteristic
393	C₂	C₅₃×#2924	1	2	yes	—
394	C₂	C₅₃×#2925	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×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
3. C₅₃×#1768	C₂	semidirect
4. C₅₃×#1767	C₂	semidirect
5. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
9. C₅₃×#1745	C₂	semidirect
10. C₅₃×#1744	C₂	semidirect
11. C₅₃×#1726	C₂	semidirect
15. C₅₃×#433	C₄	semidirect
67. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
68. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
73. #97×C₅₃	C₂×C₄	semidirect
76. C₄×C₈×C₅₃	C₂×C₄	semidirect
170. #20468	C₅₃	direct (N × H)
194. #1768	C₂×C₅₃	semidirect
195. #1767	C₂×C₅₃	semidirect
196. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
200. #1745	C₂×C₅₃	semidirect
201. #1744	C₂×C₅₃	semidirect
202. #1726	C₂×C₅₃	semidirect
212. #433	C₄×C₅₃	semidirect
237. C₅₃	#20468	direct (N × H)
265. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
266. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
271. #97	C₂×C₄×C₅₃	semidirect
274. C₄×C₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#433	2
C₄×#117×C₅₃	2
C₅₃×#1768	2
C₅₃×#1767	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1750	2
C₅₃×#1749	2
C₄×C₂²⋊C₈×C₅₃	2
C₅₃×#1745	2
C₅₃×#1744	2
C₅₃×#1726	2
#20468	53

Subgroups — C53×#20468 #27227980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20468 #27227980