Subgroups — C₅₃×#23061 #27230573

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1934	C₂	1	2	yes	maximal
7	C₅₃×#1926	C₂	1	2	yes	maximal
8	C₅₃×#1920	C₂	1	2	yes	maximal, complemented
9	C₄×#106×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	—	2	1	no	—
15	C₅₃×#476	C₄	1	8	yes	complemented
16	C₂×#106×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
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	—	2	1	no	—
27	C₅₃×#389	C₂²	1	2	yes	—
28	C₅₃×#388	—	2	2	no	—
29	C₅₃×#384	C₂²	1	1	yes	characteristic
30	C₅₃×#380	—	2	1	no	—
31	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
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	—	2	1	no	—
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₅₃	—	2	2	no	—
49	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₄×D₄×C₅₃	—	2	2	no	—
51	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
52	#117×C₅₃	—	4	4	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄²×C₅₃	—	2	2	no	—
68	C₂×C₄²×C₅₃	—	2	1	no	—
69	C₂×C₄²×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	#106×C₅₃	—	2	4	no	—
73	#106×C₅₃	C₂×C₄	1	8	yes	—
74	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
75	#102×C₅₃	—	2	2	no	—
76	#102×C₅₃	—	2	2	no	—
77	#102×C₅₃	C₂×C₄	1	8	yes	—
78	C₄×C₈×C₅₃	—	2	4	no	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	1	no	—
84	#95×C₅₃	D₄	1	2	yes	—
85	C₂⁴×C₅₃	—	2	1	no	—
86	C₂×Q₈×C₅₃	—	2	4	no	—
87	C₂×Q₈×C₅₃	—	8	1	no	—
88	C₂×Q₈×C₅₃	—	8	1	no	—
89	C₂×D₄×C₅₃	—	2	1	no	—
90	C₂×D₄×C₅₃	—	2	1	no	—
91	C₂×D₄×C₅₃	—	2	1	no	—
92	C₂×D₄×C₅₃	—	2	1	no	—
93	C₂×D₄×C₅₃	—	2	1	no	—
94	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	—	2	1	no	—
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	2	4	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	4	2	no	—
102	C₂²×C₄×C₅₃	—	4	2	no	—
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	2	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	2	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	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₅₃	#41	1	2	yes	—
117	C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	4	2	no	—
127	C₄⋊C₄×C₅₃	—	8	1	no	—
128	C₄⋊C₄×C₅₃	—	8	2	no	—
129	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	—
130	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
131	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₅₃	#41	1	2	yes	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	—	4	4	no	—
135	C₂²⋊C₄×C₅₃	—	8	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	4	4	no	—
139	C₄²×C₅₃	—	4	2	no	—
140	C₄²×C₅₃	—	8	2	no	—
141	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
142	C₄²×C₅₃	#41	1	2	yes	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	4	1	no	—
145	C₂³×C₅₃	—	4	1	no	—
146	C₂³×C₅₃	—	4	1	no	—
147	C₂³×C₅₃	—	4	1	no	—
148	C₂³×C₅₃	—	4	1	no	—
149	C₂³×C₅₃	—	4	1	no	—
150	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₅₃	—	8	4	no	—
152	D₄×C₅₃	—	2	2	no	—
153	D₄×C₅₃	—	2	2	no	—
154	D₄×C₅₃	—	4	1	no	—
155	D₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	8	no	—
173	C₂×C₄×C₅₃	—	4	1	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
180	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
181	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
182	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
183	C₂×C₄×C₅₃	#135	1	2	yes	—
184	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₅₃	—	2	4	no	—
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	4	1	no	—
193	#23061	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂²×C₅₃	—	4	2	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	—	4	2	no	—
197	C₂²×C₅₃	—	8	1	no	—
198	C₂²×C₅₃	—	8	1	no	—
199	C₂²×C₅₃	—	8	1	no	—
200	C₂²×C₅₃	—	8	1	no	—
201	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₅₃	#547	1	1	yes	characteristic
203	C₂²×C₅₃	#548	1	1	yes	characteristic
204	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
205	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
206	C₂²×C₅₃	#576	1	1	yes	characteristic
207	C₂²×C₅₃	#577	1	1	yes	characteristic
208	C₄×C₅₃	—	2	4	no	—
209	C₄×C₅₃	—	2	2	no	—
210	C₄×C₅₃	—	4	2	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	4	no	—
214	C₄×C₅₃	—	8	1	no	—
215	C₄×C₅₃	—	8	2	no	—
216	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
217	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
218	C₄×C₅₃	#519	1	1	yes	characteristic
219	C₄×C₅₃	#552	1	1	yes	characteristic
220	C₄×C₅₃	#556	1	1	yes	characteristic
221	C₂×#476	C₂×C₅₃	1	1	yes	characteristic
222	#2369	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#2354	C₂×C₅₃	1	1	yes	characteristic
224	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#1948	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#1934	C₂×C₅₃	1	2	yes	—
227	#1926	C₂×C₅₃	1	2	yes	—
228	#1920	C₂×C₅₃	1	2	yes	complemented
229	C₄×#106	C₂×C₅₃	1	2	yes	complemented
230	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
231	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
232	C₂×C₅₃	—	8	2	no	—
233	C₂×C₅₃	#2362	1	1	yes	characteristic
234	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
235	C₂×C₅₃	#2946	1	1	yes	characteristic
236	C₂×C₅₃	#3335	1	1	yes	characteristic
237	C₂×C₅₃	#3341	1	1	yes	characteristic
238	C₂×C₅₃	#3360	1	1	yes	characteristic
239	C₂×C₅₃	#3362	1	1	yes	characteristic
240	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×#117	—	2	1	no	—
243	#476	C₄×C₅₃	1	8	yes	complemented
244	C₂×#106	C₂²×C₅₃	1	2	yes	—
245	C₂×#106	C₂²×C₅₃	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×#102	C₂²×C₅₃	1	2	yes	—
250	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
252	#391	—	2	2	no	—
253	#390	C₂²×C₅₃	1	2	yes	—
254	#389	—	2	1	no	—
255	#389	C₂²×C₅₃	1	2	yes	—
256	#388	—	2	2	no	—
257	#384	C₂²×C₅₃	1	1	yes	characteristic
258	#380	—	2	1	no	—
259	C₄×C₄⋊C₄	—	2	1	no	—
260	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
261	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
262	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
264	#340	C₂²×C₅₃	1	1	yes	characteristic
265	#340	C₂²×C₅₃	1	2	yes	—
266	#340	C₂²×C₅₃	1	2	yes	—
267	#340	C₂²×C₅₃	1	2	yes	—
268	#340	C₂²×C₅₃	1	1	yes	characteristic
269	#336	C₂²×C₅₃	1	2	yes	—
270	C₅₃	#23061	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
273	C₂³×C₄	—	2	1	no	—
274	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
276	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₄⋊Q₈	—	2	2	no	—
278	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
281	#117	—	4	4	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
287	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₄²	—	2	2	no	—
297	C₂×C₄²	—	2	1	no	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
300	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
301	#106	—	2	4	no	—
302	#106	C₂×C₄×C₅₃	1	8	yes	—
303	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
304	#102	—	2	2	no	—
305	#102	—	2	2	no	—
306	#102	C₂×C₄×C₅₃	1	8	yes	—
307	C₄×C₈	—	2	4	no	—
308	C₄×C₈	—	2	2	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	1	no	—
313	#95	D₄×C₅₃	1	2	yes	—
314	C₂⁴	—	2	1	no	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	—	8	1	no	—
317	C₂×Q₈	—	8	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	2	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
333	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
345	C₂×C₈	#41×C₅₃	1	2	yes	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	8	1	no	—
357	C₄⋊C₄	—	8	2	no	—
358	C₄⋊C₄	SD₁₆×C₅₃	1	4	yes	—
359	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
360	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
361	C₄⋊C₄	#41×C₅₃	1	2	yes	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	8	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	4	4	no	—
368	C₄²	—	4	2	no	—
369	C₄²	—	8	2	no	—
370	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
371	C₄²	#41×C₅₃	1	2	yes	—
372	C₂³	—	2	2	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	1	no	—
379	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
380	Q₈	—	8	4	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	8	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
409	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
410	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
411	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
412	C₂×C₄	#135×C₅₃	1	2	yes	—
413	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
417	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
418	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
419	C₈	—	2	4	no	—
420	C₈	—	2	2	no	—
421	C₈	—	4	1	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	4	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	1	no	—
429	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
430	C₂²	C₅₃×#547	1	1	yes	characteristic
431	C₂²	C₅₃×#548	1	1	yes	characteristic
432	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
433	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
434	C₂²	C₅₃×#576	1	1	yes	characteristic
435	C₂²	C₅₃×#577	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	2	4	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	4	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
445	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
446	C₄	C₅₃×#519	1	1	yes	characteristic
447	C₄	C₅₃×#552	1	1	yes	characteristic
448	C₄	C₅₃×#556	1	1	yes	characteristic
449	C₂	—	8	2	no	—
450	C₂	C₅₃×#2362	1	1	yes	characteristic
451	C₂	C₂×C₄×SD₁₆×C₅₃	1	1	yes	characteristic
452	C₂	C₅₃×#2946	1	1	yes	characteristic
453	C₂	C₅₃×#3335	1	1	yes	characteristic
454	C₂	C₅₃×#3341	1	1	yes	characteristic
455	C₂	C₅₃×#3360	1	1	yes	characteristic
456	C₂	C₅₃×#3362	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	C₂²×D₄×C₅₃	C₂³
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
2. C₅₃×#2369	C₂	semidirect
4. C₅₃×#1990	C₂	semidirect
5. C₅₃×#1948	C₂	semidirect
8. C₅₃×#1920	C₂	semidirect
9. C₄×#106×C₅₃	C₂	semidirect
10. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
15. C₅₃×#476	C₄	semidirect
193. #23061	C₅₃	direct (N × H)
222. #2369	C₂×C₅₃	semidirect
224. #1990	C₂×C₅₃	semidirect
225. #1948	C₂×C₅₃	semidirect
228. #1920	C₂×C₅₃	semidirect
229. C₄×#106	C₂×C₅₃	semidirect
230. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
243. #476	C₄×C₅₃	semidirect
270. C₅₃	#23061	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#476	2
C₅₃×#2369	2
C₅₃×#2354	2
C₅₃×#1990	2
C₅₃×#1948	2
C₅₃×#1934	2
C₅₃×#1926	2
C₅₃×#1920	2
C₄×#106×C₅₃	2
C₈×C₄⋊C₄×C₅₃	2
C₄×#102×C₅₃	2
#23061	53

Subgroups — C53×#23061 #27230573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23061 #27230573