Subgroups — C₅₃×#21078 #27228590

Order 13568.

1118 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2911	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#418	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#418	C₂	1	1	yes	maximal, characteristic
4	C₂²×#100×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1931	C₂	1	8	yes	maximal, complemented
7	C₅₃×#1742	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#428	—	2	2	no	—
17	C₅₃×#428	C₂²	1	4	yes	complemented
18	C₅₃×#427	—	2	2	no	—
19	C₅₃×#418	—	2	2	no	—
20	C₅₃×#418	—	2	2	no	—
21	C₅₃×#418	—	2	2	no	—
22	C₅₃×#418	C₂²	1	4	yes	—
23	C₂×#102×C₅₃	—	2	2	no	—
24	C₂×#102×C₅₃	C₂²	1	4	yes	—
25	C₂×#100×C₅₃	C₄	1	8	yes	complemented
26	C₂×#100×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#405	—	2	2	no	—
28	C₅₃×#405	C₂²	1	4	yes	complemented
29	C₄×M₁₆×C₅₃	—	2	2	no	—
30	C₅₃×#343	C₂²	1	4	yes	complemented
31	C₅₃×#341	C₂²	1	4	yes	complemented
32	C₂⁵×C₅₃	—	2	1	no	—
33	C₂²×D₄×C₅₃	—	2	1	no	—
34	C₂²×D₄×C₅₃	—	2	1	no	—
35	C₂²×D₄×C₅₃	—	2	1	no	—
36	C₂²×D₄×C₅₃	—	2	1	no	—
37	C₂²×D₄×C₅₃	—	2	2	no	—
38	C₂²×D₄×C₅₃	—	2	2	no	—
39	C₂²×D₄×C₅₃	—	2	1	no	—
40	C₂²×D₄×C₅₃	—	2	2	no	—
41	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	—
42	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	—	2	1	no	—
45	C₂×M₁₆×C₅₃	—	2	1	no	—
46	C₂×M₁₆×C₅₃	—	2	1	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
56	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
57	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
64	#117×C₅₃	—	2	1	no	—
65	#117×C₅₃	—	2	1	no	—
66	#117×C₅₃	—	2	1	no	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	—	2	1	no	—
69	#117×C₅₃	—	2	1	no	—
70	#117×C₅₃	D₄	1	2	yes	—
71	#117×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
74	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	—	2	1	no	—
77	C₂×C₄²×C₅₃	—	2	1	no	—
78	#107×C₅₃	—	2	4	no	—
79	#106×C₅₃	—	2	4	no	—
80	#102×C₅₃	—	2	8	no	—
81	#102×C₅₃	—	2	8	no	—
82	#100×C₅₃	C₂×C₄	1	16	yes	complemented
83	#97×C₅₃	—	2	4	no	—
84	C₄×C₈×C₅₃	—	2	4	no	—
85	C₂⁴×C₅₃	—	2	4	no	—
86	C₂⁴×C₅₃	—	2	2	no	—
87	C₂⁴×C₅₃	—	4	1	no	—
88	C₂⁴×C₅₃	—	4	1	no	—
89	C₂⁴×C₅₃	—	4	2	no	—
90	C₂⁴×C₅₃	—	8	4	no	—
91	C₂⁴×C₅₃	#41	1	1	yes	characteristic
92	C₂×D₄×C₅₃	—	2	4	no	—
93	C₂×D₄×C₅₃	—	2	4	no	—
94	C₂×D₄×C₅₃	—	2	4	no	—
95	C₂×D₄×C₅₃	—	2	4	no	—
96	C₂×D₄×C₅₃	—	2	4	no	—
97	C₂×D₄×C₅₃	—	2	2	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	—	2	2	no	—
102	C₂×D₄×C₅₃	—	4	1	no	—
103	C₂×D₄×C₅₃	—	4	1	no	—
104	C₂×D₄×C₅₃	—	4	2	no	—
105	C₂×D₄×C₅₃	—	4	2	no	—
106	C₂×D₄×C₅₃	—	4	2	no	—
107	C₂×D₄×C₅₃	—	4	2	no	—
108	C₂×D₄×C₅₃	—	4	2	no	—
109	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	#41	1	2	yes	—
119	C₂²×C₄×C₅₃	#41	1	2	yes	—
120	C₂²×C₄×C₅₃	#41	1	2	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	M₁₆×C₅₃	—	2	4	no	—
123	M₁₆×C₅₃	—	2	4	no	—
124	M₁₆×C₅₃	—	2	4	no	—
125	M₁₆×C₅₃	—	2	4	no	—
126	M₁₆×C₅₃	—	2	2	no	—
127	M₁₆×C₅₃	—	2	2	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	2	2	no	—
130	M₁₆×C₅₃	—	4	2	no	—
131	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
132	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂×C₈×C₅₃	—	2	4	no	—
134	C₂×C₈×C₅₃	—	2	4	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	4	2	no	—
141	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
146	C₂²⋊C₄×C₅₃	—	2	2	no	—
147	C₂²⋊C₄×C₅₃	—	2	2	no	—
148	C₂²⋊C₄×C₅₃	—	4	1	no	—
149	C₂²⋊C₄×C₅₃	—	4	1	no	—
150	C₄²×C₅₃	—	2	2	no	—
151	C₄²×C₅₃	—	2	2	no	—
152	C₄²×C₅₃	—	4	1	no	—
153	C₄²×C₅₃	—	4	1	no	—
154	C₂³×C₅₃	—	2	4	no	—
155	C₂³×C₅₃	—	4	2	no	—
156	C₂³×C₅₃	—	4	2	no	—
157	C₂³×C₅₃	—	4	2	no	—
158	C₂³×C₅₃	—	4	4	no	—
159	C₂³×C₅₃	—	4	4	no	—
160	C₂³×C₅₃	—	8	4	no	—
161	C₂³×C₅₃	—	8	4	no	—
162	C₂³×C₅₃	—	8	4	no	—
163	C₂³×C₅₃	—	8	8	no	—
164	C₂³×C₅₃	—	8	8	no	—
165	C₂³×C₅₃	#117	1	2	yes	—
166	C₂³×C₅₃	#121	1	1	yes	characteristic
167	C₂³×C₅₃	#121	1	1	yes	characteristic
168	C₂³×C₅₃	#126	1	2	yes	—
169	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
170	D₄×C₅₃	—	2	4	no	—
171	D₄×C₅₃	—	2	4	no	—
172	D₄×C₅₃	—	4	2	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	4	4	no	—
175	D₄×C₅₃	—	4	4	no	—
176	C₂×C₄×C₅₃	—	2	4	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	8	2	no	—
179	C₂×C₄×C₅₃	—	8	2	no	—
180	C₂×C₄×C₅₃	#117	1	2	yes	—
181	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
186	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
187	C₂×C₄×C₅₃	#123	1	2	yes	—
188	C₂×C₄×C₅₃	#123	1	2	yes	—
189	C₂×C₄×C₅₃	#124	1	2	yes	—
190	C₂×C₄×C₅₃	#124	1	2	yes	—
191	C₂×C₄×C₅₃	#126	1	2	yes	—
192	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₈×C₅₃	—	2	4	no	—
199	C₈×C₅₃	—	2	4	no	—
200	C₈×C₅₃	—	2	4	no	—
201	C₈×C₅₃	—	4	2	no	—
202	#21078	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂²×C₅₃	—	2	2	no	—
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	4	2	no	—
207	C₂²×C₅₃	—	4	2	no	—
208	C₂²×C₅₃	—	4	4	no	—
209	C₂²×C₅₃	—	8	4	no	—
210	C₂²×C₅₃	—	8	8	no	—
211	C₂²×C₅₃	—	8	4	no	—
212	C₂²×C₅₃	—	8	8	no	—
213	C₂²×C₅₃	—	8	4	no	—
214	C₂²×C₅₃	#388	1	2	yes	—
215	C₂²×C₅₃	#388	1	2	yes	—
216	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
217	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
218	C₂²×C₅₃	C₂×#126	1	1	yes	characteristic
219	C₄×C₅₃	—	4	4	no	—
220	C₄×C₅₃	—	8	2	no	—
221	C₄×C₅₃	#388	1	2	yes	—
222	C₄×C₅₃	#388	1	2	yes	—
223	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
224	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
226	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
227	#2911	C₂×C₅₃	1	1	yes	characteristic, complemented
228	C₂×#418	C₂×C₅₃	1	1	yes	characteristic
229	C₂×#418	C₂×C₅₃	1	1	yes	characteristic
230	C₂²×#100	C₂×C₅₃	1	1	yes	characteristic
231	#2873	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1931	C₂×C₅₃	1	8	yes	complemented
233	#1742	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#1738	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₂×C₅₃	—	2	1	no	—
236	C₂×C₅₃	—	2	1	no	—
237	C₂×C₅₃	—	2	2	no	—
238	C₂×C₅₃	—	2	2	no	—
239	C₂×C₅₃	—	8	4	no	—
240	C₂×C₅₃	C₂×#388	1	1	yes	characteristic
241	C₂×C₅₃	#3068	1	2	yes	—
242	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
243	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
244	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
245	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
246	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
249	#428	—	2	2	no	—
250	#428	C₂²×C₅₃	1	4	yes	complemented
251	#427	—	2	2	no	—
252	#418	—	2	2	no	—
253	#418	—	2	2	no	—
254	#418	—	2	2	no	—
255	#418	C₂²×C₅₃	1	4	yes	—
256	C₂×#102	—	2	2	no	—
257	C₂×#102	C₂²×C₅₃	1	4	yes	—
258	C₂×#100	C₄×C₅₃	1	8	yes	complemented
259	C₂×#100	C₂²×C₅₃	1	4	yes	—
260	#405	—	2	2	no	—
261	#405	C₂²×C₅₃	1	4	yes	complemented
262	C₄×M₁₆	—	2	2	no	—
263	#343	C₂²×C₅₃	1	4	yes	complemented
264	#341	C₂²×C₅₃	1	4	yes	complemented
265	C₅₃	#21078	1	1	yes	characteristic, direct product factor, Sylow
266	C₂⁵	—	2	1	no	—
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	—
276	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
277	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
278	C₂×M₁₆	—	2	1	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂×M₁₆	—	2	2	no	—
282	C₂×M₁₆	—	2	2	no	—
283	C₂×M₁₆	—	2	2	no	—
284	C₂×M₁₆	—	2	1	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
288	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
289	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
290	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
291	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
292	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
293	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
294	C₂²×C₈	—	2	1	no	—
295	C₂²×C₈	—	2	1	no	—
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
298	#117	—	2	1	no	—
299	#117	—	2	1	no	—
300	#117	—	2	1	no	—
301	#117	—	2	1	no	—
302	#117	—	2	1	no	—
303	#117	—	2	1	no	—
304	#117	D₄×C₅₃	1	2	yes	—
305	#117	C₂³×C₅₃	1	2	yes	—
306	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	—	2	1	no	—
312	#107	—	2	4	no	—
313	#106	—	2	4	no	—
314	#102	—	2	8	no	—
315	#102	—	2	8	no	—
316	#100	C₂×C₄×C₅₃	1	16	yes	complemented
317	#97	—	2	4	no	—
318	C₄×C₈	—	2	4	no	—
319	C₂⁴	—	2	4	no	—
320	C₂⁴	—	2	2	no	—
321	C₂⁴	—	4	1	no	—
322	C₂⁴	—	4	1	no	—
323	C₂⁴	—	4	2	no	—
324	C₂⁴	—	8	4	no	—
325	C₂⁴	#41×C₅₃	1	1	yes	characteristic
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	4	1	no	—
337	C₂×D₄	—	4	1	no	—
338	C₂×D₄	—	4	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	2	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
351	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
352	C₂²×C₄	#41×C₅₃	1	2	yes	—
353	C₂²×C₄	#41×C₅₃	1	2	yes	—
354	C₂²×C₄	#41×C₅₃	1	2	yes	—
355	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
356	M₁₆	—	2	4	no	—
357	M₁₆	—	2	4	no	—
358	M₁₆	—	2	4	no	—
359	M₁₆	—	2	4	no	—
360	M₁₆	—	2	2	no	—
361	M₁₆	—	2	2	no	—
362	M₁₆	—	2	2	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	4	2	no	—
365	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
366	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
367	C₂×C₈	—	2	4	no	—
368	C₂×C₈	—	2	4	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	4	2	no	—
375	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
380	C₂²⋊C₄	—	2	2	no	—
381	C₂²⋊C₄	—	2	2	no	—
382	C₂²⋊C₄	—	4	1	no	—
383	C₂²⋊C₄	—	4	1	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	4	1	no	—
387	C₄²	—	4	1	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	4	no	—
393	C₂³	—	4	4	no	—
394	C₂³	—	8	4	no	—
395	C₂³	—	8	4	no	—
396	C₂³	—	8	4	no	—
397	C₂³	—	8	8	no	—
398	C₂³	—	8	8	no	—
399	C₂³	#117×C₅₃	1	2	yes	—
400	C₂³	#121×C₅₃	1	1	yes	characteristic
401	C₂³	#121×C₅₃	1	1	yes	characteristic
402	C₂³	#126×C₅₃	1	2	yes	—
403	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
404	D₄	—	2	4	no	—
405	D₄	—	2	4	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	4	no	—
409	D₄	—	4	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	8	2	no	—
413	C₂×C₄	—	8	2	no	—
414	C₂×C₄	#117×C₅₃	1	2	yes	—
415	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
416	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
417	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
418	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
419	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
420	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
421	C₂×C₄	#123×C₅₃	1	2	yes	—
422	C₂×C₄	#123×C₅₃	1	2	yes	—
423	C₂×C₄	#124×C₅₃	1	2	yes	—
424	C₂×C₄	#124×C₅₃	1	2	yes	—
425	C₂×C₄	#126×C₅₃	1	2	yes	—
426	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
427	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
432	C₈	—	2	4	no	—
433	C₈	—	2	4	no	—
434	C₈	—	2	4	no	—
435	C₈	—	4	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	8	4	no	—
443	C₂²	—	8	4	no	—
444	C₂²	—	8	8	no	—
445	C₂²	—	8	8	no	—
446	C₂²	—	8	4	no	—
447	C₂²	C₅₃×#388	1	2	yes	—
448	C₂²	C₅₃×#388	1	2	yes	—
449	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
450	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
451	C₂²	C₂×#126×C₅₃	1	1	yes	characteristic
452	C₄	—	4	4	no	—
453	C₄	—	8	2	no	—
454	C₄	C₅₃×#388	1	2	yes	—
455	C₄	C₅₃×#388	1	2	yes	—
456	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
457	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
458	C₄	C₂×#123×C₅₃	1	1	yes	characteristic
459	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
460	C₂	—	2	1	no	—
461	C₂	—	2	1	no	—
462	C₂	—	2	2	no	—
463	C₂	—	2	2	no	—
464	C₂	—	8	4	no	—
465	C₂	C₂×C₅₃×#388	1	1	yes	characteristic
466	C₂	C₅₃×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2911	C₂	semidirect
5. C₅₃×#2873	C₂	semidirect
6. C₅₃×#1931	C₂	semidirect
7. C₅₃×#1742	C₂	semidirect
8. C₅₃×#1738	C₂	semidirect
17. C₅₃×#428	C₂²	semidirect
25. C₂×#100×C₅₃	C₄	semidirect
28. C₅₃×#405	C₂²	semidirect
30. C₅₃×#343	C₂²	semidirect
31. C₅₃×#341	C₂²	semidirect
82. #100×C₅₃	C₂×C₄	semidirect
202. #21078	C₅₃	direct (N × H)
227. #2911	C₂×C₅₃	semidirect
231. #2873	C₂×C₅₃	semidirect
232. #1931	C₂×C₅₃	semidirect
233. #1742	C₂×C₅₃	semidirect
234. #1738	C₂×C₅₃	semidirect
250. #428	C₂²×C₅₃	semidirect
258. C₂×#100	C₄×C₅₃	semidirect
261. #405	C₂²×C₅₃	semidirect
263. #343	C₂²×C₅₃	semidirect
264. #341	C₂²×C₅₃	semidirect
265. C₅₃	#21078	direct (N × H)
316. #100	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2911	2
C₂×C₅₃×#418	2
C₂²×#100×C₅₃	2
C₅₃×#2873	2
C₅₃×#1931	2
C₅₃×#1742	2
C₅₃×#1738	2
#21078	53

Subgroups — C53×#21078 #27228590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21078 #27228590