Subgroups — C₅₃×#25435 #27232947

Order 13568.

880 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#473	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3059	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3044	C₂	1	2	yes	maximal
5	C₅₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2047	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2005	C₂	1	2	yes	maximal
8	C₅₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1886	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic
12	C₂×#137×C₅₃	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#536	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#519	C₂²	1	2	yes	—
20	C₅₃×#494	—	2	4	no	—
21	C₅₃×#473	—	2	4	no	—
22	C₅₃×#470	—	2	2	no	—
23	C₅₃×#467	—	2	2	no	—
24	C₅₃×#464	—	2	4	no	—
25	C₅₃×#462	—	2	4	no	—
26	C₅₃×#441	—	2	2	no	—
27	C₄×Q₁₆×C₅₃	—	2	2	no	—
28	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#427	C₂²	1	1	yes	characteristic
30	C₅₃×#424	C₂²	1	2	yes	—
31	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#419	C₂²	1	2	yes	—
33	C₅₃×#419	C₂²	1	2	yes	—
34	C₅₃×#419	C₂²	1	2	yes	—
35	C₅₃×#416	C₂²	1	1	yes	characteristic
36	C₂×#101×C₅₃	C₂²	1	2	yes	—
37	C₂×#101×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#405	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#341	C₂²	1	1	yes	characteristic
41	C₅₃×#339	C₂²	1	2	yes	—
42	C₅₃×#339	C₂²	1	1	yes	characteristic
43	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
46	#137×C₅₃	—	2	4	no	—
47	#137×C₅₃	—	4	2	no	—
48	C₂×Q₁₆×C₅₃	—	2	1	no	—
49	C₂×Q₁₆×C₅₃	—	2	1	no	—
50	C₂×Q₁₆×C₅₃	—	2	2	no	—
51	C₂×Q₁₆×C₅₃	—	2	4	no	—
52	C₂×Q₁₆×C₅₃	—	2	1	no	—
53	C₂×Q₁₆×C₅₃	—	2	1	no	—
54	C₂×Q₁₆×C₅₃	—	2	2	no	—
55	C₂×Q₁₆×C₅₃	—	2	2	no	—
56	C₂×Q₁₆×C₅₃	—	2	4	no	—
57	C₂×Q₁₆×C₅₃	—	2	2	no	—
58	C₂×Q₁₆×C₅₃	—	4	2	no	—
59	C₂×Q₁₆×C₅₃	D₄	1	4	yes	—
60	C₂×SD₁₆×C₅₃	—	2	2	no	—
61	#131×C₅₃	—	2	2	no	—
62	#131×C₅₃	—	2	2	no	—
63	C₂×M₁₆×C₅₃	—	2	1	no	—
64	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
66	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	—	2	1	no	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₅₃	—	2	4	no	—
71	C₄⋊Q₈×C₅₃	—	2	4	no	—
72	C₄⋊Q₈×C₅₃	—	2	2	no	—
73	#124×C₅₃	—	2	2	no	—
74	#123×C₅₃	—	4	2	no	—
75	C₂²⋊Q₈×C₅₃	—	2	4	no	—
76	C₄×Q₈×C₅₃	—	2	4	no	—
77	C₄×Q₈×C₅₃	—	2	4	no	—
78	#117×C₅₃	—	4	2	no	—
79	#117×C₅₃	C₂³	1	1	yes	characteristic
80	#117×C₅₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	#108×C₅₃	—	2	2	no	—
84	#107×C₅₃	D₄	1	2	yes	—
85	#106×C₅₃	D₄	1	2	yes	—
86	C₄⋊C₈×C₅₃	—	2	4	no	—
87	#104×C₅₃	—	2	2	no	—
88	#103×C₅₃	—	2	4	no	—
89	#103×C₅₃	—	2	4	no	—
90	#103×C₅₃	—	2	4	no	—
91	#103×C₅₃	D₄	1	2	yes	—
92	#102×C₅₃	D₄	1	2	yes	—
93	#101×C₅₃	—	2	4	no	—
94	#101×C₅₃	—	4	2	no	—
95	#97×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	2	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	#41×C₅₃	—	2	1	no	—
99	#41×C₅₃	—	2	1	no	—
100	#41×C₅₃	—	2	1	no	—
101	#41×C₅₃	—	2	1	no	—
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	—	2	2	no	—
107	C₂×Q₈×C₅₃	—	2	2	no	—
108	C₂×Q₈×C₅₃	—	2	2	no	—
109	C₂×Q₈×C₅₃	—	2	2	no	—
110	C₂×Q₈×C₅₃	—	2	2	no	—
111	C₂×Q₈×C₅₃	—	2	2	no	—
112	C₂×Q₈×C₅₃	—	2	2	no	—
113	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
114	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	Q₁₆×C₅₃	—	2	4	no	—
123	Q₁₆×C₅₃	—	2	4	no	—
124	Q₁₆×C₅₃	—	4	8	no	—
125	Q₁₆×C₅₃	—	4	4	no	—
126	Q₁₆×C₅₃	—	4	2	no	—
127	Q₁₆×C₅₃	—	4	2	no	—
128	SD₁₆×C₅₃	—	4	4	no	—
129	M₁₆×C₅₃	—	2	1	no	—
130	M₁₆×C₅₃	—	2	2	no	—
131	M₁₆×C₅₃	—	2	1	no	—
132	M₁₆×C₅₃	—	2	1	no	—
133	M₁₆×C₅₃	—	2	1	no	—
134	M₁₆×C₅₃	—	2	2	no	—
135	M₁₆×C₅₃	—	4	2	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	4	2	no	—
142	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	#41	1	2	yes	—
147	C₄⋊C₄×C₅₃	—	4	8	no	—
148	C₄⋊C₄×C₅₃	—	4	4	no	—
149	C₄⋊C₄×C₅₃	—	4	4	no	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
152	C₂²⋊C₄×C₅₃	—	2	1	no	—
153	C₂²⋊C₄×C₅₃	—	2	2	no	—
154	C₂²⋊C₄×C₅₃	—	4	4	no	—
155	C₄²×C₅₃	—	2	1	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	4	4	no	—
159	C₄²×C₅₃	#41	1	2	yes	—
160	C₂³×C₅₃	—	2	1	no	—
161	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
162	Q₈×C₅₃	—	2	4	no	—
163	Q₈×C₅₃	—	2	1	no	—
164	Q₈×C₅₃	—	2	4	no	—
165	Q₈×C₅₃	—	2	1	no	—
166	Q₈×C₅₃	—	4	2	no	—
167	Q₈×C₅₃	—	4	2	no	—
168	Q₈×C₅₃	—	4	2	no	—
169	Q₈×C₅₃	—	4	2	no	—
170	Q₈×C₅₃	—	4	2	no	—
171	Q₈×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	2	1	no	—
173	D₄×C₅₃	—	2	1	no	—
174	D₄×C₅₃	—	4	1	no	—
175	D₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	2	4	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	4	no	—
181	C₂×C₄×C₅₃	—	2	4	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
195	C₈×C₅₃	—	2	2	no	—
196	C₈×C₅₃	—	4	1	no	—
197	C₈×C₅₃	—	4	2	no	—
198	C₈×C₅₃	—	4	1	no	—
199	C₈×C₅₃	#121	1	2	yes	—
200	C₈×C₅₃	#121	1	2	yes	—
201	#25435	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂²×C₅₃	—	2	2	no	—
203	C₂²×C₅₃	—	4	1	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	#540	1	1	yes	characteristic
206	C₂²×C₅₃	D₄²	1	1	yes	characteristic
207	C₂²×C₅₃	#546	1	1	yes	characteristic
208	C₄×C₅₃	—	2	4	no	—
209	C₄×C₅₃	—	4	1	no	—
210	C₄×C₅₃	—	4	4	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	4	no	—
213	C₄×C₅₃	—	8	2	no	—
214	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
215	C₄×C₅₃	#532	1	1	yes	characteristic
216	C₄×C₅₃	D₄²	1	1	yes	characteristic
217	C₄×C₅₃	#550	1	1	yes	characteristic
218	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
219	C₂×#473	C₂×C₅₃	1	1	yes	characteristic
220	#3059	C₂×C₅₃	1	1	yes	characteristic
221	#3044	C₂×C₅₃	1	2	yes	—
222	#2950	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#2047	C₂×C₅₃	1	2	yes	complemented
224	#2005	C₂×C₅₃	1	2	yes	—
225	#1979	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#1945	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#1886	C₂×C₅₃	1	2	yes	complemented
228	#1764	C₂×C₅₃	1	1	yes	characteristic
229	C₂×C₅₃	—	2	1	no	—
230	C₂×C₅₃	—	2	1	no	—
231	C₂×C₅₃	—	8	1	no	—
232	C₂×C₅₃	#2428	1	1	yes	characteristic
233	C₂×C₅₃	#3293	1	2	yes	—
234	C₂×#137	C₂²×C₅₃	1	2	yes	—
235	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
236	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
237	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
238	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
239	#536	C₂²×C₅₃	1	2	yes	—
240	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
241	#519	C₂²×C₅₃	1	2	yes	—
242	#494	—	2	4	no	—
243	#473	—	2	4	no	—
244	#470	—	2	2	no	—
245	#467	—	2	2	no	—
246	#464	—	2	4	no	—
247	#462	—	2	4	no	—
248	#441	—	2	2	no	—
249	C₄×Q₁₆	—	2	2	no	—
250	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
251	#427	C₂²×C₅₃	1	1	yes	characteristic
252	#424	C₂²×C₅₃	1	2	yes	—
253	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
254	#419	C₂²×C₅₃	1	2	yes	—
255	#419	C₂²×C₅₃	1	2	yes	—
256	#419	C₂²×C₅₃	1	2	yes	—
257	#416	C₂²×C₅₃	1	1	yes	characteristic
258	C₂×#101	C₂²×C₅₃	1	2	yes	—
259	C₂×#101	C₂²×C₅₃	1	2	yes	—
260	#405	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
262	#341	C₂²×C₅₃	1	1	yes	characteristic
263	#339	C₂²×C₅₃	1	1	yes	characteristic
264	#339	C₂²×C₅₃	1	2	yes	—
265	C₅₃	#25435	1	1	yes	characteristic, direct product factor, Sylow
266	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
267	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
268	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
269	#137	—	2	4	no	—
270	#137	—	4	2	no	—
271	C₂×Q₁₆	—	2	4	no	—
272	C₂×Q₁₆	—	2	2	no	—
273	C₂×Q₁₆	—	2	1	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	2	4	no	—
276	C₂×Q₁₆	—	2	1	no	—
277	C₂×Q₁₆	—	2	1	no	—
278	C₂×Q₁₆	—	2	2	no	—
279	C₂×Q₁₆	—	2	2	no	—
280	C₂×Q₁₆	—	2	2	no	—
281	C₂×Q₁₆	—	4	2	no	—
282	C₂×Q₁₆	D₄×C₅₃	1	4	yes	—
283	C₂×SD₁₆	—	2	2	no	—
284	#131	—	2	2	no	—
285	#131	—	2	2	no	—
286	C₂×M₁₆	—	2	1	no	—
287	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
288	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
289	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
293	C₄⋊Q₈	—	2	4	no	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	—	2	4	no	—
296	#124	—	2	2	no	—
297	#123	—	4	2	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₄×Q₈	—	2	4	no	—
300	C₄×Q₈	—	2	4	no	—
301	#117	—	4	2	no	—
302	#117	C₂³×C₅₃	1	1	yes	characteristic
303	#117	C₂³×C₅₃	1	2	yes	—
304	C₂×C₄⋊C₄	—	4	2	no	—
305	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
306	#108	—	2	2	no	—
307	#107	D₄×C₅₃	1	2	yes	—
308	#106	D₄×C₅₃	1	2	yes	—
309	C₄⋊C₈	—	2	4	no	—
310	#104	—	2	2	no	—
311	#103	—	2	4	no	—
312	#103	—	2	4	no	—
313	#103	—	2	4	no	—
314	#103	D₄×C₅₃	1	2	yes	—
315	#102	D₄×C₅₃	1	2	yes	—
316	#101	—	2	4	no	—
317	#101	—	4	2	no	—
318	#97	—	2	1	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	—	2	1	no	—
321	#41	—	2	1	no	—
322	#41	—	2	1	no	—
323	#41	—	2	1	no	—
324	#41	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
337	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
345	Q₁₆	—	2	4	no	—
346	Q₁₆	—	2	4	no	—
347	Q₁₆	—	4	8	no	—
348	Q₁₆	—	4	4	no	—
349	Q₁₆	—	4	2	no	—
350	Q₁₆	—	4	2	no	—
351	SD₁₆	—	4	4	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	2	2	no	—
354	M₁₆	—	2	1	no	—
355	M₁₆	—	2	1	no	—
356	M₁₆	—	2	1	no	—
357	M₁₆	—	2	2	no	—
358	M₁₆	—	4	2	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	4	2	no	—
365	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
366	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
368	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₈	#41×C₅₃	1	2	yes	—
370	C₄⋊C₄	—	4	8	no	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
374	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	4	4	no	—
382	C₄²	#41×C₅₃	1	2	yes	—
383	C₂³	—	2	1	no	—
384	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
385	Q₈	—	2	4	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	2	4	no	—
388	Q₈	—	2	1	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	4	2	no	—
392	Q₈	—	4	2	no	—
393	Q₈	—	4	2	no	—
394	Q₈	—	4	2	no	—
395	D₄	—	2	1	no	—
396	D₄	—	2	1	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
417	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	4	1	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	1	no	—
422	C₈	#121×C₅₃	1	2	yes	—
423	C₈	#121×C₅₃	1	2	yes	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	2	no	—
427	C₂²	C₅₃×#540	1	1	yes	characteristic
428	C₂²	D₄²×C₅₃	1	1	yes	characteristic
429	C₂²	C₅₃×#546	1	1	yes	characteristic
430	C₄	—	2	4	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	4	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	4	no	—
435	C₄	—	8	2	no	—
436	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
437	C₄	C₅₃×#532	1	1	yes	characteristic
438	C₄	D₄²×C₅₃	1	1	yes	characteristic
439	C₄	C₅₃×#550	1	1	yes	characteristic
440	C₂	—	2	1	no	—
441	C₂	—	2	1	no	—
442	C₂	—	8	1	no	—
443	C₂	C₅₃×#2428	1	1	yes	characteristic
444	C₂	C₅₃×#3293	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#494	C₂	semidirect
5. C₅₃×#2950	C₂	semidirect
6. C₅₃×#2047	C₂	semidirect
8. C₅₃×#1979	C₂	semidirect
9. C₅₃×#1945	C₂	semidirect
10. C₅₃×#1886	C₂	semidirect
201. #25435	C₅₃	direct (N × H)
218. C₂×#494	C₂×C₅₃	semidirect
222. #2950	C₂×C₅₃	semidirect
223. #2047	C₂×C₅₃	semidirect
225. #1979	C₂×C₅₃	semidirect
226. #1945	C₂×C₅₃	semidirect
227. #1886	C₂×C₅₃	semidirect
265. C₅₃	#25435	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#494	2
C₂×C₅₃×#473	2
C₅₃×#3059	2
C₅₃×#3044	2
C₅₃×#2950	2
C₅₃×#2047	2
C₅₃×#2005	2
C₅₃×#1979	2
C₅₃×#1945	2
C₅₃×#1886	2
C₅₃×#1764	2
#25435	53

Subgroups — C53×#25435 #27232947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#25435 #27232947