Subgroups — C₅₃×#21229 #27228741

Order 13568.

1566 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#417	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2311	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1972	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1971	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1968	C₂	1	4	yes	maximal, complemented
9	C₂²×#41×C₅₃	C₂²	1	2	yes	—
10	C₂×#135×C₅₃	—	2	4	no	—
11	C₂×#135×C₅₃	—	2	4	no	—
12	C₂×#135×C₅₃	C₄	1	8	yes	complemented
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	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×#106×C₅₃	—	2	4	no	—
21	C₂×#106×C₅₃	C₂²	1	2	yes	—
22	C₂×#106×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#417	—	2	8	no	—
24	C₂×#103×C₅₃	C₂²	1	4	yes	—
25	C₂×#102×C₅₃	C₂²	1	4	yes	—
26	C₅₃×#405	—	2	4	no	—
27	C₂×#97×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#391	C₂²	1	4	yes	—
30	C₅₃×#390	C₂²	1	4	yes	—
31	C₅₃×#386	—	2	4	no	—
32	C₅₃×#380	—	2	4	no	—
33	C₂×#41×C₅₃	—	2	4	no	—
34	C₂×#41×C₅₃	—	2	4	no	—
35	C₂×#41×C₅₃	—	2	4	no	—
36	C₂×#41×C₅₃	—	2	4	no	—
37	C₂×#41×C₅₃	—	2	4	no	—
38	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
39	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×D₄×C₅₃	—	2	2	no	—
41	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
42	C₂³×C₄×C₅₃	—	2	2	no	—
43	C₂³×C₄×C₅₃	—	2	2	no	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	#135×C₅₃	—	2	32	no	—
46	C₂×Q₁₆×C₅₃	—	2	2	no	—
47	C₂×Q₁₆×C₅₃	—	2	2	no	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂×SD₁₆×C₅₃	—	2	4	no	—
50	C₂×SD₁₆×C₅₃	—	2	4	no	—
51	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
52	C₂×D₈×C₅₃	—	2	2	no	—
53	C₂×D₈×C₅₃	—	2	2	no	—
54	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
55	C₂×M₁₆×C₅₃	—	2	2	no	—
56	C₂×M₁₆×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₅₃	D₄	1	4	yes	—
60	C₂²×C₈×C₅₃	D₄	1	4	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	#117×C₅₃	—	2	2	no	—
65	#117×C₅₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
74	#106×C₅₃	—	2	4	no	—
75	#106×C₅₃	—	2	4	no	—
76	#103×C₅₃	—	2	8	no	—
77	#102×C₅₃	—	2	8	no	—
78	#97×C₅₃	—	2	4	no	—
79	C₄×C₈×C₅₃	—	2	4	no	—
80	#95×C₅₃	—	2	8	no	—
81	C₂⁴×C₅₃	—	2	2	no	—
82	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
83	#41×C₅₃	—	2	16	no	—
84	#41×C₅₃	—	2	16	no	—
85	#41×C₅₃	—	4	8	no	—
86	#41×C₅₃	—	4	8	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
93	C₂×D₄×C₅₃	—	2	8	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	—	2	2	no	—
97	C₂×D₄×C₅₃	—	2	2	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	4	4	no	—
100	C₂×D₄×C₅₃	—	4	4	no	—
101	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	8	no	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	4	4	no	—
106	C₂²×C₄×C₅₃	—	4	4	no	—
107	C₂²×C₄×C₅₃	—	4	4	no	—
108	C₂²×C₄×C₅₃	—	8	4	no	—
109	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
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	#41	1	2	yes	—
116	C₂²×C₄×C₅₃	#41	1	2	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₅₃	—	2	8	no	—
119	SD₁₆×C₅₃	—	2	16	no	—
120	D₈×C₅₃	—	2	8	no	—
121	M₁₆×C₅₃	—	2	8	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	C₂²⋊C₄	1	16	yes	complemented
125	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
128	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₅₃	—	2	4	no	—
133	C₄⋊C₄×C₅₃	—	2	4	no	—
134	C₄⋊C₄×C₅₃	—	2	4	no	—
135	C₄⋊C₄×C₅₃	—	4	4	no	—
136	C₂²⋊C₄×C₅₃	—	2	4	no	—
137	C₂²⋊C₄×C₅₃	—	8	8	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	2	4	no	—
141	C₂³×C₅₃	—	2	4	no	—
142	C₂³×C₅₃	—	4	2	no	—
143	C₂³×C₅₃	—	4	2	no	—
144	C₂³×C₅₃	—	4	4	no	—
145	C₂³×C₅₃	—	4	4	no	—
146	C₂³×C₅₃	C₄×D₄	1	2	yes	—
147	C₂³×C₅₃	#121	1	2	yes	—
148	C₂³×C₅₃	#127	1	2	yes	—
149	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	4	no	—
151	Q₈×C₅₃	—	2	4	no	—
152	Q₈×C₅₃	—	4	2	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	D₄×C₅₃	—	2	4	no	—
155	D₄×C₅₃	—	2	4	no	—
156	D₄×C₅₃	—	4	16	no	—
157	D₄×C₅₃	—	4	2	no	—
158	D₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	8	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	16	no	—
165	C₂×C₄×C₅₃	—	4	4	no	—
166	C₂×C₄×C₅₃	—	4	8	no	—
167	C₂×C₄×C₅₃	—	8	8	no	—
168	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
169	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
170	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₅₃	#121	1	4	yes	—
172	C₂×C₄×C₅₃	#121	1	2	yes	—
173	C₂×C₄×C₅₃	#124	1	4	yes	—
174	C₂×C₄×C₅₃	#127	1	2	yes	—
175	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
181	C₈×C₅₃	—	2	4	no	—
182	C₈×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
183	C₈×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
184	#21229	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂²×C₅₃	—	2	4	no	—
186	C₂²×C₅₃	—	2	4	no	—
187	C₂²×C₅₃	—	2	4	no	—
188	C₂²×C₅₃	—	4	8	no	—
189	C₂²×C₅₃	—	4	4	no	—
190	C₂²×C₅₃	—	8	2	no	—
191	C₂²×C₅₃	—	8	2	no	—
192	C₂²×C₅₃	—	8	4	no	—
193	C₂²×C₅₃	#390	1	4	yes	—
194	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
196	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
197	C₂²×C₅₃	#577	1	4	yes	—
198	C₄×C₅₃	—	4	4	no	—
199	C₄×C₅₃	—	4	4	no	—
200	C₄×C₅₃	—	8	4	no	—
201	C₄×C₅₃	#390	1	4	yes	—
202	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
203	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
205	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
206	C₂²×#135	C₂×C₅₃	1	1	yes	characteristic
207	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×#417	C₂×C₅₃	1	2	yes	complemented
209	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#2311	C₂×C₅₃	1	2	yes	complemented
211	#1972	C₂×C₅₃	1	2	yes	complemented
212	#1971	C₂×C₅₃	1	2	yes	complemented
213	#1968	C₂×C₅₃	1	4	yes	complemented
214	C₂×C₅₃	—	2	4	no	—
215	C₂×C₅₃	—	8	4	no	—
216	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
217	C₂×C₅₃	#2952	1	2	yes	—
218	C₂×C₅₃	#3060	1	2	yes	—
219	C₂×C₅₃	#3154	1	2	yes	—
220	C₂²×#41	C₂²×C₅₃	1	2	yes	—
221	C₂×#135	—	2	4	no	—
222	C₂×#135	—	2	4	no	—
223	C₂×#135	C₄×C₅₃	1	8	yes	complemented
224	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
225	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
226	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
227	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
228	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
230	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
231	C₂×#106	—	2	4	no	—
232	C₂×#106	C₂²×C₅₃	1	2	yes	—
233	C₂×#106	C₂²×C₅₃	1	2	yes	—
234	#417	—	2	8	no	—
235	C₂×#103	C₂²×C₅₃	1	4	yes	—
236	C₂×#102	C₂²×C₅₃	1	4	yes	—
237	#405	—	2	4	no	—
238	C₂×#97	C₂²×C₅₃	1	2	yes	—
239	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
240	#391	C₂²×C₅₃	1	4	yes	—
241	#390	C₂²×C₅₃	1	4	yes	—
242	#386	—	2	4	no	—
243	#380	—	2	4	no	—
244	C₅₃	#21229	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#41	—	2	4	no	—
246	C₂×#41	—	2	4	no	—
247	C₂×#41	—	2	4	no	—
248	C₂×#41	—	2	4	no	—
249	C₂×#41	—	2	4	no	—
250	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
251	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
252	C₂²×D₄	—	2	2	no	—
253	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
254	C₂³×C₄	—	2	2	no	—
255	C₂³×C₄	—	2	2	no	—
256	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
257	#135	—	2	32	no	—
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×Q₁₆	—	2	2	no	—
260	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
261	C₂×SD₁₆	—	2	4	no	—
262	C₂×SD₁₆	—	2	4	no	—
263	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
267	C₂×M₁₆	—	2	2	no	—
268	C₂×M₁₆	—	2	2	no	—
269	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
270	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
271	C₂²×C₈	D₄×C₅₃	1	4	yes	—
272	C₂²×C₈	D₄×C₅₃	1	4	yes	—
273	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
274	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
276	#117	—	2	2	no	—
277	#117	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₄⋊C₄	—	2	4	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
283	C₂×C₂²⋊C₄	—	2	8	no	—
284	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
285	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
286	#106	—	2	4	no	—
287	#106	—	2	4	no	—
288	#103	—	2	8	no	—
289	#102	—	2	8	no	—
290	#97	—	2	4	no	—
291	C₄×C₈	—	2	4	no	—
292	#95	—	2	8	no	—
293	C₂⁴	—	2	2	no	—
294	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
295	#41	—	2	16	no	—
296	#41	—	2	16	no	—
297	#41	—	4	8	no	—
298	#41	—	4	8	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
305	C₂×D₄	—	2	8	no	—
306	C₂×D₄	—	2	2	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	—	4	4	no	—
312	C₂×D₄	—	4	4	no	—
313	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	8	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	4	4	no	—
318	C₂²×C₄	—	4	4	no	—
319	C₂²×C₄	—	4	4	no	—
320	C₂²×C₄	—	8	4	no	—
321	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
323	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
327	C₂²×C₄	#41×C₅₃	1	2	yes	—
328	C₂²×C₄	#41×C₅₃	1	2	yes	—
329	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
330	Q₁₆	—	2	8	no	—
331	SD₁₆	—	2	16	no	—
332	D₈	—	2	8	no	—
333	M₁₆	—	2	8	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	C₂²⋊C₄×C₅₃	1	16	yes	complemented
337	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
338	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
339	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
340	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
342	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
343	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₂²⋊C₄	—	2	4	no	—
349	C₂²⋊C₄	—	8	8	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	2	2	no	—
352	C₂³	—	2	4	no	—
353	C₂³	—	2	4	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	4	no	—
357	C₂³	—	4	4	no	—
358	C₂³	C₄×D₄×C₅₃	1	2	yes	—
359	C₂³	#121×C₅₃	1	2	yes	—
360	C₂³	#127×C₅₃	1	2	yes	—
361	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	2	4	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	4	no	—
368	D₄	—	4	16	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	16	no	—
377	C₂×C₄	—	4	4	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	8	8	no	—
380	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
381	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
382	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
383	C₂×C₄	#121×C₅₃	1	4	yes	—
384	C₂×C₄	#121×C₅₃	1	2	yes	—
385	C₂×C₄	#124×C₅₃	1	4	yes	—
386	C₂×C₄	#127×C₅₃	1	2	yes	—
387	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
393	C₈	—	2	4	no	—
394	C₈	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
395	C₈	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	2	4	no	—
399	C₂²	—	4	8	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	8	4	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	C₅₃×#390	1	4	yes	—
405	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
406	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
407	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
408	C₂²	C₅₃×#577	1	4	yes	—
409	C₄	—	4	4	no	—
410	C₄	—	4	4	no	—
411	C₄	—	8	4	no	—
412	C₄	C₅₃×#390	1	4	yes	—
413	C₄	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
414	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
415	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
416	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
417	C₂	—	2	4	no	—
418	C₂	—	8	4	no	—
419	C₂	C₂×C₅₃×#390	1	1	yes	characteristic
420	C₂	C₅₃×#2952	1	2	yes	—
421	C₂	C₅₃×#3060	1	2	yes	—
422	C₂	C₅₃×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×#135×C₅₃	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. C₂²×#106×C₅₃	C₂	semidirect
3. C₂×C₅₃×#417	C₂	semidirect
4. C₂×C₅₃×#405	C₂	semidirect
5. C₅₃×#2311	C₂	semidirect
6. C₅₃×#1972	C₂	semidirect
7. C₅₃×#1971	C₂	semidirect
8. C₅₃×#1968	C₂	semidirect
12. C₂×#135×C₅₃	C₄	semidirect
48. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
51. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
54. C₂×D₈×C₅₃	C₂×C₄	semidirect
124. C₂×C₈×C₅₃	C₂²⋊C₄	semidirect
184. #21229	C₅₃	direct (N × H)
207. C₂²×#106	C₂×C₅₃	semidirect
208. C₂×#417	C₂×C₅₃	semidirect
209. C₂×#405	C₂×C₅₃	semidirect
210. #2311	C₂×C₅₃	semidirect
211. #1972	C₂×C₅₃	semidirect
212. #1971	C₂×C₅₃	semidirect
213. #1968	C₂×C₅₃	semidirect
223. C₂×#135	C₄×C₅₃	semidirect
244. C₅₃	#21229	direct (N × H)
260. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
263. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
266. C₂×D₈	C₂×C₄×C₅₃	semidirect
336. C₂×C₈	C₂²⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₅₃	2
C₂²×#106×C₅₃	2
C₂×C₅₃×#417	2
C₂×C₅₃×#405	2
C₅₃×#2311	2
C₅₃×#1972	2
C₅₃×#1971	2
C₅₃×#1968	2
#21229	53

Subgroups — C53×#21229 #27228741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21229 #27228741