Subgroups — C₅₃×#25433 #27232945

Order 13568.

916 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#472	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3042	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2046	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2003	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1885	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#536	C₂²	1	2	yes	—
18	C₂×#124×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#518	C₂²	1	2	yes	—
20	C₅₃×#495	—	2	4	no	—
21	C₅₃×#472	—	2	4	no	—
22	C₅₃×#470	—	2	2	no	—
23	C₅₃×#467	—	2	2	no	—
24	C₅₃×#463	—	2	4	no	—
25	C₅₃×#461	—	2	4	no	—
26	C₅₃×#442	—	2	2	no	—
27	C₄×D₈×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×#418	C₂²	1	2	yes	—
35	C₅₃×#416	C₂²	1	1	yes	characteristic
36	C₂×#101×C₅₃	C₂²	1	2	yes	—
37	C₂×#100×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×D₄×C₅₃	C₂³	1	2	yes	—
46	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
47	AGL(1,ℤ/8)×C₅₃	—	4	2	no	—
48	C₂×Q₁₆×C₅₃	—	2	2	no	—
49	C₂×Q₁₆×C₅₃	—	2	1	no	—
50	C₂×Q₁₆×C₅₃	—	2	1	no	—
51	C₂×Q₁₆×C₅₃	—	2	2	no	—
52	C₂×SD₁₆×C₅₃	—	2	4	no	—
53	C₂×SD₁₆×C₅₃	—	2	2	no	—
54	C₂×SD₁₆×C₅₃	—	2	4	no	—
55	C₂×SD₁₆×C₅₃	—	4	2	no	—
56	C₂×D₈×C₅₃	—	2	1	no	—
57	C₂×D₈×C₅₃	—	2	1	no	—
58	C₂×D₈×C₅₃	—	2	2	no	—
59	C₂×D₈×C₅₃	—	2	2	no	—
60	C₂×D₈×C₅₃	D₄	1	4	yes	—
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	2	no	—
71	#124×C₅₃	—	2	4	no	—
72	#124×C₅₃	—	2	4	no	—
73	#124×C₅₃	—	2	2	no	—
74	#123×C₅₃	—	4	2	no	—
75	C₂²⋊Q₈×C₅₃	—	2	4	no	—
76	C₄×D₄×C₅₃	—	2	4	no	—
77	C₄×D₄×C₅₃	—	2	4	no	—
78	#117×C₅₃	C₂³	1	1	yes	characteristic
79	#117×C₅₃	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₂×C₂²⋊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	D₄	1	2	yes	—
91	#102×C₅₃	—	2	4	no	—
92	#102×C₅₃	D₄	1	2	yes	—
93	#101×C₅₃	—	4	2	no	—
94	#100×C₅₃	—	2	4	no	—
95	#97×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	2	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₂⁴×C₅₃	—	2	2	no	—
99	#41×C₅₃	—	2	1	no	—
100	#41×C₅₃	—	2	1	no	—
101	#41×C₅₃	—	2	1	no	—
102	#41×C₅₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	2	2	no	—
116	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
117	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	4	4	no	—
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	Q₁₆×C₅₃	—	4	2	no	—
124	Q₁₆×C₅₃	—	4	2	no	—
125	SD₁₆×C₅₃	—	4	8	no	—
126	SD₁₆×C₅₃	—	4	4	no	—
127	D₈×C₅₃	—	2	4	no	—
128	D₈×C₅₃	—	2	4	no	—
129	D₈×C₅₃	—	4	4	no	—
130	M₁₆×C₅₃	—	2	1	no	—
131	M₁₆×C₅₃	—	2	2	no	—
132	M₁₆×C₅₃	—	2	1	no	—
133	M₁₆×C₅₃	—	2	1	no	—
134	M₁₆×C₅₃	—	2	1	no	—
135	M₁₆×C₅₃	—	2	2	no	—
136	M₁₆×C₅₃	—	4	2	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	1	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	4	2	no	—
143	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	#41	1	2	yes	—
148	C₄⋊C₄×C₅₃	—	4	4	no	—
149	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₅₃	—	2	1	no	—
152	C₂²⋊C₄×C₅₃	—	2	2	no	—
153	C₂²⋊C₄×C₅₃	—	4	4	no	—
154	C₂²⋊C₄×C₅₃	—	4	8	no	—
155	C₂²⋊C₄×C₅₃	—	4	4	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	#41	1	2	yes	—
160	C₂³×C₅₃	—	2	1	no	—
161	C₂³×C₅₃	—	2	4	no	—
162	C₂³×C₅₃	—	4	2	no	—
163	C₂³×C₅₃	—	4	2	no	—
164	C₂³×C₅₃	—	8	4	no	—
165	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₅₃	—	2	1	no	—
167	Q₈×C₅₃	—	2	1	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	4	no	—
173	D₄×C₅₃	—	2	1	no	—
174	D₄×C₅₃	—	2	4	no	—
175	D₄×C₅₃	—	2	1	no	—
176	D₄×C₅₃	—	4	1	no	—
177	D₄×C₅₃	—	4	1	no	—
178	D₄×C₅₃	—	4	2	no	—
179	D₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	4	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	16	4	no	—
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
197	C₈×C₅₃	—	2	2	no	—
198	C₈×C₅₃	—	4	1	no	—
199	C₈×C₅₃	—	4	2	no	—
200	C₈×C₅₃	—	4	1	no	—
201	C₈×C₅₃	#121	1	2	yes	—
202	C₈×C₅₃	#121	1	2	yes	—
203	#25433	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	4	1	no	—
206	C₂²×C₅₃	—	4	4	no	—
207	C₂²×C₅₃	—	4	2	no	—
208	C₂²×C₅₃	—	8	4	no	—
209	C₂²×C₅₃	—	8	4	no	—
210	C₂²×C₅₃	—	16	2	no	—
211	C₂²×C₅₃	#540	1	1	yes	characteristic
212	C₂²×C₅₃	D₄²	1	1	yes	characteristic
213	C₂²×C₅₃	#546	1	1	yes	characteristic
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	4	1	no	—
216	C₄×C₅₃	—	4	4	no	—
217	C₄×C₅₃	—	4	2	no	—
218	C₄×C₅₃	—	8	2	no	—
219	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
220	C₄×C₅₃	#532	1	1	yes	characteristic
221	C₄×C₅₃	D₄²	1	1	yes	characteristic
222	C₄×C₅₃	#550	1	1	yes	characteristic
223	C₂×#495	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×#472	C₂×C₅₃	1	1	yes	characteristic
225	#3059	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#3042	C₂×C₅₃	1	2	yes	complemented
227	#2948	C₂×C₅₃	1	1	yes	characteristic, complemented
228	#2046	C₂×C₅₃	1	2	yes	complemented
229	#2003	C₂×C₅₃	1	2	yes	complemented
230	#1979	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#1945	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1885	C₂×C₅₃	1	2	yes	complemented
233	#1764	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×C₅₃	—	2	1	no	—
235	C₂×C₅₃	—	2	1	no	—
236	C₂×C₅₃	—	8	1	no	—
237	C₂×C₅₃	—	8	4	no	—
238	C₂×C₅₃	#2428	1	1	yes	characteristic
239	C₂×C₅₃	#3290	1	2	yes	—
240	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
241	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
242	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
243	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
245	#536	C₂²×C₅₃	1	2	yes	—
246	C₂×#124	C₂²×C₅₃	1	2	yes	—
247	#518	C₂²×C₅₃	1	2	yes	—
248	#495	—	2	4	no	—
249	#472	—	2	4	no	—
250	#470	—	2	2	no	—
251	#467	—	2	2	no	—
252	#463	—	2	4	no	—
253	#461	—	2	4	no	—
254	#442	—	2	2	no	—
255	C₄×D₈	—	2	2	no	—
256	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
257	#427	C₂²×C₅₃	1	1	yes	characteristic
258	#424	C₂²×C₅₃	1	2	yes	—
259	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
260	#419	C₂²×C₅₃	1	2	yes	—
261	#419	C₂²×C₅₃	1	2	yes	—
262	#418	C₂²×C₅₃	1	2	yes	—
263	#416	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×#101	C₂²×C₅₃	1	2	yes	—
265	C₂×#100	C₂²×C₅₃	1	2	yes	—
266	#405	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
268	#341	C₂²×C₅₃	1	1	yes	characteristic
269	#339	C₂²×C₅₃	1	1	yes	characteristic
270	#339	C₂²×C₅₃	1	2	yes	—
271	C₅₃	#25433	1	1	yes	characteristic, direct product factor, Sylow
272	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
273	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
274	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
275	AGL(1,ℤ/8)	—	2	4	no	—
276	AGL(1,ℤ/8)	—	4	2	no	—
277	C₂×Q₁₆	—	2	1	no	—
278	C₂×Q₁₆	—	2	1	no	—
279	C₂×Q₁₆	—	2	2	no	—
280	C₂×Q₁₆	—	2	2	no	—
281	C₂×SD₁₆	—	2	2	no	—
282	C₂×SD₁₆	—	2	4	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	4	2	no	—
285	C₂×D₈	—	2	2	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	1	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	D₄×C₅₃	1	4	yes	—
290	#131	—	2	2	no	—
291	#131	—	2	2	no	—
292	C₂×M₁₆	—	2	1	no	—
293	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
295	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
299	C₄⋊Q₈	—	2	2	no	—
300	#124	—	2	4	no	—
301	#124	—	2	2	no	—
302	#124	—	2	4	no	—
303	#123	—	4	2	no	—
304	C₂²⋊Q₈	—	2	4	no	—
305	C₄×D₄	—	2	4	no	—
306	C₄×D₄	—	2	4	no	—
307	#117	C₂³×C₅₃	1	1	yes	characteristic
308	#117	C₂³×C₅₃	1	2	yes	—
309	C₂×C₂²⋊C₄	—	4	2	no	—
310	C₂×C₂²⋊C₄	—	4	2	no	—
311	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
312	#108	—	2	2	no	—
313	#107	D₄×C₅₃	1	2	yes	—
314	#106	D₄×C₅₃	1	2	yes	—
315	C₄⋊C₈	—	2	4	no	—
316	#104	—	2	2	no	—
317	#103	—	2	4	no	—
318	#103	—	2	4	no	—
319	#103	D₄×C₅₃	1	2	yes	—
320	#102	—	2	4	no	—
321	#102	D₄×C₅₃	1	2	yes	—
322	#101	—	4	2	no	—
323	#100	—	2	4	no	—
324	#97	—	2	1	no	—
325	C₄×C₈	—	2	2	no	—
326	C₄×C₈	—	2	1	no	—
327	C₂⁴	—	2	2	no	—
328	#41	—	2	1	no	—
329	#41	—	2	1	no	—
330	#41	—	2	1	no	—
331	#41	—	2	1	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	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
346	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
352	Q₁₆	—	4	2	no	—
353	Q₁₆	—	4	2	no	—
354	SD₁₆	—	4	8	no	—
355	SD₁₆	—	4	4	no	—
356	D₈	—	2	4	no	—
357	D₈	—	2	4	no	—
358	D₈	—	4	4	no	—
359	M₁₆	—	2	1	no	—
360	M₁₆	—	2	2	no	—
361	M₁₆	—	2	1	no	—
362	M₁₆	—	2	1	no	—
363	M₁₆	—	2	1	no	—
364	M₁₆	—	2	2	no	—
365	M₁₆	—	4	2	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	2	no	—
372	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
376	C₂×C₈	#41×C₅₃	1	2	yes	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
379	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	2	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	4	8	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	2	no	—
388	C₄²	#41×C₅₃	1	2	yes	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	8	4	no	—
394	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
395	Q₈	—	2	1	no	—
396	Q₈	—	2	1	no	—
397	Q₈	—	4	2	no	—
398	Q₈	—	4	2	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	4	no	—
402	D₄	—	2	1	no	—
403	D₄	—	2	1	no	—
404	D₄	—	2	4	no	—
405	D₄	—	4	1	no	—
406	D₄	—	4	1	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	2	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	16	4	no	—
420	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
425	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	4	1	no	—
428	C₈	—	4	2	no	—
429	C₈	—	4	1	no	—
430	C₈	#121×C₅₃	1	2	yes	—
431	C₈	#121×C₅₃	1	2	yes	—
432	C₂²	—	2	2	no	—
433	C₂²	—	4	1	no	—
434	C₂²	—	4	4	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	8	4	no	—
437	C₂²	—	8	4	no	—
438	C₂²	—	16	2	no	—
439	C₂²	C₅₃×#540	1	1	yes	characteristic
440	C₂²	D₄²×C₅₃	1	1	yes	characteristic
441	C₂²	C₅₃×#546	1	1	yes	characteristic
442	C₄	—	4	2	no	—
443	C₄	—	4	1	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	4	no	—
446	C₄	—	8	2	no	—
447	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
448	C₄	C₅₃×#532	1	1	yes	characteristic
449	C₄	D₄²×C₅₃	1	1	yes	characteristic
450	C₄	C₅₃×#550	1	1	yes	characteristic
451	C₂	—	2	1	no	—
452	C₂	—	2	1	no	—
453	C₂	—	8	1	no	—
454	C₂	—	8	4	no	—
455	C₂	C₅₃×#2428	1	1	yes	characteristic
456	C₂	C₅₃×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#472	C₂
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×#495	C₂	semidirect
3. C₅₃×#3059	C₂	semidirect
4. C₅₃×#3042	C₂	semidirect
5. C₅₃×#2948	C₂	semidirect
6. C₅₃×#2046	C₂	semidirect
7. C₅₃×#2003	C₂	semidirect
8. C₅₃×#1979	C₂	semidirect
9. C₅₃×#1945	C₂	semidirect
10. C₅₃×#1885	C₂	semidirect
11. C₅₃×#1764	C₂	semidirect
203. #25433	C₅₃	direct (N × H)
223. C₂×#495	C₂×C₅₃	semidirect
225. #3059	C₂×C₅₃	semidirect
226. #3042	C₂×C₅₃	semidirect
227. #2948	C₂×C₅₃	semidirect
228. #2046	C₂×C₅₃	semidirect
229. #2003	C₂×C₅₃	semidirect
230. #1979	C₂×C₅₃	semidirect
231. #1945	C₂×C₅₃	semidirect
232. #1885	C₂×C₅₃	semidirect
233. #1764	C₂×C₅₃	semidirect
271. C₅₃	#25433	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×#495	2
C₂×C₅₃×#472	2
C₅₃×#3059	2
C₅₃×#3042	2
C₅₃×#2948	2
C₅₃×#2046	2
C₅₃×#2003	2
C₅₃×#1979	2
C₅₃×#1945	2
C₅₃×#1885	2
C₅₃×#1764	2
#25433	53

Subgroups — C53×#25433 #27232945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#25433 #27232945