Subgroups — C₅₃×#34083 #27241595

Order 13568.

2346 conjugacy classes of subgroups.

The subgroup lattice has 636 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×#2337	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2336	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2334	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2329	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2325	C₂	1	2	yes	maximal
7	C₅₃×#2322	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2298	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2296	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2295	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2290	C₂	1	2	yes	maximal, complemented
12	C₅₃×#2290	C₂	1	2	yes	maximal, complemented
13	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄²×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×C₄⋊C₄×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	4	yes	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
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	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
33	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#391	C₂²	1	4	yes	—
35	C₅₃×#389	C₂²	1	4	yes	—
36	C₅₃×#389	C₂²	1	4	yes	—
37	C₅₃×#389	C₂²	1	4	yes	—
38	C₅₃×#388	C₂²	1	4	yes	—
39	C₅₃×#388	C₂²	1	4	yes	—
40	C₅₃×#388	C₂²	1	4	yes	—
41	C₅₃×#388	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×#387	C₂²	1	4	yes	—
46	C₅₃×#387	C₂²	1	4	yes	—
47	C₅₃×#387	C₂²	1	4	yes	—
48	C₅₃×#386	C₂²	1	4	yes	—
49	C₅₃×#386	C₂²	1	4	yes	—
50	C₅₃×#385	C₂²	1	4	yes	—
51	C₅₃×#385	C₂²	1	4	yes	—
52	C₅₃×#385	C₂²	1	4	yes	—
53	C₅₃×#385	C₂²	1	4	yes	—
54	C₅₃×#383	C₂²	1	2	yes	—
55	C₅₃×#382	C₂²	1	2	yes	—
56	C₅₃×#381	C₂²	1	2	yes	—
57	C₅₃×#380	C₂²	1	2	yes	—
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	2	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	1	yes	characteristic
65	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₅₃×#376	C₂²	1	2	yes	—
73	C₅₃×#376	C₂²	1	2	yes	—
74	C₂×#95×C₅₃	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	1	yes	characteristic
81	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
84	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
86	#126×C₅₃	C₂×C₄	1	16	yes	complemented
87	#125×C₅₃	C₂×C₄	1	8	yes	complemented
88	#124×C₅₃	C₂×C₄	1	8	yes	complemented
89	#123×C₅₃	C₂×C₄	1	16	yes	complemented
90	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
91	#121×C₅₃	C₂×C₄	1	8	yes	—
92	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
93	C₄×D₄×C₅₃	—	2	16	no	—
94	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
95	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	complemented
96	#117×C₅₃	—	4	4	no	—
97	#117×C₅₃	C₂×C₄	1	16	yes	complemented
98	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	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	4	yes	—
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	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	—	4	2	no	—
122	C₂×C₄²×C₅₃	—	4	2	no	—
123	C₂×C₄²×C₅₃	—	4	4	no	—
124	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
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	2	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	2	yes	—
133	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
136	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
139	#95×C₅₃	C₂³	1	4	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	#95×C₅₃	C₂³	1	2	yes	—
150	#95×C₅₃	C₂³	1	2	yes	—
151	#95×C₅₃	C₂³	1	4	yes	—
152	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
153	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₂×Q₈×C₅₃	—	4	2	no	—
155	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
156	C₂×D₄×C₅₃	—	2	8	no	—
157	C₂×D₄×C₅₃	—	4	2	no	—
158	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
159	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	2	2	no	—
162	C₂²×C₄×C₅₃	—	2	4	no	—
163	C₂²×C₄×C₅₃	—	2	2	no	—
164	C₂²×C₄×C₅₃	—	2	4	no	—
165	C₂²×C₄×C₅₃	—	2	4	no	—
166	C₂²×C₄×C₅₃	—	2	8	no	—
167	C₂²×C₄×C₅₃	—	4	2	no	—
168	C₂²×C₄×C₅₃	—	4	4	no	—
169	C₂²×C₄×C₅₃	—	4	2	no	—
170	C₂²×C₄×C₅₃	—	4	2	no	—
171	C₂²×C₄×C₅₃	—	4	4	no	—
172	C₂²×C₄×C₅₃	—	4	2	no	—
173	C₂²×C₄×C₅₃	—	4	2	no	—
174	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
175	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
176	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
177	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
178	C₂²×C₄×C₅₃	#41	1	8	yes	—
179	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
191	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
192	C₄⋊C₄×C₅₃	—	2	8	no	—
193	C₄⋊C₄×C₅₃	—	2	8	no	—
194	C₄⋊C₄×C₅₃	—	2	8	no	—
195	C₄⋊C₄×C₅₃	—	4	2	no	—
196	C₄⋊C₄×C₅₃	—	4	2	no	—
197	C₄⋊C₄×C₅₃	—	4	4	no	—
198	C₄⋊C₄×C₅₃	—	4	4	no	—
199	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
200	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
201	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
202	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
203	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
204	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
205	C₂²⋊C₄×C₅₃	—	2	8	no	—
206	C₂²⋊C₄×C₅₃	—	2	8	no	—
207	C₂²⋊C₄×C₅₃	—	4	4	no	—
208	C₂²⋊C₄×C₅₃	—	4	4	no	—
209	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
210	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
211	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
212	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
213	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
214	C₄²×C₅₃	—	2	8	no	—
215	C₄²×C₅₃	—	2	4	no	—
216	C₄²×C₅₃	—	2	4	no	—
217	C₄²×C₅₃	—	2	4	no	—
218	C₄²×C₅₃	—	2	4	no	—
219	C₄²×C₅₃	—	4	2	no	—
220	C₄²×C₅₃	—	4	2	no	—
221	C₄²×C₅₃	—	4	4	no	—
222	C₄²×C₅₃	—	4	4	no	—
223	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
224	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
225	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
226	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
227	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
228	C₂³×C₅₃	—	2	1	no	—
229	C₂³×C₅₃	—	2	1	no	—
230	C₂³×C₅₃	—	2	2	no	—
231	C₂³×C₅₃	—	2	4	no	—
232	C₂³×C₅₃	—	2	2	no	—
233	C₂³×C₅₃	—	2	2	no	—
234	C₂³×C₅₃	—	2	4	no	—
235	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
236	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
237	C₂³×C₅₃	C₂×#41	1	4	yes	—
238	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
239	Q₈×C₅₃	—	4	4	no	—
240	D₄×C₅₃	—	2	16	no	—
241	D₄×C₅₃	—	4	4	no	—
242	C₂×C₄×C₅₃	—	2	8	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	2	2	no	—
245	C₂×C₄×C₅₃	—	2	4	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	2	no	—
248	C₂×C₄×C₅₃	—	2	4	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	4	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	2	no	—
254	C₂×C₄×C₅₃	—	2	4	no	—
255	C₂×C₄×C₅₃	—	2	8	no	—
256	C₂×C₄×C₅₃	—	2	16	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	—	2	4	no	—
266	C₂×C₄×C₅₃	—	4	4	no	—
267	C₂×C₄×C₅₃	—	4	8	no	—
268	C₂×C₄×C₅₃	—	4	16	no	—
269	C₂×C₄×C₅₃	—	4	8	no	—
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	2	yes	—
274	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
275	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
276	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
277	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
278	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
279	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
280	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
281	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
282	#34083	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂²×C₅₃	—	2	2	no	—
284	C₂²×C₅₃	—	2	2	no	—
285	C₂²×C₅₃	—	2	4	no	—
286	C₂²×C₅₃	—	2	4	no	—
287	C₂²×C₅₃	—	2	4	no	—
288	C₂²×C₅₃	—	2	4	no	—
289	C₂²×C₅₃	—	2	4	no	—
290	C₂²×C₅₃	—	4	4	no	—
291	C₂²×C₅₃	—	4	4	no	—
292	C₂²×C₅₃	C₄×#41	1	4	yes	—
293	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
294	C₂²×C₅₃	C₂²×#41	1	2	yes	—
295	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
296	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
297	C₂²×C₅₃	#585	1	2	yes	—
298	C₄×C₅₃	—	2	2	no	—
299	C₄×C₅₃	—	2	2	no	—
300	C₄×C₅₃	—	2	4	no	—
301	C₄×C₅₃	—	2	2	no	—
302	C₄×C₅₃	—	2	4	no	—
303	C₄×C₅₃	—	2	2	no	—
304	C₄×C₅₃	—	2	2	no	—
305	C₄×C₅₃	—	2	4	no	—
306	C₄×C₅₃	—	2	8	no	—
307	C₄×C₅₃	—	4	4	no	—
308	C₄×C₅₃	—	4	4	no	—
309	C₄×C₅₃	—	4	4	no	—
310	C₄×C₅₃	#520	1	4	yes	—
311	C₂×#529	C₂×C₅₃	1	1	yes	characteristic
312	#2337	C₂×C₅₃	1	2	yes	complemented
313	#2336	C₂×C₅₃	1	2	yes	complemented
314	#2334	C₂×C₅₃	1	4	yes	complemented
315	#2329	C₂×C₅₃	1	2	yes	complemented
316	#2325	C₂×C₅₃	1	2	yes	—
317	#2322	C₂×C₅₃	1	4	yes	complemented
318	#2298	C₂×C₅₃	1	2	yes	complemented
319	#2296	C₂×C₅₃	1	2	yes	complemented
320	#2295	C₂×C₅₃	1	2	yes	complemented
321	#2290	C₂×C₅₃	1	2	yes	complemented
322	#2290	C₂×C₅₃	1	2	yes	complemented
323	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
324	C₄²×D₄	C₂×C₅₃	1	2	yes	complemented
325	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
326	C₂×C₅₃	—	2	4	no	—
327	C₂×C₅₃	—	4	4	no	—
328	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
329	C₂×C₅₃	C₂×#520	1	1	yes	characteristic
330	C₂×C₅₃	#3428	1	1	yes	characteristic
331	C₂×C₅₃	#3459	1	2	yes	—
332	C₂×C₅₃	#3461	1	2	yes	—
333	#529	C₄×C₅₃	1	16	yes	complemented
334	C₂×#126	C₂²×C₅₃	1	2	yes	—
335	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
336	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
337	C₂×#123	C₂²×C₅₃	1	2	yes	—
338	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
339	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
340	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
342	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
343	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
344	C₂×#117	C₂²×C₅₃	1	2	yes	—
345	C₂×#117	C₂²×C₅₃	1	2	yes	—
346	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
347	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
348	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
349	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
350	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
351	#391	C₂²×C₅₃	1	4	yes	—
352	#389	C₂²×C₅₃	1	4	yes	—
353	#389	C₂²×C₅₃	1	4	yes	—
354	#389	C₂²×C₅₃	1	4	yes	—
355	#388	C₂²×C₅₃	1	4	yes	—
356	#388	C₂²×C₅₃	1	4	yes	—
357	#388	C₂²×C₅₃	1	4	yes	—
358	#388	C₂²×C₅₃	1	4	yes	—
359	#387	C₂²×C₅₃	1	4	yes	—
360	#387	C₂²×C₅₃	1	4	yes	—
361	#387	C₂²×C₅₃	1	4	yes	—
362	#387	C₂²×C₅₃	1	4	yes	—
363	#387	C₂²×C₅₃	1	4	yes	—
364	#387	C₂²×C₅₃	1	4	yes	—
365	#386	C₂²×C₅₃	1	4	yes	—
366	#386	C₂²×C₅₃	1	4	yes	—
367	#385	C₂²×C₅₃	1	4	yes	—
368	#385	C₂²×C₅₃	1	4	yes	—
369	#385	C₂²×C₅₃	1	4	yes	—
370	#385	C₂²×C₅₃	1	4	yes	—
371	#383	C₂²×C₅₃	1	2	yes	—
372	#382	C₂²×C₅₃	1	2	yes	—
373	#381	C₂²×C₅₃	1	2	yes	—
374	#380	C₂²×C₅₃	1	2	yes	—
375	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
376	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	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	2	yes	—
381	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
382	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
383	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
384	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
386	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	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	C₂×#95	C₂²×C₅₃	1	2	yes	—
392	C₄³	C₂²×C₅₃	1	2	yes	—
393	C₅₃	#34083	1	1	yes	characteristic, direct product factor, Sylow
394	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
395	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
396	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
399	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
401	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
402	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
403	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
404	#126	C₂×C₄×C₅₃	1	16	yes	complemented
405	#125	C₂×C₄×C₅₃	1	8	yes	complemented
406	#124	C₂×C₄×C₅₃	1	8	yes	complemented
407	#123	C₂×C₄×C₅₃	1	16	yes	complemented
408	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
409	#121	C₂×C₄×C₅₃	1	8	yes	—
410	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
411	C₄×D₄	—	2	16	no	—
412	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	complemented
413	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
414	#117	—	4	4	no	—
415	#117	C₂×C₄×C₅₃	1	16	yes	complemented
416	C₂×C₄⋊C₄	—	2	8	no	—
417	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	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	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
428	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	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	4	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×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
439	C₂×C₄²	—	4	2	no	—
440	C₂×C₄²	—	4	4	no	—
441	C₂×C₄²	—	4	2	no	—
442	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
443	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
444	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
445	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
446	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
447	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
448	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
449	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
450	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
451	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
452	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
453	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
454	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
456	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
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	#95	C₂³×C₅₃	1	4	yes	—
467	#95	C₂³×C₅₃	1	2	yes	—
468	#95	C₂³×C₅₃	1	2	yes	—
469	#95	C₂³×C₅₃	1	4	yes	—
470	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
471	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
472	C₂×Q₈	—	4	2	no	—
473	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
474	C₂×D₄	—	2	8	no	—
475	C₂×D₄	—	4	2	no	—
476	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
477	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	8	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	4	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	4	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
493	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
494	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
495	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
496	C₂²×C₄	#41×C₅₃	1	8	yes	—
497	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
500	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
506	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
508	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
509	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	8	no	—
512	C₄⋊C₄	—	2	8	no	—
513	C₄⋊C₄	—	4	4	no	—
514	C₄⋊C₄	—	4	4	no	—
515	C₄⋊C₄	—	4	2	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
518	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
519	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
520	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
521	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
522	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
523	C₂²⋊C₄	—	2	8	no	—
524	C₂²⋊C₄	—	2	8	no	—
525	C₂²⋊C₄	—	4	4	no	—
526	C₂²⋊C₄	—	4	4	no	—
527	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
528	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
529	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
530	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
531	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
532	C₄²	—	2	8	no	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	4	no	—
535	C₄²	—	2	4	no	—
536	C₄²	—	2	4	no	—
537	C₄²	—	4	2	no	—
538	C₄²	—	4	4	no	—
539	C₄²	—	4	2	no	—
540	C₄²	—	4	4	no	—
541	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
542	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
543	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
544	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
545	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	4	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	4	no	—
553	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
554	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
555	C₂³	C₂×#41×C₅₃	1	4	yes	—
556	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
557	Q₈	—	4	4	no	—
558	D₄	—	2	16	no	—
559	D₄	—	4	4	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	16	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	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	4	16	no	—
587	C₂×C₄	—	4	8	no	—
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	2	yes	—
591	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
592	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
593	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
594	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
595	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
596	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
597	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
598	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
599	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	4	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	2	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	4	no	—
609	C₂²	C₄×#41×C₅₃	1	4	yes	—
610	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
611	C₂²	C₂²×#41×C₅₃	1	2	yes	—
612	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
613	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
614	C₂²	C₅₃×#585	1	2	yes	—
615	C₄	—	2	8	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	4	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	4	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	4	no	—
624	C₄	—	4	4	no	—
625	C₄	—	4	4	no	—
626	C₄	—	4	4	no	—
627	C₄	C₅₃×#520	1	4	yes	—
628	C₂	—	2	4	no	—
629	C₂	—	4	4	no	—
630	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
631	C₂	C₂×C₅₃×#520	1	1	yes	characteristic
632	C₂	C₅₃×#3428	1	1	yes	characteristic
633	C₂	C₅₃×#3459	1	2	yes	—
634	C₂	C₅₃×#3461	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2337	C₂	semidirect
3. C₅₃×#2336	C₂	semidirect
4. C₅₃×#2334	C₂	semidirect
5. C₅₃×#2329	C₂	semidirect
7. C₅₃×#2322	C₂	semidirect
8. C₅₃×#2298	C₂	semidirect
9. C₅₃×#2296	C₂	semidirect
10. C₅₃×#2295	C₂	semidirect
11. C₅₃×#2290	C₂	semidirect
12. C₅₃×#2290	C₂	semidirect
13. C₅₃×#2283	C₂	semidirect
14. C₄²×D₄×C₅₃	C₂	semidirect
15. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
16. C₅₃×#529	C₄	semidirect
86. #126×C₅₃	C₂×C₄	semidirect
87. #125×C₅₃	C₂×C₄	semidirect
88. #124×C₅₃	C₂×C₄	semidirect
89. #123×C₅₃	C₂×C₄	semidirect
90. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
92. C₄×Q₈×C₅₃	C₂×C₄	semidirect
94. C₄×D₄×C₅₃	C₂×C₄	semidirect
95. C₄×D₄×C₅₃	C₂×C₄	semidirect
97. #117×C₅₃	C₂×C₄	semidirect
124. C₂×C₄²×C₅₃	C₂×C₄	semidirect
282. #34083	C₅₃	direct (N × H)
312. #2337	C₂×C₅₃	semidirect
313. #2336	C₂×C₅₃	semidirect
314. #2334	C₂×C₅₃	semidirect
315. #2329	C₂×C₅₃	semidirect
317. #2322	C₂×C₅₃	semidirect
318. #2298	C₂×C₅₃	semidirect
319. #2296	C₂×C₅₃	semidirect
320. #2295	C₂×C₅₃	semidirect
321. #2290	C₂×C₅₃	semidirect
322. #2290	C₂×C₅₃	semidirect
323. #2283	C₂×C₅₃	semidirect
324. C₄²×D₄	C₂×C₅₃	semidirect
325. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
333. #529	C₄×C₅₃	semidirect
393. C₅₃	#34083	direct (N × H)
404. #126	C₂×C₄×C₅₃	semidirect
405. #125	C₂×C₄×C₅₃	semidirect
406. #124	C₂×C₄×C₅₃	semidirect
407. #123	C₂×C₄×C₅₃	semidirect
408. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
410. C₄×Q₈	C₂×C₄×C₅₃	semidirect
412. C₄×D₄	C₂×C₄×C₅₃	semidirect
413. C₄×D₄	C₂×C₄×C₅₃	semidirect
415. #117	C₂×C₄×C₅₃	semidirect
442. 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×#2337	2
C₅₃×#2336	2
C₅₃×#2334	2
C₅₃×#2329	2
C₅₃×#2325	2
C₅₃×#2322	2
C₅₃×#2298	2
C₅₃×#2296	2
C₅₃×#2295	2
C₅₃×#2290	2
C₅₃×#2283	2
C₄²×D₄×C₅₃	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#34083	53

Subgroups — C53×#34083 #27241595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34083 #27241595