Subgroups — C₅₃×#33455 #27240967

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2951	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2176	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2175	C₂	1	2	yes	maximal, complemented
6	C₄×Q₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₄×SD₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2138	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#520	—	2	1	no	—
15	C₅₃×#519	—	2	1	no	—
16	C₅₃×#510	C₄	1	8	yes	complemented
17	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
18	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#441	—	2	2	no	—
22	C₅₃×#440	—	2	2	no	—
23	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
26	C₄×D₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#427	C₂²	1	1	yes	characteristic
28	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#419	—	2	1	no	—
30	C₅₃×#417	—	2	1	no	—
31	C₅₃×#405	C₂²	1	1	yes	characteristic
32	C₅₃×#367	C₂²	1	2	yes	—
33	C₅₃×#366	C₂²	1	2	yes	—
34	C₅₃×#358	C₂²	1	2	yes	—
35	C₅₃×#358	C₂²	1	1	yes	characteristic
36	C₅₃×#358	C₂²	1	2	yes	—
37	C₅₃×#358	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	1	yes	characteristic
39	C₅₃×#357	C₂²	1	1	yes	characteristic
40	C₅₃×#346	C₂²	1	2	yes	—
41	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
42	C₂×#41×C₅₃	—	2	1	no	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	#135×C₅₃	—	2	2	no	—
45	#135×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×Q₁₆×C₅₃	—	2	1	no	—
47	C₂×Q₁₆×C₅₃	—	2	2	no	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×SD₁₆×C₅₃	—	2	1	no	—
52	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	—	2	2	no	—
58	C₄×Q₈×C₅₃	—	4	1	no	—
59	C₄×Q₈×C₅₃	—	4	1	no	—
60	C₄×D₄×C₅₃	—	2	2	no	—
61	C₄×D₄×C₅₃	—	4	2	no	—
62	#117×C₅₃	—	2	1	no	—
63	#117×C₅₃	—	4	1	no	—
64	#117×C₅₃	—	4	2	no	—
65	#117×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
68	Q₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
69	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
70	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
71	C₂×C₁₆×C₅₃	—	2	2	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#107×C₅₃	C₂³	1	1	yes	characteristic
78	#107×C₅₃	C₂³	1	2	yes	—
79	#106×C₅₃	—	2	1	no	—
80	#103×C₅₃	—	2	2	no	—
81	#103×C₅₃	—	2	2	no	—
82	#103×C₅₃	—	2	2	no	—
83	#102×C₅₃	—	2	2	no	—
84	#97×C₅₃	D₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	#41×C₅₃	—	2	4	no	—
87	#41×C₅₃	—	4	2	no	—
88	C₂×Q₈×C₅₃	—	2	4	no	—
89	C₂×Q₈×C₅₃	—	2	1	no	—
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	—	4	1	no	—
92	C₂×Q₈×C₅₃	—	4	2	no	—
93	C₂×D₄×C₅₃	—	2	1	no	—
94	C₂×D₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	—	4	1	no	—
96	C₂²×C₄×C₅₃	—	4	1	no	—
97	C₂²×C₄×C₅₃	—	4	1	no	—
98	C₂²×C₄×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	Q₁₆×C₅₃	—	2	4	no	—
101	Q₁₆×C₅₃	—	2	2	no	—
102	Q₁₆×C₅₃	—	2	1	no	—
103	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
104	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	—
105	SD₁₆×C₅₃	—	2	4	no	—
106	D₈×C₅₃	—	2	1	no	—
107	D₈×C₅₃	C₂²×C₄	1	2	yes	—
108	M₁₆×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	#41	1	2	yes	—
113	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	1	no	—
116	C₄⋊C₄×C₅₃	—	2	1	no	—
117	C₄⋊C₄×C₅₃	—	4	1	no	—
118	C₄⋊C₄×C₅₃	—	4	2	no	—
119	C₄⋊C₄×C₅₃	—	4	2	no	—
120	C₄⋊C₄×C₅₃	—	4	2	no	—
121	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
122	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₅₃	—	4	1	no	—
124	C₂²⋊C₄×C₅₃	—	4	2	no	—
125	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
126	C₄²×C₅₃	—	4	1	no	—
127	C₄²×C₅₃	—	4	2	no	—
128	C₄²×C₅₃	—	4	2	no	—
129	C₄²×C₅₃	—	4	2	no	—
130	C₄²×C₅₃	C₂×D₄	1	2	yes	—
131	C₁₆×C₅₃	—	2	1	no	—
132	C₁₆×C₅₃	—	2	1	no	—
133	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
134	C₂³×C₅₃	—	4	1	no	—
135	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
136	Q₈×C₅₃	—	2	4	no	—
137	Q₈×C₅₃	—	2	2	no	—
138	Q₈×C₅₃	—	4	2	no	—
139	Q₈×C₅₃	—	4	1	no	—
140	Q₈×C₅₃	—	4	4	no	—
141	D₄×C₅₃	—	2	2	no	—
142	D₄×C₅₃	—	4	4	no	—
143	D₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	4	no	—
147	C₂×C₄×C₅₃	—	4	4	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	16	2	no	—
152	C₂×C₄×C₅₃	—	16	1	no	—
153	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#135	1	2	yes	—
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	C₄×D₄	1	2	yes	—
161	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
163	#33455	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	4	2	no	—
167	C₂²×C₅₃	—	8	1	no	—
168	C₂²×C₅₃	—	8	1	no	—
169	C₂²×C₅₃	C₄×D₈	1	2	yes	—
170	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	—	4	4	no	—
175	C₄×C₅₃	—	8	1	no	—
176	C₄×C₅₃	—	8	2	no	—
177	C₄×C₅₃	—	8	1	no	—
178	C₄×C₅₃	C₄×D₈	1	2	yes	—
179	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
181	C₂×#510	C₂×C₅₃	1	1	yes	characteristic
182	#2951	C₂×C₅₃	1	1	yes	characteristic
183	#2950	C₂×C₅₃	1	1	yes	characteristic, complemented
184	#2176	C₂×C₅₃	1	2	yes	complemented
185	#2175	C₂×C₅₃	1	2	yes	complemented
186	C₄×Q₃₂	C₂×C₅₃	1	2	yes	complemented
187	C₄×SD₃₂	C₂×C₅₃	1	2	yes	complemented
188	#2159	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#2138	C₂×C₅₃	1	1	yes	characteristic
190	C₂×#358	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#2108	C₂×C₅₃	1	1	yes	characteristic, complemented
192	C₂×C₅₃	—	2	2	no	—
193	C₂×C₅₃	—	8	2	no	—
194	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
195	C₂×C₅₃	#3416	1	1	yes	characteristic
196	C₂×C₅₃	#3417	1	1	yes	characteristic
197	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
198	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
199	#520	—	2	1	no	—
200	#519	—	2	1	no	—
201	#510	C₄×C₅₃	1	8	yes	complemented
202	C₂×Q₃₂	C₂²×C₅₃	1	2	yes	—
203	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
204	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
205	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
206	#441	—	2	2	no	—
207	#440	—	2	2	no	—
208	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
209	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
210	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
211	C₄×D₈	C₂²×C₅₃	1	2	yes	—
212	#427	C₂²×C₅₃	1	1	yes	characteristic
213	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
214	#419	—	2	1	no	—
215	#417	—	2	1	no	—
216	#405	C₂²×C₅₃	1	1	yes	characteristic
217	#367	C₂²×C₅₃	1	2	yes	—
218	#366	C₂²×C₅₃	1	2	yes	—
219	#358	C₂²×C₅₃	1	2	yes	—
220	#358	C₂²×C₅₃	1	2	yes	—
221	#358	C₂²×C₅₃	1	1	yes	characteristic
222	#358	C₂²×C₅₃	1	1	yes	characteristic
223	#357	C₂²×C₅₃	1	1	yes	characteristic
224	#357	C₂²×C₅₃	1	1	yes	characteristic
225	#346	C₂²×C₅₃	1	2	yes	—
226	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
227	C₅₃	#33455	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×#41	—	2	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	2	2	no	—
231	#135	C₂×C₄×C₅₃	1	4	yes	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
235	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
236	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
239	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	C₄×Q₈	—	4	1	no	—
245	C₄×Q₈	—	4	1	no	—
246	C₄×D₄	—	2	2	no	—
247	C₄×D₄	—	4	2	no	—
248	#117	—	2	1	no	—
249	#117	—	4	1	no	—
250	#117	—	4	2	no	—
251	#117	D₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	4	1	no	—
254	Q₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
255	SD₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
256	M₃₂	C₂×C₄×C₅₃	1	4	yes	complemented
257	C₂×C₁₆	—	2	2	no	—
258	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
259	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
260	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
261	#107	—	2	1	no	—
262	#107	C₂³×C₅₃	1	1	yes	characteristic
263	#107	C₂³×C₅₃	1	1	yes	characteristic
264	#107	C₂³×C₅₃	1	2	yes	—
265	#106	—	2	1	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	2	2	no	—
269	#102	—	2	2	no	—
270	#97	D₄×C₅₃	1	2	yes	—
271	C₄×C₈	C₂³×C₅₃	1	2	yes	—
272	#41	—	2	4	no	—
273	#41	—	4	2	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	Q₁₆	—	2	4	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	1	no	—
289	Q₁₆	C₂²×C₄×C₅₃	1	4	yes	—
290	Q₁₆	C₂²×C₄×C₅₃	1	2	yes	—
291	SD₁₆	—	2	4	no	—
292	D₈	—	2	1	no	—
293	D₈	C₂²×C₄×C₅₃	1	2	yes	—
294	M₁₆	—	2	2	no	—
295	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
296	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
297	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
298	C₂×C₈	#41×C₅₃	1	2	yes	—
299	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
308	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	4	2	no	—
315	C₄²	—	4	2	no	—
316	C₄²	C₂×D₄×C₅₃	1	2	yes	—
317	C₁₆	—	2	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	C₂²×C₄×C₅₃	1	4	yes	—
320	C₂³	—	4	1	no	—
321	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	2	4	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	1	no	—
326	Q₈	—	4	4	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	4	no	—
329	D₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	16	2	no	—
338	C₂×C₄	—	16	1	no	—
339	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
340	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
341	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
342	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
343	C₂×C₄	#135×C₅₃	1	2	yes	—
344	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	C₄×D₄×C₅₃	1	2	yes	—
347	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
348	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	C₄×D₈×C₅₃	1	2	yes	—
355	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	4	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	C₄×D₈×C₅₃	1	2	yes	—
364	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
365	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	8	2	no	—
368	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
369	C₂	C₅₃×#3416	1	1	yes	characteristic
370	C₂	C₅₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2950	C₂	semidirect
4. C₅₃×#2176	C₂	semidirect
5. C₅₃×#2175	C₂	semidirect
6. C₄×Q₃₂×C₅₃	C₂	semidirect
7. C₄×SD₃₂×C₅₃	C₂	semidirect
8. C₅₃×#2159	C₂	semidirect
10. C₂×C₅₃×#358	C₂	semidirect
11. C₅₃×#2108	C₂	semidirect
16. C₅₃×#510	C₄	semidirect
48. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
68. Q₃₂×C₅₃	C₂×C₄	semidirect
69. SD₃₂×C₅₃	C₂×C₄	semidirect
70. M₃₂×C₅₃	C₂×C₄	semidirect
163. #33455	C₅₃	direct (N × H)
183. #2950	C₂×C₅₃	semidirect
184. #2176	C₂×C₅₃	semidirect
185. #2175	C₂×C₅₃	semidirect
186. C₄×Q₃₂	C₂×C₅₃	semidirect
187. C₄×SD₃₂	C₂×C₅₃	semidirect
188. #2159	C₂×C₅₃	semidirect
190. C₂×#358	C₂×C₅₃	semidirect
191. #2108	C₂×C₅₃	semidirect
201. #510	C₄×C₅₃	semidirect
227. C₅₃	#33455	direct (N × H)
234. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
254. Q₃₂	C₂×C₄×C₅₃	semidirect
255. SD₃₂	C₂×C₄×C₅₃	semidirect
256. M₃₂	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#510	2
C₅₃×#2951	2
C₅₃×#2950	2
C₅₃×#2176	2
C₅₃×#2175	2
C₄×Q₃₂×C₅₃	2
C₄×SD₃₂×C₅₃	2
C₅₃×#2159	2
C₅₃×#2138	2
C₂×C₅₃×#358	2
C₅₃×#2108	2
#33455	53

Subgroups — C53×#33455 #27240967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33455 #27240967