Subgroups — C₅₃×#17896 #27225408

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1592	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1590	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	2	no	—
14	C₅₃×#446	C₂²	1	4	yes	complemented
15	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
16	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
17	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
18	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#436	—	2	2	no	—
20	C₅₃×#436	C₂²	1	4	yes	—
21	C₅₃×#432	—	2	2	no	—
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₅₃×#431	C₂²	1	1	yes	characteristic
25	C₅₃×#431	C₂²	1	1	yes	characteristic
26	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
27	C₂×#103×C₅₃	C₄	1	1	yes	characteristic
28	C₂×#102×C₅₃	C₄	1	1	yes	characteristic
29	C₅₃×#407	—	2	2	no	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#335	C₂²	1	2	yes	complemented
32	C₅₃×#334	C₂²	1	2	yes	complemented
33	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
34	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
35	C₅₃×#322	C₂²	1	2	yes	complemented
36	C₈²×C₅₃	C₂²	1	2	yes	complemented
37	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	—	2	1	no	—
40	C₂×SD₁₆×C₅₃	—	2	4	no	—
41	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₅₃	—	2	4	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₅₃	—	2	1	no	—
47	C₄×Q₈×C₅₃	—	2	1	no	—
48	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
50	C₄×D₄×C₅₃	—	2	1	no	—
51	C₄×D₄×C₅₃	—	2	1	no	—
52	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	—	2	1	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#106×C₅₃	C₂×C₄	1	2	yes	—
61	#106×C₅₃	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₅₃	—	2	4	no	—
63	C₄⋊C₈×C₅₃	—	2	1	no	—
64	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
65	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
66	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
67	#103×C₅₃	C₂×C₄	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	2	yes	—
69	#102×C₅₃	C₂×C₄	1	2	yes	—
70	#102×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₅₃	—	2	4	no	—
72	C₂²⋊C₈×C₅₃	—	2	1	no	—
73	C₂²⋊C₈×C₅₃	—	2	1	no	—
74	C₂²⋊C₈×C₅₃	—	2	1	no	—
75	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
76	#97×C₅₃	—	2	4	no	—
77	C₄×C₈×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	—	2	1	no	—
79	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂³	1	2	yes	—
82	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	2	yes	—
85	C₂⁴×C₅₃	—	2	1	no	—
86	C₂×Q₈×C₅₃	—	2	1	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₅₃	—	2	1	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
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	2	no	—
99	C₂²×C₄×C₅₃	—	8	2	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
103	SD₁₆×C₅₃	—	2	4	no	—
104	SD₁₆×C₅₃	—	2	4	no	—
105	M₁₆×C₅₃	—	8	4	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
111	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	1	no	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
121	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅₃	#41	1	1	yes	characteristic
129	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂³×C₅₃	—	2	2	no	—
131	C₂³×C₅₃	—	4	2	no	—
132	C₂³×C₅₃	—	8	1	no	—
133	C₂³×C₅₃	—	8	1	no	—
134	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	—	2	2	no	—
137	Q₈×C₅₃	—	4	2	no	—
138	D₄×C₅₃	—	2	2	no	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	4	no	—
148	C₂×C₄×C₅₃	—	8	2	no	—
149	C₂×C₄×C₅₃	—	8	1	no	—
150	C₂×C₄×C₅₃	—	8	1	no	—
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
155	C₂×C₄×C₅₃	#131	1	2	yes	—
156	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₅₃	—	2	1	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	4	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	—	4	2	no	—
165	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
166	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
167	#17896	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	—	4	2	no	—
173	C₂²×C₅₃	—	8	4	no	—
174	C₂²×C₅₃	#436	1	2	yes	—
175	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	#436	1	2	yes	—
183	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
184	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
185	C₄×C₅₃	#577	1	2	yes	—
186	#2983	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#2975	C₂×C₅₃	1	1	yes	characteristic
188	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
189	#1592	C₂×C₅₃	1	4	yes	complemented
190	#1590	C₂×C₅₃	1	4	yes	complemented
191	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
192	#1479	C₂×C₅₃	1	1	yes	characteristic, complemented
193	#1478	C₂×C₅₃	1	1	yes	characteristic
194	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	—	8	2	no	—
197	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
198	C₂×C₅₃	#2949	1	1	yes	characteristic
199	C₂×C₅₃	#2957	1	1	yes	characteristic
200	C₂²×SD₁₆	C₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
203	#446	—	2	2	no	—
204	#446	C₂²×C₅₃	1	4	yes	complemented
205	C₄×SD₁₆	C₄×C₅₃	1	2	yes	—
206	C₄×SD₁₆	C₄×C₅₃	1	2	yes	—
207	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
208	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
209	#436	—	2	2	no	—
210	#436	C₂²×C₅₃	1	4	yes	—
211	#432	—	2	2	no	—
212	#431	C₂²×C₅₃	1	1	yes	characteristic
213	#431	C₂²×C₅₃	1	1	yes	characteristic
214	#431	C₂²×C₅₃	1	1	yes	characteristic
215	#431	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×#106	C₄×C₅₃	1	1	yes	characteristic
217	C₂×#103	C₄×C₅₃	1	1	yes	characteristic
218	C₂×#102	C₄×C₅₃	1	1	yes	characteristic
219	#407	—	2	2	no	—
220	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
221	#335	C₂²×C₅₃	1	2	yes	complemented
222	#334	C₂²×C₅₃	1	2	yes	complemented
223	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
224	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
225	#322	C₂²×C₅₃	1	2	yes	complemented
226	C₈²	C₂²×C₅₃	1	2	yes	complemented
227	C₅₃	#17896	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₂³×C₄	—	2	1	no	—
231	C₂×SD₁₆	—	2	4	no	—
232	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
233	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
234	C₂×M₁₆	—	2	4	no	—
235	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
240	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
244	C₄×D₄	C₂³×C₅₃	1	2	yes	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₄	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
251	#106	C₂×C₄×C₅₃	1	2	yes	—
252	#106	C₂×C₄×C₅₃	1	2	yes	—
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
256	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
257	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
258	#103	C₂×C₄×C₅₃	1	2	yes	—
259	#103	C₂×C₄×C₅₃	1	2	yes	—
260	#102	C₂×C₄×C₅₃	1	2	yes	—
261	#102	C₂×C₄×C₅₃	1	2	yes	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	2	1	no	—
264	C₂²⋊C₈	—	2	1	no	—
265	C₂²⋊C₈	—	2	4	no	—
266	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
267	#97	—	2	4	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
271	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
272	C₄×C₈	C₂³×C₅₃	1	2	yes	—
273	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₅₃	1	2	yes	—
276	C₂⁴	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	8	2	no	—
291	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
293	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
294	SD₁₆	—	2	4	no	—
295	SD₁₆	—	2	4	no	—
296	M₁₆	—	8	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	M₁₆×C₅₃	1	4	yes	complemented
302	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
305	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
312	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₄²	#41×C₅₃	1	1	yes	characteristic
320	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	8	1	no	—
324	C₂³	—	8	1	no	—
325	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
326	Q₈	—	2	2	no	—
327	Q₈	—	2	2	no	—
328	Q₈	—	4	2	no	—
329	D₄	—	2	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	4	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
343	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
344	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
346	C₂×C₄	#131×C₅₃	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	—	4	2	no	—
356	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
357	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	4	no	—
364	C₂²	C₅₃×#436	1	2	yes	—
365	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	C₅₃×#436	1	2	yes	—
373	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
374	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
375	C₄	C₅₃×#577	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	8	2	no	—
378	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
379	C₂	C₅₃×#2949	1	1	yes	characteristic
380	C₂	C₅₃×#2957	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 ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2983	C₂	semidirect
4. C₅₃×#1592	C₂	semidirect
5. C₅₃×#1590	C₂	semidirect
6. C₅₃×#1564	C₂	semidirect
7. C₅₃×#1479	C₂	semidirect
9. C₅₃×#1450	C₂	semidirect
14. C₅₃×#446	C₂²	semidirect
31. C₅₃×#335	C₂²	semidirect
32. C₅₃×#334	C₂²	semidirect
33. C₅₃×Q₈⋊C₈	C₂²	semidirect
35. C₅₃×#322	C₂²	semidirect
36. C₈²×C₅₃	C₂²	semidirect
110. C₂×C₈×C₅₃	M₁₆	semidirect
167. #17896	C₅₃	direct (N × H)
186. #2983	C₂×C₅₃	semidirect
189. #1592	C₂×C₅₃	semidirect
190. #1590	C₂×C₅₃	semidirect
191. #1564	C₂×C₅₃	semidirect
192. #1479	C₂×C₅₃	semidirect
194. #1450	C₂×C₅₃	semidirect
204. #446	C₂²×C₅₃	semidirect
221. #335	C₂²×C₅₃	semidirect
222. #334	C₂²×C₅₃	semidirect
223. Q₈⋊C₈	C₂²×C₅₃	semidirect
225. #322	C₂²×C₅₃	semidirect
226. C₈²	C₂²×C₅₃	semidirect
227. C₅₃	#17896	direct (N × H)
301. C₂×C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2983	2
C₅₃×#2975	2
C₂×C₄×SD₁₆×C₅₃	2
C₅₃×#1592	2
C₅₃×#1590	2
C₅₃×#1564	2
C₅₃×#1479	2
C₅₃×#1478	2
C₅₃×#1450	2
#17896	53

Subgroups — C53×#17896 #27225408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17896 #27225408