Subgroups — C₅₃×#34244 #27241756

Order 13568.

2186 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2374	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2372	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2371	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2355	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2354	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2352	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2352	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2333	C₂	1	2	yes	maximal
10	C₅₃×#2332	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2314	C₂	1	2	yes	maximal, complemented
12	C₄×#126×C₅₃	C₂	1	4	yes	maximal, complemented
13	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#2274	C₂	1	2	yes	maximal, complemented
15	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#529	C₄	1	16	yes	complemented
17	C₂×#126×C₅₃	C₂²	1	2	yes	—
18	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#123×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
26	C₂×#117×C₅₃	C₂²	1	2	yes	—
27	C₂×#117×C₅₃	C₂²	1	2	yes	—
28	C₂×#117×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#391	C₂²	1	4	yes	—
32	C₅₃×#390	C₂²	1	4	yes	—
33	C₅₃×#388	C₂²	1	4	yes	—
34	C₅₃×#388	C₂²	1	4	yes	—
35	C₅₃×#388	C₂²	1	4	yes	—
36	C₅₃×#388	C₂²	1	4	yes	—
37	C₅₃×#388	C₂²	1	4	yes	—
38	C₅₃×#388	C₂²	1	4	yes	—
39	C₅₃×#387	C₂²	1	4	yes	—
40	C₅₃×#387	C₂²	1	4	yes	—
41	C₅₃×#387	C₂²	1	4	yes	—
42	C₅₃×#387	C₂²	1	4	yes	—
43	C₅₃×#387	C₂²	1	4	yes	—
44	C₅₃×#387	C₂²	1	4	yes	—
45	C₅₃×#386	C₂²	1	4	yes	—
46	C₅₃×#386	C₂²	1	4	yes	—
47	C₅₃×#384	C₂²	1	2	yes	—
48	C₅₃×#383	C₂²	1	2	yes	—
49	C₅₃×#382	C₂²	1	2	yes	—
50	C₅₃×#382	C₂²	1	2	yes	—
51	C₅₃×#381	C₂²	1	2	yes	—
52	C₅₃×#380	C₂²	1	1	yes	characteristic
53	C₅₃×#380	C₂²	1	1	yes	characteristic
54	C₅₃×#380	C₂²	1	2	yes	—
55	C₅₃×#380	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
64	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
65	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
66	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
69	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
72	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
73	C₅₃×#376	C₂²	1	2	yes	—
74	C₅₃×#376	C₂²	1	2	yes	—
75	C₄³×C₅₃	C₂²	1	2	yes	—
76	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
78	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
81	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
82	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
84	#126×C₅₃	C₂×C₄	1	16	yes	complemented
85	#125×C₅₃	C₂×C₄	1	8	yes	complemented
86	#124×C₅₃	C₂×C₄	1	8	yes	—
87	#123×C₅₃	C₂×C₄	1	16	yes	complemented
88	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
89	#121×C₅₃	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
93	#117×C₅₃	—	2	8	no	—
94	#117×C₅₃	—	2	8	no	—
95	#117×C₅₃	C₂×C₄	1	16	yes	complemented
96	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
122	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
123	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
124	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
125	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
126	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
128	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
129	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
130	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
135	#95×C₅₃	C₂³	1	2	yes	—
136	#95×C₅₃	C₂³	1	2	yes	—
137	#95×C₅₃	C₂³	1	4	yes	—
138	#95×C₅₃	C₂³	1	2	yes	—
139	#95×C₅₃	C₂³	1	2	yes	—
140	#95×C₅₃	C₂³	1	4	yes	—
141	#95×C₅₃	C₂³	1	4	yes	—
142	#95×C₅₃	C₂³	1	4	yes	—
143	#95×C₅₃	C₂³	1	4	yes	—
144	#95×C₅₃	C₂³	1	4	yes	—
145	#95×C₅₃	C₂³	1	4	yes	—
146	#95×C₅₃	C₂³	1	4	yes	—
147	#95×C₅₃	C₂³	1	4	yes	—
148	#95×C₅₃	C₂³	1	4	yes	—
149	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
150	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₂×Q₈×C₅₃	—	4	1	no	—
152	C₂×Q₈×C₅₃	—	4	1	no	—
153	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₅₃	—	4	1	no	—
155	C₂×D₄×C₅₃	—	4	1	no	—
156	C₂×D₄×C₅₃	—	4	2	no	—
157	C₂×D₄×C₅₃	—	4	2	no	—
158	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
159	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	2	4	no	—
162	C₂²×C₄×C₅₃	—	2	4	no	—
163	C₂²×C₄×C₅₃	—	2	8	no	—
164	C₂²×C₄×C₅₃	—	4	1	no	—
165	C₂²×C₄×C₅₃	—	4	1	no	—
166	C₂²×C₄×C₅₃	—	4	2	no	—
167	C₂²×C₄×C₅₃	—	4	4	no	—
168	C₂²×C₄×C₅₃	—	4	4	no	—
169	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
171	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₅₃	—	2	8	no	—
187	C₄⋊C₄×C₅₃	—	2	4	no	—
188	C₄⋊C₄×C₅₃	—	2	4	no	—
189	C₄⋊C₄×C₅₃	—	2	4	no	—
190	C₄⋊C₄×C₅₃	—	4	4	no	—
191	C₄⋊C₄×C₅₃	—	4	4	no	—
192	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
194	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
195	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
196	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
197	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
198	C₄⋊C₄×C₅₃	#41	1	8	yes	—
199	C₂²⋊C₄×C₅₃	—	2	8	no	—
200	C₂²⋊C₄×C₅₃	—	2	8	no	—
201	C₂²⋊C₄×C₅₃	—	4	2	no	—
202	C₂²⋊C₄×C₅₃	—	4	4	no	—
203	C₂²⋊C₄×C₅₃	—	4	4	no	—
204	C₂²⋊C₄×C₅₃	—	4	2	no	—
205	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
207	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
208	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
209	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
210	C₄²×C₅₃	—	2	8	no	—
211	C₄²×C₅₃	—	2	8	no	—
212	C₄²×C₅₃	—	4	2	no	—
213	C₄²×C₅₃	—	4	4	no	—
214	C₄²×C₅₃	—	4	4	no	—
215	C₄²×C₅₃	—	4	4	no	—
216	C₄²×C₅₃	—	4	4	no	—
217	C₄²×C₅₃	—	4	2	no	—
218	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
219	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
220	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
221	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
222	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
223	C₂³×C₅₃	—	2	2	no	—
224	C₂³×C₅₃	—	2	2	no	—
225	C₂³×C₅₃	—	2	2	no	—
226	C₂³×C₅₃	—	2	2	no	—
227	C₂³×C₅₃	—	2	2	no	—
228	C₂³×C₅₃	—	2	2	no	—
229	C₂³×C₅₃	—	2	1	no	—
230	C₂³×C₅₃	—	2	1	no	—
231	C₂³×C₅₃	—	2	2	no	—
232	C₂³×C₅₃	—	2	2	no	—
233	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
234	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
235	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
236	Q₈×C₅₃	—	4	4	no	—
237	D₄×C₅₃	—	4	4	no	—
238	D₄×C₅₃	—	4	8	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	2	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	4	no	—
249	C₂×C₄×C₅₃	—	2	4	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	2	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	2	no	—
254	C₂×C₄×C₅₃	—	2	16	no	—
255	C₂×C₄×C₅₃	—	2	8	no	—
256	C₂×C₄×C₅₃	—	2	4	no	—
257	C₂×C₄×C₅₃	—	2	4	no	—
258	C₂×C₄×C₅₃	—	2	4	no	—
259	C₂×C₄×C₅₃	—	2	4	no	—
260	C₂×C₄×C₅₃	—	2	4	no	—
261	C₂×C₄×C₅₃	—	2	4	no	—
262	C₂×C₄×C₅₃	—	2	4	no	—
263	C₂×C₄×C₅₃	—	2	4	no	—
264	C₂×C₄×C₅₃	—	2	4	no	—
265	C₂×C₄×C₅₃	—	4	4	no	—
266	C₂×C₄×C₅₃	—	4	4	no	—
267	C₂×C₄×C₅₃	—	8	8	no	—
268	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
270	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
271	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
272	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
273	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
274	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
275	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
276	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
277	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
278	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
279	#34244	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₅₃	—	2	4	no	—
281	C₂²×C₅₃	—	2	2	no	—
282	C₂²×C₅₃	—	2	2	no	—
283	C₂²×C₅₃	—	2	2	no	—
284	C₂²×C₅₃	—	2	4	no	—
285	C₂²×C₅₃	—	2	4	no	—
286	C₂²×C₅₃	—	4	2	no	—
287	C₂²×C₅₃	—	4	2	no	—
288	C₂²×C₅₃	—	4	2	no	—
289	C₂²×C₅₃	—	4	2	no	—
290	C₂²×C₅₃	—	4	2	no	—
291	C₂²×C₅₃	—	4	2	no	—
292	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
293	C₂²×C₅₃	C₂²×#41	1	2	yes	—
294	C₂²×C₅₃	#585	1	1	yes	characteristic
295	C₂²×C₅₃	#585	1	2	yes	—
296	C₂²×C₅₃	#585	1	1	yes	characteristic
297	C₄×C₅₃	—	2	2	no	—
298	C₄×C₅₃	—	2	4	no	—
299	C₄×C₅₃	—	2	2	no	—
300	C₄×C₅₃	—	2	2	no	—
301	C₄×C₅₃	—	2	2	no	—
302	C₄×C₅₃	—	2	2	no	—
303	C₄×C₅₃	—	2	4	no	—
304	C₄×C₅₃	—	2	4	no	—
305	C₄×C₅₃	—	2	8	no	—
306	C₄×C₅₃	—	4	4	no	—
307	C₄×C₅₃	—	4	4	no	—
308	C₄×C₅₃	—	4	4	no	—
309	C₄×C₅₃	C₄×#41	1	4	yes	—
310	C₂×#529	C₂×C₅₃	1	1	yes	characteristic
311	#2374	C₂×C₅₃	1	4	yes	complemented
312	#2372	C₂×C₅₃	1	2	yes	complemented
313	#2371	C₂×C₅₃	1	2	yes	complemented
314	#2355	C₂×C₅₃	1	2	yes	complemented
315	#2354	C₂×C₅₃	1	2	yes	complemented
316	#2352	C₂×C₅₃	1	2	yes	complemented
317	#2352	C₂×C₅₃	1	2	yes	complemented
318	#2333	C₂×C₅₃	1	2	yes	—
319	#2332	C₂×C₅₃	1	2	yes	complemented
320	#2314	C₂×C₅₃	1	2	yes	complemented
321	C₄×#126	C₂×C₅₃	1	4	yes	complemented
322	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
323	#2274	C₂×C₅₃	1	2	yes	complemented
324	C₄×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
325	C₂×C₅₃	—	4	2	no	—
326	C₂×C₅₃	—	4	4	no	—
327	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
328	C₂×C₅₃	#3428	1	1	yes	characteristic
329	C₂×C₅₃	#3428	1	1	yes	characteristic
330	C₂×C₅₃	#3521	1	2	yes	—
331	C₂×C₅₃	#3524	1	2	yes	—
332	#529	C₄×C₅₃	1	16	yes	complemented
333	C₂×#126	C₂²×C₅₃	1	2	yes	—
334	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
335	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
336	C₂×#123	C₂²×C₅₃	1	2	yes	—
337	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
338	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
339	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
342	C₂×#117	C₂²×C₅₃	1	2	yes	—
343	C₂×#117	C₂²×C₅₃	1	2	yes	—
344	C₂×#117	C₂²×C₅₃	1	2	yes	—
345	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
346	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
347	#391	C₂²×C₅₃	1	4	yes	—
348	#390	C₂²×C₅₃	1	4	yes	—
349	#388	C₂²×C₅₃	1	4	yes	—
350	#388	C₂²×C₅₃	1	4	yes	—
351	#388	C₂²×C₅₃	1	4	yes	—
352	#388	C₂²×C₅₃	1	4	yes	—
353	#388	C₂²×C₅₃	1	4	yes	—
354	#388	C₂²×C₅₃	1	4	yes	—
355	#387	C₂²×C₅₃	1	4	yes	—
356	#387	C₂²×C₅₃	1	4	yes	—
357	#387	C₂²×C₅₃	1	4	yes	—
358	#387	C₂²×C₅₃	1	4	yes	—
359	#387	C₂²×C₅₃	1	4	yes	—
360	#387	C₂²×C₅₃	1	4	yes	—
361	#386	C₂²×C₅₃	1	4	yes	—
362	#386	C₂²×C₅₃	1	4	yes	—
363	#384	C₂²×C₅₃	1	2	yes	—
364	#383	C₂²×C₅₃	1	2	yes	—
365	#382	C₂²×C₅₃	1	2	yes	—
366	#382	C₂²×C₅₃	1	2	yes	—
367	#381	C₂²×C₅₃	1	2	yes	—
368	#380	C₂²×C₅₃	1	2	yes	—
369	#380	C₂²×C₅₃	1	2	yes	—
370	#380	C₂²×C₅₃	1	1	yes	characteristic
371	#380	C₂²×C₅₃	1	1	yes	characteristic
372	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
373	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
374	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
375	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
376	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
377	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
378	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
379	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
380	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
381	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
382	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
383	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
384	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
386	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
387	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
388	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
389	#376	C₂²×C₅₃	1	2	yes	—
390	#376	C₂²×C₅₃	1	2	yes	—
391	C₄³	C₂²×C₅₃	1	2	yes	—
392	C₅₃	#34244	1	1	yes	characteristic, direct product factor, Sylow
393	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
394	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
395	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
397	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
399	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
401	#126	C₂×C₄×C₅₃	1	16	yes	complemented
402	#125	C₂×C₄×C₅₃	1	8	yes	complemented
403	#124	C₂×C₄×C₅₃	1	8	yes	—
404	#123	C₂×C₄×C₅₃	1	16	yes	complemented
405	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
406	#121	C₂×C₄×C₅₃	1	8	yes	—
407	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
408	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
409	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
410	#117	—	2	8	no	—
411	#117	—	2	8	no	—
412	#117	C₂×C₄×C₅₃	1	16	yes	complemented
413	C₂×C₄⋊C₄	—	2	8	no	—
414	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
420	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
438	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
439	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
440	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
441	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
442	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
443	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
444	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
445	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
446	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
447	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
449	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
452	#95	C₂³×C₅₃	1	2	yes	—
453	#95	C₂³×C₅₃	1	2	yes	—
454	#95	C₂³×C₅₃	1	4	yes	—
455	#95	C₂³×C₅₃	1	2	yes	—
456	#95	C₂³×C₅₃	1	2	yes	—
457	#95	C₂³×C₅₃	1	4	yes	—
458	#95	C₂³×C₅₃	1	4	yes	—
459	#95	C₂³×C₅₃	1	4	yes	—
460	#95	C₂³×C₅₃	1	4	yes	—
461	#95	C₂³×C₅₃	1	4	yes	—
462	#95	C₂³×C₅₃	1	4	yes	—
463	#95	C₂³×C₅₃	1	4	yes	—
464	#95	C₂³×C₅₃	1	4	yes	—
465	#95	C₂³×C₅₃	1	4	yes	—
466	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
467	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
468	C₂×Q₈	—	4	1	no	—
469	C₂×Q₈	—	4	1	no	—
470	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	—	4	2	no	—
473	C₂×D₄	—	4	1	no	—
474	C₂×D₄	—	4	1	no	—
475	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
476	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	8	no	—
481	C₂²×C₄	—	4	2	no	—
482	C₂²×C₄	—	4	1	no	—
483	C₂²×C₄	—	4	1	no	—
484	C₂²×C₄	—	4	4	no	—
485	C₂²×C₄	—	4	4	no	—
486	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
487	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
488	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
489	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
492	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
494	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	4	4	no	—
508	C₄⋊C₄	—	4	4	no	—
509	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
510	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
511	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
512	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
513	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
514	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
515	C₄⋊C₄	#41×C₅₃	1	8	yes	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	4	4	no	—
519	C₂²⋊C₄	—	4	4	no	—
520	C₂²⋊C₄	—	4	2	no	—
521	C₂²⋊C₄	—	4	2	no	—
522	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
523	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
524	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
525	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
526	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
527	C₄²	—	2	8	no	—
528	C₄²	—	2	8	no	—
529	C₄²	—	4	2	no	—
530	C₄²	—	4	4	no	—
531	C₄²	—	4	4	no	—
532	C₄²	—	4	4	no	—
533	C₄²	—	4	4	no	—
534	C₄²	—	4	2	no	—
535	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
536	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
537	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
538	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
539	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
551	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
552	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
553	Q₈	—	4	4	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	8	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	16	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	8	8	no	—
585	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
586	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
587	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
588	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
589	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
590	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
591	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
592	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
593	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
594	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
595	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	4	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
609	C₂²	C₂²×#41×C₅₃	1	2	yes	—
610	C₂²	C₅₃×#585	1	1	yes	characteristic
611	C₂²	C₅₃×#585	1	2	yes	—
612	C₂²	C₅₃×#585	1	1	yes	characteristic
613	C₄	—	2	8	no	—
614	C₄	—	2	2	no	—
615	C₄	—	2	4	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	4	no	—
621	C₄	—	2	4	no	—
622	C₄	—	4	4	no	—
623	C₄	—	4	4	no	—
624	C₄	—	4	4	no	—
625	C₄	C₄×#41×C₅₃	1	4	yes	—
626	C₂	—	4	2	no	—
627	C₂	—	4	4	no	—
628	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
629	C₂	C₅₃×#3428	1	1	yes	characteristic
630	C₂	C₅₃×#3428	1	1	yes	characteristic
631	C₂	C₅₃×#3521	1	2	yes	—
632	C₂	C₅₃×#3524	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×#121×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2374	C₂	semidirect
3. C₅₃×#2372	C₂	semidirect
4. C₅₃×#2371	C₂	semidirect
5. C₅₃×#2355	C₂	semidirect
6. C₅₃×#2354	C₂	semidirect
7. C₅₃×#2352	C₂	semidirect
8. C₅₃×#2352	C₂	semidirect
10. C₅₃×#2332	C₂	semidirect
11. C₅₃×#2314	C₂	semidirect
12. C₄×#126×C₅₃	C₂	semidirect
13. C₅₃×#2283	C₂	semidirect
14. C₅₃×#2274	C₂	semidirect
15. C₄×#117×C₅₃	C₂	semidirect
16. C₅₃×#529	C₄	semidirect
84. #126×C₅₃	C₂×C₄	semidirect
85. #125×C₅₃	C₂×C₄	semidirect
87. #123×C₅₃	C₂×C₄	semidirect
88. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
90. C₄×Q₈×C₅₃	C₂×C₄	semidirect
92. C₄×D₄×C₅₃	C₂×C₄	semidirect
95. #117×C₅₃	C₂×C₄	semidirect
121. C₂×C₄²×C₅₃	C₂×C₄	semidirect
279. #34244	C₅₃	direct (N × H)
311. #2374	C₂×C₅₃	semidirect
312. #2372	C₂×C₅₃	semidirect
313. #2371	C₂×C₅₃	semidirect
314. #2355	C₂×C₅₃	semidirect
315. #2354	C₂×C₅₃	semidirect
316. #2352	C₂×C₅₃	semidirect
317. #2352	C₂×C₅₃	semidirect
319. #2332	C₂×C₅₃	semidirect
320. #2314	C₂×C₅₃	semidirect
321. C₄×#126	C₂×C₅₃	semidirect
322. #2283	C₂×C₅₃	semidirect
323. #2274	C₂×C₅₃	semidirect
324. C₄×#117	C₂×C₅₃	semidirect
332. #529	C₄×C₅₃	semidirect
392. C₅₃	#34244	direct (N × H)
401. #126	C₂×C₄×C₅₃	semidirect
402. #125	C₂×C₄×C₅₃	semidirect
404. #123	C₂×C₄×C₅₃	semidirect
405. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
407. C₄×Q₈	C₂×C₄×C₅₃	semidirect
409. C₄×D₄	C₂×C₄×C₅₃	semidirect
412. #117	C₂×C₄×C₅₃	semidirect
438. C₂×C₄²	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
C₂×C₅₃×#529	2
C₅₃×#2374	2
C₅₃×#2372	2
C₅₃×#2371	2
C₅₃×#2355	2
C₅₃×#2354	2
C₅₃×#2352	2
C₅₃×#2333	2
C₅₃×#2332	2
C₅₃×#2314	2
C₄×#126×C₅₃	2
C₅₃×#2283	2
C₅₃×#2274	2
C₄×#117×C₅₃	2
#34244	53

Subgroups — C53×#34244 #27241756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34244 #27241756