Subgroups — C₅₃×#60622 #27268134

Order 13568.

2178 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3498	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3141	C₂	1	3	yes	maximal, complemented
4	C₅₃×#3138	C₂	1	3	yes	maximal, complemented
5	C₅₃×#3136	C₂	1	12	yes	maximal, complemented
6	C₅₃×#3131	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₃×#486	C₂	1	3	yes	maximal, complemented
8	C₅₃×#2897	C₂	1	4	yes	maximal, complemented
9	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
11	C₂×#131×C₅₃	C₂²	1	2	yes	—
12	Q₈²×C₅₃	—	2	1	no	—
13	C₅₃×#557	—	2	1	no	—
14	C₅₃×#554	—	2	3	no	—
15	C₅₃×#550	—	2	1	no	—
16	C₅₃×#550	—	2	1	no	—
17	D₄×Q₈×C₅₃	—	2	3	no	—
18	D₄²×C₅₃	—	2	3	no	—
19	D₄²×C₅₃	—	2	3	no	—
20	C₅₃×#544	C₂²	1	6	yes	complemented
21	C₅₃×#535	C₂²	1	6	yes	—
22	C₅₃×#533	C₂²	1	3	yes	complemented
23	C₅₃×#532	C₂²	1	3	yes	—
24	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
25	C₂×#127×C₅₃	C₂²	1	3	yes	—
26	C₅₃×#520	C₂²	1	2	yes	—
27	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#490	C₂²	1	12	yes	complemented
29	C₅₃×#489	C₂²	1	12	yes	complemented
30	C₅₃×#488	C₂²	1	12	yes	complemented
31	C₅₃×#487	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#487	C₂²	1	3	yes	complemented
33	C₅₃×#486	C₂²	1	3	yes	complemented
34	C₅₃×#486	C₂²	1	12	yes	complemented
35	C₅₃×#486	C₂²	1	6	yes	complemented
36	C₅₃×#486	C₂²	1	3	yes	complemented
37	C₅₃×#418	C₂²	1	12	yes	complemented
38	C₅₃×#417	C₂²	1	12	yes	complemented
39	C₅₃×#416	C₂²	1	12	yes	complemented
40	C₂×#102×C₅₃	C₂²	1	12	yes	complemented
41	C₅₃×#405	C₂²	1	6	yes	complemented
42	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
43	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
44	H(5,𝔽₂)×C₅₃	—	2	16	no	—
45	C₂×#41×C₅₃	—	2	2	no	—
46	C₂×#41×C₅₃	C₂³	1	6	yes	—
47	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₅₃	—	2	3	no	—
49	C₂²×D₄×C₅₃	—	2	6	no	—
50	C₂²×D₄×C₅₃	—	2	3	no	—
51	C₂²×D₄×C₅₃	—	2	3	no	—
52	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
53	#131×C₅₃	—	2	8	no	—
54	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
56	C₄⋊Q₈×C₅₃	—	2	6	no	—
57	C₄⋊Q₈×C₅₃	—	2	3	no	—
58	C₄⋊Q₈×C₅₃	—	2	3	no	—
59	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	#127×C₅₃	—	2	3	no	—
64	#127×C₅₃	—	2	3	no	—
65	#127×C₅₃	C₂³	1	3	yes	—
66	#127×C₅₃	C₂³	1	6	yes	—
67	#125×C₅₃	—	2	6	no	—
68	#125×C₅₃	C₂³	1	6	yes	—
69	#124×C₅₃	C₂³	1	6	yes	—
70	C₂²⋊Q₈×C₅₃	—	2	12	no	—
71	C₂²⋊Q₈×C₅₃	—	2	3	no	—
72	C₂²⋊Q₈×C₅₃	—	2	3	no	—
73	#121×C₅₃	—	2	6	no	—
74	#121×C₅₃	—	2	6	no	—
75	#121×C₅₃	—	2	3	no	—
76	#121×C₅₃	—	2	3	no	—
77	#120×C₅₃	—	2	12	no	—
78	C₄×Q₈×C₅₃	—	2	3	no	—
79	C₄×Q₈×C₅₃	—	2	3	no	—
80	C₄×Q₈×C₅₃	—	2	2	no	—
81	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
83	C₄×D₄×C₅₃	—	2	3	no	—
84	C₄×D₄×C₅₃	—	2	3	no	—
85	C₄×D₄×C₅₃	—	2	6	no	—
86	C₄×D₄×C₅₃	D₄	1	3	yes	—
87	C₄×D₄×C₅₃	D₄	1	3	yes	—
88	#117×C₅₃	C₂³	1	6	yes	—
89	#117×C₅₃	C₂³	1	3	yes	—
90	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
93	#103×C₅₃	C₂³	1	4	yes	—
94	#103×C₅₃	C₂³	1	12	yes	—
95	#102×C₅₃	C₂³	1	12	yes	—
96	#102×C₅₃	C₂³	1	24	yes	—
97	#102×C₅₃	C₂³	1	12	yes	—
98	#97×C₅₃	C₂³	1	6	yes	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	3	yes	—
101	C₄×C₈×C₅₃	C₂³	1	6	yes	—
102	C₂⁴×C₅₃	—	2	6	no	—
103	#41×C₅₃	—	2	24	no	—
104	#41×C₅₃	—	4	8	no	—
105	#41×C₅₃	#41	1	8	yes	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	3	no	—
108	C₂×Q₈×C₅₃	—	2	6	no	—
109	C₂×Q₈×C₅₃	—	8	6	no	—
110	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
112	C₂×D₄×C₅₃	—	2	6	no	—
113	C₂×D₄×C₅₃	—	2	3	no	—
114	C₂×D₄×C₅₃	—	2	3	no	—
115	C₂×D₄×C₅₃	—	2	6	no	—
116	C₂×D₄×C₅₃	—	2	6	no	—
117	C₂×D₄×C₅₃	—	2	6	no	—
118	C₂×D₄×C₅₃	—	2	24	no	—
119	C₂×D₄×C₅₃	—	4	24	no	—
120	C₂×D₄×C₅₃	—	8	6	no	—
121	C₂×D₄×C₅₃	—	8	6	no	—
122	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
123	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₅₃	—	2	6	no	—
125	C₂²×C₄×C₅₃	—	2	6	no	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
127	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
129	M₁₆×C₅₃	—	2	12	no	—
130	C₂×C₈×C₅₃	—	2	12	no	—
131	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
132	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
133	C₄⋊C₄×C₅₃	—	2	3	no	—
134	C₄⋊C₄×C₅₃	—	2	6	no	—
135	C₄⋊C₄×C₅₃	—	2	6	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	6	no	—
140	C₄⋊C₄×C₅₃	—	2	6	no	—
141	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
142	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
143	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
144	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
145	C₂²⋊C₄×C₅₃	—	2	6	no	—
146	C₂²⋊C₄×C₅₃	—	2	6	no	—
147	C₂²⋊C₄×C₅₃	—	2	6	no	—
148	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
149	C₄²×C₅₃	—	2	6	no	—
150	C₄²×C₅₃	C₂×D₄	1	3	yes	—
151	C₄²×C₅₃	C₂×D₄	1	3	yes	—
152	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₄²×C₅₃	C₂⁴	1	3	yes	—
154	C₂³×C₅₃	—	2	6	no	—
155	C₂³×C₅₃	—	4	12	no	—
156	C₂³×C₅₃	—	8	6	no	—
157	C₂³×C₅₃	—	8	6	no	—
158	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
159	Q₈×C₅₃	—	2	4	no	—
160	Q₈×C₅₃	—	4	6	no	—
161	Q₈×C₅₃	—	4	2	no	—
162	Q₈×C₅₃	—	8	6	no	—
163	Q₈×C₅₃	#124	1	4	yes	—
164	D₄×C₅₃	—	2	12	no	—
165	D₄×C₅₃	—	4	12	no	—
166	D₄×C₅₃	—	4	12	no	—
167	D₄×C₅₃	—	4	6	no	—
168	D₄×C₅₃	—	4	6	no	—
169	D₄×C₅₃	—	8	6	no	—
170	D₄×C₅₃	—	8	6	no	—
171	D₄×C₅₃	#124	1	12	yes	—
172	C₂×C₄×C₅₃	—	2	6	no	—
173	C₂×C₄×C₅₃	—	2	6	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	6	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	4	12	no	—
178	C₂×C₄×C₅₃	—	8	6	no	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
180	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
181	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
185	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
186	C₈×C₅₃	—	2	2	no	—
187	C₈×C₅₃	—	2	6	no	—
188	#60622	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂²×C₅₃	—	2	3	no	—
190	C₂²×C₅₃	—	2	3	no	—
191	C₂²×C₅₃	—	4	6	no	—
192	C₂²×C₅₃	—	4	6	no	—
193	C₂²×C₅₃	—	4	6	no	—
194	C₂²×C₅₃	—	16	12	no	—
195	C₂²×C₅₃	C₂×#124	1	6	yes	—
196	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
197	C₄×C₅₃	—	2	3	no	—
198	C₄×C₅₃	—	2	3	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	4	6	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	C₂×#124	1	6	yes	—
203	C₄×C₅₃	#576	1	2	yes	—
204	C₄×C₅₃	#577	1	2	yes	—
205	C₄×C₅₃	C₂²×#41	1	2	yes	—
206	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#3498	C₂×C₅₃	1	1	yes	characteristic
208	#3141	C₂×C₅₃	1	3	yes	complemented
209	#3138	C₂×C₅₃	1	3	yes	complemented
210	#3136	C₂×C₅₃	1	12	yes	complemented
211	#3131	C₂×C₅₃	1	3	yes	complemented
212	C₂×#486	C₂×C₅₃	1	3	yes	complemented
213	#2897	C₂×C₅₃	1	4	yes	complemented
214	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
215	C₂×C₅₃	—	2	6	no	—
216	C₂×C₅₃	—	8	6	no	—
217	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#576	1	1	yes	characteristic
219	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
220	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
221	C₂×#131	C₂²×C₅₃	1	2	yes	—
222	Q₈²	—	2	1	no	—
223	#557	—	2	1	no	—
224	#554	—	2	3	no	—
225	#550	—	2	1	no	—
226	#550	—	2	1	no	—
227	D₄×Q₈	—	2	3	no	—
228	D₄²	—	2	3	no	—
229	D₄²	—	2	3	no	—
230	#544	C₂²×C₅₃	1	6	yes	complemented
231	#535	C₂²×C₅₃	1	6	yes	—
232	#533	C₂²×C₅₃	1	3	yes	complemented
233	#532	C₂²×C₅₃	1	3	yes	—
234	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
235	C₂×#127	C₂²×C₅₃	1	3	yes	—
236	#520	C₂²×C₅₃	1	2	yes	—
237	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
238	#490	C₂²×C₅₃	1	12	yes	complemented
239	#489	C₂²×C₅₃	1	12	yes	complemented
240	#488	C₂²×C₅₃	1	12	yes	complemented
241	#487	C₂²×C₅₃	1	1	yes	characteristic, complemented
242	#487	C₂²×C₅₃	1	3	yes	complemented
243	#486	C₂²×C₅₃	1	3	yes	complemented
244	#486	C₂²×C₅₃	1	12	yes	complemented
245	#486	C₂²×C₅₃	1	6	yes	complemented
246	#486	C₂²×C₅₃	1	3	yes	complemented
247	#418	C₂²×C₅₃	1	12	yes	complemented
248	#417	C₂²×C₅₃	1	12	yes	complemented
249	#416	C₂²×C₅₃	1	12	yes	complemented
250	C₂×#102	C₂²×C₅₃	1	12	yes	complemented
251	#405	C₂²×C₅₃	1	6	yes	complemented
252	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
253	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	complemented
254	C₅₃	#60622	1	1	yes	characteristic, direct product factor, Sylow
255	H(5,𝔽₂)	—	2	16	no	—
256	C₂×#41	—	2	2	no	—
257	C₂×#41	C₂³×C₅₃	1	6	yes	—
258	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
259	C₂²×Q₈	—	2	3	no	—
260	C₂²×D₄	—	2	6	no	—
261	C₂²×D₄	—	2	3	no	—
262	C₂²×D₄	—	2	3	no	—
263	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
264	#131	—	2	8	no	—
265	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
266	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
267	C₄⋊Q₈	—	2	6	no	—
268	C₄⋊Q₈	—	2	3	no	—
269	C₄⋊Q₈	—	2	3	no	—
270	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
272	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
273	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
274	#127	—	2	3	no	—
275	#127	—	2	3	no	—
276	#127	C₂³×C₅₃	1	3	yes	—
277	#127	C₂³×C₅₃	1	6	yes	—
278	#125	—	2	6	no	—
279	#125	C₂³×C₅₃	1	6	yes	—
280	#124	C₂³×C₅₃	1	6	yes	—
281	C₂²⋊Q₈	—	2	3	no	—
282	C₂²⋊Q₈	—	2	3	no	—
283	C₂²⋊Q₈	—	2	12	no	—
284	#121	—	2	3	no	—
285	#121	—	2	3	no	—
286	#121	—	2	6	no	—
287	#121	—	2	6	no	—
288	#120	—	2	12	no	—
289	C₄×Q₈	—	2	2	no	—
290	C₄×Q₈	—	2	3	no	—
291	C₄×Q₈	—	2	3	no	—
292	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
293	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
294	C₄×D₄	—	2	6	no	—
295	C₄×D₄	—	2	3	no	—
296	C₄×D₄	—	2	3	no	—
297	C₄×D₄	D₄×C₅₃	1	3	yes	—
298	C₄×D₄	D₄×C₅₃	1	3	yes	—
299	#117	C₂³×C₅₃	1	6	yes	—
300	#117	C₂³×C₅₃	1	3	yes	—
301	C₂×C₄⋊C₄	—	2	3	no	—
302	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
303	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
304	#103	C₂³×C₅₃	1	4	yes	—
305	#103	C₂³×C₅₃	1	12	yes	—
306	#102	C₂³×C₅₃	1	12	yes	—
307	#102	C₂³×C₅₃	1	24	yes	—
308	#102	C₂³×C₅₃	1	12	yes	—
309	#97	C₂³×C₅₃	1	6	yes	—
310	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₅₃	1	3	yes	—
312	C₄×C₈	C₂³×C₅₃	1	6	yes	—
313	C₂⁴	—	2	6	no	—
314	#41	—	2	24	no	—
315	#41	—	4	8	no	—
316	#41	#41×C₅₃	1	8	yes	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	3	no	—
319	C₂×Q₈	—	2	6	no	—
320	C₂×Q₈	—	8	6	no	—
321	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
323	C₂×D₄	—	2	6	no	—
324	C₂×D₄	—	2	3	no	—
325	C₂×D₄	—	2	3	no	—
326	C₂×D₄	—	2	6	no	—
327	C₂×D₄	—	2	6	no	—
328	C₂×D₄	—	2	6	no	—
329	C₂×D₄	—	2	24	no	—
330	C₂×D₄	—	4	24	no	—
331	C₂×D₄	—	8	6	no	—
332	C₂×D₄	—	8	6	no	—
333	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
334	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
335	C₂²×C₄	—	2	6	no	—
336	C₂²×C₄	—	2	6	no	—
337	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
338	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
339	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
340	M₁₆	—	2	12	no	—
341	C₂×C₈	—	2	12	no	—
342	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
343	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
344	C₄⋊C₄	—	2	3	no	—
345	C₄⋊C₄	—	2	6	no	—
346	C₄⋊C₄	—	2	6	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	6	no	—
351	C₄⋊C₄	—	2	6	no	—
352	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
353	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
354	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
355	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
356	C₂²⋊C₄	—	2	6	no	—
357	C₂²⋊C₄	—	2	6	no	—
358	C₂²⋊C₄	—	2	6	no	—
359	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
360	C₄²	—	2	6	no	—
361	C₄²	C₂×D₄×C₅₃	1	3	yes	—
362	C₄²	C₂×D₄×C₅₃	1	3	yes	—
363	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
364	C₄²	C₂⁴×C₅₃	1	3	yes	—
365	C₂³	—	2	6	no	—
366	C₂³	—	4	12	no	—
367	C₂³	—	8	6	no	—
368	C₂³	—	8	6	no	—
369	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
370	Q₈	—	2	4	no	—
371	Q₈	—	4	6	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	8	6	no	—
374	Q₈	#124×C₅₃	1	4	yes	—
375	D₄	—	2	12	no	—
376	D₄	—	4	12	no	—
377	D₄	—	4	12	no	—
378	D₄	—	4	6	no	—
379	D₄	—	4	6	no	—
380	D₄	—	8	6	no	—
381	D₄	—	8	6	no	—
382	D₄	#124×C₅₃	1	12	yes	—
383	C₂×C₄	—	2	6	no	—
384	C₂×C₄	—	2	6	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	6	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	12	no	—
389	C₂×C₄	—	8	6	no	—
390	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
391	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
392	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
393	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
396	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	6	no	—
399	C₂²	—	2	3	no	—
400	C₂²	—	2	3	no	—
401	C₂²	—	4	6	no	—
402	C₂²	—	4	6	no	—
403	C₂²	—	4	6	no	—
404	C₂²	—	16	12	no	—
405	C₂²	C₂×#124×C₅₃	1	6	yes	—
406	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
407	C₄	—	2	3	no	—
408	C₄	—	2	3	no	—
409	C₄	—	4	6	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	2	no	—
412	C₄	C₂×#124×C₅₃	1	6	yes	—
413	C₄	C₅₃×#576	1	2	yes	—
414	C₄	C₅₃×#577	1	2	yes	—
415	C₄	C₂²×#41×C₅₃	1	2	yes	—
416	C₂	—	2	6	no	—
417	C₂	—	8	6	no	—
418	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
419	C₂	C₂×C₅₃×#576	1	1	yes	characteristic
420	C₂	C₂×C₅₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3503	C₂	semidirect
3. C₅₃×#3141	C₂	semidirect
4. C₅₃×#3138	C₂	semidirect
5. C₅₃×#3136	C₂	semidirect
6. C₅₃×#3131	C₂	semidirect
7. C₂×C₅₃×#486	C₂	semidirect
8. C₅₃×#2897	C₂	semidirect
9. C₄×#131×C₅₃	C₂	semidirect
20. C₅₃×#544	C₂²	semidirect
22. C₅₃×#533	C₂²	semidirect
24. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
28. C₅₃×#490	C₂²	semidirect
29. C₅₃×#489	C₂²	semidirect
30. C₅₃×#488	C₂²	semidirect
31. C₅₃×#487	C₂²	semidirect
32. C₅₃×#487	C₂²	semidirect
33. C₅₃×#486	C₂²	semidirect
34. C₅₃×#486	C₂²	semidirect
35. C₅₃×#486	C₂²	semidirect
36. C₅₃×#486	C₂²	semidirect
37. C₅₃×#418	C₂²	semidirect
38. C₅₃×#417	C₂²	semidirect
39. C₅₃×#416	C₂²	semidirect
40. C₂×#102×C₅₃	C₂²	semidirect
41. C₅₃×#405	C₂²	semidirect
42. C₄×M₁₆×C₅₃	C₂²	semidirect
43. C₂×C₄×C₈×C₅₃	C₂²	semidirect
188. #60622	C₅₃	direct (N × H)
206. #3503	C₂×C₅₃	semidirect
208. #3141	C₂×C₅₃	semidirect
209. #3138	C₂×C₅₃	semidirect
210. #3136	C₂×C₅₃	semidirect
211. #3131	C₂×C₅₃	semidirect
212. C₂×#486	C₂×C₅₃	semidirect
213. #2897	C₂×C₅₃	semidirect
214. C₄×#131	C₂×C₅₃	semidirect
230. #544	C₂²×C₅₃	semidirect
232. #533	C₂²×C₅₃	semidirect
234. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
238. #490	C₂²×C₅₃	semidirect
239. #489	C₂²×C₅₃	semidirect
240. #488	C₂²×C₅₃	semidirect
241. #487	C₂²×C₅₃	semidirect
242. #487	C₂²×C₅₃	semidirect
243. #486	C₂²×C₅₃	semidirect
244. #486	C₂²×C₅₃	semidirect
245. #486	C₂²×C₅₃	semidirect
246. #486	C₂²×C₅₃	semidirect
247. #418	C₂²×C₅₃	semidirect
248. #417	C₂²×C₅₃	semidirect
249. #416	C₂²×C₅₃	semidirect
250. C₂×#102	C₂²×C₅₃	semidirect
251. #405	C₂²×C₅₃	semidirect
252. C₄×M₁₆	C₂²×C₅₃	semidirect
253. C₂×C₄×C₈	C₂²×C₅₃	semidirect
254. C₅₃	#60622	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3503	2
C₅₃×#3498	2
C₅₃×#3141	2
C₅₃×#3138	2
C₅₃×#3136	2
C₅₃×#3131	2
C₂×C₅₃×#486	2
C₅₃×#2897	2
C₄×#131×C₅₃	2
#60622	53

Subgroups — C53×#60622 #27268134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60622 #27268134