Subgroups — C₅₃×#22234 #27229746

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3131	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1929	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1866	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#533	C₂²	1	1	yes	characteristic
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#488	C₄	1	4	yes	complemented
14	C₅₃×#487	C₄	1	1	yes	characteristic, complemented
15	C₅₃×#487	C₄	1	1	yes	characteristic, complemented
16	C₅₃×#486	C₄	1	1	yes	characteristic, complemented
17	C₅₃×#486	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#446	—	2	1	no	—
19	C₅₃×#446	—	2	1	no	—
20	C₈×Q₈×C₅₃	—	2	1	no	—
21	C₈×Q₈×C₅₃	—	2	1	no	—
22	C₅₃×#436	—	2	1	no	—
23	C₅₃×#436	—	2	1	no	—
24	C₈×D₄×C₅₃	—	2	1	no	—
25	C₈×D₄×C₅₃	—	2	1	no	—
26	C₂×#108×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#424	C₂²	1	2	yes	—
28	C₅₃×#423	C₂²	1	1	yes	characteristic
29	C₅₃×#423	C₂²	1	1	yes	characteristic
30	C₂×#104×C₅₃	C₂²	1	4	yes	—
31	C₅₃×#416	C₂²	1	4	yes	—
32	C₅₃×#408	C₂²	1	2	yes	—
33	C₅₃×#405	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄³×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#339	C₂²	1	4	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₂×#41×C₅₃	C₂³	1	2	yes	—
41	#131×C₅₃	—	4	4	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
53	#125×C₅₃	C₂×C₄	1	2	yes	—
54	#124×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²⋊Q₈×C₅₃	—	2	1	no	—
56	C₂²⋊Q₈×C₅₃	—	2	1	no	—
57	#121×C₅₃	—	2	1	no	—
58	#121×C₅₃	—	2	1	no	—
59	C₄×Q₈×C₅₃	—	2	1	no	—
60	C₄×Q₈×C₅₃	—	2	1	no	—
61	C₄×D₄×C₅₃	—	2	1	no	—
62	C₄×D₄×C₅₃	—	2	1	no	—
63	#117×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₅₃	—	2	1	no	—
65	C₂×C₄²×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#108×C₅₃	—	4	4	no	—
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	—	2	1	no	—
73	C₄⋊C₈×C₅₃	—	2	1	no	—
74	C₄⋊C₈×C₅₃	—	2	1	no	—
75	#104×C₅₃	—	4	4	no	—
76	#103×C₅₃	C₂×C₄	1	4	yes	—
77	#103×C₅₃	C₂×C₄	1	4	yes	—
78	#102×C₅₃	C₂×C₄	1	4	yes	—
79	#102×C₅₃	C₂×C₄	1	4	yes	—
80	C₂²⋊C₈×C₅₃	—	2	2	no	—
81	C₂²⋊C₈×C₅₃	—	2	2	no	—
82	#97×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
87	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
91	#41×C₅₃	—	2	4	no	—
92	#41×C₅₃	—	2	4	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₅₃	—	2	2	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₅₃	—	2	4	no	—
106	M₁₆×C₅₃	—	2	4	no	—
107	M₁₆×C₅₃	—	4	4	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	4	4	no	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	1	no	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂²⋊C₄×C₅₃	—	2	2	no	—
130	C₂²⋊C₄×C₅₃	—	2	2	no	—
131	C₂²⋊C₄×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	—	2	4	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	4	2	no	—
137	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
143	Q₈×C₅₃	—	2	4	no	—
144	Q₈×C₅₃	—	4	2	no	—
145	Q₈×C₅₃	—	4	2	no	—
146	D₄×C₅₃	—	2	4	no	—
147	D₄×C₅₃	—	4	4	no	—
148	D₄×C₅₃	—	4	2	no	—
149	D₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
167	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
168	C₈×C₅₃	—	2	2	no	—
169	C₈×C₅₃	—	2	2	no	—
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	2	1	no	—
173	C₈×C₅₃	—	2	1	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	#22234	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	4	2	no	—
182	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂²×C₅₃	#542	1	2	yes	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	4	4	no	—
187	C₄×C₅₃	—	4	2	no	—
188	C₄×C₅₃	—	4	2	no	—
189	C₄×C₅₃	—	4	2	no	—
190	C₄×C₅₃	#443	1	2	yes	—
191	C₄×C₅₃	#444	1	2	yes	—
192	C₄×C₅₃	#520	1	2	yes	—
193	C₄×C₅₃	#535	1	1	yes	characteristic
194	C₄×C₅₃	#536	1	1	yes	characteristic
195	#3131	C₂×C₅₃	1	1	yes	characteristic
196	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
197	#2931	C₂×C₅₃	1	1	yes	characteristic
198	#1994	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#1962	C₂×C₅₃	1	1	yes	characteristic
200	#1929	C₂×C₅₃	1	4	yes	complemented
201	#1866	C₂×C₅₃	1	4	yes	complemented
202	#1848	C₂×C₅₃	1	1	yes	characteristic, complemented
203	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
204	C₂×C₅₃	—	2	2	no	—
205	C₂×C₅₃	—	8	2	no	—
206	C₂×C₅₃	#2326	1	1	yes	characteristic
207	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
208	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
209	C₂×#131	C₂²×C₅₃	1	2	yes	—
210	#533	C₂²×C₅₃	1	1	yes	characteristic
211	#532	C₂²×C₅₃	1	1	yes	characteristic
212	#488	C₄×C₅₃	1	4	yes	complemented
213	#487	C₄×C₅₃	1	1	yes	characteristic, complemented
214	#487	C₄×C₅₃	1	1	yes	characteristic, complemented
215	#486	C₄×C₅₃	1	1	yes	characteristic, complemented
216	#486	C₄×C₅₃	1	1	yes	characteristic, complemented
217	#446	—	2	1	no	—
218	#446	—	2	1	no	—
219	C₈×Q₈	—	2	1	no	—
220	C₈×Q₈	—	2	1	no	—
221	#436	—	2	1	no	—
222	#436	—	2	1	no	—
223	C₈×D₄	—	2	1	no	—
224	C₈×D₄	—	2	1	no	—
225	C₂×#108	C₂²×C₅₃	1	4	yes	—
226	#424	C₂²×C₅₃	1	2	yes	—
227	#423	C₂²×C₅₃	1	1	yes	characteristic
228	#423	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×#104	C₂²×C₅₃	1	4	yes	—
230	#416	C₂²×C₅₃	1	4	yes	—
231	#408	C₂²×C₅₃	1	2	yes	—
232	#405	C₂²×C₅₃	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
235	C₄³	C₂²×C₅₃	1	1	yes	characteristic
236	#339	C₂²×C₅₃	1	4	yes	—
237	#336	C₂²×C₅₃	1	2	yes	—
238	#336	C₂²×C₅₃	1	2	yes	—
239	C₅₃	#22234	1	1	yes	characteristic, direct product factor, Sylow
240	C₂×#41	C₂³×C₅₃	1	2	yes	—
241	#131	—	4	4	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
244	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
247	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
249	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
251	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
252	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
253	#125	C₂×C₄×C₅₃	1	2	yes	—
254	#124	C₂×C₄×C₅₃	1	2	yes	—
255	C₂²⋊Q₈	—	2	1	no	—
256	C₂²⋊Q₈	—	2	1	no	—
257	#121	—	2	1	no	—
258	#121	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×D₄	—	2	1	no	—
262	C₄×D₄	—	2	1	no	—
263	#117	C₂³×C₅₃	1	2	yes	—
264	C₂×C₄²	—	2	1	no	—
265	C₂×C₄²	—	2	1	no	—
266	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
267	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
268	#108	—	4	4	no	—
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	2	1	no	—
275	#104	—	4	4	no	—
276	#103	C₂×C₄×C₅₃	1	4	yes	—
277	#103	C₂×C₄×C₅₃	1	4	yes	—
278	#102	C₂×C₄×C₅₃	1	4	yes	—
279	#102	C₂×C₄×C₅₃	1	4	yes	—
280	C₂²⋊C₈	—	2	2	no	—
281	C₂²⋊C₈	—	2	2	no	—
282	#97	—	2	2	no	—
283	C₄×C₈	—	2	2	no	—
284	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
287	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
288	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
289	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
290	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
291	#41	—	2	4	no	—
292	#41	—	2	4	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
305	M₁₆	—	2	4	no	—
306	M₁₆	—	2	4	no	—
307	M₁₆	—	4	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	4	4	no	—
315	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
316	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
317	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
328	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	4	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	2	no	—
337	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
338	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
343	Q₈	—	2	4	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	4	2	no	—
346	D₄	—	2	4	no	—
347	D₄	—	4	4	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
361	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
362	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
363	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
364	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
367	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
368	C₈	—	2	2	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
382	C₂²	C₅₃×#542	1	2	yes	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	4	2	no	—
389	C₄	C₅₃×#443	1	2	yes	—
390	C₄	C₅₃×#444	1	2	yes	—
391	C₄	C₅₃×#520	1	2	yes	—
392	C₄	C₅₃×#535	1	1	yes	characteristic
393	C₄	C₅₃×#536	1	1	yes	characteristic
394	C₂	—	2	2	no	—
395	C₂	—	8	2	no	—
396	C₂	C₅₃×#2326	1	1	yes	characteristic
397	C₂	C₂×C₅₃×#443	1	1	yes	characteristic
398	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	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#532	C₂²
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×#2960	C₂	semidirect
4. C₅₃×#1994	C₂	semidirect
6. C₅₃×#1929	C₂	semidirect
7. C₅₃×#1866	C₂	semidirect
8. C₅₃×#1848	C₂	semidirect
9. C₅₃×#1744	C₂	semidirect
13. C₅₃×#488	C₄	semidirect
14. C₅₃×#487	C₄	semidirect
15. C₅₃×#487	C₄	semidirect
16. C₅₃×#486	C₄	semidirect
17. C₅₃×#486	C₄	semidirect
176. #22234	C₅₃	direct (N × H)
196. #2960	C₂×C₅₃	semidirect
198. #1994	C₂×C₅₃	semidirect
200. #1929	C₂×C₅₃	semidirect
201. #1866	C₂×C₅₃	semidirect
202. #1848	C₂×C₅₃	semidirect
203. #1744	C₂×C₅₃	semidirect
212. #488	C₄×C₅₃	semidirect
213. #487	C₄×C₅₃	semidirect
214. #487	C₄×C₅₃	semidirect
215. #486	C₄×C₅₃	semidirect
216. #486	C₄×C₅₃	semidirect
239. C₅₃	#22234	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×#3131	2
C₅₃×#2960	2
C₅₃×#2931	2
C₅₃×#1994	2
C₅₃×#1962	2
C₅₃×#1929	2
C₅₃×#1866	2
C₅₃×#1848	2
C₅₃×#1744	2
#22234	53

Subgroups — C53×#22234 #27229746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22234 #27229746