Subgroups — C₅₃×#21139 #27228651

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2943	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2923	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1929	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1922	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1876	C₂	1	2	yes	maximal
7	C₅₃×#1866	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#131×C₅₃	C₂²	1	2	yes	—
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#442	C₄	1	2	yes	complemented
17	C₅₃×#441	C₄	1	2	yes	complemented
18	C₅₃×#440	C₄	1	2	yes	complemented
19	C₅₃×#440	C₄	1	2	yes	complemented
20	C₅₃×#433	—	2	1	no	—
21	C₅₃×#433	—	2	1	no	—
22	C₅₃×#432	—	2	2	no	—
23	C₂×#108×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#427	C₂²	1	1	yes	characteristic
25	C₅₃×#424	C₂²	1	2	yes	—
26	C₂×#104×C₅₃	C₂²	1	2	yes	—
27	C₂×#104×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#416	C₂²	1	1	yes	characteristic
29	C₅₃×#416	C₂²	1	1	yes	characteristic
30	C₅₃×#408	C₂²	1	2	yes	—
31	C₅₃×#405	—	2	2	no	—
32	C₅₃×#405	C₂²	1	2	yes	—
33	C₄×M₁₆×C₅₃	—	2	1	no	—
34	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₅₃	—	2	1	no	—
37	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
39	C₄³×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#341	C₂²	1	2	yes	—
41	C₅₃×#339	C₂²	1	2	yes	—
42	C₅₃×#339	C₂²	1	2	yes	—
43	C₅₃×#336	C₂²	1	2	yes	—
44	C₅₃×#336	C₂²	1	2	yes	—
45	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
47	#135×C₅₃	—	2	4	no	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#131×C₅₃	—	2	4	no	—
53	#131×C₅₃	—	2	4	no	—
54	C₂×M₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
56	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	—	2	2	no	—
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×D₄×C₅₃	—	2	2	no	—
64	C₄×D₄×C₅₃	—	2	2	no	—
65	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	—	2	1	no	—
69	#117×C₅₃	C₂³	1	1	yes	characteristic
70	#117×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#108×C₅₃	—	2	4	no	—
77	#107×C₅₃	C₂×C₄	1	2	yes	—
78	#106×C₅₃	C₂×C₄	1	2	yes	—
79	C₄⋊C₈×C₅₃	—	2	2	no	—
80	C₄⋊C₈×C₅₃	—	2	2	no	—
81	#104×C₅₃	—	2	4	no	—
82	#104×C₅₃	—	2	4	no	—
83	#103×C₅₃	C₂×C₄	1	2	yes	—
84	#103×C₅₃	C₂×C₄	1	2	yes	—
85	#102×C₅₃	C₂×C₄	1	2	yes	—
86	#102×C₅₃	C₂×C₄	1	2	yes	—
87	C₂²⋊C₈×C₅₃	—	2	2	no	—
88	C₂²⋊C₈×C₅₃	—	2	2	no	—
89	#97×C₅₃	—	2	2	no	—
90	#97×C₅₃	—	2	1	no	—
91	#97×C₅₃	—	2	1	no	—
92	#97×C₅₃	C₂×C₄	1	2	yes	—
93	#97×C₅₃	C₂×C₄	1	2	yes	—
94	#97×C₅₃	D₄	1	1	yes	characteristic
95	#97×C₅₃	D₄	1	1	yes	characteristic
96	#97×C₅₃	D₄	1	2	yes	—
97	C₄×C₈×C₅₃	—	2	2	no	—
98	C₄×C₈×C₅₃	—	2	1	no	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	#41×C₅₃	—	2	2	no	—
101	#41×C₅₃	—	2	2	no	—
102	#41×C₅₃	—	2	2	no	—
103	#41×C₅₃	#41	1	4	yes	—
104	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	#41	1	2	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
117	Q₁₆×C₅₃	—	2	2	no	—
118	SD₁₆×C₅₃	—	2	2	no	—
119	SD₁₆×C₅₃	—	2	2	no	—
120	D₈×C₅₃	—	2	2	no	—
121	M₁₆×C₅₃	—	2	8	no	—
122	M₁₆×C₅₃	—	2	2	no	—
123	M₁₆×C₅₃	—	2	2	no	—
124	M₁₆×C₅₃	—	2	2	no	—
125	M₁₆×C₅₃	#41	1	4	yes	—
126	C₂×C₈×C₅₃	—	2	8	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²⋊C₄×C₅₃	—	2	1	no	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₂²⋊C₄×C₅₃	—	2	2	no	—
145	C₂²⋊C₄×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	2	no	—
148	C₄²×C₅₃	—	2	1	no	—
149	C₄²×C₅₃	—	2	1	no	—
150	C₄²×C₅₃	—	2	2	no	—
151	C₄²×C₅₃	—	2	2	no	—
152	C₄²×C₅₃	—	4	4	no	—
153	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
154	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂³×C₅₃	—	2	1	no	—
157	C₂³×C₅₃	—	2	1	no	—
158	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
159	Q₈×C₅₃	—	2	1	no	—
160	Q₈×C₅₃	—	2	1	no	—
161	Q₈×C₅₃	—	2	1	no	—
162	Q₈×C₅₃	#117	1	2	yes	—
163	D₄×C₅₃	—	2	4	no	—
164	D₄×C₅₃	—	2	1	no	—
165	D₄×C₅₃	—	2	1	no	—
166	D₄×C₅₃	—	2	1	no	—
167	D₄×C₅₃	—	4	2	no	—
168	D₄×C₅₃	#117	1	2	yes	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	8	2	no	—
180	C₂×C₄×C₅₃	—	8	2	no	—
181	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
186	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
187	C₈×C₅₃	—	2	1	no	—
188	C₈×C₅₃	—	2	1	no	—
189	C₈×C₅₃	—	2	2	no	—
190	C₈×C₅₃	—	2	2	no	—
191	C₈×C₅₃	—	2	4	no	—
192	C₈×C₅₃	—	2	2	no	—
193	C₈×C₅₃	—	4	2	no	—
194	#21139	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	4	2	no	—
198	C₂²×C₅₃	—	4	2	no	—
199	C₂²×C₅₃	—	4	1	no	—
200	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₅₃	#529	1	2	yes	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	2	2	no	—
204	C₄×C₅₃	—	2	2	no	—
205	C₄×C₅₃	—	4	4	no	—
206	C₄×C₅₃	—	4	2	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	4	1	no	—
209	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
210	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
211	C₄×C₅₃	#525	1	2	yes	—
212	#2943	C₂×C₅₃	1	1	yes	characteristic
213	#2923	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#2875	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1929	C₂×C₅₃	1	2	yes	complemented
216	#1922	C₂×C₅₃	1	2	yes	complemented
217	#1876	C₂×C₅₃	1	2	yes	—
218	#1866	C₂×C₅₃	1	2	yes	complemented
219	#1778	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₄×#104	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₄×#104	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂×C₅₃	—	2	2	no	—
224	C₂×C₅₃	—	4	2	no	—
225	C₂×C₅₃	—	8	1	no	—
226	C₂×C₅₃	#2290	1	1	yes	characteristic
227	C₂×C₅₃	#2956	1	2	yes	—
228	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×#131	C₂²×C₅₃	1	2	yes	—
230	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
231	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
232	#442	C₄×C₅₃	1	2	yes	complemented
233	#441	C₄×C₅₃	1	2	yes	complemented
234	#440	C₄×C₅₃	1	2	yes	complemented
235	#440	C₄×C₅₃	1	2	yes	complemented
236	#433	—	2	1	no	—
237	#433	—	2	1	no	—
238	#432	—	2	2	no	—
239	C₂×#108	C₂²×C₅₃	1	2	yes	—
240	#427	C₂²×C₅₃	1	1	yes	characteristic
241	#424	C₂²×C₅₃	1	2	yes	—
242	C₂×#104	C₂²×C₅₃	1	2	yes	—
243	C₂×#104	C₂²×C₅₃	1	2	yes	—
244	#416	C₂²×C₅₃	1	1	yes	characteristic
245	#416	C₂²×C₅₃	1	1	yes	characteristic
246	#408	C₂²×C₅₃	1	2	yes	—
247	#405	—	2	2	no	—
248	#405	C₂²×C₅₃	1	2	yes	—
249	C₄×M₁₆	—	2	1	no	—
250	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
251	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×#97	—	2	1	no	—
253	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
255	C₄³	C₂²×C₅₃	1	1	yes	characteristic
256	#341	C₂²×C₅₃	1	2	yes	—
257	#339	C₂²×C₅₃	1	2	yes	—
258	#339	C₂²×C₅₃	1	2	yes	—
259	#336	C₂²×C₅₃	1	2	yes	—
260	#336	C₂²×C₅₃	1	2	yes	—
261	C₅₃	#21139	1	1	yes	characteristic, direct product factor, Sylow
262	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
263	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
264	#135	—	2	4	no	—
265	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
266	C₂×SD₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₂×SD₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
268	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
269	#131	—	2	4	no	—
270	#131	—	2	4	no	—
271	C₂×M₁₆	—	2	2	no	—
272	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
273	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
274	C₂²×C₈	—	2	2	no	—
275	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
278	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
279	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
283	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
284	#117	—	2	1	no	—
285	#117	—	2	1	no	—
286	#117	C₂³×C₅₃	1	1	yes	characteristic
287	#117	C₂³×C₅₃	1	1	yes	characteristic
288	C₂×C₄²	—	2	1	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
292	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
293	#108	—	2	4	no	—
294	#107	C₂×C₄×C₅₃	1	2	yes	—
295	#106	C₂×C₄×C₅₃	1	2	yes	—
296	C₄⋊C₈	—	2	2	no	—
297	C₄⋊C₈	—	2	2	no	—
298	#104	—	2	4	no	—
299	#104	—	2	4	no	—
300	#103	C₂×C₄×C₅₃	1	2	yes	—
301	#103	C₂×C₄×C₅₃	1	2	yes	—
302	#102	C₂×C₄×C₅₃	1	2	yes	—
303	#102	C₂×C₄×C₅₃	1	2	yes	—
304	C₂²⋊C₈	—	2	2	no	—
305	C₂²⋊C₈	—	2	2	no	—
306	#97	—	2	2	no	—
307	#97	—	2	1	no	—
308	#97	—	2	1	no	—
309	#97	C₂×C₄×C₅₃	1	2	yes	—
310	#97	C₂×C₄×C₅₃	1	2	yes	—
311	#97	D₄×C₅₃	1	1	yes	characteristic
312	#97	D₄×C₅₃	1	1	yes	characteristic
313	#97	D₄×C₅₃	1	2	yes	—
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	—	2	1	no	—
316	C₄×C₈	—	2	1	no	—
317	#41	—	2	2	no	—
318	#41	—	2	2	no	—
319	#41	—	2	2	no	—
320	#41	#41×C₅₃	1	4	yes	—
321	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
322	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
326	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	#41×C₅₃	1	2	yes	—
333	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
334	Q₁₆	—	2	2	no	—
335	SD₁₆	—	2	2	no	—
336	SD₁₆	—	2	2	no	—
337	D₈	—	2	2	no	—
338	M₁₆	—	2	8	no	—
339	M₁₆	—	2	2	no	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	#41×C₅₃	1	4	yes	—
343	C₂×C₈	—	2	8	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
351	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
352	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
358	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
359	C₂²⋊C₄	—	2	1	no	—
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	2	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	1	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	4	4	no	—
370	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
371	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
376	Q₈	—	2	1	no	—
377	Q₈	—	2	1	no	—
378	Q₈	—	2	1	no	—
379	Q₈	#117×C₅₃	1	2	yes	—
380	D₄	—	2	4	no	—
381	D₄	—	2	1	no	—
382	D₄	—	2	1	no	—
383	D₄	—	2	1	no	—
384	D₄	—	4	2	no	—
385	D₄	#117×C₅₃	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	8	2	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
399	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
400	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
403	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
404	C₈	—	2	1	no	—
405	C₈	—	2	1	no	—
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	2	4	no	—
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	1	no	—
416	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
417	C₂²	C₅₃×#529	1	2	yes	—
418	C₄	—	2	2	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	2	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	2	no	—
424	C₄	—	4	1	no	—
425	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
426	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
427	C₄	C₅₃×#525	1	2	yes	—
428	C₂	—	2	2	no	—
429	C₂	—	4	2	no	—
430	C₂	—	8	1	no	—
431	C₂	C₅₃×#2290	1	1	yes	characteristic
432	C₂	C₅₃×#2956	1	2	yes	—

	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	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2923	C₂	semidirect
3. C₅₃×#2875	C₂	semidirect
4. C₅₃×#1929	C₂	semidirect
5. C₅₃×#1922	C₂	semidirect
7. C₅₃×#1866	C₂	semidirect
8. C₅₃×#1778	C₂	semidirect
9. C₄×#104×C₅₃	C₂	semidirect
10. C₄×#104×C₅₃	C₂	semidirect
11. C₅₃×#1745	C₂	semidirect
16. C₅₃×#442	C₄	semidirect
17. C₅₃×#441	C₄	semidirect
18. C₅₃×#440	C₄	semidirect
19. C₅₃×#440	C₄	semidirect
194. #21139	C₅₃	direct (N × H)
213. #2923	C₂×C₅₃	semidirect
214. #2875	C₂×C₅₃	semidirect
215. #1929	C₂×C₅₃	semidirect
216. #1922	C₂×C₅₃	semidirect
218. #1866	C₂×C₅₃	semidirect
219. #1778	C₂×C₅₃	semidirect
220. C₄×#104	C₂×C₅₃	semidirect
221. C₄×#104	C₂×C₅₃	semidirect
222. #1745	C₂×C₅₃	semidirect
232. #442	C₄×C₅₃	semidirect
233. #441	C₄×C₅₃	semidirect
234. #440	C₄×C₅₃	semidirect
235. #440	C₄×C₅₃	semidirect
261. C₅₃	#21139	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×#2943	2
C₅₃×#2923	2
C₅₃×#2875	2
C₅₃×#1929	2
C₅₃×#1922	2
C₅₃×#1876	2
C₅₃×#1866	2
C₅₃×#1778	2
C₄×#104×C₅₃	2
C₅₃×#1745	2
#21139	53

Subgroups — C53×#21139 #27228651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21139 #27228651