Subgroups — C₅₃×#22723 #27230235

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	2	yes	maximal
2	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1940	C₂	1	2	yes	maximal
4	C₅₃×#1940	C₂	1	2	yes	maximal
5	C₅₃×#1939	C₂	1	2	yes	maximal
6	C₅₃×#1843	C₂	1	2	yes	maximal
7	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1768	C₂	1	2	yes	maximal
9	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#446	—	2	8	no	—
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	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#97×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#391	C₄	1	1	yes	characteristic
26	C₅₃×#391	C₄	1	1	yes	characteristic
27	C₅₃×#389	C₄	1	1	yes	characteristic
28	C₅₃×#389	C₄	1	1	yes	characteristic
29	C₅₃×#387	C₄	1	2	yes	—
30	C₅₃×#387	C₄	1	2	yes	—
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#376	C₂²	1	2	yes	—
34	C₄³×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₅₃×#336	C₂²	1	2	yes	—
38	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄×Q₈×C₅₃	D₄	1	4	yes	—
44	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
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	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×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	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	4	no	—
61	C₄⋊C₈×C₅₃	—	2	4	no	—
62	C₄⋊C₈×C₅₃	—	2	4	no	—
63	C₄⋊C₈×C₅₃	—	2	2	no	—
64	C₄⋊C₈×C₅₃	—	2	4	no	—
65	C₄⋊C₈×C₅₃	—	4	1	no	—
66	C₄⋊C₈×C₅₃	—	4	2	no	—
67	C₄⋊C₈×C₅₃	—	4	1	no	—
68	C₄⋊C₈×C₅₃	—	4	2	no	—
69	C₄⋊C₈×C₅₃	Q₈	1	4	yes	complemented
70	#97×C₅₃	—	2	4	no	—
71	#97×C₅₃	—	2	4	no	—
72	C₄×C₈×C₅₃	—	2	4	no	—
73	#95×C₅₃	C₂×C₄	1	2	yes	—
74	#95×C₅₃	C₂×C₄	1	2	yes	—
75	#95×C₅₃	C₂×C₄	1	2	yes	—
76	#95×C₅₃	C₂×C₄	1	2	yes	—
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	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
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	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	C₂×Q₈	1	4	yes	—
106	C₄⋊C₄×C₅₃	—	2	4	no	—
107	C₄⋊C₄×C₅₃	—	4	1	no	—
108	C₄⋊C₄×C₅₃	—	4	1	no	—
109	C₄⋊C₄×C₅₃	—	4	2	no	—
110	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
111	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	—	2	1	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	1	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	4	1	no	—
118	C₄²×C₅₃	—	4	1	no	—
119	C₄²×C₅₃	—	4	1	no	—
120	C₄²×C₅₃	C₂×D₄	1	4	yes	—
121	C₄²×C₅₃	C₂×D₄	1	2	yes	—
122	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
123	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
124	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
125	C₄²×C₅₃	#41	1	2	yes	—
126	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	Q₈×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	2	1	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	2	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	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	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	1	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
157	C₂×C₄×C₅₃	#131	1	2	yes	—
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
164	C₈×C₅₃	—	2	4	no	—
165	C₈×C₅₃	—	4	2	no	—
166	C₈×C₅₃	—	4	2	no	—
167	C₈×C₅₃	—	4	2	no	—
168	#22723	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₂²×C₅₃	#520	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
173	C₂²×C₅₃	#568	1	1	yes	characteristic
174	C₂²×C₅₃	#568	1	1	yes	characteristic
175	C₂²×C₅₃	#568	1	1	yes	characteristic
176	C₄×C₅₃	—	2	4	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	2	4	no	—
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	2	2	no	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	—	4	1	no	—
185	C₄×C₅₃	—	4	2	no	—
186	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
187	C₄×C₅₃	#550	1	1	yes	characteristic
188	C₄×C₅₃	#557	1	1	yes	characteristic
189	C₄×C₅₃	Q₈²	1	1	yes	characteristic
190	C₂×#446	C₂×C₅₃	1	2	yes	—
191	#2295	C₂×C₅₃	1	1	yes	characteristic
192	#1940	C₂×C₅₃	1	2	yes	—
193	#1940	C₂×C₅₃	1	2	yes	—
194	#1939	C₂×C₅₃	1	2	yes	—
195	#1843	C₂×C₅₃	1	2	yes	—
196	#1842	C₂×C₅₃	1	1	yes	characteristic
197	#1768	C₂×C₅₃	1	2	yes	—
198	#1767	C₂×C₅₃	1	1	yes	characteristic
199	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₅₃	#2905	1	1	yes	characteristic
201	C₂×C₅₃	#2906	1	1	yes	characteristic
202	C₂×C₅₃	#2961	1	1	yes	characteristic
203	C₂×C₅₃	#2962	1	1	yes	characteristic
204	C₂×C₅₃	#2992	1	1	yes	characteristic
205	C₂×C₅₃	#2995	1	1	yes	characteristic
206	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
207	#446	—	2	8	no	—
208	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
209	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
210	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
212	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
213	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
215	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
216	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
217	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
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	#391	C₄×C₅₃	1	1	yes	characteristic
222	#391	C₄×C₅₃	1	1	yes	characteristic
223	#389	C₄×C₅₃	1	1	yes	characteristic
224	#389	C₄×C₅₃	1	1	yes	characteristic
225	#387	C₄×C₅₃	1	2	yes	—
226	#387	C₄×C₅₃	1	2	yes	—
227	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
228	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
229	#376	C₂²×C₅₃	1	2	yes	—
230	C₄³	C₂²×C₅₃	1	1	yes	characteristic
231	#336	C₂²×C₅₃	1	2	yes	—
232	#336	C₂²×C₅₃	1	2	yes	—
233	#336	C₂²×C₅₃	1	2	yes	—
234	C₅₃	#22723	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
237	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
238	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
239	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
240	C₄×Q₈	D₄×C₅₃	1	4	yes	—
241	C₄×Q₈	Q₈×C₅₃	1	4	yes	—
242	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
244	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
247	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
253	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
254	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
256	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
257	C₄⋊C₈	—	2	4	no	—
258	C₄⋊C₈	—	2	4	no	—
259	C₄⋊C₈	—	2	4	no	—
260	C₄⋊C₈	—	2	4	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	4	1	no	—
263	C₄⋊C₈	—	4	1	no	—
264	C₄⋊C₈	—	4	2	no	—
265	C₄⋊C₈	—	4	2	no	—
266	C₄⋊C₈	Q₈×C₅₃	1	4	yes	complemented
267	#97	—	2	4	no	—
268	#97	—	2	4	no	—
269	C₄×C₈	—	2	4	no	—
270	#95	C₂×C₄×C₅₃	1	2	yes	—
271	#95	C₂×C₄×C₅₃	1	2	yes	—
272	#95	C₂×C₄×C₅₃	1	2	yes	—
273	#95	C₂×C₄×C₅₃	1	2	yes	—
274	#95	C₂×C₄×C₅₃	1	2	yes	—
275	#95	C₂×C₄×C₅₃	1	2	yes	—
276	#95	C₂×C₄×C₅₃	1	2	yes	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
280	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
281	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
282	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
284	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
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	C₂⁴×C₅₃	1	1	yes	characteristic
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
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	C₂×Q₈×C₅₃	1	4	yes	—
303	C₄⋊C₄	—	2	4	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
308	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	4	1	no	—
315	C₄²	—	4	1	no	—
316	C₄²	—	4	1	no	—
317	C₄²	C₂×D₄×C₅₃	1	4	yes	—
318	C₄²	C₂×D₄×C₅₃	1	2	yes	—
319	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
320	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
321	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
322	C₄²	#41×C₅₃	1	2	yes	—
323	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	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	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
349	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
350	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
351	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
352	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
353	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
354	C₂×C₄	#131×C₅₃	1	2	yes	—
355	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
359	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
360	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
361	C₈	—	2	4	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₈	—	4	2	no	—
365	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
366	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
367	C₂²	C₅₃×#520	1	1	yes	characteristic
368	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
369	C₂²	C₅₃×#568	1	1	yes	characteristic
370	C₂²	C₅₃×#568	1	1	yes	characteristic
371	C₂²	C₅₃×#568	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	4	no	—
377	C₄	—	2	2	no	—
378	C₄	—	2	4	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	2	no	—
382	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
383	C₄	C₅₃×#550	1	1	yes	characteristic
384	C₄	C₅₃×#557	1	1	yes	characteristic
385	C₄	Q₈²×C₅₃	1	1	yes	characteristic
386	C₂	Q₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
387	C₂	C₅₃×#2905	1	1	yes	characteristic
388	C₂	C₅₃×#2906	1	1	yes	characteristic
389	C₂	C₅₃×#2961	1	1	yes	characteristic
390	C₂	C₅₃×#2962	1	1	yes	characteristic
391	C₂	C₅₃×#2992	1	1	yes	characteristic
392	C₂	C₅₃×#2995	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
69. C₄⋊C₈×C₅₃	Q₈	semidirect
168. #22723	C₅₃	direct (N × H)
234. C₅₃	#22723	direct (N × H)
266. C₄⋊C₈	Q₈×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×#446	2
C₅₃×#2295	2
C₅₃×#1940	2
C₅₃×#1939	2
C₅₃×#1843	2
C₅₃×#1842	2
C₅₃×#1768	2
C₅₃×#1767	2
#22723	53

Subgroups — C53×#22723 #27230235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22723 #27230235