Subgroups — C₅₃×#17998 #27225510

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3104	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2962	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1591	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1584	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1584	C₂	1	2	yes	maximal, complemented
7	C₈×Q₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1571	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1493	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#484	C₄	1	2	yes	—
15	C₅₃×#477	C₄	1	1	yes	characteristic
16	C₅₃×#477	C₄	1	1	yes	characteristic
17	C₅₃×#462	C₄	1	1	yes	characteristic
18	C₅₃×#462	C₄	1	1	yes	characteristic
19	C₅₃×#451	C₄	1	2	yes	—
20	C₅₃×#446	—	2	2	no	—
21	C₅₃×#446	—	2	2	no	—
22	C₅₃×#446	C₂²	1	2	yes	—
23	C₅₃×#446	C₂²	1	2	yes	—
24	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₅₃×#424	—	2	1	no	—
30	C₅₃×#423	—	2	1	no	—
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#405	—	2	1	no	—
34	C₄×M₁₆×C₅₃	—	2	1	no	—
35	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	2	yes	—
38	C₅₃×#335	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	1	yes	characteristic
42	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₅₃×#322	C₂²	1	2	yes	—
46	C₈²×C₅₃	C₂²	1	2	yes	—
47	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	#125×C₅₃	—	2	1	no	—
58	C₂²⋊Q₈×C₅₃	—	2	1	no	—
59	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₅₃	—	2	1	no	—
61	C₄×Q₈×C₅₃	—	2	1	no	—
62	C₄×Q₈×C₅₃	—	2	1	no	—
63	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
68	#117×C₅₃	—	2	1	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	#107×C₅₃	C₂×C₄	1	2	yes	—
74	#107×C₅₃	C₂×C₄	1	2	yes	—
75	C₄⋊C₈×C₅₃	—	2	2	no	—
76	C₄⋊C₈×C₅₃	—	2	2	no	—
77	C₄⋊C₈×C₅₃	—	2	2	no	—
78	C₄⋊C₈×C₅₃	—	2	2	no	—
79	C₄⋊C₈×C₅₃	—	2	1	no	—
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
86	#103×C₅₃	C₂×C₄	1	2	yes	—
87	#103×C₅₃	C₂×C₄	1	2	yes	—
88	#103×C₅₃	C₂×C₄	1	2	yes	—
89	#103×C₅₃	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
91	#97×C₅₃	—	2	2	no	—
92	#97×C₅₃	—	2	2	no	—
93	#97×C₅₃	D₄	1	2	yes	—
94	C₄×C₈×C₅₃	—	2	2	no	—
95	C₄×C₈×C₅₃	—	2	2	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	C₂×Q₈×C₅₃	—	2	1	no	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	4	1	no	—
105	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
107	C₂×Q₈×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	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
113	Q₁₆×C₅₃	—	2	4	no	—
114	Q₁₆×C₅₃	—	4	2	no	—
115	M₁₆×C₅₃	—	2	2	no	—
116	M₁₆×C₅₃	—	4	2	no	—
117	M₁₆×C₅₃	Q₁₆	1	4	yes	complemented
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	4	2	no	—
124	C₂×C₈×C₅₃	—	4	2	no	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
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	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₅₃	#41	1	1	yes	characteristic
151	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
152	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
153	Q₈×C₅₃	—	2	1	no	—
154	Q₈×C₅₃	—	2	1	no	—
155	Q₈×C₅₃	—	2	2	no	—
156	Q₈×C₅₃	—	2	1	no	—
157	Q₈×C₅₃	—	2	2	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	8	1	no	—
160	Q₈×C₅₃	#131	1	2	yes	—
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	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	4	1	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	8	1	no	—
172	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
176	C₂×C₄×C₅₃	#135	1	2	yes	—
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	—	4	1	no	—
185	C₈×C₅₃	—	4	1	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	—	4	2	no	—
188	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
189	#17998	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
193	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	4	1	no	—
201	C₄×C₅₃	—	4	1	no	—
202	C₄×C₅₃	—	4	2	no	—
203	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
204	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
205	C₄×C₅₃	#568	1	1	yes	characteristic
206	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
207	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
208	#3104	C₂×C₅₃	1	1	yes	characteristic
209	Q₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
210	#2962	C₂×C₅₃	1	1	yes	characteristic
211	#1591	C₂×C₅₃	1	2	yes	complemented
212	#1584	C₂×C₅₃	1	2	yes	complemented
213	#1584	C₂×C₅₃	1	2	yes	complemented
214	C₈×Q₁₆	C₂×C₅₃	1	2	yes	complemented
215	#1571	C₂×C₅₃	1	1	yes	characteristic
216	#1493	C₂×C₅₃	1	1	yes	characteristic
217	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic
218	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
219	C₂×C₅₃	—	2	2	no	—
220	C₂×C₅₃	#2933	1	1	yes	characteristic
221	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
222	C₂×C₅₃	#2956	1	1	yes	characteristic
223	#533	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
225	#484	C₄×C₅₃	1	2	yes	—
226	#477	C₄×C₅₃	1	1	yes	characteristic
227	#477	C₄×C₅₃	1	1	yes	characteristic
228	#462	C₄×C₅₃	1	1	yes	characteristic
229	#462	C₄×C₅₃	1	1	yes	characteristic
230	#451	C₄×C₅₃	1	2	yes	—
231	#446	—	2	2	no	—
232	#446	—	2	2	no	—
233	#446	C₂²×C₅₃	1	2	yes	—
234	#446	C₂²×C₅₃	1	2	yes	—
235	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
236	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
237	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
238	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
239	#431	C₂²×C₅₃	1	1	yes	characteristic
240	#424	—	2	1	no	—
241	#423	—	2	1	no	—
242	#423	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
244	#405	—	2	1	no	—
245	C₄×M₁₆	—	2	1	no	—
246	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
248	#335	C₂²×C₅₃	1	2	yes	—
249	#335	C₂²×C₅₃	1	2	yes	—
250	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
251	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
252	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
253	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
254	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
255	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
256	#322	C₂²×C₅₃	1	2	yes	—
257	C₈²	C₂²×C₅₃	1	2	yes	—
258	C₅₃	#17998	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
260	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
261	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
262	C₂×M₁₆	—	2	1	no	—
263	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
264	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
268	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
269	#125	—	2	1	no	—
270	C₂²⋊Q₈	—	2	1	no	—
271	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
272	C₄×Q₈	—	2	1	no	—
273	C₄×Q₈	—	2	1	no	—
274	C₄×Q₈	—	2	1	no	—
275	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
276	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
277	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
280	#117	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
283	C₂×C₄²	—	2	1	no	—
284	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
285	#107	C₂×C₄×C₅₃	1	2	yes	—
286	#107	C₂×C₄×C₅₃	1	2	yes	—
287	C₄⋊C₈	—	2	2	no	—
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	—	2	1	no	—
293	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
294	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
295	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
296	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
297	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
298	#103	C₂×C₄×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	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
303	#97	—	2	2	no	—
304	#97	—	2	2	no	—
305	#97	D₄×C₅₃	1	2	yes	—
306	C₄×C₈	—	2	2	no	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	—	2	1	no	—
309	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₅₃	1	2	yes	—
312	C₄×C₈	C₂³×C₅₃	1	2	yes	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	4	1	no	—
317	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
318	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
319	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
325	Q₁₆	—	2	4	no	—
326	Q₁₆	—	4	2	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	4	2	no	—
329	M₁₆	Q₁₆×C₅₃	1	4	yes	complemented
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
338	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
339	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
340	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	1	no	—
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	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
352	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
353	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
354	C₂²⋊C₄	—	2	2	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	1	no	—
360	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
361	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
362	C₄²	#41×C₅₃	1	1	yes	characteristic
363	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
364	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
365	Q₈	—	2	1	no	—
366	Q₈	—	2	2	no	—
367	Q₈	—	2	1	no	—
368	Q₈	—	2	1	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	8	1	no	—
372	Q₈	#131×C₅₃	1	2	yes	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
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	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	8	1	no	—
384	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
386	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
388	C₂×C₄	#135×C₅₃	1	2	yes	—
389	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
392	C₈	—	2	2	no	—
393	C₈	—	2	2	no	—
394	C₈	—	2	2	no	—
395	C₈	—	2	2	no	—
396	C₈	—	4	1	no	—
397	C₈	—	4	1	no	—
398	C₈	—	4	1	no	—
399	C₈	—	4	2	no	—
400	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
401	C₂²	—	2	1	no	—
402	C₂²	—	2	1	no	—
403	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
404	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	2	2	no	—
409	C₄	—	4	1	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	2	no	—
414	C₄	C₄×Q₁₆×C₅₃	1	2	yes	—
415	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
416	C₄	C₅₃×#568	1	1	yes	characteristic
417	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
418	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
419	C₂	—	2	2	no	—
420	C₂	C₅₃×#2933	1	1	yes	characteristic
421	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
422	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	C₂³×C₅₃	C₄×D₄
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
4. C₅₃×#1591	C₂	semidirect
5. C₅₃×#1584	C₂	semidirect
6. C₅₃×#1584	C₂	semidirect
7. C₈×Q₁₆×C₅₃	C₂	semidirect
117. M₁₆×C₅₃	Q₁₆	semidirect
189. #17998	C₅₃	direct (N × H)
211. #1591	C₂×C₅₃	semidirect
212. #1584	C₂×C₅₃	semidirect
213. #1584	C₂×C₅₃	semidirect
214. C₈×Q₁₆	C₂×C₅₃	semidirect
258. C₅₃	#17998	direct (N × H)
329. M₁₆	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3104	2
Q₈×M₁₆×C₅₃	2
C₅₃×#2962	2
C₅₃×#1591	2
C₅₃×#1584	2
C₈×Q₁₆×C₅₃	2
C₅₃×#1571	2
C₅₃×#1493	2
C₂×C₅₃×Q₈⋊C₈	2
C₈×M₁₆×C₅₃	2
#17998	53

Subgroups — C53×#17998 #27225510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17998 #27225510