Subgroups — C₅₃×#18016 #27225528

Order 13568.

600 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×#3107	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2990	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1591	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1587	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1585	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#485	C₄	1	2	yes	—
15	C₅₃×#475	C₄	1	1	yes	characteristic
16	C₅₃×#474	C₄	1	1	yes	characteristic
17	C₅₃×#462	C₄	1	1	yes	characteristic
18	C₅₃×#461	C₄	1	1	yes	characteristic
19	C₅₃×#452	C₄	1	2	yes	—
20	C₅₃×#446	—	2	1	no	—
21	C₅₃×#446	C₂²	1	2	yes	—
22	C₅₃×#446	C₂²	1	2	yes	—
23	C₈×Q₈×C₅₃	—	2	1	no	—
24	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
26	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#435	—	2	2	no	—
28	C₈×D₄×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#433	—	2	2	no	—
30	C₅₃×#432	—	2	1	no	—
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#431	C₂²	1	1	yes	characteristic
34	C₅₃×#423	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	2	yes	—
38	C₅₃×#334	C₂²	1	2	yes	—
39	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
40	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
41	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
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	—
46	C₈²×C₅₃	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#125×C₅₃	—	2	1	no	—
56	C₂²⋊Q₈×C₅₃	—	2	1	no	—
57	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×Q₈×C₅₃	—	2	1	no	—
59	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
62	C₄×D₄×C₅₃	—	2	1	no	—
63	C₄×D₄×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
66	#117×C₅₃	—	2	1	no	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	—	2	1	no	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	C₂×C₄	1	2	yes	—
72	#106×C₅₃	C₂×C₄	1	2	yes	—
73	C₄⋊C₈×C₅₃	—	2	2	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	C₄⋊C₈×C₅₃	—	2	2	no	—
76	C₄⋊C₈×C₅₃	—	2	1	no	—
77	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
81	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
84	#103×C₅₃	C₂×C₄	1	2	yes	—
85	#103×C₅₃	C₂×C₄	1	2	yes	—
86	#103×C₅₃	C₂×C₄	1	2	yes	—
87	#102×C₅₃	C₂×C₄	1	2	yes	—
88	C₂²⋊C₈×C₅₃	—	2	2	no	—
89	C₂²⋊C₈×C₅₃	—	2	1	no	—
90	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
92	#97×C₅₃	—	2	2	no	—
93	#97×C₅₃	—	2	2	no	—
94	C₄×C₈×C₅₃	—	2	2	no	—
95	C₄×C₈×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	#41×C₅₃	—	2	2	no	—
102	#41×C₅₃	—	4	1	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
114	Q₁₆×C₅₃	—	4	2	no	—
115	SD₁₆×C₅₃	—	2	4	no	—
116	M₁₆×C₅₃	—	8	1	no	—
117	C₂×C₈×C₅₃	—	2	4	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	8	2	no	—
124	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
136	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₂²⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅₃	#41	1	1	yes	characteristic
149	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
152	Q₈×C₅₃	—	2	1	no	—
153	Q₈×C₅₃	—	2	2	no	—
154	Q₈×C₅₃	—	2	1	no	—
155	Q₈×C₅₃	—	2	2	no	—
156	Q₈×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	2	1	no	—
158	D₄×C₅₃	—	4	2	no	—
159	D₄×C₅₃	#131	1	2	yes	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	2	2	no	—
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	2	2	no	—
185	C₈×C₅₃	—	2	2	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	—	4	2	no	—
188	C₈×C₅₃	—	4	1	no	—
189	#18016	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	2	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	—	4	1	no	—
196	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	4	1	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	—	4	1	no	—
203	C₄×C₅₃	—	4	1	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	#443	1	2	yes	—
206	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
207	C₄×C₅₃	#568	1	1	yes	characteristic
208	C₄×C₅₃	#577	1	1	yes	characteristic
209	C₄×C₅₃	#578	1	1	yes	characteristic
210	#3107	C₂×C₅₃	1	1	yes	characteristic
211	#2990	C₂×C₅₃	1	1	yes	characteristic
212	#2986	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#1591	C₂×C₅₃	1	2	yes	complemented
214	#1587	C₂×C₅₃	1	2	yes	complemented
215	#1585	C₂×C₅₃	1	2	yes	complemented
216	C₈×SD₁₆	C₂×C₅₃	1	2	yes	complemented
217	#1574	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#1493	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#1476	C₂×C₅₃	1	1	yes	characteristic
220	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₂×C₅₃	—	4	1	no	—
222	C₂×C₅₃	—	4	2	no	—
223	C₂×C₅₃	#2933	1	1	yes	characteristic
224	C₂×C₅₃	#2947	1	1	yes	characteristic
225	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
226	#533	C₂²×C₅₃	1	1	yes	characteristic
227	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
228	#485	C₄×C₅₃	1	2	yes	—
229	#475	C₄×C₅₃	1	1	yes	characteristic
230	#474	C₄×C₅₃	1	1	yes	characteristic
231	#462	C₄×C₅₃	1	1	yes	characteristic
232	#461	C₄×C₅₃	1	1	yes	characteristic
233	#452	C₄×C₅₃	1	2	yes	—
234	#446	—	2	1	no	—
235	#446	C₂²×C₅₃	1	2	yes	—
236	#446	C₂²×C₅₃	1	2	yes	—
237	C₈×Q₈	—	2	1	no	—
238	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
239	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
240	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
241	#435	—	2	2	no	—
242	C₈×D₄	C₂²×C₅₃	1	2	yes	—
243	#433	—	2	2	no	—
244	#432	—	2	1	no	—
245	#431	C₂²×C₅₃	1	1	yes	characteristic
246	#431	C₂²×C₅₃	1	1	yes	characteristic
247	#431	C₂²×C₅₃	1	1	yes	characteristic
248	#423	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₈	—	2	1	no	—
250	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
251	#335	C₂²×C₅₃	1	2	yes	—
252	#334	C₂²×C₅₃	1	2	yes	—
253	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
254	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
255	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
256	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
257	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
258	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
259	#322	C₂²×C₅₃	1	2	yes	—
260	C₈²	C₂²×C₅₃	1	2	yes	—
261	C₅₃	#18016	1	1	yes	characteristic, direct product factor, Sylow
262	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
263	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
264	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
265	C₂×M₁₆	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	2	no	—
268	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
269	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
270	#125	—	2	1	no	—
271	C₂²⋊Q₈	—	2	1	no	—
272	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
273	C₄×Q₈	—	2	1	no	—
274	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
275	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
280	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
281	#117	—	2	1	no	—
282	#117	—	2	1	no	—
283	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
286	#107	C₂×C₄×C₅₃	1	2	yes	—
287	#106	C₂×C₄×C₅₃	1	2	yes	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
293	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
294	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
295	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
296	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
297	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
299	#103	C₂×C₄×C₅₃	1	2	yes	—
300	#103	C₂×C₄×C₅₃	1	2	yes	—
301	#103	C₂×C₄×C₅₃	1	2	yes	—
302	#102	C₂×C₄×C₅₃	1	2	yes	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	2	2	no	—
305	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
306	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
307	#97	—	2	2	no	—
308	#97	—	2	2	no	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₅₃	1	2	yes	—
315	C₄×C₈	C₂³×C₅₃	1	2	yes	—
316	#41	—	2	2	no	—
317	#41	—	4	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
320	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
328	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
329	Q₁₆	—	4	2	no	—
330	SD₁₆	—	2	4	no	—
331	M₁₆	—	8	1	no	—
332	C₂×C₈	—	2	4	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	8	2	no	—
339	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
340	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
342	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
351	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
352	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
353	C₂²⋊C₄	—	2	2	no	—
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	2	1	no	—
356	C₂²⋊C₄	—	2	1	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
362	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
363	C₄²	#41×C₅₃	1	1	yes	characteristic
364	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
365	C₂³	—	2	1	no	—
366	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
367	Q₈	—	2	2	no	—
368	Q₈	—	2	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	2	2	no	—
371	Q₈	—	4	2	no	—
372	D₄	—	2	1	no	—
373	D₄	—	4	2	no	—
374	D₄	#131×C₅₃	1	2	yes	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
395	C₈	—	2	2	no	—
396	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	1	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	1	no	—
410	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	1	no	—
415	C₄	—	4	1	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	2	no	—
419	C₄	C₅₃×#443	1	2	yes	—
420	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
421	C₄	C₅₃×#568	1	1	yes	characteristic
422	C₄	C₅₃×#577	1	1	yes	characteristic
423	C₄	C₅₃×#578	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	4	2	no	—
426	C₂	C₅₃×#2933	1	1	yes	characteristic
427	C₂	C₅₃×#2947	1	1	yes	characteristic
428	C₂	C₂×C₅₃×#443	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	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2986	C₂	semidirect
4. C₅₃×#1591	C₂	semidirect
5. C₅₃×#1587	C₂	semidirect
6. C₅₃×#1585	C₂	semidirect
7. C₈×SD₁₆×C₅₃	C₂	semidirect
8. C₅₃×#1574	C₂	semidirect
9. C₅₃×#1493	C₂	semidirect
11. C₅₃×#1450	C₂	semidirect
189. #18016	C₅₃	direct (N × H)
212. #2986	C₂×C₅₃	semidirect
213. #1591	C₂×C₅₃	semidirect
214. #1587	C₂×C₅₃	semidirect
215. #1585	C₂×C₅₃	semidirect
216. C₈×SD₁₆	C₂×C₅₃	semidirect
217. #1574	C₂×C₅₃	semidirect
218. #1493	C₂×C₅₃	semidirect
220. #1450	C₂×C₅₃	semidirect
261. C₅₃	#18016	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×#3107	2
C₅₃×#2990	2
C₅₃×#2986	2
C₅₃×#1591	2
C₅₃×#1587	2
C₅₃×#1585	2
C₈×SD₁₆×C₅₃	2
C₅₃×#1574	2
C₅₃×#1493	2
C₅₃×#1476	2
C₅₃×#1450	2
#18016	53

Subgroups — C53×#18016 #27225528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18016 #27225528