Subgroups — C₅₃×#20867 #27228379

Order 13568.

1312 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2905	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#421	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#421	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1861	C₂	1	8	yes	maximal, complemented
7	C₂×C₅₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1733	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#568	—	2	2	no	—
11	C₅₃×#568	—	2	2	no	—
12	C₅₃×#568	—	2	2	no	—
13	C₅₃×#568	C₂²	1	4	yes	—
14	C₂×#131×C₅₃	—	2	2	no	—
15	C₂×#131×C₅₃	—	2	2	no	—
16	C₂²×M₁₆×C₅₃	—	2	2	no	—
17	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#430	—	2	2	no	—
24	C₅₃×#430	C₂²	1	4	yes	complemented
25	C₂×#108×C₅₃	—	2	2	no	—
26	C₅₃×#424	—	2	2	no	—
27	C₅₃×#424	—	2	2	no	—
28	C₅₃×#423	C₂²	1	4	yes	complemented
29	C₅₃×#421	—	2	2	no	—
30	C₅₃×#421	C₄	1	4	yes	complemented
31	C₅₃×#421	C₄	1	4	yes	complemented
32	C₅₃×#421	C₂²	1	4	yes	—
33	C₂×#104×C₅₃	—	2	2	no	—
34	C₂×#104×C₅₃	C₂²	1	4	yes	—
35	C₅₃×#344	C₂²	1	4	yes	complemented
36	C₅₃×#339	C₂²	1	4	yes	complemented
37	C₂×#41×C₅₃	—	2	1	no	—
38	C₂×#41×C₅₃	—	2	1	no	—
39	C₂×#41×C₅₃	—	2	1	no	—
40	C₂×#41×C₅₃	—	2	1	no	—
41	C₂×#41×C₅₃	—	2	1	no	—
42	C₂×#41×C₅₃	—	2	1	no	—
43	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×#41×C₅₃	D₄	1	2	yes	—
46	C₂×#41×C₅₃	D₄	1	2	yes	—
47	C₂×#41×C₅₃	Q₈	1	2	yes	—
48	C₂×#41×C₅₃	C₂³	1	2	yes	—
49	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	—	2	1	no	—
53	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	#131×C₅₃	—	2	8	no	—
55	#131×C₅₃	—	2	8	no	—
56	#131×C₅₃	—	2	8	no	—
57	#131×C₅₃	—	2	8	no	—
58	C₂×M₁₆×C₅₃	—	2	4	no	—
59	C₂×M₁₆×C₅₃	—	2	8	no	—
60	C₂×M₁₆×C₅₃	—	2	8	no	—
61	C₂×M₁₆×C₅₃	—	2	1	no	—
62	C₂×M₁₆×C₅₃	—	2	1	no	—
63	C₂×M₁₆×C₅₃	—	2	1	no	—
64	C₂×M₁₆×C₅₃	—	2	1	no	—
65	C₂×M₁₆×C₅₃	—	2	1	no	—
66	C₂×M₁₆×C₅₃	—	2	1	no	—
67	C₂×M₁₆×C₅₃	—	2	1	no	—
68	C₂×M₁₆×C₅₃	—	2	1	no	—
69	C₂×M₁₆×C₅₃	—	2	1	no	—
70	C₂×M₁₆×C₅₃	—	2	1	no	—
71	C₂×M₁₆×C₅₃	—	2	1	no	—
72	C₂×M₁₆×C₅₃	—	2	1	no	—
73	C₂×M₁₆×C₅₃	—	2	1	no	—
74	C₂×M₁₆×C₅₃	—	2	1	no	—
75	C₂×M₁₆×C₅₃	—	2	1	no	—
76	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
77	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
78	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
79	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
80	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
81	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
82	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
83	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
84	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
85	C₂²×C₈×C₅₃	—	2	4	no	—
86	C₂²×C₈×C₅₃	—	2	1	no	—
87	C₂²×C₈×C₅₃	—	2	1	no	—
88	C₂²×C₈×C₅₃	—	2	1	no	—
89	C₂²×C₈×C₅₃	D₄	1	2	yes	—
90	#117×C₅₃	—	2	1	no	—
91	#117×C₅₃	—	2	1	no	—
92	#117×C₅₃	—	2	1	no	—
93	#117×C₅₃	—	2	1	no	—
94	#117×C₅₃	—	2	1	no	—
95	#117×C₅₃	C₂×C₄	1	2	yes	—
96	#117×C₅₃	C₂×C₄	1	2	yes	—
97	#117×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
102	C₂×C₄²×C₅₃	—	2	1	no	—
103	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
104	#108×C₅₃	—	2	8	no	—
105	C₄⋊C₈×C₅₃	—	2	8	no	—
106	#104×C₅₃	—	2	8	no	—
107	#104×C₅₃	C₂×C₄	1	16	yes	complemented
108	C₂⁴×C₅₃	—	2	1	no	—
109	C₂⁴×C₅₃	#41	1	1	yes	characteristic
110	#41×C₅₃	—	2	2	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	—	2	2	no	—
113	#41×C₅₃	—	2	2	no	—
114	#41×C₅₃	—	2	2	no	—
115	#41×C₅₃	—	2	2	no	—
116	#41×C₅₃	—	2	2	no	—
117	#41×C₅₃	—	2	2	no	—
118	#41×C₅₃	—	2	2	no	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	—	2	2	no	—
121	#41×C₅₃	—	2	2	no	—
122	#41×C₅₃	C₄⋊C₄	1	4	yes	—
123	#41×C₅₃	C₄⋊C₄	1	4	yes	—
124	#41×C₅₃	C₂²×C₄	1	4	yes	—
125	#41×C₅₃	C₂×D₄	1	4	yes	—
126	C₂×Q₈×C₅₃	—	2	1	no	—
127	C₂×Q₈×C₅₃	—	2	1	no	—
128	C₂×Q₈×C₅₃	—	2	1	no	—
129	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
130	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
131	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
132	C₂×D₄×C₅₃	—	2	2	no	—
133	C₂×D₄×C₅₃	—	2	2	no	—
134	C₂×D₄×C₅₃	—	2	2	no	—
135	C₂×D₄×C₅₃	—	2	2	no	—
136	C₂×D₄×C₅₃	—	2	2	no	—
137	C₂×D₄×C₅₃	—	2	2	no	—
138	C₂×D₄×C₅₃	—	2	1	no	—
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	1	no	—
141	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
142	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
143	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
144	C₂²×C₄×C₅₃	—	2	1	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	2	no	—
147	C₂²×C₄×C₅₃	—	2	2	no	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	2	no	—
151	C₂²×C₄×C₅₃	—	2	2	no	—
152	C₂²×C₄×C₅₃	—	4	1	no	—
153	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
167	M₁₆×C₅₃	—	2	16	no	—
168	M₁₆×C₅₃	—	2	2	no	—
169	M₁₆×C₅₃	—	2	2	no	—
170	M₁₆×C₅₃	—	2	2	no	—
171	M₁₆×C₅₃	—	2	2	no	—
172	M₁₆×C₅₃	—	2	2	no	—
173	M₁₆×C₅₃	—	2	2	no	—
174	M₁₆×C₅₃	—	2	2	no	—
175	M₁₆×C₅₃	—	2	2	no	—
176	M₁₆×C₅₃	—	2	2	no	—
177	M₁₆×C₅₃	—	2	2	no	—
178	M₁₆×C₅₃	—	2	2	no	—
179	M₁₆×C₅₃	—	2	2	no	—
180	M₁₆×C₅₃	—	4	1	no	—
181	M₁₆×C₅₃	—	4	1	no	—
182	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
183	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
184	C₂×C₈×C₅₃	—	2	16	no	—
185	C₂×C₈×C₅₃	—	2	2	no	—
186	C₂×C₈×C₅₃	—	2	2	no	—
187	C₂×C₈×C₅₃	—	2	2	no	—
188	C₂×C₈×C₅₃	—	2	2	no	—
189	C₂×C₈×C₅₃	—	2	2	no	—
190	C₂×C₈×C₅₃	—	2	2	no	—
191	C₂×C₈×C₅₃	—	2	2	no	—
192	C₂×C₈×C₅₃	—	2	2	no	—
193	C₂×C₈×C₅₃	—	2	2	no	—
194	C₂×C₈×C₅₃	—	4	1	no	—
195	C₂×C₈×C₅₃	—	4	1	no	—
196	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₅₃	—	2	2	no	—
198	C₄⋊C₄×C₅₃	—	2	2	no	—
199	C₄⋊C₄×C₅₃	—	2	2	no	—
200	C₄⋊C₄×C₅₃	—	4	1	no	—
201	C₂²⋊C₄×C₅₃	—	2	2	no	—
202	C₂²⋊C₄×C₅₃	—	2	2	no	—
203	C₂²⋊C₄×C₅₃	—	2	2	no	—
204	C₂²⋊C₄×C₅₃	—	4	1	no	—
205	C₄²×C₅₃	—	2	2	no	—
206	C₄²×C₅₃	—	2	2	no	—
207	C₄²×C₅₃	—	4	1	no	—
208	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
209	C₂³×C₅₃	—	2	2	no	—
210	C₂³×C₅₃	—	2	2	no	—
211	C₂³×C₅₃	—	2	2	no	—
212	C₂³×C₅₃	—	8	1	no	—
213	C₂³×C₅₃	—	8	2	no	—
214	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
215	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
216	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
217	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
218	C₂³×C₅₃	#123	1	1	yes	characteristic
219	C₂³×C₅₃	#123	1	1	yes	characteristic
220	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
221	Q₈×C₅₃	—	2	2	no	—
222	Q₈×C₅₃	—	2	2	no	—
223	Q₈×C₅₃	—	2	2	no	—
224	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
225	D₄×C₅₃	—	2	4	no	—
226	D₄×C₅₃	—	2	4	no	—
227	D₄×C₅₃	—	2	4	no	—
228	D₄×C₅₃	—	2	4	no	—
229	D₄×C₅₃	—	2	2	no	—
230	D₄×C₅₃	—	2	2	no	—
231	D₄×C₅₃	—	2	2	no	—
232	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	4	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	4	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	4	2	no	—
242	C₂×C₄×C₅₃	—	8	2	no	—
243	C₂×C₄×C₅₃	—	8	2	no	—
244	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
254	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
255	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
256	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
257	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
264	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
265	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
266	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
267	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
271	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
272	C₈×C₅₃	—	2	4	no	—
273	C₈×C₅₃	—	2	4	no	—
274	C₈×C₅₃	—	2	4	no	—
275	C₈×C₅₃	—	4	2	no	—
276	#20867	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂²×C₅₃	—	2	4	no	—
278	C₂²×C₅₃	—	4	1	no	—
279	C₂²×C₅₃	—	4	1	no	—
280	C₂²×C₅₃	—	4	1	no	—
281	C₂²×C₅₃	—	4	1	no	—
282	C₂²×C₅₃	—	4	1	no	—
283	C₂²×C₅₃	—	4	1	no	—
284	C₂²×C₅₃	—	4	1	no	—
285	C₂²×C₅₃	—	4	4	no	—
286	C₂²×C₅₃	—	8	2	no	—
287	C₂²×C₅₃	—	8	2	no	—
288	C₂²×C₅₃	#385	1	1	yes	characteristic
289	C₂²×C₅₃	#385	1	1	yes	characteristic
290	C₂²×C₅₃	#385	1	1	yes	characteristic
291	C₂²×C₅₃	#385	1	1	yes	characteristic
292	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
293	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
294	C₂²×C₅₃	C₂×#123	1	1	yes	characteristic
295	C₄×C₅₃	—	2	4	no	—
296	C₄×C₅₃	—	4	4	no	—
297	C₄×C₅₃	—	8	2	no	—
298	C₄×C₅₃	#385	1	1	yes	characteristic
299	C₄×C₅₃	#385	1	1	yes	characteristic
300	C₄×C₅₃	#385	1	1	yes	characteristic
301	C₄×C₅₃	#385	1	1	yes	characteristic
302	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
303	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
304	C₄×C₅₃	C₂×#120	1	1	yes	characteristic
305	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
306	C₂×#568	C₂×C₅₃	1	1	yes	characteristic
307	C₂×#430	C₂×C₅₃	1	1	yes	characteristic, complemented
308	#2905	C₂×C₅₃	1	1	yes	characteristic, complemented
309	C₂×#421	C₂×C₅₃	1	1	yes	characteristic
310	C₂×#421	C₂×C₅₃	1	1	yes	characteristic
311	#1861	C₂×C₅₃	1	8	yes	complemented
312	C₂×#344	C₂×C₅₃	1	1	yes	characteristic, complemented
313	#1733	C₂×C₅₃	1	1	yes	characteristic, complemented
314	C₂×C₅₃	—	2	1	no	—
315	C₂×C₅₃	—	2	1	no	—
316	C₂×C₅₃	—	2	1	no	—
317	C₂×C₅₃	—	2	1	no	—
318	C₂×C₅₃	—	2	1	no	—
319	C₂×C₅₃	—	2	1	no	—
320	C₂×C₅₃	—	2	1	no	—
321	C₂×C₅₃	—	4	4	no	—
322	C₂×C₅₃	C₂×#385	1	1	yes	characteristic
323	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
324	#568	—	2	2	no	—
325	#568	—	2	2	no	—
326	#568	—	2	2	no	—
327	#568	C₂²×C₅₃	1	4	yes	—
328	C₂×#131	—	2	2	no	—
329	C₂×#131	—	2	2	no	—
330	C₂²×M₁₆	—	2	2	no	—
331	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
332	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
333	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
334	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
335	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
336	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
337	#430	—	2	2	no	—
338	#430	C₂²×C₅₃	1	4	yes	complemented
339	C₂×#108	—	2	2	no	—
340	#424	—	2	2	no	—
341	#424	—	2	2	no	—
342	#423	C₂²×C₅₃	1	4	yes	complemented
343	#421	—	2	2	no	—
344	#421	C₄×C₅₃	1	4	yes	complemented
345	#421	C₄×C₅₃	1	4	yes	complemented
346	#421	C₂²×C₅₃	1	4	yes	—
347	C₂×#104	—	2	2	no	—
348	C₂×#104	C₂²×C₅₃	1	4	yes	—
349	#344	C₂²×C₅₃	1	4	yes	complemented
350	#339	C₂²×C₅₃	1	4	yes	complemented
351	C₅₃	#20867	1	1	yes	characteristic, direct product factor, Sylow
352	C₂×#41	—	2	1	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	—	2	1	no	—
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
359	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
360	C₂×#41	D₄×C₅₃	1	2	yes	—
361	C₂×#41	D₄×C₅₃	1	2	yes	—
362	C₂×#41	Q₈×C₅₃	1	2	yes	—
363	C₂×#41	C₂³×C₅₃	1	2	yes	—
364	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
365	C₂²×D₄	—	2	1	no	—
366	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
369	#131	—	2	8	no	—
370	#131	—	2	8	no	—
371	#131	—	2	8	no	—
372	#131	—	2	8	no	—
373	C₂×M₁₆	—	2	4	no	—
374	C₂×M₁₆	—	2	8	no	—
375	C₂×M₁₆	—	2	1	no	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	—	2	1	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	2	1	no	—
383	C₂×M₁₆	—	2	1	no	—
384	C₂×M₁₆	—	2	1	no	—
385	C₂×M₁₆	—	2	8	no	—
386	C₂×M₁₆	—	2	1	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂×M₁₆	—	2	1	no	—
389	C₂×M₁₆	—	2	1	no	—
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
392	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
393	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
394	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
395	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
396	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
397	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
398	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
399	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
400	C₂²×C₈	—	2	4	no	—
401	C₂²×C₈	—	2	1	no	—
402	C₂²×C₈	—	2	1	no	—
403	C₂²×C₈	—	2	1	no	—
404	C₂²×C₈	D₄×C₅₃	1	2	yes	—
405	#117	—	2	1	no	—
406	#117	—	2	1	no	—
407	#117	—	2	1	no	—
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#117	C₂×C₄×C₅₃	1	2	yes	—
411	#117	C₂×C₄×C₅₃	1	2	yes	—
412	#117	C₂³×C₅₃	1	2	yes	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
419	#108	—	2	8	no	—
420	C₄⋊C₈	—	2	8	no	—
421	#104	—	2	8	no	—
422	#104	C₂×C₄×C₅₃	1	16	yes	complemented
423	C₂⁴	—	2	1	no	—
424	C₂⁴	#41×C₅₃	1	1	yes	characteristic
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	2	no	—
429	#41	—	2	2	no	—
430	#41	—	2	2	no	—
431	#41	—	2	2	no	—
432	#41	—	2	2	no	—
433	#41	—	2	2	no	—
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	#41	—	2	2	no	—
437	#41	C₄⋊C₄×C₅₃	1	4	yes	—
438	#41	C₄⋊C₄×C₅₃	1	4	yes	—
439	#41	C₂²×C₄×C₅₃	1	4	yes	—
440	#41	C₂×D₄×C₅₃	1	4	yes	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
445	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
446	C₂×Q₈	C₂×Q₈×C₅₃	1	2	yes	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
457	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
458	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	4	1	no	—
468	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
476	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
478	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
479	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
480	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
482	M₁₆	—	2	16	no	—
483	M₁₆	—	2	2	no	—
484	M₁₆	—	2	2	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	—	2	2	no	—
492	M₁₆	—	2	2	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	—	2	2	no	—
495	M₁₆	—	4	1	no	—
496	M₁₆	—	4	1	no	—
497	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
498	M₁₆	C₂×D₄×C₅₃	1	4	yes	—
499	C₂×C₈	—	2	16	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	2	2	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	2	2	no	—
508	C₂×C₈	—	2	2	no	—
509	C₂×C₈	—	4	1	no	—
510	C₂×C₈	—	4	1	no	—
511	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	4	1	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	4	1	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	4	1	no	—
523	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	8	1	no	—
528	C₂³	—	8	2	no	—
529	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
530	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
531	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
532	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
533	C₂³	#123×C₅₃	1	1	yes	characteristic
534	C₂³	#123×C₅₃	1	1	yes	characteristic
535	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
536	Q₈	—	2	2	no	—
537	Q₈	—	2	2	no	—
538	Q₈	—	2	2	no	—
539	Q₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	—	2	2	no	—
546	D₄	—	2	2	no	—
547	D₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	8	2	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
560	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
561	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
562	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
563	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
564	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
565	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
566	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
567	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
568	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
569	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
570	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
571	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
572	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
573	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
574	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
575	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
576	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
577	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
578	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
579	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
580	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
581	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
585	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
587	C₈	—	2	4	no	—
588	C₈	—	2	4	no	—
589	C₈	—	2	4	no	—
590	C₈	—	4	2	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	4	1	no	—
598	C₂²	—	4	1	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	8	2	no	—
601	C₂²	—	8	2	no	—
602	C₂²	C₅₃×#385	1	1	yes	characteristic
603	C₂²	C₅₃×#385	1	1	yes	characteristic
604	C₂²	C₅₃×#385	1	1	yes	characteristic
605	C₂²	C₅₃×#385	1	1	yes	characteristic
606	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
607	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
608	C₂²	C₂×#123×C₅₃	1	1	yes	characteristic
609	C₄	—	2	4	no	—
610	C₄	—	4	4	no	—
611	C₄	—	8	2	no	—
612	C₄	C₅₃×#385	1	1	yes	characteristic
613	C₄	C₅₃×#385	1	1	yes	characteristic
614	C₄	C₅₃×#385	1	1	yes	characteristic
615	C₄	C₅₃×#385	1	1	yes	characteristic
616	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
617	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
618	C₄	C₂×#120×C₅₃	1	1	yes	characteristic
619	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
620	C₂	—	2	1	no	—
621	C₂	—	2	1	no	—
622	C₂	—	2	1	no	—
623	C₂	—	2	1	no	—
624	C₂	—	2	1	no	—
625	C₂	—	2	1	no	—
626	C₂	—	2	1	no	—
627	C₂	—	4	4	no	—
628	C₂	C₂×C₅₃×#385	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	C₂×#385
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#430	C₂	semidirect
3. C₅₃×#2905	C₂	semidirect
6. C₅₃×#1861	C₂	semidirect
7. C₂×C₅₃×#344	C₂	semidirect
8. C₅₃×#1733	C₂	semidirect
24. C₅₃×#430	C₂²	semidirect
28. C₅₃×#423	C₂²	semidirect
30. C₅₃×#421	C₄	semidirect
31. C₅₃×#421	C₄	semidirect
35. C₅₃×#344	C₂²	semidirect
36. C₅₃×#339	C₂²	semidirect
107. #104×C₅₃	C₂×C₄	semidirect
276. #20867	C₅₃	direct (N × H)
307. C₂×#430	C₂×C₅₃	semidirect
308. #2905	C₂×C₅₃	semidirect
311. #1861	C₂×C₅₃	semidirect
312. C₂×#344	C₂×C₅₃	semidirect
313. #1733	C₂×C₅₃	semidirect
338. #430	C₂²×C₅₃	semidirect
342. #423	C₂²×C₅₃	semidirect
344. #421	C₄×C₅₃	semidirect
345. #421	C₄×C₅₃	semidirect
349. #344	C₂²×C₅₃	semidirect
350. #339	C₂²×C₅₃	semidirect
351. C₅₃	#20867	direct (N × H)
422. #104	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×#568	2
C₂×C₅₃×#430	2
C₅₃×#2905	2
C₂×C₅₃×#421	2
C₅₃×#1861	2
C₂×C₅₃×#344	2
C₅₃×#1733	2
#20867	53

Subgroups — C53×#20867 #27228379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20867 #27228379