Subgroups — C₅₃×#23095 #27230607

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#460	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2366	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1968	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1926	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1877	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#103×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#102×C₅₃	C₂	1	2	yes	maximal, complemented
12	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	—	2	1	no	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#460	C₄	1	8	yes	complemented
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	2	yes	—
24	C₂×#102×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#391	C₂²	1	2	yes	—
28	C₅₃×#390	—	2	1	no	—
29	C₅₃×#390	C₂²	1	2	yes	—
30	C₅₃×#387	—	2	2	no	—
31	C₅₃×#386	—	2	2	no	—
32	C₅₃×#384	C₂²	1	1	yes	characteristic
33	C₅₃×#382	—	2	1	no	—
34	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	1	yes	characteristic
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	—	2	1	no	—
44	C₂²×D₄×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	—	2	2	no	—
47	C₂×SD₁₆×C₅₃	—	4	2	no	—
48	C₂×SD₁₆×C₅₃	—	4	2	no	—
49	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	#127×C₅₃	—	2	2	no	—
54	#127×C₅₃	C₂×C₄	1	4	yes	—
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
57	C₄×D₄×C₅₃	—	2	4	no	—
58	C₄×D₄×C₅₃	—	4	2	no	—
59	C₄×D₄×C₅₃	—	4	4	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	—	2	1	no	—
74	C₂×C₄²×C₅₃	—	2	2	no	—
75	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	#106×C₅₃	—	4	2	no	—
78	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
79	#103×C₅₃	—	2	2	no	—
80	#103×C₅₃	—	2	2	no	—
81	#102×C₅₃	—	2	4	no	—
82	#102×C₅₃	—	4	2	no	—
83	#102×C₅₃	C₂×C₄	1	8	yes	complemented
84	C₄×C₈×C₅₃	—	2	2	no	—
85	C₄×C₈×C₅₃	—	4	2	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	1	no	—
89	#95×C₅₃	—	2	2	no	—
90	#95×C₅₃	D₄	1	2	yes	—
91	C₂⁴×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	—	2	1	no	—
97	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×D₄×C₅₃	—	2	4	no	—
99	C₂×D₄×C₅₃	—	2	4	no	—
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	—	2	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	4	2	no	—
106	C₂×D₄×C₅₃	—	4	2	no	—
107	C₂×D₄×C₅₃	—	8	1	no	—
108	C₂×D₄×C₅₃	—	8	1	no	—
109	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	4	4	no	—
116	C₂²×C₄×C₅₃	—	4	4	no	—
117	C₂²×C₄×C₅₃	—	8	2	no	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	SD₁₆×C₅₃	—	4	8	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
132	C₂×C₈×C₅₃	#41	1	2	yes	—
133	C₄⋊C₄×C₅₃	—	2	4	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	4	2	no	—
142	C₄⋊C₄×C₅₃	—	8	1	no	—
143	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	complemented
144	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
145	C₄⋊C₄×C₅₃	#41	1	2	yes	—
146	C₂²⋊C₄×C₅₃	—	4	4	no	—
147	C₂²⋊C₄×C₅₃	—	4	4	no	—
148	C₂²⋊C₄×C₅₃	—	8	2	no	—
149	C₄²×C₅₃	—	2	4	no	—
150	C₄²×C₅₃	—	2	2	no	—
151	C₄²×C₅₃	—	2	2	no	—
152	C₄²×C₅₃	—	4	2	no	—
153	C₄²×C₅₃	—	8	2	no	—
154	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
155	C₄²×C₅₃	#41	1	2	yes	—
156	C₂³×C₅₃	—	2	4	no	—
157	C₂³×C₅₃	—	4	2	no	—
158	C₂³×C₅₃	—	4	2	no	—
159	C₂³×C₅₃	—	4	2	no	—
160	C₂³×C₅₃	—	4	2	no	—
161	C₂³×C₅₃	—	4	2	no	—
162	C₂³×C₅₃	—	4	2	no	—
163	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₅₃	—	2	2	no	—
165	Q₈×C₅₃	—	2	2	no	—
166	Q₈×C₅₃	—	4	1	no	—
167	Q₈×C₅₃	—	4	1	no	—
168	D₄×C₅₃	—	2	4	no	—
169	D₄×C₅₃	—	2	4	no	—
170	D₄×C₅₃	—	4	2	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	4	8	no	—
173	D₄×C₅₃	—	8	4	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	8	8	no	—
197	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
198	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
199	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
200	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
201	C₂×C₄×C₅₃	#135	1	2	yes	—
202	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
208	C₈×C₅₃	—	2	4	no	—
209	C₈×C₅₃	—	2	2	no	—
210	C₈×C₅₃	—	4	1	no	—
211	#23095	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂²×C₅₃	—	4	4	no	—
213	C₂²×C₅₃	—	4	4	no	—
214	C₂²×C₅₃	—	4	4	no	—
215	C₂²×C₅₃	—	8	2	no	—
216	C₂²×C₅₃	—	8	2	no	—
217	C₂²×C₅₃	—	8	2	no	—
218	C₂²×C₅₃	—	8	2	no	—
219	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₅₃	#546	1	1	yes	characteristic
221	C₂²×C₅₃	#550	1	1	yes	characteristic
222	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
223	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
224	C₂²×C₅₃	#576	1	1	yes	characteristic
225	C₂²×C₅₃	#577	1	1	yes	characteristic
226	C₄×C₅₃	—	2	2	no	—
227	C₄×C₅₃	—	2	4	no	—
228	C₄×C₅₃	—	4	2	no	—
229	C₄×C₅₃	—	4	2	no	—
230	C₄×C₅₃	—	4	2	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	8	1	no	—
233	C₄×C₅₃	—	8	2	no	—
234	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
235	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
236	C₄×C₅₃	#518	1	1	yes	characteristic
237	C₄×C₅₃	#553	1	1	yes	characteristic
238	C₄×C₅₃	#555	1	1	yes	characteristic
239	C₂×#460	C₂×C₅₃	1	1	yes	characteristic
240	#2366	C₂×C₅₃	1	1	yes	characteristic
241	#2355	C₂×C₅₃	1	1	yes	characteristic, complemented
242	#1968	C₂×C₅₃	1	2	yes	complemented
243	#1957	C₂×C₅₃	1	1	yes	characteristic
244	#1948	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#1926	C₂×C₅₃	1	2	yes	complemented
246	#1877	C₂×C₅₃	1	2	yes	complemented
247	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
248	C₄×#103	C₂×C₅₃	1	1	yes	characteristic, complemented
249	C₄×#102	C₂×C₅₃	1	2	yes	complemented
250	C₂×C₅₃	—	8	4	no	—
251	C₂×C₅₃	#2363	1	1	yes	characteristic
252	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
253	C₂×C₅₃	#2946	1	1	yes	characteristic
254	C₂×C₅₃	#3295	1	1	yes	characteristic
255	C₂×C₅₃	#3304	1	1	yes	characteristic
256	C₂×C₅₃	#3392	1	1	yes	characteristic
257	C₂×C₅₃	#3396	1	1	yes	characteristic
258	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
259	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄×D₄	—	2	1	no	—
262	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
263	#460	C₄×C₅₃	1	8	yes	complemented
264	C₂×#106	C₂²×C₅₃	1	2	yes	—
265	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
266	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×#102	C₂²×C₅₃	1	2	yes	—
269	C₂×#102	C₂²×C₅₃	1	2	yes	—
270	C₂×#102	C₂²×C₅₃	1	2	yes	—
271	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
273	#391	C₂²×C₅₃	1	2	yes	—
274	#390	—	2	1	no	—
275	#390	C₂²×C₅₃	1	2	yes	—
276	#387	—	2	2	no	—
277	#386	—	2	2	no	—
278	#384	C₂²×C₅₃	1	1	yes	characteristic
279	#382	—	2	1	no	—
280	C₄×C₄⋊C₄	—	2	1	no	—
281	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
282	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
283	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
284	#340	C₂²×C₅₃	1	2	yes	—
285	#340	C₂²×C₅₃	1	1	yes	characteristic
286	#340	C₂²×C₅₃	1	1	yes	characteristic
287	#336	C₂²×C₅₃	1	2	yes	—
288	C₅₃	#23095	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
290	C₂²×D₄	—	2	1	no	—
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
293	C₂³×C₄	—	2	2	no	—
294	C₂×SD₁₆	—	4	2	no	—
295	C₂×SD₁₆	—	4	2	no	—
296	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
297	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
299	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
300	#127	—	2	2	no	—
301	#127	C₂×C₄×C₅₃	1	4	yes	—
302	C₄×Q₈	—	2	2	no	—
303	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	—	4	2	no	—
306	C₄×D₄	—	4	4	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
311	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
323	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
324	#106	—	4	2	no	—
325	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
326	#103	—	2	2	no	—
327	#103	—	2	2	no	—
328	#102	—	2	4	no	—
329	#102	—	4	2	no	—
330	#102	C₂×C₄×C₅₃	1	8	yes	complemented
331	C₄×C₈	—	2	2	no	—
332	C₄×C₈	—	4	2	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	#95	—	2	1	no	—
336	#95	—	2	2	no	—
337	#95	D₄×C₅₃	1	2	yes	—
338	C₂⁴	—	2	2	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	1	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂×D₄	—	8	1	no	—
355	C₂×D₄	—	8	1	no	—
356	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	4	4	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	8	2	no	—
365	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
366	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
370	SD₁₆	—	4	8	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
379	C₂×C₈	#41×C₅₃	1	2	yes	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	8	1	no	—
390	C₄⋊C₄	SD₁₆×C₅₃	1	4	yes	complemented
391	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
392	C₄⋊C₄	#41×C₅₃	1	2	yes	—
393	C₂²⋊C₄	—	4	4	no	—
394	C₂²⋊C₄	—	4	4	no	—
395	C₂²⋊C₄	—	8	2	no	—
396	C₄²	—	2	4	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	4	2	no	—
400	C₄²	—	8	2	no	—
401	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
402	C₄²	#41×C₅₃	1	2	yes	—
403	C₂³	—	2	4	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
411	Q₈	—	2	2	no	—
412	Q₈	—	2	2	no	—
413	Q₈	—	4	1	no	—
414	Q₈	—	4	1	no	—
415	D₄	—	2	4	no	—
416	D₄	—	2	4	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	8	no	—
420	D₄	—	8	4	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	8	8	no	—
444	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
445	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
446	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
447	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
448	C₂×C₄	#135×C₅₃	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
451	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
453	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
454	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
455	C₈	—	2	4	no	—
456	C₈	—	2	2	no	—
457	C₈	—	4	1	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	4	4	no	—
460	C₂²	—	4	4	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	8	2	no	—
465	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
466	C₂²	C₅₃×#546	1	1	yes	characteristic
467	C₂²	C₅₃×#550	1	1	yes	characteristic
468	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
469	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
470	C₂²	C₅₃×#576	1	1	yes	characteristic
471	C₂²	C₅₃×#577	1	1	yes	characteristic
472	C₄	—	2	2	no	—
473	C₄	—	2	4	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	8	1	no	—
479	C₄	—	8	2	no	—
480	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
481	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
482	C₄	C₅₃×#518	1	1	yes	characteristic
483	C₄	C₅₃×#553	1	1	yes	characteristic
484	C₄	C₅₃×#555	1	1	yes	characteristic
485	C₂	—	8	4	no	—
486	C₂	C₅₃×#2363	1	1	yes	characteristic
487	C₂	C₂×C₄×SD₁₆×C₅₃	1	1	yes	characteristic
488	C₂	C₅₃×#2946	1	1	yes	characteristic
489	C₂	C₅₃×#3295	1	1	yes	characteristic
490	C₂	C₅₃×#3304	1	1	yes	characteristic
491	C₂	C₅₃×#3392	1	1	yes	characteristic
492	C₂	C₅₃×#3396	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×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2355	C₂	semidirect
4. C₅₃×#1968	C₂	semidirect
6. C₅₃×#1948	C₂	semidirect
7. C₅₃×#1926	C₂	semidirect
8. C₅₃×#1877	C₂	semidirect
9. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₄×#103×C₅₃	C₂	semidirect
11. C₄×#102×C₅₃	C₂	semidirect
17. C₅₃×#460	C₄	semidirect
49. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
56. C₄×Q₈×C₅₃	C₂×C₄	semidirect
78. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
83. #102×C₅₃	C₂×C₄	semidirect
143. C₄⋊C₄×C₅₃	SD₁₆	semidirect
211. #23095	C₅₃	direct (N × H)
241. #2355	C₂×C₅₃	semidirect
242. #1968	C₂×C₅₃	semidirect
244. #1948	C₂×C₅₃	semidirect
245. #1926	C₂×C₅₃	semidirect
246. #1877	C₂×C₅₃	semidirect
247. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
248. C₄×#103	C₂×C₅₃	semidirect
249. C₄×#102	C₂×C₅₃	semidirect
263. #460	C₄×C₅₃	semidirect
288. C₅₃	#23095	direct (N × H)
296. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
303. C₄×Q₈	C₂×C₄×C₅₃	semidirect
325. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
330. #102	C₂×C₄×C₅₃	semidirect
390. C₄⋊C₄	SD₁₆×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×#460	2
C₅₃×#2366	2
C₅₃×#2355	2
C₅₃×#1968	2
C₅₃×#1957	2
C₅₃×#1948	2
C₅₃×#1926	2
C₅₃×#1877	2
C₈×C₄⋊C₄×C₅₃	2
C₄×#103×C₅₃	2
C₄×#102×C₅₃	2
#23095	53

Subgroups — C53×#23095 #27230607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23095 #27230607