Subgroups — C₅₃×#17995 #27225507

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1590	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1582	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₅₃	—	2	1	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#483	C₄	1	2	yes	—
16	C₅₃×#476	C₄	1	1	yes	characteristic
17	C₅₃×#476	C₄	1	1	yes	characteristic
18	C₅₃×#461	C₄	1	1	yes	characteristic
19	C₅₃×#461	C₄	1	1	yes	characteristic
20	C₅₃×#450	C₄	1	2	yes	—
21	C₅₃×#446	—	2	2	no	—
22	C₅₃×#446	C₂²	1	2	yes	complemented
23	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
24	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#436	C₂²	1	2	yes	—
27	C₅₃×#435	—	2	2	no	—
28	C₈×D₄×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₅₃×#424	—	2	1	no	—
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#407	—	2	1	no	—
34	C₅₃×#405	—	2	1	no	—
35	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#334	C₂²	1	2	yes	complemented
38	C₅₃×#334	C₂²	1	2	yes	complemented
39	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
40	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
41	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₅₃	C₂²	1	2	yes	complemented
47	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	1	no	—
49	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	1	no	—
53	C₂×M₁₆×C₅₃	—	4	4	no	—
54	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	#125×C₅₃	—	2	1	no	—
62	C₂²⋊Q₈×C₅₃	—	2	1	no	—
63	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×Q₈×C₅₃	—	2	1	no	—
65	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
72	#117×C₅₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#106×C₅₃	C₂×C₄	1	2	yes	—
77	#106×C₅₃	C₂×C₄	1	2	yes	—
78	C₄⋊C₈×C₅₃	—	2	2	no	—
79	C₄⋊C₈×C₅₃	—	2	2	no	—
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	—	2	1	no	—
82	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
87	#103×C₅₃	C₂×C₄	1	2	yes	—
88	#103×C₅₃	C₂×C₄	1	2	yes	—
89	#102×C₅₃	C₂×C₄	1	2	yes	—
90	#102×C₅₃	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₅₃	—	2	2	no	—
92	C₂²⋊C₈×C₅₃	—	2	2	no	—
93	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
94	#97×C₅₃	—	2	2	no	—
95	#97×C₅₃	D₄	1	2	yes	—
96	C₄×C₈×C₅₃	—	2	2	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂⁴×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	—	4	1	no	—
108	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	4	2	no	—
116	C₂²×C₄×C₅₃	—	4	2	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
120	SD₁₆×C₅₃	—	2	4	no	—
121	SD₁₆×C₅₃	—	4	2	no	—
122	M₁₆×C₅₃	—	2	2	no	—
123	M₁₆×C₅₃	—	4	2	no	—
124	M₁₆×C₅₃	—	4	4	no	—
125	M₁₆×C₅₃	SD₁₆	1	4	yes	complemented
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	4	2	no	—
132	C₂×C₈×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	—	4	4	no	—
134	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
135	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
145	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²⋊C₄×C₅₃	—	2	2	no	—
148	C₂²⋊C₄×C₅₃	—	2	1	no	—
149	C₂²⋊C₄×C₅₃	—	2	1	no	—
150	C₂²⋊C₄×C₅₃	—	2	1	no	—
151	C₂²⋊C₄×C₅₃	—	2	1	no	—
152	C₄²×C₅₃	—	2	2	no	—
153	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₅₃	#41	1	1	yes	characteristic
156	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂³×C₅₃	—	2	1	no	—
158	C₂³×C₅₃	—	2	1	no	—
159	C₂³×C₅₃	—	4	2	no	—
160	C₂³×C₅₃	—	4	2	no	—
161	C₂³×C₅₃	—	8	1	no	—
162	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
163	Q₈×C₅₃	—	2	2	no	—
164	Q₈×C₅₃	—	2	2	no	—
165	Q₈×C₅₃	—	8	1	no	—
166	D₄×C₅₃	—	2	1	no	—
167	D₄×C₅₃	—	2	1	no	—
168	D₄×C₅₃	—	2	1	no	—
169	D₄×C₅₃	—	4	2	no	—
170	D₄×C₅₃	#131	1	2	yes	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
189	C₂×C₄×C₅₃	#135	1	2	yes	—
190	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	2	2	no	—
195	C₈×C₅₃	—	2	2	no	—
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	—	4	1	no	—
198	C₈×C₅₃	—	4	2	no	—
199	C₈×C₅₃	—	4	1	no	—
200	C₈×C₅₃	—	4	1	no	—
201	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
202	#17995	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	—	2	1	no	—
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	4	1	no	—
207	C₂²×C₅₃	—	4	1	no	—
208	C₂²×C₅₃	—	4	1	no	—
209	C₂²×C₅₃	—	4	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	—	8	4	no	—
212	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
213	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	4	2	no	—
218	C₄×C₅₃	—	4	1	no	—
219	C₄×C₅₃	—	4	1	no	—
220	C₄×C₅₃	—	4	2	no	—
221	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
222	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
223	C₄×C₅₃	#568	1	1	yes	characteristic
224	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
225	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
226	#3101	C₂×C₅₃	1	1	yes	characteristic
227	#2962	C₂×C₅₃	1	1	yes	characteristic, complemented
228	D₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
229	#1590	C₂×C₅₃	1	2	yes	complemented
230	#1583	C₂×C₅₃	1	2	yes	complemented
231	#1582	C₂×C₅₃	1	2	yes	complemented
232	C₈×SD₁₆	C₂×C₅₃	1	2	yes	complemented
233	#1570	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#1493	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
236	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
237	C₂×C₅₃	—	2	2	no	—
238	C₂×C₅₃	—	4	2	no	—
239	C₂×C₅₃	—	8	1	no	—
240	C₂×C₅₃	#2933	1	1	yes	characteristic
241	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
242	C₂×C₅₃	#2956	1	1	yes	characteristic
243	C₂²×M₁₆	—	2	1	no	—
244	#533	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
246	#483	C₄×C₅₃	1	2	yes	—
247	#476	C₄×C₅₃	1	1	yes	characteristic
248	#476	C₄×C₅₃	1	1	yes	characteristic
249	#461	C₄×C₅₃	1	1	yes	characteristic
250	#461	C₄×C₅₃	1	1	yes	characteristic
251	#450	C₄×C₅₃	1	2	yes	—
252	#446	—	2	2	no	—
253	#446	C₂²×C₅₃	1	2	yes	complemented
254	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
255	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
256	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
257	#436	C₂²×C₅₃	1	2	yes	—
258	#435	—	2	2	no	—
259	C₈×D₄	C₂²×C₅₃	1	2	yes	—
260	#431	C₂²×C₅₃	1	1	yes	characteristic
261	#424	—	2	1	no	—
262	#423	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
264	#407	—	2	1	no	—
265	#405	—	2	1	no	—
266	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
268	#334	C₂²×C₅₃	1	2	yes	complemented
269	#334	C₂²×C₅₃	1	2	yes	complemented
270	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
271	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
272	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
273	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
274	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
275	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
276	#322	C₂²×C₅₃	1	2	yes	complemented
277	C₈²	C₂²×C₅₃	1	2	yes	complemented
278	C₅₃	#17995	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
280	C₂³×C₄	—	2	1	no	—
281	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
282	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
283	C₂×M₁₆	—	2	1	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	4	4	no	—
286	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
287	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	2	2	no	—
291	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
292	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
293	#125	—	2	1	no	—
294	C₂²⋊Q₈	—	2	1	no	—
295	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
296	C₄×Q₈	—	2	1	no	—
297	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
301	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
302	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
303	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
304	#117	—	2	1	no	—
305	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
308	#106	C₂×C₄×C₅₃	1	2	yes	—
309	#106	C₂×C₄×C₅₃	1	2	yes	—
310	C₄⋊C₈	—	2	2	no	—
311	C₄⋊C₈	—	2	2	no	—
312	C₄⋊C₈	—	2	1	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
315	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
317	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
319	#103	C₂×C₄×C₅₃	1	2	yes	—
320	#103	C₂×C₄×C₅₃	1	2	yes	—
321	#102	C₂×C₄×C₅₃	1	2	yes	—
322	#102	C₂×C₄×C₅₃	1	2	yes	—
323	C₂²⋊C₈	—	2	2	no	—
324	C₂²⋊C₈	—	2	2	no	—
325	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
326	#97	—	2	2	no	—
327	#97	D₄×C₅₃	1	2	yes	—
328	C₄×C₈	—	2	2	no	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
331	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
332	C₄×C₈	C₂³×C₅₃	1	2	yes	—
333	C₄×C₈	C₂³×C₅₃	1	2	yes	—
334	C₂⁴	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	4	1	no	—
340	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
341	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
351	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
352	SD₁₆	—	2	4	no	—
353	SD₁₆	—	4	2	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	4	2	no	—
356	M₁₆	—	4	4	no	—
357	M₁₆	SD₁₆×C₅₃	1	4	yes	complemented
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	4	2	no	—
364	C₂×C₈	—	4	2	no	—
365	C₂×C₈	—	4	4	no	—
366	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
367	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
368	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
369	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
377	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
378	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
379	C₂²⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	1	no	—
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	1	no	—
384	C₄²	—	2	2	no	—
385	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
386	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
387	C₄²	#41×C₅₃	1	1	yes	characteristic
388	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
389	C₂³	—	2	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	8	1	no	—
394	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
395	Q₈	—	2	2	no	—
396	Q₈	—	2	2	no	—
397	Q₈	—	8	1	no	—
398	D₄	—	2	1	no	—
399	D₄	—	2	1	no	—
400	D₄	—	2	1	no	—
401	D₄	—	4	2	no	—
402	D₄	#131×C₅₃	1	2	yes	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
418	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
419	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
420	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
421	C₂×C₄	#135×C₅₃	1	2	yes	—
422	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
425	C₈	—	2	2	no	—
426	C₈	—	2	2	no	—
427	C₈	—	2	2	no	—
428	C₈	—	2	2	no	—
429	C₈	—	4	1	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	1	no	—
432	C₈	—	4	1	no	—
433	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	1	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	1	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	8	4	no	—
443	C₂²	C₄×SD₁₆×C₅₃	1	2	yes	—
444	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	C₄×SD₁₆×C₅₃	1	2	yes	—
453	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
454	C₄	C₅₃×#568	1	1	yes	characteristic
455	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
456	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
457	C₂	—	2	2	no	—
458	C₂	—	4	2	no	—
459	C₂	—	8	1	no	—
460	C₂	C₅₃×#2933	1	1	yes	characteristic
461	C₂	C₂×C₄×SD₁₆×C₅₃	1	1	yes	characteristic
462	C₂	C₅₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2962	C₂	semidirect
4. C₅₃×#1590	C₂	semidirect
5. C₅₃×#1583	C₂	semidirect
6. C₅₃×#1582	C₂	semidirect
7. C₈×SD₁₆×C₅₃	C₂	semidirect
8. C₅₃×#1570	C₂	semidirect
9. C₅₃×#1493	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
22. C₅₃×#446	C₂²	semidirect
23. C₈×Q₈×C₅₃	C₂²	semidirect
37. C₅₃×#334	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
39. C₅₃×Q₈⋊C₈	C₂²	semidirect
40. C₅₃×Q₈⋊C₈	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₈²×C₅₃	C₂²	semidirect
125. M₁₆×C₅₃	SD₁₆	semidirect
202. #17995	C₅₃	direct (N × H)
227. #2962	C₂×C₅₃	semidirect
229. #1590	C₂×C₅₃	semidirect
230. #1583	C₂×C₅₃	semidirect
231. #1582	C₂×C₅₃	semidirect
232. C₈×SD₁₆	C₂×C₅₃	semidirect
233. #1570	C₂×C₅₃	semidirect
234. #1493	C₂×C₅₃	semidirect
236. C₈×M₁₆	C₂×C₅₃	semidirect
253. #446	C₂²×C₅₃	semidirect
254. C₈×Q₈	C₂²×C₅₃	semidirect
268. #334	C₂²×C₅₃	semidirect
269. #334	C₂²×C₅₃	semidirect
270. Q₈⋊C₈	C₂²×C₅₃	semidirect
271. Q₈⋊C₈	C₂²×C₅₃	semidirect
276. #322	C₂²×C₅₃	semidirect
277. C₈²	C₂²×C₅₃	semidirect
278. C₅₃	#17995	direct (N × H)
357. M₁₆	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3101	2
C₅₃×#2962	2
D₄×M₁₆×C₅₃	2
C₅₃×#1590	2
C₅₃×#1583	2
C₅₃×#1582	2
C₈×SD₁₆×C₅₃	2
C₅₃×#1570	2
C₅₃×#1493	2
C₂×C₅₃×D₄⋊C₈	2
C₈×M₁₆×C₅₃	2
#17995	53

Subgroups — C53×#17995 #27225507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17995 #27225507