Subgroups — C₅₃×#22199 #27229711

Order 13568.

934 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×#469	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2324	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1919	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1893	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1879	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1864	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1809	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#121×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#469	C₄	1	8	yes	complemented
17	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#390	—	2	2	no	—
24	C₅₃×#390	C₂²	1	2	yes	—
25	C₅₃×#389	C₂²	1	2	yes	—
26	C₅₃×#388	—	2	2	no	—
27	C₅₃×#386	—	2	2	no	—
28	C₅₃×#385	—	2	2	no	—
29	C₅₃×#380	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₅₃×#340	C₂²	1	2	yes	—
32	C₅₃×#340	C₂²	1	2	yes	—
33	C₅₃×#340	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₂²×D₄×C₅₃	—	2	2	no	—
37	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₅₃	—	2	2	no	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×D₈×C₅₃	—	4	1	no	—
41	C₂×D₈×C₅₃	—	4	1	no	—
42	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₅₃	—	4	1	no	—
44	C₂×M₁₆×C₅₃	—	4	1	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	—	4	1	no	—
48	C₂²×C₈×C₅₃	—	4	1	no	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#121×C₅₃	—	2	4	no	—
53	#121×C₅₃	C₂×C₄	1	8	yes	complemented
54	C₄×D₄×C₅₃	—	4	2	no	—
55	C₄×D₄×C₅₃	—	4	2	no	—
56	#117×C₅₃	—	4	1	no	—
57	#117×C₅₃	—	4	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
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×C₂²⋊C₄×C₅₃	—	2	2	no	—
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	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	4	1	no	—
73	#106×C₅₃	—	4	1	no	—
74	#106×C₅₃	C₂×C₄	1	4	yes	complemented
75	#102×C₅₃	—	4	2	no	—
76	#102×C₅₃	—	4	2	no	—
77	#102×C₅₃	C₂×C₄	1	8	yes	complemented
78	#95×C₅₃	—	2	2	no	—
79	#95×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	2	no	—
81	C₂⁴×C₅₃	—	2	2	no	—
82	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×D₄×C₅₃	—	2	8	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	2	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	8	2	no	—
90	C₂×D₄×C₅₃	—	8	2	no	—
91	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₅₃	—	2	1	no	—
93	C₂²×C₄×C₅₃	—	2	1	no	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	—	2	1	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	4	4	no	—
100	C₂²×C₄×C₅₃	—	8	2	no	—
101	C₂²×C₄×C₅₃	—	8	2	no	—
102	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	#41	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	D₈×C₅₃	—	4	4	no	—
109	M₁₆×C₅₃	—	4	4	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	4	4	no	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
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	—	4	2	no	—
126	C₄⋊C₄×C₅₃	—	8	2	no	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
128	C₂²⋊C₄×C₅₃	—	2	8	no	—
129	C₂²⋊C₄×C₅₃	—	4	1	no	—
130	C₂²⋊C₄×C₅₃	—	4	1	no	—
131	C₂²⋊C₄×C₅₃	—	4	4	no	—
132	C₂²⋊C₄×C₅₃	—	8	2	no	—
133	C₂²⋊C₄×C₅₃	—	8	2	no	—
134	C₂²⋊C₄×C₅₃	—	8	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	8	2	no	—
138	C₂³×C₅₃	—	2	1	no	—
139	C₂³×C₅₃	—	2	1	no	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	2	4	no	—
143	C₂³×C₅₃	—	4	2	no	—
144	C₂³×C₅₃	—	4	2	no	—
145	C₂³×C₅₃	—	4	2	no	—
146	C₂³×C₅₃	—	4	2	no	—
147	C₂³×C₅₃	—	4	2	no	—
148	C₂³×C₅₃	—	4	2	no	—
149	C₂³×C₅₃	C₄×D₄	1	2	yes	—
150	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	D₄×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	2	4	no	—
153	D₄×C₅₃	—	4	2	no	—
154	D₄×C₅₃	—	4	2	no	—
155	D₄×C₅₃	—	8	8	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	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	—	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	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	8	8	no	—
178	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
185	C₈×C₅₃	—	2	2	no	—
186	C₈×C₅₃	—	2	2	no	—
187	C₈×C₅₃	—	4	2	no	—
188	#22199	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂²×C₅₃	—	2	2	no	—
190	C₂²×C₅₃	—	2	4	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	4	1	no	—
193	C₂²×C₅₃	—	4	1	no	—
194	C₂²×C₅₃	—	4	4	no	—
195	C₂²×C₅₃	—	4	4	no	—
196	C₂²×C₅₃	—	4	4	no	—
197	C₂²×C₅₃	—	8	2	no	—
198	C₂²×C₅₃	—	8	2	no	—
199	C₂²×C₅₃	—	8	2	no	—
200	C₂²×C₅₃	—	8	2	no	—
201	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₅₃	#534	1	1	yes	characteristic
203	C₂²×C₅₃	#537	1	1	yes	characteristic
204	C₂²×C₅₃	#576	1	2	yes	—
205	C₂²×C₅₃	#577	1	2	yes	—
206	C₄×C₅₃	—	2	2	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	4	2	no	—
209	C₄×C₅₃	—	4	4	no	—
210	C₄×C₅₃	—	8	2	no	—
211	C₄×C₅₃	—	8	2	no	—
212	C₄×C₅₃	#518	1	1	yes	characteristic
213	C₄×C₅₃	#520	1	1	yes	characteristic
214	C₄×C₅₃	#538	1	1	yes	characteristic
215	C₄×C₅₃	#539	1	1	yes	characteristic
216	C₂×#469	C₂×C₅₃	1	1	yes	characteristic
217	#2324	C₂×C₅₃	1	2	yes	complemented
218	#1925	C₂×C₅₃	1	2	yes	complemented
219	#1919	C₂×C₅₃	1	2	yes	complemented
220	#1893	C₂×C₅₃	1	2	yes	complemented
221	#1879	C₂×C₅₃	1	2	yes	complemented
222	#1864	C₂×C₅₃	1	2	yes	complemented
223	#1809	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#1729	C₂×C₅₃	1	1	yes	characteristic, complemented
225	C₂×C₅₃	—	4	2	no	—
226	C₂×C₅₃	—	8	4	no	—
227	C₂×C₅₃	#2325	1	1	yes	characteristic
228	C₂×C₅₃	#2946	1	2	yes	—
229	C₂×C₅₃	#3198	1	2	yes	—
230	C₂×C₅₃	#3259	1	1	yes	characteristic
231	C₂×C₅₃	#3263	1	1	yes	characteristic
232	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
233	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
234	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
235	C₂×#121	C₂²×C₅₃	1	2	yes	—
236	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
237	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
238	#469	C₄×C₅₃	1	8	yes	complemented
239	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×#102	C₂²×C₅₃	1	2	yes	—
243	C₂×#102	C₂²×C₅₃	1	2	yes	—
244	C₂×#102	C₂²×C₅₃	1	2	yes	—
245	#390	—	2	2	no	—
246	#390	C₂²×C₅₃	1	2	yes	—
247	#389	C₂²×C₅₃	1	2	yes	—
248	#388	—	2	2	no	—
249	#386	—	2	2	no	—
250	#385	—	2	2	no	—
251	#380	C₂²×C₅₃	1	2	yes	—
252	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
253	#340	C₂²×C₅₃	1	2	yes	—
254	#340	C₂²×C₅₃	1	2	yes	—
255	#340	C₂²×C₅₃	1	2	yes	—
256	#336	C₂²×C₅₃	1	2	yes	—
257	#336	C₂²×C₅₃	1	2	yes	—
258	C₅₃	#22199	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²×D₄	—	2	2	no	—
260	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
261	C₂³×C₄	—	2	2	no	—
262	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
263	C₂×D₈	—	4	1	no	—
264	C₂×D₈	—	4	1	no	—
265	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
266	C₂×M₁₆	—	4	1	no	—
267	C₂×M₁₆	—	4	1	no	—
268	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
269	C₂²×C₈	—	2	2	no	—
270	C₂²×C₈	—	4	1	no	—
271	C₂²×C₈	—	4	1	no	—
272	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
273	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	#121	—	2	4	no	—
276	#121	C₂×C₄×C₅₃	1	8	yes	complemented
277	C₄×D₄	—	4	2	no	—
278	C₄×D₄	—	4	2	no	—
279	#117	—	4	1	no	—
280	#117	—	4	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
295	#107	—	4	1	no	—
296	#106	—	4	1	no	—
297	#106	C₂×C₄×C₅₃	1	4	yes	complemented
298	#102	—	4	2	no	—
299	#102	—	4	2	no	—
300	#102	C₂×C₄×C₅₃	1	8	yes	complemented
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	C₂⁴	—	2	2	no	—
305	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂×D₄	—	2	8	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	8	2	no	—
313	C₂×D₄	—	8	2	no	—
314	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	4	4	no	—
323	C₂²×C₄	—	8	2	no	—
324	C₂²×C₄	—	8	2	no	—
325	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
328	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	#41×C₅₃	1	2	yes	—
330	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
331	D₈	—	4	4	no	—
332	M₁₆	—	4	4	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	4	no	—
341	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
342	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	8	2	no	—
350	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
351	C₂²⋊C₄	—	2	8	no	—
352	C₂²⋊C₄	—	4	1	no	—
353	C₂²⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	4	no	—
355	C₂²⋊C₄	—	8	2	no	—
356	C₂²⋊C₄	—	8	2	no	—
357	C₂²⋊C₄	—	8	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	8	2	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	1	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	2	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	C₄×D₄×C₅₃	1	2	yes	—
373	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
374	D₄	—	2	4	no	—
375	D₄	—	2	4	no	—
376	D₄	—	4	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	8	8	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	8	8	no	—
401	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
402	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
404	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
405	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
406	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
407	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
408	C₈	—	2	2	no	—
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₂²	—	2	4	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	1	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
424	C₂²	C₅₃×#534	1	1	yes	characteristic
425	C₂²	C₅₃×#537	1	1	yes	characteristic
426	C₂²	C₅₃×#576	1	2	yes	—
427	C₂²	C₅₃×#577	1	2	yes	—
428	C₄	—	2	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	4	no	—
432	C₄	—	8	2	no	—
433	C₄	—	8	2	no	—
434	C₄	C₅₃×#518	1	1	yes	characteristic
435	C₄	C₅₃×#520	1	1	yes	characteristic
436	C₄	C₅₃×#538	1	1	yes	characteristic
437	C₄	C₅₃×#539	1	1	yes	characteristic
438	C₂	—	4	2	no	—
439	C₂	—	8	4	no	—
440	C₂	C₅₃×#2325	1	1	yes	characteristic
441	C₂	C₅₃×#2946	1	2	yes	—
442	C₂	C₅₃×#3198	1	2	yes	—
443	C₂	C₅₃×#3259	1	1	yes	characteristic
444	C₂	C₅₃×#3263	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	C₂×C₅₃×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2324	C₂	semidirect
3. C₅₃×#1925	C₂	semidirect
4. C₅₃×#1919	C₂	semidirect
5. C₅₃×#1893	C₂	semidirect
6. C₅₃×#1879	C₂	semidirect
7. C₅₃×#1864	C₂	semidirect
8. C₅₃×#1809	C₂	semidirect
9. C₅₃×#1729	C₂	semidirect
16. C₅₃×#469	C₄	semidirect
45. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
53. #121×C₅₃	C₂×C₄	semidirect
74. #106×C₅₃	C₂×C₄	semidirect
77. #102×C₅₃	C₂×C₄	semidirect
188. #22199	C₅₃	direct (N × H)
217. #2324	C₂×C₅₃	semidirect
218. #1925	C₂×C₅₃	semidirect
219. #1919	C₂×C₅₃	semidirect
220. #1893	C₂×C₅₃	semidirect
221. #1879	C₂×C₅₃	semidirect
222. #1864	C₂×C₅₃	semidirect
223. #1809	C₂×C₅₃	semidirect
224. #1729	C₂×C₅₃	semidirect
238. #469	C₄×C₅₃	semidirect
258. C₅₃	#22199	direct (N × H)
268. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
276. #121	C₂×C₄×C₅₃	semidirect
297. #106	C₂×C₄×C₅₃	semidirect
300. #102	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×C₅₃×#469	2
C₅₃×#2324	2
C₅₃×#1925	2
C₅₃×#1919	2
C₅₃×#1893	2
C₅₃×#1879	2
C₅₃×#1864	2
C₅₃×#1809	2
C₅₃×#1729	2
#22199	53

Subgroups — C53×#22199 #27229711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22199 #27229711