Subgroups — C₅₃×#22230 #27229742

Order 13568.

846 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	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2331	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2328	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1961	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1926	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1864	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1862	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#486	C₄	1	8	yes	complemented
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×#102×C₅₃	C₂²	1	2	yes	—
19	C₂×#102×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#391	—	2	2	no	—
25	C₅₃×#390	C₂²	1	2	yes	—
26	C₅₃×#389	C₂²	1	2	yes	—
27	C₅₃×#387	—	2	2	no	—
28	C₅₃×#386	—	2	2	no	—
29	C₅₃×#386	—	2	2	no	—
30	C₅₃×#382	C₂²	1	1	yes	characteristic
31	C₅₃×#382	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₅₃×#376	C₂²	1	1	yes	characteristic
34	C₅₃×#340	C₂²	1	2	yes	—
35	C₅₃×#340	C₂²	1	2	yes	—
36	C₅₃×#336	C₂²	1	1	yes	characteristic
37	C₅₃×#336	C₂²	1	1	yes	characteristic
38	C₅₃×#336	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	1	yes	characteristic
40	C₂²×Q₈×C₅₃	—	2	1	no	—
41	C₂²×D₄×C₅₃	—	2	1	no	—
42	C₂²×D₄×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
44	C₂³×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	—	2	2	no	—
50	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
51	#127×C₅₃	—	2	2	no	—
52	#127×C₅₃	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₅₃	—	4	2	no	—
54	C₄×D₄×C₅₃	—	4	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	#107×C₅₃	—	4	2	no	—
68	#106×C₅₃	—	4	2	no	—
69	#102×C₅₃	—	4	2	no	—
70	#102×C₅₃	—	4	2	no	—
71	#102×C₅₃	C₂×C₄	1	8	yes	complemented
72	#102×C₅₃	C₂×C₄	1	8	yes	complemented
73	#97×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
75	#95×C₅₃	—	2	2	no	—
76	#95×C₅₃	—	2	1	no	—
77	#95×C₅₃	—	2	1	no	—
78	#95×C₅₃	—	2	1	no	—
79	#95×C₅₃	—	2	1	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	D₄	1	2	yes	—
82	#95×C₅₃	D₄	1	2	yes	—
83	C₂⁴×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	4	no	—
85	C₂×Q₈×C₅₃	—	8	2	no	—
86	C₂×D₄×C₅₃	—	2	4	no	—
87	C₂×D₄×C₅₃	—	2	4	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	—	8	2	no	—
94	C₂×D₄×C₅₃	—	8	2	no	—
95	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
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	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	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	1	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₅₃	—	2	4	no	—
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	—	8	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
127	C₂²⋊C₄×C₅₃	—	4	4	no	—
128	C₂²⋊C₄×C₅₃	—	8	2	no	—
129	C₂²⋊C₄×C₅₃	—	8	2	no	—
130	C₄²×C₅₃	—	2	1	no	—
131	C₄²×C₅₃	—	2	1	no	—
132	C₄²×C₅₃	—	2	1	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	—	8	2	no	—
135	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
136	C₄²×C₅₃	#41	1	2	yes	—
137	C₂³×C₅₃	—	2	4	no	—
138	C₂³×C₅₃	—	4	2	no	—
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	4	2	no	—
141	C₂³×C₅₃	—	4	2	no	—
142	C₂³×C₅₃	—	4	2	no	—
143	C₂³×C₅₃	—	4	2	no	—
144	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₅₃	—	8	4	no	—
146	D₄×C₅₃	—	2	4	no	—
147	D₄×C₅₃	—	2	4	no	—
148	D₄×C₅₃	—	4	2	no	—
149	D₄×C₅₃	—	4	2	no	—
150	D₄×C₅₃	—	8	4	no	—
151	D₄×C₅₃	—	8	4	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	4	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	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	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	8	8	no	—
170	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
177	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
178	C₈×C₅₃	—	2	2	no	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	4	1	no	—
181	C₈×C₅₃	—	4	1	no	—
182	#22230	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	—	4	4	no	—
185	C₂²×C₅₃	—	4	4	no	—
186	C₂²×C₅₃	—	8	2	no	—
187	C₂²×C₅₃	—	8	2	no	—
188	C₂²×C₅₃	—	8	2	no	—
189	C₂²×C₅₃	—	8	2	no	—
190	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂²×C₅₃	#535	1	1	yes	characteristic
192	C₂²×C₅₃	#536	1	1	yes	characteristic
193	C₂²×C₅₃	#576	1	2	yes	—
194	C₂²×C₅₃	#577	1	2	yes	—
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	2	no	—
197	C₄×C₅₃	—	4	4	no	—
198	C₄×C₅₃	—	4	2	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	—	8	2	no	—
202	C₄×C₅₃	#520	1	1	yes	characteristic
203	C₄×C₅₃	#520	1	1	yes	characteristic
204	C₄×C₅₃	#542	1	1	yes	characteristic
205	C₄×C₅₃	#542	1	1	yes	characteristic
206	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
207	#2331	C₂×C₅₃	1	1	yes	characteristic, complemented
208	#2328	C₂×C₅₃	1	1	yes	characteristic
209	#1993	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1961	C₂×C₅₃	1	1	yes	characteristic
211	#1926	C₂×C₅₃	1	2	yes	complemented
212	#1925	C₂×C₅₃	1	2	yes	complemented
213	#1864	C₂×C₅₃	1	2	yes	complemented
214	#1862	C₂×C₅₃	1	2	yes	complemented
215	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
217	C₂×C₅₃	—	8	4	no	—
218	C₂×C₅₃	#2326	1	1	yes	characteristic
219	C₂×C₅₃	#2948	1	1	yes	characteristic
220	C₂×C₅₃	#2949	1	1	yes	characteristic
221	C₂×C₅₃	#3224	1	2	yes	—
222	C₂×C₅₃	#3241	1	2	yes	—
223	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
226	#486	C₄×C₅₃	1	8	yes	complemented
227	C₂×#107	C₂²×C₅₃	1	2	yes	—
228	C₂×#106	C₂²×C₅₃	1	2	yes	—
229	C₂×#102	C₂²×C₅₃	1	2	yes	—
230	C₂×#102	C₂²×C₅₃	1	2	yes	—
231	C₂×#102	C₂²×C₅₃	1	2	yes	—
232	C₂×#102	C₂²×C₅₃	1	2	yes	—
233	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
235	#391	—	2	2	no	—
236	#390	C₂²×C₅₃	1	2	yes	—
237	#389	C₂²×C₅₃	1	2	yes	—
238	#387	—	2	2	no	—
239	#386	—	2	2	no	—
240	#386	—	2	2	no	—
241	#382	C₂²×C₅₃	1	1	yes	characteristic
242	#382	C₂²×C₅₃	1	1	yes	characteristic
243	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
244	#376	C₂²×C₅₃	1	1	yes	characteristic
245	#340	C₂²×C₅₃	1	2	yes	—
246	#340	C₂²×C₅₃	1	2	yes	—
247	#336	C₂²×C₅₃	1	1	yes	characteristic
248	#336	C₂²×C₅₃	1	1	yes	characteristic
249	#336	C₂²×C₅₃	1	1	yes	characteristic
250	#336	C₂²×C₅₃	1	1	yes	characteristic
251	C₅₃	#22230	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²×Q₈	—	2	1	no	—
253	C₂²×D₄	—	2	1	no	—
254	C₂²×D₄	—	2	1	no	—
255	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
256	C₂³×C₄	—	2	2	no	—
257	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
263	#127	—	2	2	no	—
264	#127	C₂×C₄×C₅₃	1	4	yes	—
265	C₄×D₄	—	4	2	no	—
266	C₄×D₄	—	4	2	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
271	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₄²	—	2	2	no	—
276	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
279	#107	—	4	2	no	—
280	#106	—	4	2	no	—
281	#102	—	4	2	no	—
282	#102	—	4	2	no	—
283	#102	C₂×C₄×C₅₃	1	8	yes	complemented
284	#102	C₂×C₄×C₅₃	1	8	yes	complemented
285	#97	—	2	2	no	—
286	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
287	#95	—	2	2	no	—
288	#95	—	2	1	no	—
289	#95	—	2	1	no	—
290	#95	—	2	1	no	—
291	#95	—	2	1	no	—
292	#95	—	2	2	no	—
293	#95	D₄×C₅₃	1	2	yes	—
294	#95	D₄×C₅₃	1	2	yes	—
295	C₂⁴	—	2	2	no	—
296	C₂×Q₈	—	2	4	no	—
297	C₂×Q₈	—	8	2	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	8	2	no	—
306	C₂×D₄	—	8	2	no	—
307	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	—	8	2	no	—
313	C₂²×C₄	—	8	2	no	—
314	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
330	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	8	2	no	—
338	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
339	C₂²⋊C₄	—	4	4	no	—
340	C₂²⋊C₄	—	8	2	no	—
341	C₂²⋊C₄	—	8	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	8	2	no	—
347	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
348	C₄²	#41×C₅₃	1	2	yes	—
349	C₂³	—	2	4	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
357	Q₈	—	8	4	no	—
358	D₄	—	2	4	no	—
359	D₄	—	2	4	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	2	no	—
362	D₄	—	8	4	no	—
363	D₄	—	8	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	8	no	—
382	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
383	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
384	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
386	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
387	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
388	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
389	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	2	2	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	1	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
402	C₂²	C₅₃×#535	1	1	yes	characteristic
403	C₂²	C₅₃×#536	1	1	yes	characteristic
404	C₂²	C₅₃×#576	1	2	yes	—
405	C₂²	C₅₃×#577	1	2	yes	—
406	C₄	—	2	2	no	—
407	C₄	—	2	2	no	—
408	C₄	—	4	4	no	—
409	C₄	—	4	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	2	no	—
412	C₄	—	8	2	no	—
413	C₄	C₅₃×#520	1	1	yes	characteristic
414	C₄	C₅₃×#520	1	1	yes	characteristic
415	C₄	C₅₃×#542	1	1	yes	characteristic
416	C₄	C₅₃×#542	1	1	yes	characteristic
417	C₂	—	8	4	no	—
418	C₂	C₅₃×#2326	1	1	yes	characteristic
419	C₂	C₅₃×#2948	1	1	yes	characteristic
420	C₂	C₅₃×#2949	1	1	yes	characteristic
421	C₂	C₅₃×#3224	1	2	yes	—
422	C₂	C₅₃×#3241	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	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2331	C₂	semidirect
4. C₅₃×#1993	C₂	semidirect
6. C₅₃×#1926	C₂	semidirect
7. C₅₃×#1925	C₂	semidirect
8. C₅₃×#1864	C₂	semidirect
9. C₅₃×#1862	C₂	semidirect
10. C₅₃×#1846	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
15. C₅₃×#486	C₄	semidirect
50. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
71. #102×C₅₃	C₂×C₄	semidirect
72. #102×C₅₃	C₂×C₄	semidirect
74. C₄×C₈×C₅₃	C₂×C₄	semidirect
182. #22230	C₅₃	direct (N × H)
207. #2331	C₂×C₅₃	semidirect
209. #1993	C₂×C₅₃	semidirect
211. #1926	C₂×C₅₃	semidirect
212. #1925	C₂×C₅₃	semidirect
213. #1864	C₂×C₅₃	semidirect
214. #1862	C₂×C₅₃	semidirect
215. #1846	C₂×C₅₃	semidirect
216. #1746	C₂×C₅₃	semidirect
226. #486	C₄×C₅₃	semidirect
251. C₅₃	#22230	direct (N × H)
262. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
283. #102	C₂×C₄×C₅₃	semidirect
284. #102	C₂×C₄×C₅₃	semidirect
286. C₄×C₈	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×#486	2
C₅₃×#2331	2
C₅₃×#2328	2
C₅₃×#1993	2
C₅₃×#1961	2
C₅₃×#1926	2
C₅₃×#1925	2
C₅₃×#1864	2
C₅₃×#1862	2
C₅₃×#1846	2
C₅₃×#1746	2
#22230	53

Subgroups — C53×#22230 #27229742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22230 #27229742