Subgroups — C₅₃×#23094 #27230606

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#461	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1968	C₂	1	2	yes	maximal
5	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1927	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1880	C₂	1	2	yes	maximal
9	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#103×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	—	2	1	no	—
17	C₅₃×#461	C₄	1	8	yes	complemented
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#103×C₅₃	C₂²	1	2	yes	—
22	C₂×#103×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#391	C₂²	1	2	yes	—
28	C₅₃×#390	C₂²	1	2	yes	—
29	C₅₃×#389	—	2	1	no	—
30	C₅₃×#388	—	2	2	no	—
31	C₅₃×#387	—	2	2	no	—
32	C₅₃×#384	C₂²	1	1	yes	characteristic
33	C₅₃×#380	—	2	1	no	—
34	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	1	yes	characteristic
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂×SD₁₆×C₅₃	—	4	2	no	—
46	C₂×SD₁₆×C₅₃	—	4	2	no	—
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	—	2	2	no	—
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
53	C₄×Q₈×C₅₃	—	2	4	no	—
54	C₄×Q₈×C₅₃	—	4	2	no	—
55	C₄×D₄×C₅₃	—	2	2	no	—
56	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
57	#117×C₅₃	—	4	4	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	—	2	2	no	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	—	2	1	no	—
75	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	#106×C₅₃	—	4	2	no	—
78	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
79	#103×C₅₃	—	2	4	no	—
80	#103×C₅₃	—	4	2	no	—
81	#103×C₅₃	C₂×C₄	1	8	yes	—
82	#102×C₅₃	—	2	2	no	—
83	#102×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	—	2	2	no	—
85	C₄×C₈×C₅₃	—	4	2	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	1	no	—
89	#95×C₅₃	—	2	2	no	—
90	#95×C₅₃	D₄	1	2	yes	—
91	C₂⁴×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	2	2	no	—
95	C₂×Q₈×C₅₃	—	2	2	no	—
96	C₂×Q₈×C₅₃	—	2	2	no	—
97	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	—	2	1	no	—
102	C₂×D₄×C₅₃	—	2	1	no	—
103	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	2	4	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	4	2	no	—
111	C₂²×C₄×C₅₃	—	4	2	no	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₅₃	—	4	8	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₅₃	#41	1	2	yes	—
127	C₄⋊C₄×C₅₃	—	2	4	no	—
128	C₄⋊C₄×C₅₃	—	2	4	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	4	4	no	—
137	C₄⋊C₄×C₅₃	—	4	2	no	—
138	C₄⋊C₄×C₅₃	—	8	1	no	—
139	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	complemented
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₅₃	#41	1	2	yes	—
142	C₂²⋊C₄×C₅₃	—	2	4	no	—
143	C₂²⋊C₄×C₅₃	—	4	4	no	—
144	C₂²⋊C₄×C₅₃	—	8	2	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	4	4	no	—
148	C₄²×C₅₃	—	4	4	no	—
149	C₄²×C₅₃	—	4	2	no	—
150	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
151	C₄²×C₅₃	#41	1	2	yes	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	—	4	1	no	—
154	C₂³×C₅₃	—	4	1	no	—
155	C₂³×C₅₃	—	4	1	no	—
156	C₂³×C₅₃	—	4	1	no	—
157	C₂³×C₅₃	—	4	1	no	—
158	C₂³×C₅₃	—	4	1	no	—
159	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₅₃	—	2	4	no	—
161	Q₈×C₅₃	—	2	4	no	—
162	Q₈×C₅₃	—	4	2	no	—
163	Q₈×C₅₃	—	4	2	no	—
164	D₄×C₅₃	—	2	2	no	—
165	D₄×C₅₃	—	2	2	no	—
166	D₄×C₅₃	—	4	1	no	—
167	D₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	8	no	—
185	C₂×C₄×C₅₃	—	4	2	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
194	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
195	C₂×C₄×C₅₃	#135	1	2	yes	—
196	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
202	C₈×C₅₃	—	2	4	no	—
203	C₈×C₅₃	—	2	2	no	—
204	C₈×C₅₃	—	4	1	no	—
205	#23094	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂²×C₅₃	—	4	2	no	—
207	C₂²×C₅₃	—	4	2	no	—
208	C₂²×C₅₃	—	4	2	no	—
209	C₂²×C₅₃	—	8	1	no	—
210	C₂²×C₅₃	—	8	1	no	—
211	C₂²×C₅₃	—	8	1	no	—
212	C₂²×C₅₃	—	8	1	no	—
213	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂²×C₅₃	#546	1	1	yes	characteristic
215	C₂²×C₅₃	#550	1	1	yes	characteristic
216	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
217	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
218	C₂²×C₅₃	#577	1	1	yes	characteristic
219	C₂²×C₅₃	#578	1	1	yes	characteristic
220	C₄×C₅₃	—	2	2	no	—
221	C₄×C₅₃	—	2	4	no	—
222	C₄×C₅₃	—	4	4	no	—
223	C₄×C₅₃	—	4	2	no	—
224	C₄×C₅₃	—	4	2	no	—
225	C₄×C₅₃	—	4	2	no	—
226	C₄×C₅₃	—	8	2	no	—
227	C₄×C₅₃	—	8	1	no	—
228	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
229	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
230	C₄×C₅₃	#518	1	1	yes	characteristic
231	C₄×C₅₃	#553	1	1	yes	characteristic
232	C₄×C₅₃	#555	1	1	yes	characteristic
233	C₂×#461	C₂×C₅₃	1	1	yes	characteristic
234	#2365	C₂×C₅₃	1	1	yes	characteristic, complemented
235	#2354	C₂×C₅₃	1	1	yes	characteristic
236	#1968	C₂×C₅₃	1	2	yes	—
237	#1956	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#1948	C₂×C₅₃	1	1	yes	characteristic, complemented
239	#1927	C₂×C₅₃	1	2	yes	complemented
240	#1880	C₂×C₅₃	1	2	yes	—
241	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
242	C₄×#103	C₂×C₅₃	1	2	yes	complemented
243	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
244	C₂×C₅₃	—	8	2	no	—
245	C₂×C₅₃	#2363	1	1	yes	characteristic
246	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
247	C₂×C₅₃	#2947	1	1	yes	characteristic
248	C₂×C₅₃	#3298	1	1	yes	characteristic
249	C₂×C₅₃	#3303	1	1	yes	characteristic
250	C₂×C₅₃	#3392	1	1	yes	characteristic
251	C₂×C₅₃	#3393	1	1	yes	characteristic
252	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
253	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
255	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×#117	—	2	1	no	—
257	#461	C₄×C₅₃	1	8	yes	complemented
258	C₂×#106	C₂²×C₅₃	1	2	yes	—
259	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×#103	C₂²×C₅₃	1	2	yes	—
261	C₂×#103	C₂²×C₅₃	1	2	yes	—
262	C₂×#103	C₂²×C₅₃	1	2	yes	—
263	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
267	#391	C₂²×C₅₃	1	2	yes	—
268	#390	C₂²×C₅₃	1	2	yes	—
269	#389	—	2	1	no	—
270	#388	—	2	2	no	—
271	#387	—	2	2	no	—
272	#384	C₂²×C₅₃	1	1	yes	characteristic
273	#380	—	2	1	no	—
274	C₄×C₄⋊C₄	—	2	1	no	—
275	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
276	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
277	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
278	#340	C₂²×C₅₃	1	2	yes	—
279	#340	C₂²×C₅₃	1	1	yes	characteristic
280	#340	C₂²×C₅₃	1	1	yes	characteristic
281	#336	C₂²×C₅₃	1	2	yes	—
282	C₅₃	#23094	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
284	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
285	C₂³×C₄	—	2	1	no	—
286	C₂×SD₁₆	—	4	2	no	—
287	C₂×SD₁₆	—	4	2	no	—
288	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	—
289	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
291	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
294	C₄×Q₈	—	2	4	no	—
295	C₄×Q₈	—	4	2	no	—
296	C₄×D₄	—	2	2	no	—
297	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
298	#117	—	4	4	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₄²	—	2	2	no	—
314	C₂×C₄²	—	2	2	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
317	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
318	#106	—	4	2	no	—
319	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
320	#103	—	2	4	no	—
321	#103	—	4	2	no	—
322	#103	C₂×C₄×C₅₃	1	8	yes	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	C₄×C₈	—	2	2	no	—
326	C₄×C₈	—	4	2	no	—
327	#95	—	2	2	no	—
328	#95	—	2	2	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	D₄×C₅₃	1	2	yes	—
332	C₂⁴	—	2	1	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	—	4	2	no	—
353	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
358	SD₁₆	—	4	8	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
367	C₂×C₈	#41×C₅₃	1	2	yes	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	8	1	no	—
380	C₄⋊C₄	SD₁₆×C₅₃	1	4	yes	complemented
381	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
382	C₄⋊C₄	#41×C₅₃	1	2	yes	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₂²⋊C₄	—	8	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	4	4	no	—
389	C₄²	—	4	4	no	—
390	C₄²	—	4	2	no	—
391	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
392	C₄²	#41×C₅₃	1	2	yes	—
393	C₂³	—	2	2	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
401	Q₈	—	2	4	no	—
402	Q₈	—	2	4	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	4	2	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	4	1	no	—
408	D₄	—	4	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	8	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
433	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
434	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
435	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
436	C₂×C₄	#135×C₅₃	1	2	yes	—
437	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
441	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
442	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
443	C₈	—	2	4	no	—
444	C₈	—	2	2	no	—
445	C₈	—	4	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
454	C₂²	C₅₃×#546	1	1	yes	characteristic
455	C₂²	C₅₃×#550	1	1	yes	characteristic
456	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
457	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
458	C₂²	C₅₃×#577	1	1	yes	characteristic
459	C₂²	C₅₃×#578	1	1	yes	characteristic
460	C₄	—	2	2	no	—
461	C₄	—	2	4	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	4	no	—
466	C₄	—	8	2	no	—
467	C₄	—	8	1	no	—
468	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
469	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
470	C₄	C₅₃×#518	1	1	yes	characteristic
471	C₄	C₅₃×#553	1	1	yes	characteristic
472	C₄	C₅₃×#555	1	1	yes	characteristic
473	C₂	—	8	2	no	—
474	C₂	C₅₃×#2363	1	1	yes	characteristic
475	C₂	C₂×C₄×SD₁₆×C₅₃	1	1	yes	characteristic
476	C₂	C₅₃×#2947	1	1	yes	characteristic
477	C₂	C₅₃×#3298	1	1	yes	characteristic
478	C₂	C₅₃×#3303	1	1	yes	characteristic
479	C₂	C₅₃×#3392	1	1	yes	characteristic
480	C₂	C₅₃×#3393	1	1	yes	characteristic

	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	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2365	C₂	semidirect
5. C₅₃×#1956	C₂	semidirect
6. C₅₃×#1948	C₂	semidirect
7. C₅₃×#1927	C₂	semidirect
9. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₄×#103×C₅₃	C₂	semidirect
17. C₅₃×#461	C₄	semidirect
139. C₄⋊C₄×C₅₃	SD₁₆	semidirect
205. #23094	C₅₃	direct (N × H)
234. #2365	C₂×C₅₃	semidirect
237. #1956	C₂×C₅₃	semidirect
238. #1948	C₂×C₅₃	semidirect
239. #1927	C₂×C₅₃	semidirect
241. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
242. C₄×#103	C₂×C₅₃	semidirect
257. #461	C₄×C₅₃	semidirect
282. C₅₃	#23094	direct (N × H)
380. C₄⋊C₄	SD₁₆×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×#461	2
C₅₃×#2365	2
C₅₃×#2354	2
C₅₃×#1968	2
C₅₃×#1956	2
C₅₃×#1948	2
C₅₃×#1927	2
C₅₃×#1880	2
C₈×C₄⋊C₄×C₅₃	2
C₄×#103×C₅₃	2
C₄×#102×C₅₃	2
#23094	53

Subgroups — C53×#23094 #27230606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23094 #27230606