Subgroups — C₅₃×#22706 #27230218

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2352	C₂	1	2	yes	maximal
3	C₅₃×#1949	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1854	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1853	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₈×D₄×C₅₃	C₄	1	8	yes	complemented
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#106×C₅₃	C₂²	1	2	yes	—
20	C₂×#106×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#388	—	2	2	no	—
26	C₅₃×#387	—	2	2	no	—
27	C₅₃×#386	—	2	2	no	—
28	C₅₃×#382	C₂²	1	2	yes	—
29	C₅₃×#380	C₂²	1	2	yes	—
30	C₅₃×#380	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₅₃×#340	C₂²	1	2	yes	—
34	C₅₃×#340	C₂²	1	2	yes	—
35	C₅₃×#340	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	1	yes	characteristic
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	—	2	2	no	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	—	4	2	no	—
45	C₂²×C₈×C₅₃	—	4	2	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₅₃	Q₈	1	4	yes	—
54	#117×C₅₃	—	2	4	no	—
55	#117×C₅₃	—	4	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	#107×C₅₃	—	4	2	no	—
73	#106×C₅₃	—	2	4	no	—
74	#106×C₅₃	—	4	2	no	—
75	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
76	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
77	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
78	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
79	#95×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	2	no	—
84	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	4	4	no	—
98	C₂²×C₄×C₅₃	—	4	4	no	—
99	C₂²×C₄×C₅₃	—	8	2	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
113	C₂×C₈×C₅₃	—	4	4	no	—
114	C₂×C₈×C₅₃	—	4	4	no	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	4	4	no	—
127	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
128	C₄⋊C₄×C₅₃	#41	1	4	yes	—
129	C₂²⋊C₄×C₅₃	—	4	4	no	—
130	C₂²⋊C₄×C₅₃	—	4	4	no	—
131	C₂²⋊C₄×C₅₃	—	8	2	no	—
132	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
133	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
134	C₄²×C₅₃	—	4	4	no	—
135	C₄²×C₅₃	—	8	2	no	—
136	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
137	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	1	no	—
142	C₂³×C₅₃	—	2	1	no	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
145	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₂³×C₅₃	#135	1	2	yes	—
147	C₂³×C₅₃	#135	1	2	yes	—
148	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	D₄×C₅₃	—	2	8	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	8	8	no	—
174	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
175	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
177	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
181	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	2	2	no	—
185	C₈×C₅₃	—	2	2	no	—
186	#22706	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂²×C₅₃	—	2	2	no	—
188	C₂²×C₅₃	—	2	2	no	—
189	C₂²×C₅₃	—	2	4	no	—
190	C₂²×C₅₃	—	2	4	no	—
191	C₂²×C₅₃	—	2	2	no	—
192	C₂²×C₅₃	—	2	2	no	—
193	C₂²×C₅₃	—	2	2	no	—
194	C₂²×C₅₃	—	4	2	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	#427	1	4	yes	—
197	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₅₃	#546	1	2	yes	—
199	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
200	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
201	C₂²×C₅₃	#577	1	2	yes	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	2	2	no	—
204	C₄×C₅₃	—	2	2	no	—
205	C₄×C₅₃	—	2	2	no	—
206	C₄×C₅₃	—	4	4	no	—
207	C₄×C₅₃	—	8	2	no	—
208	C₄×C₅₃	—	8	2	no	—
209	C₄×C₅₃	—	8	2	no	—
210	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
211	C₄×C₅₃	#518	1	1	yes	characteristic
212	C₄×C₅₃	#554	1	2	yes	—
213	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
214	#2352	C₂×C₅₃	1	2	yes	—
215	#1949	C₂×C₅₃	1	2	yes	complemented
216	#1854	C₂×C₅₃	1	2	yes	complemented
217	#1853	C₂×C₅₃	1	2	yes	complemented
218	#1835	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#1808	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#1807	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₄×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1735	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂×#340	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×C₅₃	—	2	4	no	—
225	C₂×C₅₃	—	4	2	no	—
226	C₂×C₅₃	#2349	1	1	yes	characteristic
227	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
228	C₂×C₅₃	#2912	1	1	yes	characteristic
229	C₂×C₅₃	#3296	1	2	yes	—
230	C₂×C₅₃	#3301	1	2	yes	—
231	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
232	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×#117	C₂²×C₅₃	1	2	yes	—
235	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
236	C₈×D₄	C₄×C₅₃	1	8	yes	complemented
237	C₂×#107	C₂²×C₅₃	1	2	yes	—
238	C₂×#106	C₂²×C₅₃	1	2	yes	—
239	C₂×#106	C₂²×C₅₃	1	2	yes	—
240	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
244	#388	—	2	2	no	—
245	#387	—	2	2	no	—
246	#386	—	2	2	no	—
247	#382	C₂²×C₅₃	1	2	yes	—
248	#380	C₂²×C₅₃	1	2	yes	—
249	#380	C₂²×C₅₃	1	2	yes	—
250	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
251	C₄×C₂²⋊C₄	—	2	2	no	—
252	#340	C₂²×C₅₃	1	1	yes	characteristic
253	#340	C₂²×C₅₃	1	1	yes	characteristic
254	#340	C₂²×C₅₃	1	1	yes	characteristic
255	#340	C₂²×C₅₃	1	2	yes	—
256	#340	C₂²×C₅₃	1	2	yes	—
257	#340	C₂²×C₅₃	1	2	yes	—
258	#340	C₂²×C₅₃	1	1	yes	characteristic
259	C₅₃	#22706	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
261	C₂³×C₄	—	2	2	no	—
262	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
263	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
264	C₂²×C₈	—	4	2	no	—
265	C₂²×C₈	—	4	2	no	—
266	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
267	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
268	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
272	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
273	C₄×D₄	Q₈×C₅₃	1	4	yes	—
274	#117	—	2	4	no	—
275	#117	—	4	2	no	—
276	C₂×C₄⋊C₄	—	2	4	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	4	2	no	—
279	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
288	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
292	#107	—	4	2	no	—
293	#106	—	2	4	no	—
294	#106	—	4	2	no	—
295	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
296	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
297	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
298	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
305	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	4	no	—
318	C₂²×C₄	—	4	4	no	—
319	C₂²×C₄	—	8	2	no	—
320	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
327	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
328	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	4	4	no	—
334	C₂×C₈	—	4	4	no	—
335	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
338	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
348	C₄⋊C₄	#41×C₅₃	1	4	yes	—
349	C₂²⋊C₄	—	4	4	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₂²⋊C₄	—	8	2	no	—
352	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
353	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
354	C₄²	—	4	4	no	—
355	C₄²	—	8	2	no	—
356	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
357	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	1	no	—
363	C₂³	—	2	2	no	—
364	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
365	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
366	C₂³	#135×C₅₃	1	2	yes	—
367	C₂³	#135×C₅₃	1	2	yes	—
368	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
369	D₄	—	2	8	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	8	8	no	—
394	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
395	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
396	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
397	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
398	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
401	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
402	C₈	—	2	2	no	—
403	C₈	—	2	2	no	—
404	C₈	—	2	2	no	—
405	C₈	—	2	2	no	—
406	C₂²	—	2	4	no	—
407	C₂²	—	2	4	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	C₅₃×#427	1	4	yes	—
416	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
417	C₂²	C₅₃×#546	1	2	yes	—
418	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
419	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
420	C₂²	C₅₃×#577	1	2	yes	—
421	C₄	—	2	2	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	4	4	no	—
426	C₄	—	8	2	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	2	no	—
429	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
430	C₄	C₅₃×#518	1	1	yes	characteristic
431	C₄	C₅₃×#554	1	2	yes	—
432	C₂	—	2	4	no	—
433	C₂	—	4	2	no	—
434	C₂	C₅₃×#2349	1	1	yes	characteristic
435	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
436	C₂	C₅₃×#2912	1	1	yes	characteristic
437	C₂	C₅₃×#3296	1	2	yes	—
438	C₂	C₅₃×#3301	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1949	C₂	semidirect
4. C₅₃×#1854	C₂	semidirect
5. C₅₃×#1853	C₂	semidirect
6. C₅₃×#1835	C₂	semidirect
7. C₅₃×#1808	C₂	semidirect
8. C₅₃×#1807	C₂	semidirect
9. C₄×#106×C₅₃	C₂	semidirect
10. C₅₃×#1735	C₂	semidirect
11. C₂×C₅₃×#340	C₂	semidirect
17. C₈×D₄×C₅₃	C₄	semidirect
47. C₂²×C₈×C₅₃	C₂×C₄	semidirect
75. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
77. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
78. C₄×C₈×C₅₃	C₂×C₄	semidirect
186. #22706	C₅₃	direct (N × H)
215. #1949	C₂×C₅₃	semidirect
216. #1854	C₂×C₅₃	semidirect
217. #1853	C₂×C₅₃	semidirect
218. #1835	C₂×C₅₃	semidirect
219. #1808	C₂×C₅₃	semidirect
220. #1807	C₂×C₅₃	semidirect
221. C₄×#106	C₂×C₅₃	semidirect
222. #1735	C₂×C₅₃	semidirect
223. C₂×#340	C₂×C₅₃	semidirect
236. C₈×D₄	C₄×C₅₃	semidirect
259. C₅₃	#22706	direct (N × H)
267. C₂²×C₈	C₂×C₄×C₅₃	semidirect
295. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
297. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
298. 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×C₈×D₄×C₅₃	2
C₅₃×#2352	2
C₅₃×#1949	2
C₅₃×#1854	2
C₅₃×#1853	2
C₅₃×#1835	2
C₅₃×#1808	2
C₅₃×#1807	2
C₄×#106×C₅₃	2
C₅₃×#1735	2
C₂×C₅₃×#340	2
#22706	53

Subgroups — C53×#22706 #27230218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22706 #27230218