Subgroups — C₅₃×#26083 #27233595

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3083	C₂	1	2	yes	maximal
3	C₅₃×#2913	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2089	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2060	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1859	C₂	1	2	yes	maximal
9	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1816	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic
12	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#544	C₂²	1	2	yes	—
14	C₅₃×#520	C₂²	1	2	yes	—
15	C₅₃×#501	—	2	2	no	—
16	C₅₃×#500	—	2	1	no	—
17	C₅₃×#499	—	2	1	no	—
18	C₅₃×#479	—	2	2	no	—
19	C₅₃×#478	—	2	2	no	—
20	C₅₃×#477	—	2	2	no	—
21	C₅₃×#474	—	2	2	no	—
22	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#428	—	2	2	no	—
24	C₅₃×#428	C₂²	1	2	yes	—
25	C₅₃×#428	C₂²	1	2	yes	—
26	C₅₃×#428	C₂²	1	2	yes	—
27	C₅₃×#427	—	2	1	no	—
28	C₅₃×#427	C₂²	1	1	yes	characteristic
29	C₂×#107×C₅₃	—	2	1	no	—
30	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#107×C₅₃	C₂²	1	2	yes	—
32	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#424	C₂²	1	2	yes	—
34	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#417	C₂²	1	2	yes	—
36	C₅₃×#416	C₂²	1	1	yes	characteristic
37	C₅₃×#405	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#384	C₂²	1	2	yes	—
40	C₅₃×#343	C₂²	1	2	yes	—
41	C₅₃×#343	C₂²	1	2	yes	—
42	C₅₃×#341	C₂²	1	1	yes	characteristic
43	C₅₃×#339	C₂²	1	2	yes	—
44	C₅₃×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
46	#131×C₅₃	—	2	2	no	—
47	#131×C₅₃	Q₈	1	4	yes	complemented
48	C₂×M₁₆×C₅₃	—	2	1	no	—
49	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	—	2	2	no	—
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	#125×C₅₃	—	2	2	no	—
58	#125×C₅₃	—	2	2	no	—
59	C₂²⋊Q₈×C₅₃	—	4	2	no	—
60	C₄×Q₈×C₅₃	—	2	2	no	—
61	C₄×D₄×C₅₃	—	2	2	no	—
62	C₄×D₄×C₅₃	—	4	2	no	—
63	#117×C₅₃	—	2	2	no	—
64	#117×C₅₃	C₂³	1	2	yes	—
65	#117×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#108×C₅₃	—	2	2	no	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	—	2	1	no	—
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	—	2	2	no	—
77	#107×C₅₃	—	2	2	no	—
78	#107×C₅₃	—	2	2	no	—
79	#107×C₅₃	—	2	2	no	—
80	#107×C₅₃	—	2	2	no	—
81	#107×C₅₃	D₄	1	1	yes	characteristic
82	#107×C₅₃	D₄	1	1	yes	characteristic
83	#106×C₅₃	—	2	2	no	—
84	#106×C₅₃	—	2	2	no	—
85	#106×C₅₃	—	2	2	no	—
86	#106×C₅₃	D₄	1	2	yes	—
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	—	2	2	no	—
89	#104×C₅₃	Q₈	1	4	yes	complemented
90	#103×C₅₃	—	2	2	no	—
91	#103×C₅₃	D₄	1	1	yes	characteristic
92	#103×C₅₃	D₄	1	1	yes	characteristic
93	#102×C₅₃	—	2	2	no	—
94	#102×C₅₃	D₄	1	1	yes	characteristic
95	#102×C₅₃	D₄	1	1	yes	characteristic
96	#97×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	—	2	2	no	—
99	#41×C₅₃	—	2	1	no	—
100	#41×C₅₃	—	2	1	no	—
101	#41×C₅₃	C₂×Q₈	1	2	yes	—
102	#41×C₅₃	C₂×Q₈	1	2	yes	—
103	C₂×Q₈×C₅₃	—	4	2	no	—
104	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	4	2	no	—
112	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₅₃	—	2	1	no	—
114	M₁₆×C₅₃	—	2	1	no	—
115	M₁₆×C₅₃	—	2	4	no	—
116	M₁₆×C₅₃	—	2	2	no	—
117	M₁₆×C₅₃	—	2	2	no	—
118	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
119	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	4	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₅₃	#41	1	2	yes	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	4	2	no	—
137	C₄⋊C₄×C₅₃	—	4	2	no	—
138	C₄⋊C₄×C₅₃	—	4	2	no	—
139	C₄⋊C₄×C₅₃	—	4	2	no	—
140	C₄⋊C₄×C₅₃	—	8	4	no	—
141	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
145	C₂²⋊C₄×C₅₃	—	2	2	no	—
146	C₂²⋊C₄×C₅₃	—	2	1	no	—
147	C₂²⋊C₄×C₅₃	—	4	2	no	—
148	C₄²×C₅₃	—	2	1	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	2	1	no	—
151	C₄²×C₅₃	—	4	2	no	—
152	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
153	C₂³×C₅₃	—	2	1	no	—
154	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
155	Q₈×C₅₃	—	2	1	no	—
156	Q₈×C₅₃	—	16	2	no	—
157	Q₈×C₅₃	C₄⋊Q₈	1	2	yes	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	2	2	no	—
160	D₄×C₅₃	—	2	1	no	—
161	D₄×C₅₃	C₄⋊Q₈	1	2	yes	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	8	2	no	—
175	C₂×C₄×C₅₃	—	8	2	no	—
176	C₂×C₄×C₅₃	—	16	2	no	—
177	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	2	1	no	—
185	C₈×C₅₃	—	2	1	no	—
186	C₈×C₅₃	—	2	2	no	—
187	C₈×C₅₃	—	4	2	no	—
188	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
189	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
190	#26083	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	2	no	—
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	—	4	1	no	—
196	C₂²×C₅₃	#544	1	1	yes	characteristic
197	C₂²×C₅₃	D₄²	1	1	yes	characteristic
198	C₂²×C₅₃	#554	1	1	yes	characteristic
199	C₄×C₅₃	—	2	4	no	—
200	C₄×C₅₃	—	2	2	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	—	4	1	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	4	1	no	—
205	C₄×C₅₃	—	8	2	no	—
206	C₄×C₅₃	—	8	2	no	—
207	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
208	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
209	C₄×C₅₃	#547	1	1	yes	characteristic
210	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
211	#3164	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#3083	C₂×C₅₃	1	2	yes	—
213	#2913	C₂×C₅₃	1	1	yes	characteristic
214	#2089	C₂×C₅₃	1	2	yes	complemented
215	#2060	C₂×C₅₃	1	2	yes	complemented
216	#1952	C₂×C₅₃	1	1	yes	characteristic, complemented
217	#1945	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#1859	C₂×C₅₃	1	2	yes	—
219	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#1816	C₂×C₅₃	1	1	yes	characteristic
221	#1764	C₂×C₅₃	1	1	yes	characteristic
222	C₂×C₅₃	—	2	1	no	—
223	C₂×C₅₃	—	2	1	no	—
224	C₂×C₅₃	—	4	2	no	—
225	C₂×C₅₃	#2440	1	1	yes	characteristic
226	C₂×C₅₃	#3292	1	1	yes	characteristic
227	C₂×C₅₃	#3293	1	1	yes	characteristic
228	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
229	#544	C₂²×C₅₃	1	2	yes	—
230	#520	C₂²×C₅₃	1	2	yes	—
231	#501	—	2	2	no	—
232	#500	—	2	1	no	—
233	#499	—	2	1	no	—
234	#479	—	2	2	no	—
235	#478	—	2	2	no	—
236	#477	—	2	2	no	—
237	#474	—	2	2	no	—
238	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
239	#428	—	2	2	no	—
240	#428	C₂²×C₅₃	1	2	yes	—
241	#428	C₂²×C₅₃	1	2	yes	—
242	#428	C₂²×C₅₃	1	2	yes	—
243	#427	—	2	1	no	—
244	#427	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×#107	—	2	1	no	—
246	C₂×#107	C₂²×C₅₃	1	2	yes	—
247	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
249	#424	C₂²×C₅₃	1	2	yes	—
250	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
251	#417	C₂²×C₅₃	1	2	yes	—
252	#416	C₂²×C₅₃	1	1	yes	characteristic
253	#405	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
255	#384	C₂²×C₅₃	1	2	yes	—
256	#343	C₂²×C₅₃	1	2	yes	—
257	#343	C₂²×C₅₃	1	2	yes	—
258	#341	C₂²×C₅₃	1	1	yes	characteristic
259	#339	C₂²×C₅₃	1	1	yes	characteristic
260	#339	C₂²×C₅₃	1	2	yes	—
261	C₅₃	#26083	1	1	yes	characteristic, direct product factor, Sylow
262	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
263	#131	—	2	2	no	—
264	#131	Q₈×C₅₃	1	4	yes	complemented
265	C₂×M₁₆	—	2	1	no	—
266	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
267	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
268	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
272	C₄⋊Q₈	—	2	2	no	—
273	C₄⋊Q₈	—	2	2	no	—
274	#125	—	2	2	no	—
275	#125	—	2	2	no	—
276	C₂²⋊Q₈	—	4	2	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	—	4	2	no	—
280	#117	—	2	2	no	—
281	#117	C₂³×C₅₃	1	2	yes	—
282	#117	C₂³×C₅₃	1	1	yes	characteristic
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	4	2	no	—
285	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
287	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
288	#108	—	2	2	no	—
289	#107	—	2	1	no	—
290	#107	—	2	1	no	—
291	#107	—	2	2	no	—
292	#107	—	2	2	no	—
293	#107	—	2	1	no	—
294	#107	—	2	1	no	—
295	#107	—	2	2	no	—
296	#107	—	2	2	no	—
297	#107	—	2	2	no	—
298	#107	D₄×C₅₃	1	1	yes	characteristic
299	#107	D₄×C₅₃	1	1	yes	characteristic
300	#106	—	2	2	no	—
301	#106	—	2	2	no	—
302	#106	—	2	2	no	—
303	#106	D₄×C₅₃	1	2	yes	—
304	C₄⋊C₈	—	2	2	no	—
305	C₄⋊C₈	—	2	2	no	—
306	#104	Q₈×C₅₃	1	4	yes	complemented
307	#103	—	2	2	no	—
308	#103	D₄×C₅₃	1	1	yes	characteristic
309	#103	D₄×C₅₃	1	1	yes	characteristic
310	#102	—	2	2	no	—
311	#102	D₄×C₅₃	1	1	yes	characteristic
312	#102	D₄×C₅₃	1	1	yes	characteristic
313	#97	—	2	1	no	—
314	C₄×C₈	—	2	1	no	—
315	C₄×C₈	—	2	2	no	—
316	#41	—	2	1	no	—
317	#41	—	2	1	no	—
318	#41	C₂×Q₈×C₅₃	1	2	yes	—
319	#41	C₂×Q₈×C₅₃	1	2	yes	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
330	M₁₆	—	2	1	no	—
331	M₁₆	—	2	1	no	—
332	M₁₆	—	2	4	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	—	2	2	no	—
335	M₁₆	C₂×Q₈×C₅₃	1	2	yes	—
336	M₁₆	C₂×Q₈×C₅₃	1	2	yes	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	4	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
346	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
347	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
348	C₂×C₈	#41×C₅₃	1	2	yes	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	8	4	no	—
358	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
360	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
361	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	1	no	—
368	C₄²	—	4	2	no	—
369	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
370	C₂³	—	2	1	no	—
371	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
372	Q₈	—	2	1	no	—
373	Q₈	—	16	2	no	—
374	Q₈	C₄⋊Q₈×C₅₃	1	2	yes	—
375	D₄	—	2	2	no	—
376	D₄	—	2	2	no	—
377	D₄	—	2	1	no	—
378	D₄	C₄⋊Q₈×C₅₃	1	2	yes	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	—	16	2	no	—
394	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
397	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
399	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
400	C₈	—	2	2	no	—
401	C₈	—	2	1	no	—
402	C₈	—	2	1	no	—
403	C₈	—	2	2	no	—
404	C₈	—	4	2	no	—
405	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
406	C₈	C₂²⋊Q₈×C₅₃	1	2	yes	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	C₅₃×#544	1	1	yes	characteristic
413	C₂²	D₄²×C₅₃	1	1	yes	characteristic
414	C₂²	C₅₃×#554	1	1	yes	characteristic
415	C₄	—	2	4	no	—
416	C₄	—	2	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	1	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	1	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	2	no	—
423	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
424	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
425	C₄	C₅₃×#547	1	1	yes	characteristic
426	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	4	2	no	—
430	C₂	C₅₃×#2440	1	1	yes	characteristic
431	C₂	C₅₃×#3292	1	1	yes	characteristic
432	C₂	C₅₃×#3293	1	1	yes	characteristic

	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. C₅₃×#3164	C₂	semidirect
4. C₅₃×#2089	C₂	semidirect
5. C₅₃×#2060	C₂	semidirect
6. C₅₃×#1952	C₂	semidirect
7. C₅₃×#1945	C₂	semidirect
9. C₅₃×#1845	C₂	semidirect
47. #131×C₅₃	Q₈	semidirect
89. #104×C₅₃	Q₈	semidirect
190. #26083	C₅₃	direct (N × H)
211. #3164	C₂×C₅₃	semidirect
214. #2089	C₂×C₅₃	semidirect
215. #2060	C₂×C₅₃	semidirect
216. #1952	C₂×C₅₃	semidirect
217. #1945	C₂×C₅₃	semidirect
219. #1845	C₂×C₅₃	semidirect
261. C₅₃	#26083	direct (N × H)
264. #131	Q₈×C₅₃	semidirect
306. #104	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3164	2
C₅₃×#3083	2
C₅₃×#2913	2
C₅₃×#2089	2
C₅₃×#2060	2
C₅₃×#1952	2
C₅₃×#1945	2
C₅₃×#1859	2
C₅₃×#1845	2
C₅₃×#1816	2
C₅₃×#1764	2
#26083	53

Subgroups — C53×#26083 #27233595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#26083 #27233595