Subgroups — C₅₃×#33702 #27241214

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3155	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2244	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2242	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2240	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
13	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#544	—	2	1	no	—
16	C₅₃×#535	—	2	1	no	—
17	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#501	—	2	2	no	—
20	C₅₃×#500	C₂²	1	2	yes	complemented
21	C₅₃×#499	C₂²	1	2	yes	complemented
22	C₅₃×#496	—	2	2	no	—
23	C₅₃×#495	C₂²	1	2	yes	—
24	C₅₃×#493	C₂²	1	2	yes	—
25	C₅₃×#428	—	2	1	no	—
26	C₅₃×#427	C₂²	1	1	yes	characteristic
27	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#405	C₂²	1	1	yes	characteristic
29	C₅₃×#358	C₂²	1	1	yes	characteristic, complemented
30	C₅₃×#358	C₂²	1	2	yes	complemented
31	C₅₃×#358	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#357	C₂²	1	2	yes	—
33	C₅₃×#357	C₂²	1	2	yes	—
34	C₅₃×#357	C₂²	1	2	yes	—
35	C₅₃×#357	C₂²	1	2	yes	—
36	C₅₃×#357	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#357	C₂²	1	2	yes	complemented
38	C₅₃×#357	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#346	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
41	C₂×#41×C₅₃	—	2	1	no	—
42	C₂²×D₄×C₅₃	—	2	1	no	—
43	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
44	#135×C₅₃	—	2	2	no	—
45	#135×C₅₃	—	2	2	no	—
46	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆×C₅₃	—	2	2	no	—
48	C₂×SD₁₆×C₅₃	—	2	1	no	—
49	C₂×D₈×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	—	2	1	no	—
51	C₂×D₈×C₅₃	—	2	2	no	—
52	C₂×D₈×C₅₃	—	2	2	no	—
53	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×D₈×C₅₃	C₂³	1	2	yes	—
55	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	—	4	1	no	—
59	#127×C₅₃	—	2	2	no	—
60	#125×C₅₃	—	2	2	no	—
61	#125×C₅₃	—	4	1	no	—
62	#124×C₅₃	—	2	2	no	—
63	C₂²⋊Q₈×C₅₃	—	4	2	no	—
64	#121×C₅₃	—	4	1	no	—
65	#121×C₅₃	—	4	1	no	—
66	#120×C₅₃	—	4	2	no	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	M₃₂×C₅₃	—	2	2	no	—
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	2	no	—
76	#107×C₅₃	—	2	1	no	—
77	#107×C₅₃	C₂³	1	1	yes	characteristic
78	#107×C₅₃	C₂³	1	2	yes	—
79	#107×C₅₃	C₂³	1	1	yes	characteristic
80	#106×C₅₃	—	2	2	no	—
81	#106×C₅₃	—	2	1	no	—
82	#97×C₅₃	D₄	1	2	yes	—
83	C₄×C₈×C₅₃	C₂³	1	2	yes	—
84	C₂⁴×C₅₃	—	4	1	no	—
85	#41×C₅₃	—	4	2	no	—
86	#41×C₅₃	—	4	2	no	—
87	C₂×Q₈×C₅₃	—	2	1	no	—
88	C₂×Q₈×C₅₃	—	4	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	1	no	—
91	C₂×D₄×C₅₃	—	4	1	no	—
92	C₂×D₄×C₅₃	—	4	2	no	—
93	C₂×D₄×C₅₃	—	4	1	no	—
94	C₂×D₄×C₅₃	—	4	2	no	—
95	C₂×D₄×C₅₃	—	4	2	no	—
96	C₂×D₄×C₅₃	—	4	2	no	—
97	C₂²×C₄×C₅₃	—	4	1	no	—
98	C₂²×C₄×C₅₃	—	4	1	no	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	Q₁₆×C₅₃	—	2	1	no	—
101	Q₁₆×C₅₃	—	2	1	no	—
102	SD₁₆×C₅₃	—	4	2	no	—
103	SD₁₆×C₅₃	—	4	4	no	—
104	D₈×C₅₃	—	2	2	no	—
105	D₈×C₅₃	—	2	2	no	—
106	D₈×C₅₃	—	2	1	no	—
107	D₈×C₅₃	—	2	1	no	—
108	D₈×C₅₃	—	4	2	no	—
109	D₈×C₅₃	—	4	4	no	—
110	M₁₆×C₅₃	SD₁₆	1	4	yes	—
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₅₃	#41	1	2	yes	—
114	C₂×C₈×C₅₃	#41	1	2	yes	—
115	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	—	2	1	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	4	1	no	—
120	C₄⋊C₄×C₅₃	—	4	2	no	—
121	C₄⋊C₄×C₅₃	—	4	2	no	—
122	C₄⋊C₄×C₅₃	—	4	2	no	—
123	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
124	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₅₃	—	4	1	no	—
126	C₂²⋊C₄×C₅₃	—	4	2	no	—
127	C₂²⋊C₄×C₅₃	—	4	2	no	—
128	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
129	C₄²×C₅₃	—	4	1	no	—
130	C₄²×C₅₃	C₂×D₄	1	2	yes	—
131	C₁₆×C₅₃	—	2	1	no	—
132	C₁₆×C₅₃	—	2	1	no	—
133	C₁₆×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	—	4	1	no	—
135	C₂³×C₅₃	—	4	2	no	—
136	C₂³×C₅₃	—	8	2	no	—
137	C₂³×C₅₃	—	16	1	no	—
138	C₂³×C₅₃	—	16	1	no	—
139	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₅₃	—	4	1	no	—
141	Q₈×C₅₃	—	4	1	no	—
142	Q₈×C₅₃	—	16	2	no	—
143	D₄×C₅₃	—	4	2	no	—
144	D₄×C₅₃	—	4	1	no	—
145	D₄×C₅₃	—	4	2	no	—
146	D₄×C₅₃	—	4	1	no	—
147	D₄×C₅₃	—	8	2	no	—
148	D₄×C₅₃	—	8	2	no	—
149	D₄×C₅₃	—	16	2	no	—
150	D₄×C₅₃	—	16	2	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	8	2	no	—
156	C₂×C₄×C₅₃	—	16	1	no	—
157	C₂×C₄×C₅₃	#124	1	2	yes	—
158	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₅₃	#124	1	2	yes	—
164	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
165	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
167	#33702	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	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	1	no	—
175	C₂²×C₅₃	—	8	2	no	—
176	C₂²×C₅₃	—	16	4	no	—
177	C₂²×C₅₃	#486	1	2	yes	—
178	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	8	1	no	—
182	C₄×C₅₃	—	8	2	no	—
183	C₄×C₅₃	—	8	1	no	—
184	C₄×C₅₃	—	8	1	no	—
185	C₄×C₅₃	#486	1	2	yes	—
186	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
187	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
188	#3164	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#3155	C₂×C₅₃	1	1	yes	characteristic
190	#2244	C₂×C₅₃	1	2	yes	complemented
191	#2243	C₂×C₅₃	1	2	yes	complemented
192	#2242	C₂×C₅₃	1	2	yes	complemented
193	#2240	C₂×C₅₃	1	2	yes	complemented
194	#2140	C₂×C₅₃	1	1	yes	characteristic, complemented
195	#2139	C₂×C₅₃	1	1	yes	characteristic, complemented
196	#2138	C₂×C₅₃	1	1	yes	characteristic, complemented
197	C₂×#357	C₂×C₅₃	1	1	yes	characteristic, complemented
198	#2108	C₂×C₅₃	1	1	yes	characteristic, complemented
199	C₂×C₅₃	—	2	2	no	—
200	C₂×C₅₃	—	16	1	no	—
201	C₂×C₅₃	—	16	2	no	—
202	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
203	C₂×C₅₃	#3415	1	1	yes	characteristic
204	C₂×C₅₃	#3416	1	1	yes	characteristic
205	C₂×AGL(1,ℤ/8)	—	2	1	no	—
206	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
207	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
208	#544	—	2	1	no	—
209	#535	—	2	1	no	—
210	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
211	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
212	#501	—	2	2	no	—
213	#500	C₂²×C₅₃	1	2	yes	complemented
214	#499	C₂²×C₅₃	1	2	yes	complemented
215	#496	—	2	2	no	—
216	#495	C₂²×C₅₃	1	2	yes	—
217	#493	C₂²×C₅₃	1	2	yes	—
218	#428	—	2	1	no	—
219	#427	C₂²×C₅₃	1	1	yes	characteristic
220	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
221	#405	C₂²×C₅₃	1	1	yes	characteristic
222	#358	C₂²×C₅₃	1	1	yes	characteristic, complemented
223	#358	C₂²×C₅₃	1	1	yes	characteristic, complemented
224	#358	C₂²×C₅₃	1	2	yes	complemented
225	#357	C₂²×C₅₃	1	2	yes	—
226	#357	C₂²×C₅₃	1	2	yes	—
227	#357	C₂²×C₅₃	1	2	yes	—
228	#357	C₂²×C₅₃	1	2	yes	—
229	#357	C₂²×C₅₃	1	1	yes	characteristic, complemented
230	#357	C₂²×C₅₃	1	1	yes	characteristic, complemented
231	#357	C₂²×C₅₃	1	2	yes	complemented
232	#346	C₂²×C₅₃	1	2	yes	complemented
233	C₄×C₁₆	C₂²×C₅₃	1	2	yes	complemented
234	C₅₃	#33702	1	1	yes	characteristic, direct product factor, Sylow
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	2	1	no	—
237	AGL(1,ℤ/8)	—	4	4	no	—
238	#135	—	2	2	no	—
239	#135	—	2	2	no	—
240	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×SD₁₆	—	2	2	no	—
243	C₂×D₈	—	2	1	no	—
244	C₂×D₈	—	2	2	no	—
245	C₂×D₈	—	2	2	no	—
246	C₂×D₈	—	2	2	no	—
247	C₂×D₈	C₂³×C₅₃	1	2	yes	—
248	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
249	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
251	C₄⋊Q₈	—	2	2	no	—
252	C₄⋊Q₈	—	4	1	no	—
253	#127	—	2	2	no	—
254	#125	—	2	2	no	—
255	#125	—	4	1	no	—
256	#124	—	2	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	#121	—	4	1	no	—
259	#121	—	4	1	no	—
260	#120	—	4	2	no	—
261	#117	—	2	1	no	—
262	#117	D₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	2	1	no	—
264	M₃₂	—	2	2	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
267	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
268	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
269	#107	—	2	1	no	—
270	#107	—	2	2	no	—
271	#107	C₂³×C₅₃	1	2	yes	—
272	#107	C₂³×C₅₃	1	1	yes	characteristic
273	#107	C₂³×C₅₃	1	1	yes	characteristic
274	#106	—	2	1	no	—
275	#106	—	2	2	no	—
276	#97	D₄×C₅₃	1	2	yes	—
277	C₄×C₈	C₂³×C₅₃	1	2	yes	—
278	C₂⁴	—	4	1	no	—
279	#41	—	4	2	no	—
280	#41	—	4	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	4	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	2	1	no	—
296	SD₁₆	—	4	2	no	—
297	SD₁₆	—	4	4	no	—
298	D₈	—	2	2	no	—
299	D₈	—	2	2	no	—
300	D₈	—	2	1	no	—
301	D₈	—	2	1	no	—
302	D₈	—	4	2	no	—
303	D₈	—	4	4	no	—
304	M₁₆	SD₁₆×C₅₃	1	4	yes	—
305	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
307	C₂×C₈	#41×C₅₃	1	2	yes	—
308	C₂×C₈	#41×C₅₃	1	2	yes	—
309	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
318	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	4	2	no	—
322	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
323	C₄²	—	4	1	no	—
324	C₄²	C₂×D₄×C₅₃	1	2	yes	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	2	2	no	—
328	C₂³	—	4	2	no	—
329	C₂³	—	4	1	no	—
330	C₂³	—	8	2	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	16	1	no	—
333	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	16	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	1	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	D₄	—	16	2	no	—
344	D₄	—	16	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	16	1	no	—
351	C₂×C₄	#124×C₅₃	1	2	yes	—
352	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
353	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
354	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
356	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
357	C₈	#124×C₅₃	1	2	yes	—
358	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
359	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
360	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	16	4	no	—
370	C₂²	C₅₃×#486	1	2	yes	—
371	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	8	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	C₅₃×#486	1	2	yes	—
379	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
380	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	16	2	no	—
383	C₂	—	16	1	no	—
384	C₂	C₂×C₅₃×#486	1	1	yes	characteristic
385	C₂	C₅₃×#3415	1	1	yes	characteristic
386	C₂	C₅₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₅₃×#3155	C₂
LCS	G ▷ C₂×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
1. C₅₃×#3164	C₂	semidirect
3. C₅₃×#2244	C₂	semidirect
4. C₅₃×#2243	C₂	semidirect
5. C₅₃×#2242	C₂	semidirect
6. C₅₃×#2240	C₂	semidirect
7. C₅₃×#2140	C₂	semidirect
8. C₅₃×#2139	C₂	semidirect
9. C₅₃×#2138	C₂	semidirect
10. C₂×C₅₃×#357	C₂	semidirect
11. C₅₃×#2108	C₂	semidirect
20. C₅₃×#500	C₂²	semidirect
21. C₅₃×#499	C₂²	semidirect
29. C₅₃×#358	C₂²	semidirect
30. C₅₃×#358	C₂²	semidirect
31. C₅₃×#358	C₂²	semidirect
36. C₅₃×#357	C₂²	semidirect
37. C₅₃×#357	C₂²	semidirect
38. C₅₃×#357	C₂²	semidirect
39. C₅₃×#346	C₂²	semidirect
40. C₄×C₁₆×C₅₃	C₂²	semidirect
167. #33702	C₅₃	direct (N × H)
188. #3164	C₂×C₅₃	semidirect
190. #2244	C₂×C₅₃	semidirect
191. #2243	C₂×C₅₃	semidirect
192. #2242	C₂×C₅₃	semidirect
193. #2240	C₂×C₅₃	semidirect
194. #2140	C₂×C₅₃	semidirect
195. #2139	C₂×C₅₃	semidirect
196. #2138	C₂×C₅₃	semidirect
197. C₂×#357	C₂×C₅₃	semidirect
198. #2108	C₂×C₅₃	semidirect
213. #500	C₂²×C₅₃	semidirect
214. #499	C₂²×C₅₃	semidirect
222. #358	C₂²×C₅₃	semidirect
223. #358	C₂²×C₅₃	semidirect
224. #358	C₂²×C₅₃	semidirect
229. #357	C₂²×C₅₃	semidirect
230. #357	C₂²×C₅₃	semidirect
231. #357	C₂²×C₅₃	semidirect
232. #346	C₂²×C₅₃	semidirect
233. C₄×C₁₆	C₂²×C₅₃	semidirect
234. C₅₃	#33702	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×#3164	2
C₅₃×#3155	2
C₅₃×#2244	2
C₅₃×#2243	2
C₅₃×#2242	2
C₅₃×#2240	2
C₅₃×#2140	2
C₅₃×#2139	2
C₅₃×#2138	2
C₂×C₅₃×#357	2
C₅₃×#2108	2
#33702	53

Subgroups — C53×#33702 #27241214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33702 #27241214