Subgroups — C₅₃×#18054 #27225566

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2974	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1589	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1586	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1490	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
11	C₄×D₈×C₅₃	C₄	1	2	yes	—
12	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
13	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
14	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×D₈×C₅₃	C₂²	1	2	yes	—
16	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#436	C₂²	1	4	yes	complemented
18	C₅₃×#435	—	2	4	no	—
19	C₈×D₄×C₅₃	C₂²	1	4	yes	complemented
20	C₅₃×#432	C₂²	1	2	yes	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
23	C₂×#102×C₅₃	C₄	1	2	yes	—
24	C₅₃×#407	—	2	2	no	—
25	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
26	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#335	C₂²	1	2	yes	complemented
29	C₅₃×#334	C₂²	1	2	yes	complemented
30	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
31	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
32	C₅₃×#322	C₂²	1	2	yes	complemented
33	C₈²×C₅₃	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₅₃	—	2	2	no	—
36	C₂×D₈×C₅₃	—	2	4	no	—
37	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	4	no	—
41	C₂×M₁₆×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	—	2	4	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×D₄×C₅₃	—	2	2	no	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₅₃	C₂³	1	2	yes	—
50	C₄×D₄×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#107×C₅₃	C₂×C₄	1	2	yes	—
55	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
56	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
58	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
59	#102×C₅₃	C₂×C₄	1	2	yes	—
60	#102×C₅₃	C₂×C₄	1	2	yes	—
61	#102×C₅₃	C₂×C₄	1	2	yes	—
62	#102×C₅₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₅₃	—	2	4	no	—
64	C₂²⋊C₈×C₅₃	—	2	4	no	—
65	C₂²⋊C₈×C₅₃	—	2	2	no	—
66	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
68	#97×C₅₃	—	2	2	no	—
69	#97×C₅₃	—	2	1	no	—
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	C₂⁴×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	2	4	no	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	—	4	2	no	—
80	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	—	8	4	no	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
91	D₈×C₅₃	—	2	4	no	—
92	D₈×C₅₃	—	2	2	no	—
93	D₈×C₅₃	M₁₆	1	4	yes	complemented
94	M₁₆×C₅₃	—	2	4	no	—
95	M₁₆×C₅₃	—	8	4	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	4	1	no	—
100	C₂×C₈×C₅₃	—	8	4	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²⋊C₄×C₅₃	—	2	2	no	—
111	C₂²⋊C₄×C₅₃	—	2	2	no	—
112	C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₂²⋊C₄×C₅₃	—	2	2	no	—
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	#41	1	1	yes	characteristic
117	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	—	2	2	no	—
120	C₂³×C₅₃	—	4	4	no	—
121	C₂³×C₅₃	—	4	4	no	—
122	C₂³×C₅₃	—	8	2	no	—
123	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
124	D₄×C₅₃	—	2	2	no	—
125	D₄×C₅₃	—	2	2	no	—
126	D₄×C₅₃	—	2	2	no	—
127	D₄×C₅₃	—	4	4	no	—
128	D₄×C₅₃	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	4	4	no	—
136	C₂×C₄×C₅₃	—	4	4	no	—
137	C₂×C₄×C₅₃	—	8	2	no	—
138	C₂×C₄×C₅₃	—	8	2	no	—
139	C₂×C₄×C₅₃	—	8	2	no	—
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₃	—	2	4	no	—
147	C₈×C₅₃	—	2	1	no	—
148	C₈×C₅₃	—	2	1	no	—
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	—	2	2	no	—
151	C₈×C₅₃	—	2	2	no	—
152	C₈×C₅₃	—	4	4	no	—
153	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
154	#18054	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	4	no	—
158	C₂²×C₅₃	—	4	2	no	—
159	C₂²×C₅₃	—	4	2	no	—
160	C₂²×C₅₃	—	4	2	no	—
161	C₂²×C₅₃	—	4	2	no	—
162	C₂²×C₅₃	—	4	2	no	—
163	C₂²×C₅₃	—	8	8	no	—
164	C₂²×C₅₃	#443	1	2	yes	—
165	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
173	C₄×C₅₃	#568	1	1	yes	characteristic
174	C₄×C₅₃	#576	1	2	yes	—
175	#2974	C₂×C₅₃	1	2	yes	complemented
176	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
177	#1589	C₂×C₅₃	1	2	yes	complemented
178	#1586	C₂×C₅₃	1	4	yes	complemented
179	C₈×D₈	C₂×C₅₃	1	2	yes	complemented
180	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
181	#1490	C₂×C₅₃	1	2	yes	complemented
182	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
183	C₂×C₅₃	—	2	2	no	—
184	C₂×C₅₃	—	4	4	no	—
185	C₂×C₅₃	—	8	2	no	—
186	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
187	C₂×C₅₃	#2948	1	1	yes	characteristic
188	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
189	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
191	C₄×D₈	C₄×C₅₃	1	2	yes	—
192	C₄×D₈	C₄×C₅₃	1	1	yes	characteristic
193	C₄×D₈	C₄×C₅₃	1	1	yes	characteristic
194	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
195	C₄×D₈	C₂²×C₅₃	1	2	yes	—
196	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
197	#436	C₂²×C₅₃	1	4	yes	complemented
198	#435	—	2	4	no	—
199	C₈×D₄	C₂²×C₅₃	1	4	yes	complemented
200	#432	C₂²×C₅₃	1	2	yes	—
201	#431	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
203	C₂×#102	C₄×C₅₃	1	2	yes	—
204	#407	—	2	2	no	—
205	C₂×C₂²⋊C₈	—	2	2	no	—
206	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
208	#335	C₂²×C₅₃	1	2	yes	complemented
209	#334	C₂²×C₅₃	1	2	yes	complemented
210	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
211	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
212	#322	C₂²×C₅₃	1	2	yes	complemented
213	C₈²	C₂²×C₅₃	1	2	yes	complemented
214	C₅₃	#18054	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
216	C₂³×C₄	—	2	2	no	—
217	C₂×D₈	—	2	4	no	—
218	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
219	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
220	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	2	4	no	—
223	C₂²×C₈	—	2	4	no	—
224	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
229	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
230	C₄×D₄	C₂³×C₅₃	1	2	yes	—
231	C₄×D₄	C₂³×C₅₃	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
235	#107	C₂×C₄×C₅₃	1	2	yes	—
236	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
237	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
239	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
240	#102	C₂×C₄×C₅₃	1	2	yes	—
241	#102	C₂×C₄×C₅₃	1	2	yes	—
242	#102	C₂×C₄×C₅₃	1	2	yes	—
243	#102	C₂×C₄×C₅₃	1	2	yes	—
244	C₂²⋊C₈	—	2	2	no	—
245	C₂²⋊C₈	—	2	4	no	—
246	C₂²⋊C₈	—	2	4	no	—
247	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
248	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
249	#97	—	2	2	no	—
250	#97	—	2	1	no	—
251	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
252	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₅₃	1	2	yes	—
254	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₅₃	1	2	yes	—
257	C₂⁴	—	2	2	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
262	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	8	4	no	—
269	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
272	D₈	—	2	4	no	—
273	D₈	—	2	2	no	—
274	D₈	M₁₆×C₅₃	1	4	yes	complemented
275	M₁₆	—	2	4	no	—
276	M₁₆	—	8	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	8	4	no	—
282	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
283	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
284	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
287	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
290	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
296	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
297	C₄²	#41×C₅₃	1	1	yes	characteristic
298	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	4	no	—
303	C₂³	—	8	2	no	—
304	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
305	D₄	—	2	2	no	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	4	4	no	—
309	D₄	C₂×M₁₆×C₅₃	1	4	yes	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
322	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
323	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	4	no	—
334	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	4	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	8	no	—
344	C₂²	C₅₃×#443	1	2	yes	—
345	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
353	C₄	C₅₃×#568	1	1	yes	characteristic
354	C₄	C₅₃×#576	1	2	yes	—
355	C₂	—	2	2	no	—
356	C₂	—	4	4	no	—
357	C₂	—	8	2	no	—
358	C₂	D₄×M₁₆×C₅₃	1	1	yes	characteristic
359	C₂	C₅₃×#2948	1	1	yes	characteristic
360	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	C₂²×D₈×C₅₃	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×#2974	C₂	semidirect
3. C₅₃×#1589	C₂	semidirect
4. C₅₃×#1586	C₂	semidirect
5. C₈×D₈×C₅₃	C₂	semidirect
6. C₅₃×#1568	C₂	semidirect
7. C₅₃×#1490	C₂	semidirect
8. C₈×M₁₆×C₅₃	C₂	semidirect
17. C₅₃×#436	C₂²	semidirect
19. C₈×D₄×C₅₃	C₂²	semidirect
28. C₅₃×#335	C₂²	semidirect
29. C₅₃×#334	C₂²	semidirect
30. C₅₃×D₄⋊C₈	C₂²	semidirect
31. C₅₃×D₄⋊C₈	C₂²	semidirect
32. C₅₃×#322	C₂²	semidirect
33. C₈²×C₅₃	C₂²	semidirect
93. D₈×C₅₃	M₁₆	semidirect
154. #18054	C₅₃	direct (N × H)
175. #2974	C₂×C₅₃	semidirect
177. #1589	C₂×C₅₃	semidirect
178. #1586	C₂×C₅₃	semidirect
179. C₈×D₈	C₂×C₅₃	semidirect
180. #1568	C₂×C₅₃	semidirect
181. #1490	C₂×C₅₃	semidirect
182. C₈×M₁₆	C₂×C₅₃	semidirect
197. #436	C₂²×C₅₃	semidirect
199. C₈×D₄	C₂²×C₅₃	semidirect
208. #335	C₂²×C₅₃	semidirect
209. #334	C₂²×C₅₃	semidirect
210. D₄⋊C₈	C₂²×C₅₃	semidirect
211. D₄⋊C₈	C₂²×C₅₃	semidirect
212. #322	C₂²×C₅₃	semidirect
213. C₈²	C₂²×C₅₃	semidirect
214. C₅₃	#18054	direct (N × H)
274. D₈	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×#2974	2
C₂×C₄×D₈×C₅₃	2
C₅₃×#1589	2
C₅₃×#1586	2
C₈×D₈×C₅₃	2
C₅₃×#1568	2
C₅₃×#1490	2
C₈×M₁₆×C₅₃	2
#18054	53

Subgroups — C53×#18054 #27225566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18054 #27225566