Subgroups — C₅₃×#60624 #27268136

Order 13568.

2038 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3500	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3142	C₂	1	3	yes	maximal, complemented
4	C₅₃×#3140	C₂	1	3	yes	maximal, complemented
5	C₅₃×#3137	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3136	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3131	C₂	1	3	yes	maximal, complemented
8	C₂×C₅₃×#488	C₂	1	3	yes	maximal, complemented
9	C₅₃×#2898	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2897	C₂	1	2	yes	maximal, complemented
11	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#557	—	2	2	no	—
16	C₅₃×#554	—	2	6	no	—
17	C₅₃×#548	—	2	2	no	—
18	C₅₃×#546	—	2	6	no	—
19	C₅₃×#544	C₂²	1	6	yes	complemented
20	C₅₃×#536	C₂²	1	3	yes	—
21	C₅₃×#535	C₂²	1	3	yes	—
22	C₅₃×#533	C₂²	1	3	yes	complemented
23	C₅₃×#532	C₂²	1	3	yes	—
24	C₂×#125×C₅₃	C₂²	1	3	yes	complemented
25	C₂×#124×C₅₃	C₂²	1	3	yes	—
26	C₅₃×#520	C₂²	1	2	yes	—
27	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#491	C₂²	1	6	yes	complemented
29	C₅₃×#490	C₂²	1	6	yes	complemented
30	C₅₃×#489	C₂²	1	12	yes	complemented
31	C₅₃×#488	C₂²	1	3	yes	complemented
32	C₅₃×#488	C₂²	1	12	yes	complemented
33	C₅₃×#488	C₂²	1	6	yes	complemented
34	C₅₃×#488	C₂²	1	1	yes	characteristic, complemented
35	C₅₃×#488	C₂²	1	3	yes	complemented
36	C₅₃×#488	C₂²	1	3	yes	complemented
37	C₅₃×#487	C₂²	1	6	yes	complemented
38	C₅₃×#486	C₂²	1	6	yes	complemented
39	C₅₃×#419	C₂²	1	6	yes	complemented
40	C₅₃×#418	C₂²	1	6	yes	—
41	C₅₃×#417	C₂²	1	6	yes	—
42	C₅₃×#417	C₂²	1	6	yes	complemented
43	C₅₃×#416	C₂²	1	6	yes	—
44	C₅₃×#416	C₂²	1	6	yes	complemented
45	C₂×#103×C₅₃	C₂²	1	6	yes	complemented
46	C₂×#102×C₅₃	C₂²	1	6	yes	—
47	C₅₃×#405	C₂²	1	6	yes	complemented
48	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
49	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
50	2₋¹⁺⁴×C₅₃	—	2	8	no	—
51	H(5,𝔽₂)×C₅₃	—	2	8	no	—
52	C₂×#41×C₅₃	—	2	1	no	—
53	C₂×#41×C₅₃	—	2	3	no	—
54	C₂×#41×C₅₃	C₂³	1	3	yes	—
55	C₂×#41×C₅₃	C₂³	1	3	yes	—
56	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₅₃	—	2	1	no	—
58	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
59	C₂²×D₄×C₅₃	—	2	3	no	—
60	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
61	#131×C₅₃	—	2	8	no	—
62	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
63	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
64	C₄⋊Q₈×C₅₃	—	2	6	no	—
65	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
66	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
67	#127×C₅₃	C₂³	1	3	yes	—
68	#125×C₅₃	—	2	6	no	—
69	#125×C₅₃	—	2	6	no	—
70	#125×C₅₃	C₂³	1	3	yes	—
71	#125×C₅₃	C₂³	1	1	yes	characteristic
72	#125×C₅₃	C₂³	1	6	yes	—
73	#124×C₅₃	—	2	6	no	—
74	#124×C₅₃	C₂³	1	3	yes	—
75	#124×C₅₃	C₂³	1	1	yes	characteristic
76	#124×C₅₃	C₂³	1	6	yes	—
77	#123×C₅₃	—	2	6	no	—
78	C₂²⋊Q₈×C₅₃	—	2	6	no	—
79	C₂²⋊Q₈×C₅₃	—	2	12	no	—
80	C₂²⋊Q₈×C₅₃	—	2	6	no	—
81	#121×C₅₃	—	2	6	no	—
82	#121×C₅₃	—	2	6	no	—
83	#120×C₅₃	—	2	6	no	—
84	C₄×Q₈×C₅₃	—	2	6	no	—
85	C₄×Q₈×C₅₃	—	2	2	no	—
86	C₄×Q₈×C₅₃	D₄	1	2	yes	—
87	C₄×D₄×C₅₃	—	2	6	no	—
88	C₄×D₄×C₅₃	—	2	6	no	—
89	C₄×D₄×C₅₃	D₄	1	6	yes	—
90	#117×C₅₃	C₂³	1	6	yes	—
91	#117×C₅₃	C₂³	1	3	yes	—
92	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
94	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
95	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
96	#103×C₅₃	C₂³	1	2	yes	—
97	#103×C₅₃	C₂³	1	6	yes	—
98	#103×C₅₃	C₂³	1	6	yes	—
99	#103×C₅₃	C₂³	1	6	yes	—
100	#103×C₅₃	C₂³	1	12	yes	—
101	#102×C₅₃	C₂³	1	2	yes	—
102	#102×C₅₃	C₂³	1	6	yes	—
103	#102×C₅₃	C₂³	1	6	yes	—
104	#102×C₅₃	C₂³	1	6	yes	—
105	#102×C₅₃	C₂³	1	12	yes	—
106	#97×C₅₃	C₂³	1	6	yes	—
107	C₄×C₈×C₅₃	C₂³	1	6	yes	—
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₅₃	C₂³	1	3	yes	—
110	C₂⁴×C₅₃	—	2	3	no	—
111	#41×C₅₃	—	2	12	no	—
112	#41×C₅₃	—	2	12	no	—
113	#41×C₅₃	—	4	4	no	—
114	#41×C₅₃	—	4	12	no	—
115	#41×C₅₃	#41	1	8	yes	—
116	C₂×Q₈×C₅₃	—	2	3	no	—
117	C₂×Q₈×C₅₃	—	2	6	no	—
118	C₂×Q₈×C₅₃	—	2	12	no	—
119	C₂×Q₈×C₅₃	—	4	4	no	—
120	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
122	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂×D₄×C₅₃	—	2	3	no	—
124	C₂×D₄×C₅₃	—	2	3	no	—
125	C₂×D₄×C₅₃	—	2	3	no	—
126	C₂×D₄×C₅₃	—	2	6	no	—
127	C₂×D₄×C₅₃	—	2	12	no	—
128	C₂×D₄×C₅₃	—	4	12	no	—
129	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
130	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
131	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	—	2	3	no	—
133	C₂²×C₄×C₅₃	—	2	6	no	—
134	C₂²×C₄×C₅₃	—	2	3	no	—
135	C₂²×C₄×C₅₃	—	2	3	no	—
136	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
138	M₁₆×C₅₃	—	2	12	no	—
139	C₂×C₈×C₅₃	—	2	12	no	—
140	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
141	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
142	C₄⋊C₄×C₅₃	—	2	6	no	—
143	C₄⋊C₄×C₅₃	—	2	12	no	—
144	C₄⋊C₄×C₅₃	—	2	6	no	—
145	C₄⋊C₄×C₅₃	—	2	6	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	6	no	—
148	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
149	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
150	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
151	C₂²⋊C₄×C₅₃	—	2	6	no	—
152	C₂²⋊C₄×C₅₃	—	2	6	no	—
153	C₂²⋊C₄×C₅₃	—	2	12	no	—
154	C₂²⋊C₄×C₅₃	—	2	2	no	—
155	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
156	C₄²×C₅₃	—	2	6	no	—
157	C₄²×C₅₃	C₂×D₄	1	6	yes	—
158	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
159	C₄²×C₅₃	C₂⁴	1	3	yes	—
160	C₂³×C₅₃	—	2	1	no	—
161	C₂³×C₅₃	—	2	3	no	—
162	C₂³×C₅₃	—	4	6	no	—
163	C₂³×C₅₃	—	8	6	no	—
164	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
165	Q₈×C₅₃	—	2	6	no	—
166	Q₈×C₅₃	—	2	2	no	—
167	Q₈×C₅₃	—	4	6	no	—
168	Q₈×C₅₃	—	8	6	no	—
169	Q₈×C₅₃	#124	1	4	yes	—
170	D₄×C₅₃	—	2	6	no	—
171	D₄×C₅₃	—	2	2	no	—
172	D₄×C₅₃	—	4	6	no	—
173	D₄×C₅₃	—	4	6	no	—
174	D₄×C₅₃	—	4	6	no	—
175	D₄×C₅₃	—	8	6	no	—
176	D₄×C₅₃	#124	1	12	yes	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	3	no	—
179	C₂×C₄×C₅₃	—	2	6	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	4	6	no	—
182	C₂×C₄×C₅₃	—	4	6	no	—
183	C₂×C₄×C₅₃	—	4	6	no	—
184	C₂×C₄×C₅₃	—	4	6	no	—
185	C₂×C₄×C₅₃	—	8	6	no	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
189	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
190	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	2	6	no	—
193	#60624	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂²×C₅₃	—	2	6	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	—	4	6	no	—
197	C₂²×C₅₃	—	4	1	no	—
198	C₂²×C₅₃	—	4	3	no	—
199	C₂²×C₅₃	—	16	6	no	—
200	C₂²×C₅₃	C₂×#124	1	6	yes	—
201	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	2	6	no	—
204	C₄×C₅₃	—	4	6	no	—
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	4	1	no	—
207	C₄×C₅₃	—	4	3	no	—
208	C₄×C₅₃	C₂×#124	1	6	yes	—
209	C₄×C₅₃	C₂²×#41	1	2	yes	—
210	#3504	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#3500	C₂×C₅₃	1	1	yes	characteristic
212	#3142	C₂×C₅₃	1	3	yes	complemented
213	#3140	C₂×C₅₃	1	3	yes	complemented
214	#3137	C₂×C₅₃	1	6	yes	complemented
215	#3136	C₂×C₅₃	1	6	yes	complemented
216	#3131	C₂×C₅₃	1	3	yes	complemented
217	C₂×#488	C₂×C₅₃	1	3	yes	complemented
218	#2898	C₂×C₅₃	1	2	yes	complemented
219	#2897	C₂×C₅₃	1	2	yes	complemented
220	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₂×C₅₃	—	2	6	no	—
222	C₂×C₅₃	—	8	1	no	—
223	C₂×C₅₃	—	8	3	no	—
224	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
225	C₂×C₅₃	#3584	1	2	yes	—
226	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
227	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×#131	C₂²×C₅₃	1	2	yes	—
229	#557	—	2	2	no	—
230	#554	—	2	6	no	—
231	#548	—	2	2	no	—
232	#546	—	2	6	no	—
233	#544	C₂²×C₅₃	1	6	yes	complemented
234	#536	C₂²×C₅₃	1	3	yes	—
235	#535	C₂²×C₅₃	1	3	yes	—
236	#533	C₂²×C₅₃	1	3	yes	complemented
237	#532	C₂²×C₅₃	1	3	yes	—
238	C₂×#125	C₂²×C₅₃	1	3	yes	complemented
239	C₂×#124	C₂²×C₅₃	1	3	yes	—
240	#520	C₂²×C₅₃	1	2	yes	—
241	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
242	#491	C₂²×C₅₃	1	6	yes	complemented
243	#490	C₂²×C₅₃	1	6	yes	complemented
244	#489	C₂²×C₅₃	1	12	yes	complemented
245	#488	C₂²×C₅₃	1	3	yes	complemented
246	#488	C₂²×C₅₃	1	12	yes	complemented
247	#488	C₂²×C₅₃	1	6	yes	complemented
248	#488	C₂²×C₅₃	1	1	yes	characteristic, complemented
249	#488	C₂²×C₅₃	1	3	yes	complemented
250	#488	C₂²×C₅₃	1	3	yes	complemented
251	#487	C₂²×C₅₃	1	6	yes	complemented
252	#486	C₂²×C₅₃	1	6	yes	complemented
253	#419	C₂²×C₅₃	1	6	yes	complemented
254	#418	C₂²×C₅₃	1	6	yes	—
255	#417	C₂²×C₅₃	1	6	yes	—
256	#417	C₂²×C₅₃	1	6	yes	complemented
257	#416	C₂²×C₅₃	1	6	yes	—
258	#416	C₂²×C₅₃	1	6	yes	complemented
259	C₂×#103	C₂²×C₅₃	1	6	yes	complemented
260	C₂×#102	C₂²×C₅₃	1	6	yes	—
261	#405	C₂²×C₅₃	1	6	yes	complemented
262	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
263	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	complemented
264	C₅₃	#60624	1	1	yes	characteristic, direct product factor, Sylow
265	2₋¹⁺⁴	—	2	8	no	—
266	H(5,𝔽₂)	—	2	8	no	—
267	C₂×#41	—	2	1	no	—
268	C₂×#41	—	2	3	no	—
269	C₂×#41	C₂³×C₅₃	1	3	yes	—
270	C₂×#41	C₂³×C₅₃	1	3	yes	—
271	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
272	C₂²×Q₈	—	2	1	no	—
273	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
274	C₂²×D₄	—	2	3	no	—
275	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
276	#131	—	2	8	no	—
277	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
278	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
279	C₄⋊Q₈	—	2	6	no	—
280	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
281	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
282	#127	C₂³×C₅₃	1	3	yes	—
283	#125	—	2	6	no	—
284	#125	—	2	6	no	—
285	#125	C₂³×C₅₃	1	3	yes	—
286	#125	C₂³×C₅₃	1	6	yes	—
287	#125	C₂³×C₅₃	1	1	yes	characteristic
288	#124	—	2	6	no	—
289	#124	C₂³×C₅₃	1	3	yes	—
290	#124	C₂³×C₅₃	1	6	yes	—
291	#124	C₂³×C₅₃	1	1	yes	characteristic
292	#123	—	2	6	no	—
293	C₂²⋊Q₈	—	2	6	no	—
294	C₂²⋊Q₈	—	2	6	no	—
295	C₂²⋊Q₈	—	2	12	no	—
296	#121	—	2	6	no	—
297	#121	—	2	6	no	—
298	#120	—	2	6	no	—
299	C₄×Q₈	—	2	2	no	—
300	C₄×Q₈	—	2	6	no	—
301	C₄×Q₈	D₄×C₅₃	1	2	yes	—
302	C₄×D₄	—	2	6	no	—
303	C₄×D₄	—	2	6	no	—
304	C₄×D₄	D₄×C₅₃	1	6	yes	—
305	#117	C₂³×C₅₃	1	6	yes	—
306	#117	C₂³×C₅₃	1	3	yes	—
307	C₂×C₄⋊C₄	—	2	6	no	—
308	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
309	C₂×C₂²⋊C₄	—	2	6	no	—
310	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
311	#103	C₂³×C₅₃	1	6	yes	—
312	#103	C₂³×C₅₃	1	12	yes	—
313	#103	C₂³×C₅₃	1	2	yes	—
314	#103	C₂³×C₅₃	1	6	yes	—
315	#103	C₂³×C₅₃	1	6	yes	—
316	#102	C₂³×C₅₃	1	6	yes	—
317	#102	C₂³×C₅₃	1	12	yes	—
318	#102	C₂³×C₅₃	1	6	yes	—
319	#102	C₂³×C₅₃	1	2	yes	—
320	#102	C₂³×C₅₃	1	6	yes	—
321	#97	C₂³×C₅₃	1	6	yes	—
322	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
323	C₄×C₈	C₂³×C₅₃	1	3	yes	—
324	C₄×C₈	C₂³×C₅₃	1	6	yes	—
325	C₂⁴	—	2	3	no	—
326	#41	—	2	12	no	—
327	#41	—	2	12	no	—
328	#41	—	4	4	no	—
329	#41	—	4	12	no	—
330	#41	#41×C₅₃	1	8	yes	—
331	C₂×Q₈	—	2	3	no	—
332	C₂×Q₈	—	2	6	no	—
333	C₂×Q₈	—	2	12	no	—
334	C₂×Q₈	—	4	4	no	—
335	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
337	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
338	C₂×D₄	—	2	3	no	—
339	C₂×D₄	—	2	3	no	—
340	C₂×D₄	—	2	3	no	—
341	C₂×D₄	—	2	6	no	—
342	C₂×D₄	—	2	12	no	—
343	C₂×D₄	—	4	12	no	—
344	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
345	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
346	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
347	C₂²×C₄	—	2	3	no	—
348	C₂²×C₄	—	2	6	no	—
349	C₂²×C₄	—	2	3	no	—
350	C₂²×C₄	—	2	3	no	—
351	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
352	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
353	M₁₆	—	2	12	no	—
354	C₂×C₈	—	2	12	no	—
355	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
356	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
357	C₄⋊C₄	—	2	6	no	—
358	C₄⋊C₄	—	2	12	no	—
359	C₄⋊C₄	—	2	6	no	—
360	C₄⋊C₄	—	2	6	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	6	no	—
363	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
364	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
365	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
366	C₂²⋊C₄	—	2	6	no	—
367	C₂²⋊C₄	—	2	6	no	—
368	C₂²⋊C₄	—	2	12	no	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
371	C₄²	—	2	6	no	—
372	C₄²	C₂×D₄×C₅₃	1	6	yes	—
373	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
374	C₄²	C₂⁴×C₅₃	1	3	yes	—
375	C₂³	—	2	3	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	4	6	no	—
378	C₂³	—	8	6	no	—
379	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
380	Q₈	—	2	6	no	—
381	Q₈	—	2	2	no	—
382	Q₈	—	4	6	no	—
383	Q₈	—	8	6	no	—
384	Q₈	#124×C₅₃	1	4	yes	—
385	D₄	—	2	6	no	—
386	D₄	—	2	2	no	—
387	D₄	—	4	6	no	—
388	D₄	—	4	6	no	—
389	D₄	—	4	6	no	—
390	D₄	—	8	6	no	—
391	D₄	#124×C₅₃	1	12	yes	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	6	no	—
394	C₂×C₄	—	2	3	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	4	6	no	—
397	C₂×C₄	—	4	6	no	—
398	C₂×C₄	—	4	6	no	—
399	C₂×C₄	—	4	6	no	—
400	C₂×C₄	—	8	6	no	—
401	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
402	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
403	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
404	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
405	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	6	no	—
408	C₂²	—	2	6	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	6	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	3	no	—
413	C₂²	—	16	6	no	—
414	C₂²	C₂×#124×C₅₃	1	6	yes	—
415	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	6	no	—
418	C₄	—	4	6	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	1	no	—
421	C₄	—	4	3	no	—
422	C₄	C₂×#124×C₅₃	1	6	yes	—
423	C₄	C₂²×#41×C₅₃	1	2	yes	—
424	C₂	—	2	6	no	—
425	C₂	—	8	3	no	—
426	C₂	—	8	1	no	—
427	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
428	C₂	C₅₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3500	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×#3504	C₂	semidirect
3. C₅₃×#3142	C₂	semidirect
4. C₅₃×#3140	C₂	semidirect
5. C₅₃×#3137	C₂	semidirect
6. C₅₃×#3136	C₂	semidirect
7. C₅₃×#3131	C₂	semidirect
8. C₂×C₅₃×#488	C₂	semidirect
9. C₅₃×#2898	C₂	semidirect
10. C₅₃×#2897	C₂	semidirect
11. C₄×#131×C₅₃	C₂	semidirect
19. C₅₃×#544	C₂²	semidirect
22. C₅₃×#533	C₂²	semidirect
24. C₂×#125×C₅₃	C₂²	semidirect
28. C₅₃×#491	C₂²	semidirect
29. C₅₃×#490	C₂²	semidirect
30. C₅₃×#489	C₂²	semidirect
31. C₅₃×#488	C₂²	semidirect
32. C₅₃×#488	C₂²	semidirect
33. C₅₃×#488	C₂²	semidirect
34. C₅₃×#488	C₂²	semidirect
35. C₅₃×#488	C₂²	semidirect
36. C₅₃×#488	C₂²	semidirect
37. C₅₃×#487	C₂²	semidirect
38. C₅₃×#486	C₂²	semidirect
39. C₅₃×#419	C₂²	semidirect
42. C₅₃×#417	C₂²	semidirect
44. C₅₃×#416	C₂²	semidirect
45. C₂×#103×C₅₃	C₂²	semidirect
47. C₅₃×#405	C₂²	semidirect
48. C₄×M₁₆×C₅₃	C₂²	semidirect
49. C₂×C₄×C₈×C₅₃	C₂²	semidirect
193. #60624	C₅₃	direct (N × H)
210. #3504	C₂×C₅₃	semidirect
212. #3142	C₂×C₅₃	semidirect
213. #3140	C₂×C₅₃	semidirect
214. #3137	C₂×C₅₃	semidirect
215. #3136	C₂×C₅₃	semidirect
216. #3131	C₂×C₅₃	semidirect
217. C₂×#488	C₂×C₅₃	semidirect
218. #2898	C₂×C₅₃	semidirect
219. #2897	C₂×C₅₃	semidirect
220. C₄×#131	C₂×C₅₃	semidirect
233. #544	C₂²×C₅₃	semidirect
236. #533	C₂²×C₅₃	semidirect
238. C₂×#125	C₂²×C₅₃	semidirect
242. #491	C₂²×C₅₃	semidirect
243. #490	C₂²×C₅₃	semidirect
244. #489	C₂²×C₅₃	semidirect
245. #488	C₂²×C₅₃	semidirect
246. #488	C₂²×C₅₃	semidirect
247. #488	C₂²×C₅₃	semidirect
248. #488	C₂²×C₅₃	semidirect
249. #488	C₂²×C₅₃	semidirect
250. #488	C₂²×C₅₃	semidirect
251. #487	C₂²×C₅₃	semidirect
252. #486	C₂²×C₅₃	semidirect
253. #419	C₂²×C₅₃	semidirect
256. #417	C₂²×C₅₃	semidirect
258. #416	C₂²×C₅₃	semidirect
259. C₂×#103	C₂²×C₅₃	semidirect
261. #405	C₂²×C₅₃	semidirect
262. C₄×M₁₆	C₂²×C₅₃	semidirect
263. C₂×C₄×C₈	C₂²×C₅₃	semidirect
264. C₅₃	#60624	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×#3504	2
C₅₃×#3500	2
C₅₃×#3142	2
C₅₃×#3140	2
C₅₃×#3137	2
C₅₃×#3136	2
C₅₃×#3131	2
C₂×C₅₃×#488	2
C₅₃×#2898	2
C₅₃×#2897	2
C₄×#131×C₅₃	2
#60624	53

Subgroups — C53×#60624 #27268136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60624 #27268136