Subgroups — C₅₃×#17978 #27225490

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2966	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1584	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1582	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1581	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1494	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#529	C₂²	1	2	yes	—
11	C₅₃×#499	C₄	1	1	yes	characteristic
12	C₅₃×#495	C₄	1	1	yes	characteristic
13	C₅₃×#488	C₄	1	2	yes	—
14	C₅₃×#467	C₄	1	1	yes	characteristic
15	C₅₃×#466	C₄	1	1	yes	characteristic
16	C₅₃×#465	C₄	1	2	yes	—
17	C₅₃×#446	C₂²	1	2	yes	—
18	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
21	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#436	C₂²	1	2	yes	—
23	C₅₃×#435	—	2	2	no	—
24	C₈×D₄×C₅₃	—	2	2	no	—
25	C₈×D₄×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#433	—	2	2	no	—
27	C₅₃×#432	C₂²	1	2	yes	—
28	C₅₃×#405	—	2	2	no	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	1	yes	characteristic
34	C₅₃×#334	C₂²	1	1	yes	characteristic
35	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
38	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
39	C₅₃×#322	C₂²	1	2	yes	—
40	C₅₃×#322	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	2	yes	—
50	#126×C₅₃	—	2	2	no	—
51	#125×C₅₃	C₂×C₄	1	2	yes	—
52	#124×C₅₃	C₂×C₄	1	2	yes	—
53	#123×C₅₃	—	2	2	no	—
54	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
55	#121×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	C₂³	1	2	yes	—
59	#117×C₅₃	—	2	2	no	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
63	#106×C₅₃	C₂×C₄	1	2	yes	—
64	C₄⋊C₈×C₅₃	—	2	2	no	—
65	C₄⋊C₈×C₅₃	—	2	2	no	—
66	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	2	yes	—
69	#103×C₅₃	C₂×C₄	1	2	yes	—
70	#102×C₅₃	C₂×C₄	1	2	yes	—
71	#102×C₅₃	C₂×C₄	1	2	yes	—
72	C₂²⋊C₈×C₅₃	—	2	2	no	—
73	C₂²⋊C₈×C₅₃	—	2	2	no	—
74	C₂²⋊C₈×C₅₃	—	2	2	no	—
75	#97×C₅₃	—	2	2	no	—
76	#97×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	—	2	2	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	D₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂³	1	2	yes	—
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
94	Q₁₆×C₅₃	—	4	1	no	—
95	SD₁₆×C₅₃	—	4	2	no	—
96	D₈×C₅₃	—	4	1	no	—
97	M₁₆×C₅₃	—	4	4	no	—
98	C₂×C₈×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	4	1	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₂×C₈×C₅₃	—	4	2	no	—
105	C₂×C₈×C₅₃	—	4	4	no	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
117	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₂²⋊C₄×C₅₃	—	2	2	no	—
120	C₂²⋊C₄×C₅₃	—	2	2	no	—
121	C₂²⋊C₄×C₅₃	—	2	2	no	—
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅₃	#41	1	1	yes	characteristic
127	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂³×C₅₃	—	2	2	no	—
129	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	2	2	no	—
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	—	2	2	no	—
134	D₄×C₅₃	—	4	4	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	—	4	2	no	—
158	C₈×C₅₃	#135	1	4	yes	—
159	#17978	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	4	2	no	—
163	C₂²×C₅₃	—	4	2	no	—
164	C₂²×C₅₃	—	4	2	no	—
165	C₂²×C₅₃	#444	1	2	yes	—
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	#568	1	1	yes	characteristic
175	C₄×C₅₃	#568	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#135	1	2	yes	—
177	#3168	C₂×C₅₃	1	1	yes	characteristic
178	#2966	C₂×C₅₃	1	2	yes	complemented
179	#1584	C₂×C₅₃	1	2	yes	complemented
180	#1583	C₂×C₅₃	1	2	yes	complemented
181	#1582	C₂×C₅₃	1	2	yes	complemented
182	#1581	C₂×C₅₃	1	2	yes	complemented
183	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
184	#1494	C₂×C₅₃	1	2	yes	complemented
185	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
186	C₂×C₅₃	—	2	2	no	—
187	C₂×C₅₃	—	8	2	no	—
188	C₂×C₅₃	#2932	1	1	yes	characteristic
189	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
191	#529	C₂²×C₅₃	1	2	yes	—
192	#499	C₄×C₅₃	1	1	yes	characteristic
193	#495	C₄×C₅₃	1	1	yes	characteristic
194	#488	C₄×C₅₃	1	2	yes	—
195	#467	C₄×C₅₃	1	1	yes	characteristic
196	#466	C₄×C₅₃	1	1	yes	characteristic
197	#465	C₄×C₅₃	1	2	yes	—
198	#446	C₂²×C₅₃	1	2	yes	—
199	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
200	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
201	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
202	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
203	#436	C₂²×C₅₃	1	2	yes	—
204	#435	—	2	2	no	—
205	C₈×D₄	—	2	2	no	—
206	C₈×D₄	C₂²×C₅₃	1	2	yes	—
207	#433	—	2	2	no	—
208	#432	C₂²×C₅₃	1	2	yes	—
209	#405	—	2	2	no	—
210	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
213	#335	C₂²×C₅₃	1	2	yes	—
214	#334	C₂²×C₅₃	1	1	yes	characteristic
215	#334	C₂²×C₅₃	1	1	yes	characteristic
216	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
217	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
218	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
219	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
220	#322	C₂²×C₅₃	1	2	yes	—
221	#322	C₂²×C₅₃	1	2	yes	—
222	C₅₃	#17978	1	1	yes	characteristic, direct product factor, Sylow
223	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
224	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
225	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×M₁₆	—	2	2	no	—
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₈	D₄×C₅₃	1	2	yes	—
232	#126	—	2	2	no	—
233	#125	C₂×C₄×C₅₃	1	2	yes	—
234	#124	C₂×C₄×C₅₃	1	2	yes	—
235	#123	—	2	2	no	—
236	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
237	#121	C₂×C₄×C₅₃	1	2	yes	—
238	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	C₂³×C₅₃	1	2	yes	—
241	#117	—	2	2	no	—
242	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
243	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
244	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
245	#106	C₂×C₄×C₅₃	1	2	yes	—
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
249	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
250	#103	C₂×C₄×C₅₃	1	2	yes	—
251	#103	C₂×C₄×C₅₃	1	2	yes	—
252	#102	C₂×C₄×C₅₃	1	2	yes	—
253	#102	C₂×C₄×C₅₃	1	2	yes	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	#97	—	2	2	no	—
258	#97	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
262	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
263	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
264	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
265	C₄×C₈	C₂³×C₅₃	1	2	yes	—
266	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
268	C₄×C₈	C₂³×C₅₃	1	2	yes	—
269	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
276	Q₁₆	—	4	1	no	—
277	SD₁₆	—	4	2	no	—
278	D₈	—	4	1	no	—
279	M₁₆	—	4	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	4	4	no	—
288	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
293	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
299	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₄²	#41×C₅₃	1	1	yes	characteristic
309	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
328	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	#135×C₅₃	1	4	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₅₃×#444	1	2	yes	—
347	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	C₅₃×#568	1	1	yes	characteristic
356	C₄	C₅₃×#568	1	1	yes	characteristic
357	C₄	C₂×#135×C₅₃	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	8	2	no	—
360	C₂	C₅₃×#2932	1	1	yes	characteristic
361	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
362	C₂	C₂×C₅₃×#444	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	C₅₃×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2966	C₂	semidirect
3. C₅₃×#1584	C₂	semidirect
4. C₅₃×#1583	C₂	semidirect
5. C₅₃×#1582	C₂	semidirect
6. C₅₃×#1581	C₂	semidirect
7. C₅₃×#1568	C₂	semidirect
8. C₅₃×#1494	C₂	semidirect
9. C₂×C₅₃×#322	C₂	semidirect
159. #17978	C₅₃	direct (N × H)
178. #2966	C₂×C₅₃	semidirect
179. #1584	C₂×C₅₃	semidirect
180. #1583	C₂×C₅₃	semidirect
181. #1582	C₂×C₅₃	semidirect
182. #1581	C₂×C₅₃	semidirect
183. #1568	C₂×C₅₃	semidirect
184. #1494	C₂×C₅₃	semidirect
185. C₂×#322	C₂×C₅₃	semidirect
222. C₅₃	#17978	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×#3168	2
C₅₃×#2966	2
C₅₃×#1584	2
C₅₃×#1583	2
C₅₃×#1582	2
C₅₃×#1581	2
C₅₃×#1568	2
C₅₃×#1494	2
C₂×C₅₃×#322	2
#17978	53

Subgroups — C53×#17978 #27225490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17978 #27225490