Subgroups — C₅₃×#21087 #27228599

Order 13568.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#417	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#416	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2317	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2311	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1936	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1935	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1934	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1933	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#417	—	2	4	no	—
20	C₅₃×#416	C₄	1	8	yes	complemented
21	C₂×#103×C₅₃	C₂²	1	2	yes	—
22	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#102×C₅₃	C₂²	1	2	yes	—
25	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#405	—	2	4	no	—
28	C₂×#97×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#391	C₂²	1	2	yes	—
31	C₅₃×#390	C₂²	1	2	yes	—
32	C₅₃×#389	C₂²	1	2	yes	—
33	C₅₃×#389	C₂²	1	2	yes	—
34	C₅₃×#386	—	2	2	no	—
35	C₅₃×#385	—	2	2	no	—
36	C₅₃×#384	—	2	1	no	—
37	C₅₃×#382	—	2	1	no	—
38	C₅₃×#380	—	2	1	no	—
39	C₅₃×#380	—	2	1	no	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#340	C₂²	1	2	yes	—
43	C₅₃×#340	C₂²	1	2	yes	—
44	C₂×#41×C₅₃	—	2	2	no	—
45	C₂×#41×C₅₃	—	2	2	no	—
46	C₂×#41×C₅₃	—	2	2	no	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×#41×C₅₃	—	2	2	no	—
49	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₅₃	—	2	1	no	—
52	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	—	2	1	no	—
54	C₂³×C₄×C₅₃	—	2	1	no	—
55	C₂³×C₄×C₅₃	—	2	1	no	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₅₃	—	2	2	no	—
58	C₂×M₁₆×C₅₃	—	2	2	no	—
59	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
60	C₂²×C₈×C₅₃	—	2	2	no	—
61	C₂²×C₈×C₅₃	—	2	2	no	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
66	#117×C₅₃	—	2	2	no	—
67	#117×C₅₃	—	2	2	no	—
68	#117×C₅₃	—	2	2	no	—
69	#117×C₅₃	C₂×C₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
88	C₂×C₄²×C₅₃	—	2	1	no	—
89	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
90	#103×C₅₃	—	2	4	no	—
91	#103×C₅₃	C₂×C₄	1	4	yes	complemented
92	#103×C₅₃	C₂×C₄	1	4	yes	complemented
93	#102×C₅₃	—	2	4	no	—
94	#102×C₅₃	C₂×C₄	1	4	yes	complemented
95	#102×C₅₃	C₂×C₄	1	4	yes	complemented
96	#97×C₅₃	—	2	4	no	—
97	C₄×C₈×C₅₃	—	2	4	no	—
98	#95×C₅₃	—	2	2	no	—
99	#95×C₅₃	—	2	2	no	—
100	#95×C₅₃	—	2	2	no	—
101	#95×C₅₃	—	2	2	no	—
102	C₂⁴×C₅₃	—	2	1	no	—
103	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
104	#41×C₅₃	—	2	8	no	—
105	#41×C₅₃	—	2	8	no	—
106	#41×C₅₃	—	4	4	no	—
107	#41×C₅₃	—	4	4	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	—	2	1	no	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
114	C₂×D₄×C₅₃	—	2	4	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	1	no	—
119	C₂×D₄×C₅₃	—	2	1	no	—
120	C₂×D₄×C₅₃	—	4	2	no	—
121	C₂×D₄×C₅₃	—	4	2	no	—
122	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	4	2	no	—
130	C₂²×C₄×C₅₃	—	4	2	no	—
131	C₂²×C₄×C₅₃	—	4	2	no	—
132	C₂²×C₄×C₅₃	—	4	2	no	—
133	C₂²×C₄×C₅₃	—	8	1	no	—
134	C₂²×C₄×C₅₃	—	8	1	no	—
135	C₂²×C₄×C₅₃	—	8	1	no	—
136	C₂²×C₄×C₅₃	—	8	1	no	—
137	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₅₃	#41	1	2	yes	—
143	C₂²×C₄×C₅₃	#41	1	2	yes	—
144	C₂²×C₄×C₅₃	#41	1	2	yes	—
145	C₂²×C₄×C₅₃	#41	1	2	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
147	M₁₆×C₅₃	—	2	8	no	—
148	C₂×C₈×C₅₃	—	2	8	no	—
149	C₂×C₈×C₅₃	—	2	1	no	—
150	C₂×C₈×C₅₃	—	2	1	no	—
151	C₂×C₈×C₅₃	—	2	1	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	2	2	no	—
154	C₂×C₈×C₅₃	—	2	2	no	—
155	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
158	C₄⋊C₄×C₅₃	—	2	2	no	—
159	C₄⋊C₄×C₅₃	—	2	2	no	—
160	C₄⋊C₄×C₅₃	—	2	1	no	—
161	C₄⋊C₄×C₅₃	—	2	1	no	—
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	2	no	—
165	C₄⋊C₄×C₅₃	—	2	2	no	—
166	C₄⋊C₄×C₅₃	—	4	2	no	—
167	C₄⋊C₄×C₅₃	—	4	2	no	—
168	C₄⋊C₄×C₅₃	—	8	2	no	—
169	C₄⋊C₄×C₅₃	—	8	2	no	—
170	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
171	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
172	C₂²⋊C₄×C₅₃	—	2	4	no	—
173	C₂²⋊C₄×C₅₃	—	2	4	no	—
174	C₂²⋊C₄×C₅₃	—	8	2	no	—
175	C₂²⋊C₄×C₅₃	—	8	2	no	—
176	C₄²×C₅₃	—	2	4	no	—
177	C₄²×C₅₃	—	2	2	no	—
178	C₄²×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	—	2	1	no	—
180	C₂³×C₅₃	—	2	1	no	—
181	C₂³×C₅₃	—	2	1	no	—
182	C₂³×C₅₃	—	2	1	no	—
183	C₂³×C₅₃	—	2	2	no	—
184	C₂³×C₅₃	—	4	1	no	—
185	C₂³×C₅₃	—	4	1	no	—
186	C₂³×C₅₃	—	4	1	no	—
187	C₂³×C₅₃	—	4	1	no	—
188	C₂³×C₅₃	—	4	1	no	—
189	C₂³×C₅₃	—	4	1	no	—
190	C₂³×C₅₃	C₄×D₄	1	2	yes	—
191	C₂³×C₅₃	#121	1	2	yes	—
192	C₂³×C₅₃	#124	1	2	yes	—
193	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₅₃	—	2	2	no	—
195	Q₈×C₅₃	—	2	2	no	—
196	Q₈×C₅₃	—	4	1	no	—
197	Q₈×C₅₃	—	4	1	no	—
198	D₄×C₅₃	—	2	2	no	—
199	D₄×C₅₃	—	2	2	no	—
200	D₄×C₅₃	—	4	8	no	—
201	D₄×C₅₃	—	4	1	no	—
202	D₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	4	1	no	—
213	C₂×C₄×C₅₃	—	4	1	no	—
214	C₂×C₄×C₅₃	—	4	1	no	—
215	C₂×C₄×C₅₃	—	4	1	no	—
216	C₂×C₄×C₅₃	—	4	1	no	—
217	C₂×C₄×C₅₃	—	4	1	no	—
218	C₂×C₄×C₅₃	—	4	8	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	—	4	2	no	—
223	C₂×C₄×C₅₃	—	4	2	no	—
224	C₂×C₄×C₅₃	—	4	2	no	—
225	C₂×C₄×C₅₃	—	8	4	no	—
226	C₂×C₄×C₅₃	—	8	4	no	—
227	C₂×C₄×C₅₃	#117	1	4	yes	—
228	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
229	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
230	C₂×C₄×C₅₃	#121	1	2	yes	—
231	C₂×C₄×C₅₃	#123	1	4	yes	—
232	C₂×C₄×C₅₃	#124	1	2	yes	—
233	C₂×C₄×C₅₃	#126	1	4	yes	—
234	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
240	C₈×C₅₃	—	2	2	no	—
241	C₈×C₅₃	—	2	2	no	—
242	C₈×C₅₃	—	2	4	no	—
243	#21087	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂²×C₅₃	—	2	2	no	—
245	C₂²×C₅₃	—	2	2	no	—
246	C₂²×C₅₃	—	2	2	no	—
247	C₂²×C₅₃	—	2	2	no	—
248	C₂²×C₅₃	—	2	2	no	—
249	C₂²×C₅₃	—	2	2	no	—
250	C₂²×C₅₃	—	4	2	no	—
251	C₂²×C₅₃	—	4	2	no	—
252	C₂²×C₅₃	—	4	2	no	—
253	C₂²×C₅₃	—	8	1	no	—
254	C₂²×C₅₃	—	8	1	no	—
255	C₂²×C₅₃	—	8	1	no	—
256	C₂²×C₅₃	—	8	1	no	—
257	C₂²×C₅₃	#388	1	4	yes	—
258	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
259	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
260	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
261	C₂²×C₅₃	#576	1	1	yes	characteristic
262	C₂²×C₅₃	#577	1	1	yes	characteristic
263	C₂²×C₅₃	#577	1	1	yes	characteristic
264	C₂²×C₅₃	#578	1	1	yes	characteristic
265	C₄×C₅₃	—	4	2	no	—
266	C₄×C₅₃	—	4	2	no	—
267	C₄×C₅₃	—	4	2	no	—
268	C₄×C₅₃	—	4	2	no	—
269	C₄×C₅₃	—	4	2	no	—
270	C₄×C₅₃	—	8	2	no	—
271	C₄×C₅₃	—	8	2	no	—
272	C₄×C₅₃	#388	1	4	yes	—
273	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
274	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
275	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
276	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
277	C₂×#417	C₂×C₅₃	1	1	yes	characteristic
278	C₂×#416	C₂×C₅₃	1	1	yes	characteristic
279	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#2317	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#2311	C₂×C₅₃	1	1	yes	characteristic
282	#1936	C₂×C₅₃	1	2	yes	complemented
283	#1935	C₂×C₅₃	1	2	yes	complemented
284	#1934	C₂×C₅₃	1	2	yes	complemented
285	#1933	C₂×C₅₃	1	2	yes	complemented
286	#1736	C₂×C₅₃	1	1	yes	characteristic, complemented
287	C₂×#340	C₂×C₅₃	1	1	yes	characteristic, complemented
288	C₂×C₅₃	—	2	4	no	—
289	C₂×C₅₃	—	8	2	no	—
290	C₂×C₅₃	C₂×#388	1	1	yes	characteristic
291	C₂×C₅₃	#2951	1	1	yes	characteristic
292	C₂×C₅₃	#2952	1	1	yes	characteristic
293	C₂×C₅₃	#3062	1	1	yes	characteristic
294	C₂×C₅₃	#3063	1	1	yes	characteristic
295	C₂×C₅₃	#3136	1	1	yes	characteristic
296	C₂×C₅₃	#3137	1	1	yes	characteristic
297	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
298	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
299	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
300	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
301	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
304	#417	—	2	4	no	—
305	#416	C₄×C₅₃	1	8	yes	complemented
306	C₂×#103	C₂²×C₅₃	1	2	yes	—
307	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
308	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
309	C₂×#102	C₂²×C₅₃	1	2	yes	—
310	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
311	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
312	#405	—	2	4	no	—
313	C₂×#97	C₂²×C₅₃	1	2	yes	—
314	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
315	#391	C₂²×C₅₃	1	2	yes	—
316	#390	C₂²×C₅₃	1	2	yes	—
317	#389	C₂²×C₅₃	1	2	yes	—
318	#389	C₂²×C₅₃	1	2	yes	—
319	#386	—	2	2	no	—
320	#385	—	2	2	no	—
321	#384	—	2	1	no	—
322	#382	—	2	1	no	—
323	#380	—	2	1	no	—
324	#380	—	2	1	no	—
325	#340	C₂²×C₅₃	1	2	yes	—
326	#340	C₂²×C₅₃	1	2	yes	—
327	#340	C₂²×C₅₃	1	2	yes	—
328	#340	C₂²×C₅₃	1	2	yes	—
329	C₅₃	#21087	1	1	yes	characteristic, direct product factor, Sylow
330	C₂×#41	—	2	2	no	—
331	C₂×#41	—	2	2	no	—
332	C₂×#41	—	2	2	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
336	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
337	C₂²×D₄	—	2	1	no	—
338	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂³×C₄	—	2	1	no	—
342	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	—	2	2	no	—
345	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
346	C₂²×C₈	—	2	2	no	—
347	C₂²×C₈	—	2	2	no	—
348	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
349	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
350	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
352	#117	—	2	2	no	—
353	#117	—	2	2	no	—
354	#117	—	2	2	no	—
355	#117	C₂×C₄×C₅₃	1	4	yes	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	—	2	2	no	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
365	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
370	C₂×C₂²⋊C₄	—	2	1	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
374	C₂×C₄²	—	2	1	no	—
375	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
376	#103	—	2	4	no	—
377	#103	C₂×C₄×C₅₃	1	4	yes	complemented
378	#103	C₂×C₄×C₅₃	1	4	yes	complemented
379	#102	—	2	4	no	—
380	#102	C₂×C₄×C₅₃	1	4	yes	complemented
381	#102	C₂×C₄×C₅₃	1	4	yes	complemented
382	#97	—	2	4	no	—
383	C₄×C₈	—	2	4	no	—
384	#95	—	2	2	no	—
385	#95	—	2	2	no	—
386	#95	—	2	2	no	—
387	#95	—	2	2	no	—
388	C₂⁴	—	2	1	no	—
389	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
390	#41	—	2	8	no	—
391	#41	—	2	8	no	—
392	#41	—	4	4	no	—
393	#41	—	4	4	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	—	4	2	no	—
408	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	4	2	no	—
416	C₂²×C₄	—	4	2	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	8	1	no	—
420	C₂²×C₄	—	8	1	no	—
421	C₂²×C₄	—	8	1	no	—
422	C₂²×C₄	—	8	1	no	—
423	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
428	C₂²×C₄	#41×C₅₃	1	2	yes	—
429	C₂²×C₄	#41×C₅₃	1	2	yes	—
430	C₂²×C₄	#41×C₅₃	1	2	yes	—
431	C₂²×C₄	#41×C₅₃	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
433	M₁₆	—	2	8	no	—
434	C₂×C₈	—	2	8	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
442	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
443	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	4	2	no	—
453	C₄⋊C₄	—	4	2	no	—
454	C₄⋊C₄	—	8	2	no	—
455	C₄⋊C₄	—	8	2	no	—
456	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
457	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	8	2	no	—
461	C₂²⋊C₄	—	8	2	no	—
462	C₄²	—	2	4	no	—
463	C₄²	—	2	2	no	—
464	C₄²	—	2	2	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	2	1	no	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	1	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	1	no	—
476	C₂³	C₄×D₄×C₅₃	1	2	yes	—
477	C₂³	#121×C₅₃	1	2	yes	—
478	C₂³	#124×C₅₃	1	2	yes	—
479	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
480	Q₈	—	2	2	no	—
481	Q₈	—	2	2	no	—
482	Q₈	—	4	1	no	—
483	Q₈	—	4	1	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	1	no	—
488	D₄	—	4	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	8	4	no	—
512	C₂×C₄	—	8	4	no	—
513	C₂×C₄	#117×C₅₃	1	4	yes	—
514	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
515	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
516	C₂×C₄	#121×C₅₃	1	2	yes	—
517	C₂×C₄	#123×C₅₃	1	4	yes	—
518	C₂×C₄	#124×C₅₃	1	2	yes	—
519	C₂×C₄	#126×C₅₃	1	4	yes	—
520	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
523	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
526	C₈	—	2	2	no	—
527	C₈	—	2	2	no	—
528	C₈	—	2	4	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	8	1	no	—
539	C₂²	—	8	1	no	—
540	C₂²	—	8	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	C₅₃×#388	1	4	yes	—
543	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
544	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
545	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
546	C₂²	C₅₃×#576	1	1	yes	characteristic
547	C₂²	C₅₃×#577	1	1	yes	characteristic
548	C₂²	C₅₃×#577	1	1	yes	characteristic
549	C₂²	C₅₃×#578	1	1	yes	characteristic
550	C₄	—	4	2	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	8	2	no	—
556	C₄	—	8	2	no	—
557	C₄	C₅₃×#388	1	4	yes	—
558	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
559	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
560	C₄	C₂×#123×C₅₃	1	1	yes	characteristic
561	C₄	C₂×#126×C₅₃	1	1	yes	characteristic
562	C₂	—	2	4	no	—
563	C₂	—	8	2	no	—
564	C₂	C₂×C₅₃×#388	1	1	yes	characteristic
565	C₂	C₅₃×#2951	1	1	yes	characteristic
566	C₂	C₅₃×#2952	1	1	yes	characteristic
567	C₂	C₅₃×#3062	1	1	yes	characteristic
568	C₂	C₅₃×#3063	1	1	yes	characteristic
569	C₂	C₅₃×#3136	1	1	yes	characteristic
570	C₂	C₅₃×#3137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#405	C₂	semidirect
4. C₅₃×#2317	C₂	semidirect
6. C₅₃×#1936	C₂	semidirect
7. C₅₃×#1935	C₂	semidirect
8. C₅₃×#1934	C₂	semidirect
9. C₅₃×#1933	C₂	semidirect
10. C₅₃×#1736	C₂	semidirect
11. C₂×C₅₃×#340	C₂	semidirect
20. C₅₃×#416	C₄	semidirect
91. #103×C₅₃	C₂×C₄	semidirect
92. #103×C₅₃	C₂×C₄	semidirect
94. #102×C₅₃	C₂×C₄	semidirect
95. #102×C₅₃	C₂×C₄	semidirect
243. #21087	C₅₃	direct (N × H)
279. C₂×#405	C₂×C₅₃	semidirect
280. #2317	C₂×C₅₃	semidirect
282. #1936	C₂×C₅₃	semidirect
283. #1935	C₂×C₅₃	semidirect
284. #1934	C₂×C₅₃	semidirect
285. #1933	C₂×C₅₃	semidirect
286. #1736	C₂×C₅₃	semidirect
287. C₂×#340	C₂×C₅₃	semidirect
305. #416	C₄×C₅₃	semidirect
329. C₅₃	#21087	direct (N × H)
377. #103	C₂×C₄×C₅₃	semidirect
378. #103	C₂×C₄×C₅₃	semidirect
380. #102	C₂×C₄×C₅₃	semidirect
381. #102	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×#417	2
C₂×C₅₃×#416	2
C₂×C₅₃×#405	2
C₅₃×#2317	2
C₅₃×#2311	2
C₅₃×#1936	2
C₅₃×#1935	2
C₅₃×#1934	2
C₅₃×#1933	2
C₅₃×#1736	2
C₂×C₅₃×#340	2
#21087	53

Subgroups — C53×#21087 #27228599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21087 #27228599