Subgroups — C₅₃×#19884 #27227396

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3294	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3272	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1685	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1681	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1647	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1619	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
11	C₅₃×#547	C₂²	1	2	yes	—
12	D₄²×C₅₃	C₂²	1	2	yes	complemented
13	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅₃×#493	C₂²	1	1	yes	characteristic
15	C₅₃×#492	C₂²	1	2	yes	complemented
16	C₅₃×#486	C₂²	1	2	yes	complemented
17	C₅₃×#480	—	2	2	no	—
18	C₅₃×#474	C₂²	1	2	yes	complemented
19	C₅₃×#468	—	2	2	no	—
20	C₅₃×#466	—	2	2	no	—
21	C₅₃×#461	C₂²	1	2	yes	complemented
22	C₅₃×#459	C₂²	1	2	yes	—
23	C₅₃×#459	C₂²	1	2	yes	—
24	C₅₃×#450	—	2	2	no	—
25	C₅₃×#449	—	2	2	no	—
26	C₅₃×#447	—	2	2	no	—
27	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#436	C₂²	1	2	yes	complemented
30	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
31	C₂×#102×C₅₃	—	2	2	no	—
32	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
33	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#332	C₂²	1	2	yes	complemented
35	C₅₃×#331	C₂²	1	2	yes	complemented
36	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
37	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
38	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₅₃	—	2	2	no	—
40	C₂²×D₄×C₅₃	—	2	2	no	—
41	C₂²×D₄×C₅₃	—	2	2	no	—
42	AGL(1,ℤ/8)×C₅₃	—	4	8	no	—
43	C₂×SD₁₆×C₅₃	—	2	2	no	—
44	C₂×SD₁₆×C₅₃	—	2	2	no	—
45	C₂×SD₁₆×C₅₃	—	2	2	no	—
46	C₂×D₈×C₅₃	—	2	2	no	—
47	C₂×D₈×C₅₃	—	2	2	no	—
48	C₂×D₈×C₅₃	—	4	1	no	—
49	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	—	2	2	no	—
53	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
54	#127×C₅₃	C₂³	1	2	yes	—
55	#123×C₅₃	—	4	2	no	—
56	C₂²⋊Q₈×C₅₃	—	2	2	no	—
57	#121×C₅₃	—	2	2	no	—
58	#121×C₅₃	—	2	2	no	—
59	#121×C₅₃	—	2	2	no	—
60	#120×C₅₃	—	4	2	no	—
61	C₄×D₄×C₅₃	C₂³	1	2	yes	—
62	C₄×D₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
64	#107×C₅₃	—	2	2	no	—
65	#107×C₅₃	—	4	1	no	—
66	#107×C₅₃	D₄	1	1	yes	characteristic
67	#107×C₅₃	D₄	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
70	#103×C₅₃	—	2	2	no	—
71	#102×C₅₃	—	2	2	no	—
72	#102×C₅₃	—	2	2	no	—
73	#102×C₅₃	—	2	2	no	—
74	#102×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	2	2	no	—
76	#102×C₅₃	D₄	1	2	yes	—
77	#102×C₅₃	D₄	1	2	yes	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	C₂²⋊C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₂⁴×C₅₃	—	4	2	no	—
84	#41×C₅₃	—	4	4	no	—
85	#41×C₅₃	—	8	2	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	4	2	no	—
91	C₂×D₄×C₅₃	—	4	2	no	—
92	C₂×D₄×C₅₃	—	4	4	no	—
93	C₂×D₄×C₅₃	—	4	4	no	—
94	C₂×D₄×C₅₃	—	4	2	no	—
95	C₂×D₄×C₅₃	—	8	2	no	—
96	C₂×D₄×C₅₃	—	8	2	no	—
97	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	4	2	no	—
102	SD₁₆×C₅₃	—	4	4	no	—
103	SD₁₆×C₅₃	—	4	4	no	—
104	SD₁₆×C₅₃	—	8	2	no	—
105	D₈×C₅₃	—	4	1	no	—
106	D₈×C₅₃	—	4	4	no	—
107	D₈×C₅₃	—	8	2	no	—
108	D₈×C₅₃	—	8	2	no	—
109	D₈×C₅₃	D₈	1	4	yes	complemented
110	M₁₆×C₅₃	—	4	4	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	8	2	no	—
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	4	2	no	—
122	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
124	C₂²⋊C₄×C₅₃	—	2	2	no	—
125	C₂²⋊C₄×C₅₃	—	2	2	no	—
126	C₂²⋊C₄×C₅₃	—	4	2	no	—
127	C₂²⋊C₄×C₅₃	—	4	2	no	—
128	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂³×C₅₃	—	2	2	no	—
130	C₂³×C₅₃	—	2	2	no	—
131	C₂³×C₅₃	—	4	2	no	—
132	C₂³×C₅₃	—	8	4	no	—
133	C₂³×C₅₃	—	8	4	no	—
134	C₂³×C₅₃	—	16	2	no	—
135	Q₈×C₅₃	—	4	2	no	—
136	Q₈×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	2	2	no	—
138	D₄×C₅₃	—	2	2	no	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	—	4	2	no	—
141	D₄×C₅₃	—	4	2	no	—
142	D₄×C₅₃	—	4	2	no	—
143	D₄×C₅₃	—	4	2	no	—
144	D₄×C₅₃	—	8	2	no	—
145	D₄×C₅₃	—	8	4	no	—
146	D₄×C₅₃	—	8	4	no	—
147	D₄×C₅₃	—	8	4	no	—
148	D₄×C₅₃	C₂×D₈	1	4	yes	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	—	8	4	no	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	4	2	no	—
165	C₈×C₅₃	—	4	2	no	—
166	C₈×C₅₃	C₂×D₈	1	2	yes	—
167	#19884	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂²×C₅₃	—	2	4	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	—	4	2	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	—	8	2	no	—
175	C₂²×C₅₃	—	8	2	no	—
176	C₂²×C₅₃	—	8	2	no	—
177	C₂²×C₅₃	—	16	8	no	—
178	C₂²×C₅₃	D₄²	1	1	yes	characteristic
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	—	8	2	no	—
183	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
184	C₄×C₅₃	#575	1	1	yes	characteristic
185	C₄×C₅₃	#576	1	1	yes	characteristic
186	C₄×C₅₃	#576	1	1	yes	characteristic
187	#3294	C₂×C₅₃	1	2	yes	complemented
188	#3272	C₂×C₅₃	1	2	yes	complemented
189	#1706	C₂×C₅₃	1	1	yes	characteristic, complemented
190	#1685	C₂×C₅₃	1	2	yes	complemented
191	#1681	C₂×C₅₃	1	2	yes	complemented
192	#1647	C₂×C₅₃	1	2	yes	complemented
193	#1619	C₂×C₅₃	1	2	yes	complemented
194	#1589	C₂×C₅₃	1	1	yes	characteristic, complemented
195	C₈×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
196	C₂×C₅₃	—	4	4	no	—
197	C₂×C₅₃	—	8	2	no	—
198	C₂×C₅₃	—	16	2	no	—
199	C₂×C₅₃	#3273	1	1	yes	characteristic
200	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
201	C₂×C₅₃	#3291	1	1	yes	characteristic
202	C₂×AGL(1,ℤ/8)	—	2	2	no	—
203	#547	C₂²×C₅₃	1	2	yes	—
204	D₄²	C₂²×C₅₃	1	2	yes	complemented
205	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
206	#493	C₂²×C₅₃	1	1	yes	characteristic
207	#492	C₂²×C₅₃	1	2	yes	complemented
208	#486	C₂²×C₅₃	1	2	yes	complemented
209	#480	—	2	2	no	—
210	#474	C₂²×C₅₃	1	2	yes	complemented
211	#468	—	2	2	no	—
212	#466	—	2	2	no	—
213	#461	C₂²×C₅₃	1	2	yes	complemented
214	#459	C₂²×C₅₃	1	2	yes	—
215	#459	C₂²×C₅₃	1	2	yes	—
216	#450	—	2	2	no	—
217	#449	—	2	2	no	—
218	#447	—	2	2	no	—
219	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
220	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
221	#436	C₂²×C₅₃	1	2	yes	complemented
222	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
223	C₂×#102	—	2	2	no	—
224	#335	C₂²×C₅₃	1	1	yes	characteristic, complemented
225	#335	C₂²×C₅₃	1	1	yes	characteristic, complemented
226	#332	C₂²×C₅₃	1	2	yes	complemented
227	#331	C₂²×C₅₃	1	2	yes	complemented
228	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
229	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
230	C₈²	C₂²×C₅₃	1	1	yes	characteristic, complemented
231	C₅₃	#19884	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×#41	—	2	2	no	—
233	C₂²×D₄	—	2	2	no	—
234	C₂²×D₄	—	2	2	no	—
235	AGL(1,ℤ/8)	—	4	8	no	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	2	2	no	—
238	C₂×SD₁₆	—	2	2	no	—
239	C₂×D₈	—	2	2	no	—
240	C₂×D₈	—	2	2	no	—
241	C₂×D₈	—	4	1	no	—
242	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
243	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
244	C₂×M₁₆	—	2	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
247	#127	C₂³×C₅₃	1	2	yes	—
248	#123	—	4	2	no	—
249	C₂²⋊Q₈	—	2	2	no	—
250	#121	—	2	2	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#120	—	4	2	no	—
254	C₄×D₄	C₂³×C₅₃	1	2	yes	—
255	C₄×D₄	C₂³×C₅₃	1	2	yes	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	#107	—	2	2	no	—
258	#107	—	4	1	no	—
259	#107	D₄×C₅₃	1	1	yes	characteristic
260	#107	D₄×C₅₃	1	1	yes	characteristic
261	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
262	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
263	#103	—	2	2	no	—
264	#102	—	2	2	no	—
265	#102	—	2	2	no	—
266	#102	—	2	2	no	—
267	#102	—	2	2	no	—
268	#102	—	2	2	no	—
269	#102	D₄×C₅₃	1	2	yes	—
270	#102	D₄×C₅₃	1	2	yes	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₂²⋊C₈	—	2	2	no	—
273	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₂⁴	—	4	2	no	—
277	#41	—	4	4	no	—
278	#41	—	8	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	2	no	—
285	C₂×D₄	—	4	4	no	—
286	C₂×D₄	—	4	4	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	—	8	2	no	—
289	C₂×D₄	—	8	2	no	—
290	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
291	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	SD₁₆	—	4	4	no	—
296	SD₁₆	—	4	4	no	—
297	SD₁₆	—	8	2	no	—
298	D₈	—	4	1	no	—
299	D₈	—	4	4	no	—
300	D₈	—	8	2	no	—
301	D₈	—	8	2	no	—
302	D₈	D₈×C₅₃	1	4	yes	complemented
303	M₁₆	—	4	4	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	8	2	no	—
308	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	—	2	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	8	4	no	—
326	C₂³	—	8	4	no	—
327	C₂³	—	16	2	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	2	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	8	2	no	—
338	D₄	—	8	4	no	—
339	D₄	—	8	4	no	—
340	D₄	—	8	4	no	—
341	D₄	C₂×D₈×C₅₃	1	4	yes	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	4	2	no	—
359	C₈	C₂×D₈×C₅₃	1	2	yes	—
360	C₂²	—	2	4	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	16	8	no	—
370	C₂²	D₄²×C₅₃	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	4	2	no	—
373	C₄	—	4	2	no	—
374	C₄	—	8	2	no	—
375	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
376	C₄	C₅₃×#575	1	1	yes	characteristic
377	C₄	C₅₃×#576	1	1	yes	characteristic
378	C₄	C₅₃×#576	1	1	yes	characteristic
379	C₂	—	4	4	no	—
380	C₂	—	8	2	no	—
381	C₂	—	16	2	no	—
382	C₂	C₅₃×#3273	1	1	yes	characteristic
383	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic
384	C₂	C₅₃×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3294	C₂	semidirect
2. C₅₃×#3272	C₂	semidirect
3. C₅₃×#1706	C₂	semidirect
4. C₅₃×#1685	C₂	semidirect
5. C₅₃×#1681	C₂	semidirect
6. C₅₃×#1647	C₂	semidirect
7. C₅₃×#1619	C₂	semidirect
8. C₅₃×#1589	C₂	semidirect
9. C₈×D₈×C₅₃	C₂	semidirect
12. D₄²×C₅₃	C₂²	semidirect
13. C₅₃×#500	C₂²	semidirect
15. C₅₃×#492	C₂²	semidirect
16. C₅₃×#486	C₂²	semidirect
18. C₅₃×#474	C₂²	semidirect
21. C₅₃×#461	C₂²	semidirect
29. C₅₃×#436	C₂²	semidirect
30. C₈×D₄×C₅₃	C₂²	semidirect
32. C₅₃×#335	C₂²	semidirect
33. C₅₃×#335	C₂²	semidirect
34. C₅₃×#332	C₂²	semidirect
35. C₅₃×#331	C₂²	semidirect
36. C₅₃×D₄⋊C₈	C₂²	semidirect
37. C₅₃×D₄⋊C₈	C₂²	semidirect
38. C₈²×C₅₃	C₂²	semidirect
109. D₈×C₅₃	D₈	semidirect
167. #19884	C₅₃	direct (N × H)
187. #3294	C₂×C₅₃	semidirect
188. #3272	C₂×C₅₃	semidirect
189. #1706	C₂×C₅₃	semidirect
190. #1685	C₂×C₅₃	semidirect
191. #1681	C₂×C₅₃	semidirect
192. #1647	C₂×C₅₃	semidirect
193. #1619	C₂×C₅₃	semidirect
194. #1589	C₂×C₅₃	semidirect
195. C₈×D₈	C₂×C₅₃	semidirect
204. D₄²	C₂²×C₅₃	semidirect
205. #500	C₂²×C₅₃	semidirect
207. #492	C₂²×C₅₃	semidirect
208. #486	C₂²×C₅₃	semidirect
210. #474	C₂²×C₅₃	semidirect
213. #461	C₂²×C₅₃	semidirect
221. #436	C₂²×C₅₃	semidirect
222. C₈×D₄	C₂²×C₅₃	semidirect
224. #335	C₂²×C₅₃	semidirect
225. #335	C₂²×C₅₃	semidirect
226. #332	C₂²×C₅₃	semidirect
227. #331	C₂²×C₅₃	semidirect
228. D₄⋊C₈	C₂²×C₅₃	semidirect
229. D₄⋊C₈	C₂²×C₅₃	semidirect
230. C₈²	C₂²×C₅₃	semidirect
231. C₅₃	#19884	direct (N × H)
302. D₈	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3294	2
C₅₃×#3272	2
C₅₃×#1706	2
C₅₃×#1685	2
C₅₃×#1681	2
C₅₃×#1647	2
C₅₃×#1619	2
C₅₃×#1589	2
C₈×D₈×C₅₃	2
#19884	53

Subgroups — C53×#19884 #27227396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19884 #27227396