Subgroups — C₅₃×#33892 #27241404

Order 13568.

620 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×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2234	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2218	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2184	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2140	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#555	—	2	1	no	—
14	D₄²×C₅₃	—	2	1	no	—
15	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×C₅₃	C₂²	1	2	yes	complemented
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	complemented
18	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#493	C₂²	1	1	yes	characteristic
20	C₅₃×#482	—	2	2	no	—
21	C₅₃×#476	—	2	1	no	—
22	C₅₃×#466	C₂²	1	2	yes	—
23	C₅₃×#466	C₂²	1	2	yes	complemented
24	C₅₃×#459	—	2	1	no	—
25	C₅₃×#447	—	2	2	no	—
26	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#427	C₂²	1	2	yes	complemented
30	C₅₃×#367	C₂²	1	2	yes	complemented
31	C₅₃×#366	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
33	C₅₃×#357	C₂²	1	1	yes	characteristic
34	C₅₃×#357	C₂²	1	2	yes	—
35	C₅₃×#357	C₂²	1	1	yes	characteristic, complemented
36	C₅₃×#357	C₂²	1	2	yes	complemented
37	C₅₃×#357	C₂²	1	2	yes	complemented
38	C₅₃×#357	C₂²	1	2	yes	complemented
39	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₄×C₅₃	—	2	2	no	—
42	C₂²×D₄×C₅₃	—	4	1	no	—
43	C₂×D₈×C₅₃	—	2	2	no	—
44	C₂×D₈×C₅₃	—	2	1	no	—
45	C₂×D₈×C₅₃	—	2	4	no	—
46	C₂×D₈×C₅₃	—	4	2	no	—
47	C₂×D₈×C₅₃	C₂³	1	2	yes	—
48	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	#127×C₅₃	—	2	1	no	—
53	#124×C₅₃	—	4	2	no	—
54	#121×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	2	no	—
56	#121×C₅₃	—	4	1	no	—
57	#121×C₅₃	—	4	1	no	—
58	#120×C₅₃	—	4	2	no	—
59	C₄×D₄×C₅₃	—	2	1	no	—
60	C₄×D₄×C₅₃	—	2	1	no	—
61	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
62	#117×C₅₃	—	2	2	no	—
63	#117×C₅₃	—	4	1	no	—
64	D₁₆×C₅₃	—	2	2	no	—
65	D₁₆×C₅₃	—	4	2	no	—
66	M₃₂×C₅₃	—	4	2	no	—
67	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	#107×C₅₃	—	2	1	no	—
71	#107×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	C₂³	1	1	yes	characteristic
73	#107×C₅₃	C₂³	1	1	yes	characteristic
74	#106×C₅₃	—	2	2	no	—
75	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
76	#102×C₅₃	—	2	1	no	—
77	#102×C₅₃	—	2	1	no	—
78	#102×C₅₃	—	2	2	no	—
79	#102×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂⁴×C₅₃	—	4	2	no	—
83	C₂×Q₈×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	—	2	1	no	—
85	C₂×D₄×C₅₃	—	2	1	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	4	4	no	—
88	C₂×D₄×C₅₃	—	4	2	no	—
89	C₂×D₄×C₅₃	—	4	2	no	—
90	C₂×D₄×C₅₃	—	4	2	no	—
91	C₂×D₄×C₅₃	—	4	1	no	—
92	C₂×D₄×C₅₃	—	4	1	no	—
93	C₂×D₄×C₅₃	—	8	2	no	—
94	C₂×D₄×C₅₃	—	8	4	no	—
95	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	—	4	1	no	—
97	C₂²×C₄×C₅₃	—	4	1	no	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	D₈×C₅₃	—	2	1	no	—
100	D₈×C₅₃	—	2	2	no	—
101	D₈×C₅₃	—	2	1	no	—
102	D₈×C₅₃	—	2	1	no	—
103	D₈×C₅₃	—	2	2	no	—
104	D₈×C₅₃	—	4	4	no	—
105	D₈×C₅₃	—	8	1	no	—
106	D₈×C₅₃	#41	1	2	yes	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	4	1	no	—
116	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
117	C₂²⋊C₄×C₅₃	—	4	2	no	—
118	C₂²⋊C₄×C₅₃	—	4	2	no	—
119	C₂²⋊C₄×C₅₃	—	4	1	no	—
120	C₂²⋊C₄×C₅₃	—	4	1	no	—
121	C₂²⋊C₄×C₅₃	—	4	2	no	—
122	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
123	C₄²×C₅₃	—	4	2	no	—
124	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₅₃	—	2	1	no	—
126	C₁₆×C₅₃	—	2	1	no	—
127	C₁₆×C₅₃	—	2	2	no	—
128	C₂³×C₅₃	—	4	1	no	—
129	C₂³×C₅₃	—	4	1	no	—
130	C₂³×C₅₃	—	4	2	no	—
131	C₂³×C₅₃	—	8	4	no	—
132	C₂³×C₅₃	—	8	4	no	—
133	C₂³×C₅₃	—	16	2	no	—
134	C₂³×C₅₃	C₂×D₈	1	2	yes	—
135	Q₈×C₅₃	—	16	1	no	—
136	D₄×C₅₃	—	2	2	no	—
137	D₄×C₅₃	—	2	2	no	—
138	D₄×C₅₃	—	4	2	no	—
139	D₄×C₅₃	—	4	1	no	—
140	D₄×C₅₃	—	4	1	no	—
141	D₄×C₅₃	—	4	1	no	—
142	D₄×C₅₃	—	4	2	no	—
143	D₄×C₅₃	—	8	4	no	—
144	D₄×C₅₃	—	8	4	no	—
145	D₄×C₅₃	—	8	2	no	—
146	D₄×C₅₃	—	16	1	no	—
147	D₄×C₅₃	—	16	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	8	2	no	—
154	C₂×C₄×C₅₃	—	8	2	no	—
155	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
162	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
163	#33892	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂²×C₅₃	—	2	2	no	—
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	—	2	2	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	8	1	no	—
169	C₂²×C₅₃	—	8	1	no	—
170	C₂²×C₅₃	—	8	1	no	—
171	C₂²×C₅₃	—	8	2	no	—
172	C₂²×C₅₃	—	8	1	no	—
173	C₂²×C₅₃	—	8	2	no	—
174	C₂²×C₅₃	—	8	2	no	—
175	C₂²×C₅₃	—	8	1	no	—
176	C₂²×C₅₃	—	16	8	no	—
177	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	8	1	no	—
181	C₄×C₅₃	—	8	1	no	—
182	C₄×C₅₃	—	8	2	no	—
183	C₄×C₅₃	#509	1	2	yes	—
184	C₄×C₅₃	#546	1	1	yes	characteristic
185	C₄×C₅₃	#577	1	1	yes	characteristic
186	#3341	C₂×C₅₃	1	1	yes	characteristic, complemented
187	D₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#2240	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#2234	C₂×C₅₃	1	2	yes	complemented
190	#2226	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#2218	C₂×C₅₃	1	2	yes	complemented
192	#2206	C₂×C₅₃	1	1	yes	characteristic, complemented
193	#2184	C₂×C₅₃	1	2	yes	complemented
194	C₄×D₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
195	#2169	C₂×C₅₃	1	1	yes	characteristic, complemented
196	#2140	C₂×C₅₃	1	2	yes	complemented
197	C₂×C₅₃	—	4	2	no	—
198	C₂×C₅₃	—	8	2	no	—
199	C₂×C₅₃	—	16	1	no	—
200	C₂×C₅₃	—	16	2	no	—
201	C₂×C₅₃	#3294	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
203	C₂×C₅₃	#3415	1	1	yes	characteristic
204	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
205	#555	—	2	1	no	—
206	D₄²	—	2	1	no	—
207	C₂×D₁₆	C₂²×C₅₃	1	1	yes	characteristic
208	C₂×D₁₆	C₂²×C₅₃	1	2	yes	complemented
209	C₂×M₃₂	C₂²×C₅₃	1	2	yes	complemented
210	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
211	#493	C₂²×C₅₃	1	1	yes	characteristic
212	#482	—	2	2	no	—
213	#476	—	2	1	no	—
214	#466	C₂²×C₅₃	1	2	yes	—
215	#466	C₂²×C₅₃	1	2	yes	complemented
216	#459	—	2	1	no	—
217	#447	—	2	2	no	—
218	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
219	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
220	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
221	#427	C₂²×C₅₃	1	2	yes	complemented
222	#367	C₂²×C₅₃	1	2	yes	complemented
223	#366	C₂²×C₅₃	1	1	yes	characteristic, complemented
224	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
225	#357	C₂²×C₅₃	1	1	yes	characteristic
226	#357	C₂²×C₅₃	1	2	yes	—
227	#357	C₂²×C₅₃	1	2	yes	complemented
228	#357	C₂²×C₅₃	1	2	yes	complemented
229	#357	C₂²×C₅₃	1	1	yes	characteristic, complemented
230	#357	C₂²×C₅₃	1	2	yes	complemented
231	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	complemented
232	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
233	C₅₃	#33892	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²×D₄	—	2	2	no	—
235	C₂²×D₄	—	4	1	no	—
236	C₂×D₈	—	2	4	no	—
237	C₂×D₈	—	2	2	no	—
238	C₂×D₈	—	2	1	no	—
239	C₂×D₈	—	4	2	no	—
240	C₂×D₈	C₂³×C₅₃	1	2	yes	—
241	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
242	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
243	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
244	C₄⋊Q₈	—	2	1	no	—
245	#127	—	2	1	no	—
246	#124	—	4	2	no	—
247	#121	—	2	2	no	—
248	#121	—	2	2	no	—
249	#121	—	4	1	no	—
250	#121	—	4	1	no	—
251	#120	—	4	2	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	—	2	1	no	—
254	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
255	#117	—	2	2	no	—
256	#117	—	4	1	no	—
257	D₁₆	—	2	2	no	—
258	D₁₆	—	4	2	no	—
259	M₃₂	—	4	2	no	—
260	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
261	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
262	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
263	#107	—	2	1	no	—
264	#107	C₂³×C₅₃	1	1	yes	characteristic
265	#107	C₂³×C₅₃	1	2	yes	—
266	#107	C₂³×C₅₃	1	1	yes	characteristic
267	#106	—	2	2	no	—
268	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
269	#102	—	2	1	no	—
270	#102	—	2	1	no	—
271	#102	—	2	2	no	—
272	#102	—	2	2	no	—
273	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
274	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₂⁴	—	4	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	4	4	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	8	2	no	—
287	C₂×D₄	—	8	4	no	—
288	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
292	D₈	—	2	1	no	—
293	D₈	—	2	2	no	—
294	D₈	—	2	1	no	—
295	D₈	—	2	1	no	—
296	D₈	—	2	2	no	—
297	D₈	—	4	4	no	—
298	D₈	—	8	1	no	—
299	D₈	#41×C₅₃	1	2	yes	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
304	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	1	no	—
313	C₂²⋊C₄	—	4	1	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	8	4	no	—
325	C₂³	—	8	4	no	—
326	C₂³	—	16	2	no	—
327	C₂³	C₂×D₈×C₅₃	1	2	yes	—
328	Q₈	—	16	1	no	—
329	D₄	—	2	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	1	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	1	no	—
336	D₄	—	8	4	no	—
337	D₄	—	8	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	16	1	no	—
340	D₄	—	16	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
349	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
355	C₈	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	8	1	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	1	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	16	8	no	—
369	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
370	C₄	—	2	2	no	—
371	C₄	—	2	1	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	2	no	—
375	C₄	C₅₃×#509	1	2	yes	—
376	C₄	C₅₃×#546	1	1	yes	characteristic
377	C₄	C₅₃×#577	1	1	yes	characteristic
378	C₂	—	4	2	no	—
379	C₂	—	8	2	no	—
380	C₂	—	16	2	no	—
381	C₂	—	16	1	no	—
382	C₂	C₅₃×#3294	1	1	yes	characteristic
383	C₂	C₂×C₅₃×#509	1	1	yes	characteristic
384	C₂	C₅₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×D₄×C₅₃ ◁ C₈×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3341	C₂	semidirect
2. D₄×D₈×C₅₃	C₂	semidirect
3. C₅₃×#2240	C₂	semidirect
4. C₅₃×#2234	C₂	semidirect
5. C₅₃×#2226	C₂	semidirect
6. C₅₃×#2218	C₂	semidirect
7. C₅₃×#2206	C₂	semidirect
8. C₅₃×#2184	C₂	semidirect
9. C₄×D₁₆×C₅₃	C₂	semidirect
10. C₅₃×#2169	C₂	semidirect
11. C₅₃×#2140	C₂	semidirect
16. C₂×D₁₆×C₅₃	C₂²	semidirect
17. C₂×M₃₂×C₅₃	C₂²	semidirect
18. C₅₃×#500	C₂²	semidirect
23. C₅₃×#466	C₂²	semidirect
27. C₄×D₈×C₅₃	C₂²	semidirect
29. C₅₃×#427	C₂²	semidirect
30. C₅₃×#367	C₂²	semidirect
31. C₅₃×#366	C₂²	semidirect
32. C₅₃×C₄⋊C₁₆	C₂²	semidirect
35. C₅₃×#357	C₂²	semidirect
36. C₅₃×#357	C₂²	semidirect
37. C₅₃×#357	C₂²	semidirect
38. C₅₃×#357	C₂²	semidirect
39. C₅₃×C₂²⋊C₁₆	C₂²	semidirect
40. C₄×C₁₆×C₅₃	C₂²	semidirect
163. #33892	C₅₃	direct (N × H)
186. #3341	C₂×C₅₃	semidirect
187. D₄×D₈	C₂×C₅₃	semidirect
188. #2240	C₂×C₅₃	semidirect
189. #2234	C₂×C₅₃	semidirect
190. #2226	C₂×C₅₃	semidirect
191. #2218	C₂×C₅₃	semidirect
192. #2206	C₂×C₅₃	semidirect
193. #2184	C₂×C₅₃	semidirect
194. C₄×D₁₆	C₂×C₅₃	semidirect
195. #2169	C₂×C₅₃	semidirect
196. #2140	C₂×C₅₃	semidirect
208. C₂×D₁₆	C₂²×C₅₃	semidirect
209. C₂×M₃₂	C₂²×C₅₃	semidirect
210. #500	C₂²×C₅₃	semidirect
215. #466	C₂²×C₅₃	semidirect
219. C₄×D₈	C₂²×C₅₃	semidirect
221. #427	C₂²×C₅₃	semidirect
222. #367	C₂²×C₅₃	semidirect
223. #366	C₂²×C₅₃	semidirect
224. C₄⋊C₁₆	C₂²×C₅₃	semidirect
227. #357	C₂²×C₅₃	semidirect
228. #357	C₂²×C₅₃	semidirect
229. #357	C₂²×C₅₃	semidirect
230. #357	C₂²×C₅₃	semidirect
231. C₂²⋊C₁₆	C₂²×C₅₃	semidirect
232. C₄×C₁₆	C₂²×C₅₃	semidirect
233. C₅₃	#33892	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×#3341	2
D₄×D₈×C₅₃	2
C₅₃×#2240	2
C₅₃×#2234	2
C₅₃×#2226	2
C₅₃×#2218	2
C₅₃×#2206	2
C₅₃×#2184	2
C₄×D₁₆×C₅₃	2
C₅₃×#2169	2
C₅₃×#2140	2
#33892	53

Subgroups — C53×#33892 #27241404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33892 #27241404