Subgroups — C₅₃×#18032 #27225544

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3104	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2984	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1584	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1572	C₂	1	1	yes	maximal, characteristic
13	C₅₃×#1497	C₂	1	1	yes	maximal, characteristic
14	C₅₃×#1479	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1454	C₂	1	1	yes	maximal, characteristic
16	C₅₃×#533	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#484	C₄	1	1	yes	characteristic
19	C₅₃×#484	C₄	1	1	yes	characteristic
20	C₅₃×#477	C₄	1	1	yes	characteristic
21	C₅₃×#477	C₄	1	1	yes	characteristic
22	C₅₃×#462	C₄	1	1	yes	characteristic
23	C₅₃×#462	C₄	1	1	yes	characteristic
24	C₅₃×#451	C₄	1	1	yes	characteristic
25	C₅₃×#451	C₄	1	1	yes	characteristic
26	C₅₃×#446	—	2	1	no	—
27	C₅₃×#446	—	2	1	no	—
28	C₅₃×#446	—	2	1	no	—
29	C₅₃×#446	C₂²	1	1	yes	characteristic
30	C₅₃×#446	C₂²	1	1	yes	characteristic
31	C₅₃×#446	C₂²	1	1	yes	characteristic
32	C₅₃×#446	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₅₃	—	2	1	no	—
34	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#433	—	2	1	no	—
43	C₅₃×#433	—	2	1	no	—
44	C₅₃×#432	—	2	1	no	—
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₅₃×#431	—	2	1	no	—
47	C₅₃×#431	C₂²	1	1	yes	characteristic
48	C₅₃×#431	C₂²	1	1	yes	characteristic
49	C₅₃×#431	C₂²	1	1	yes	characteristic
50	C₅₃×#431	C₂²	1	1	yes	characteristic
51	C₅₃×#335	C₂²	1	1	yes	characteristic
52	C₅₃×#335	C₂²	1	1	yes	characteristic
53	C₅₃×#334	C₂²	1	1	yes	characteristic
54	C₅₃×#334	C₂²	1	1	yes	characteristic
55	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
62	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₅₃×#322	C₂²	1	1	yes	characteristic
64	C₅₃×#322	C₂²	1	1	yes	characteristic
65	C₅₃×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₅₃	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#125×C₅₃	—	2	1	no	—
75	C₂²⋊Q₈×C₅₃	—	2	1	no	—
76	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₅₃	—	2	1	no	—
79	C₄×Q₈×C₅₃	—	2	1	no	—
80	C₄×Q₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
87	#117×C₅₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	—	2	1	no	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	—	2	1	no	—
98	C₄⋊C₈×C₅₃	—	2	1	no	—
99	C₄⋊C₈×C₅₃	—	2	1	no	—
100	C₄⋊C₈×C₅₃	—	2	1	no	—
101	C₄⋊C₈×C₅₃	—	2	1	no	—
102	C₄⋊C₈×C₅₃	—	2	1	no	—
103	C₄⋊C₈×C₅₃	—	2	1	no	—
104	C₄⋊C₈×C₅₃	—	2	1	no	—
105	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
108	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
113	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
114	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
115	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
116	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
117	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
118	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
120	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
121	C₂²⋊C₈×C₅₃	—	2	1	no	—
122	C₂²⋊C₈×C₅₃	—	2	1	no	—
123	C₂²⋊C₈×C₅₃	—	2	1	no	—
124	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
125	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
126	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
127	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
128	#97×C₅₃	—	2	1	no	—
129	#97×C₅₃	—	2	1	no	—
130	#97×C₅₃	—	2	1	no	—
131	#97×C₅₃	—	2	1	no	—
132	#97×C₅₃	—	2	1	no	—
133	C₄×C₈×C₅₃	—	2	1	no	—
134	C₄×C₈×C₅₃	—	2	1	no	—
135	C₄×C₈×C₅₃	—	2	1	no	—
136	C₄×C₈×C₅₃	—	2	1	no	—
137	C₄×C₈×C₅₃	—	2	1	no	—
138	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
144	C₂×Q₈×C₅₃	—	2	1	no	—
145	C₂×Q₈×C₅₃	—	2	1	no	—
146	C₂×Q₈×C₅₃	—	2	2	no	—
147	C₂×Q₈×C₅₃	—	4	1	no	—
148	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
157	Q₁₆×C₅₃	—	2	2	no	—
158	Q₁₆×C₅₃	—	2	2	no	—
159	Q₁₆×C₅₃	—	4	1	no	—
160	Q₁₆×C₅₃	—	4	1	no	—
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
175	C₄⋊C₄×C₅₃	—	2	1	no	—
176	C₄⋊C₄×C₅₃	—	2	1	no	—
177	C₄⋊C₄×C₅₃	—	2	1	no	—
178	C₄⋊C₄×C₅₃	—	2	1	no	—
179	C₄⋊C₄×C₅₃	—	2	1	no	—
180	C₄⋊C₄×C₅₃	—	2	1	no	—
181	C₄⋊C₄×C₅₃	—	2	1	no	—
182	C₄⋊C₄×C₅₃	—	2	1	no	—
183	C₄⋊C₄×C₅₃	—	2	1	no	—
184	C₄⋊C₄×C₅₃	—	2	1	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	2	1	no	—
187	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
188	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
191	C₂²⋊C₄×C₅₃	—	2	1	no	—
192	C₂²⋊C₄×C₅₃	—	2	1	no	—
193	C₄²×C₅₃	—	2	1	no	—
194	C₄²×C₅₃	—	2	1	no	—
195	C₄²×C₅₃	—	2	1	no	—
196	C₄²×C₅₃	—	2	1	no	—
197	C₄²×C₅₃	—	2	1	no	—
198	C₄²×C₅₃	—	2	1	no	—
199	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
200	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₅₃	#41	1	1	yes	characteristic
202	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
203	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
204	Q₈×C₅₃	—	2	1	no	—
205	Q₈×C₅₃	—	2	1	no	—
206	Q₈×C₅₃	—	2	1	no	—
207	Q₈×C₅₃	—	2	1	no	—
208	Q₈×C₅₃	—	2	1	no	—
209	Q₈×C₅₃	—	2	1	no	—
210	Q₈×C₅₃	—	2	1	no	—
211	Q₈×C₅₃	—	4	2	no	—
212	Q₈×C₅₃	—	4	2	no	—
213	Q₈×C₅₃	#131	1	2	yes	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	4	1	no	—
227	C₂×C₄×C₅₃	—	4	2	no	—
228	C₂×C₄×C₅₃	—	4	2	no	—
229	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
235	C₈×C₅₃	—	2	1	no	—
236	C₈×C₅₃	—	2	1	no	—
237	C₈×C₅₃	—	2	1	no	—
238	C₈×C₅₃	—	2	1	no	—
239	C₈×C₅₃	—	2	1	no	—
240	C₈×C₅₃	—	2	1	no	—
241	C₈×C₅₃	—	2	1	no	—
242	C₈×C₅₃	—	2	1	no	—
243	C₈×C₅₃	—	4	1	no	—
244	C₈×C₅₃	—	4	1	no	—
245	C₈×C₅₃	—	4	1	no	—
246	C₈×C₅₃	—	4	1	no	—
247	C₈×C₅₃	—	4	1	no	—
248	C₈×C₅₃	—	4	1	no	—
249	#18032	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂²×C₅₃	—	2	1	no	—
251	C₂²×C₅₃	—	2	1	no	—
252	C₂²×C₅₃	—	2	1	no	—
253	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
254	C₄×C₅₃	—	2	1	no	—
255	C₄×C₅₃	—	2	2	no	—
256	C₄×C₅₃	—	2	1	no	—
257	C₄×C₅₃	—	2	1	no	—
258	C₄×C₅₃	—	2	1	no	—
259	C₄×C₅₃	—	4	1	no	—
260	C₄×C₅₃	—	4	1	no	—
261	C₄×C₅₃	—	4	1	no	—
262	C₄×C₅₃	—	4	1	no	—
263	C₄×C₅₃	—	4	1	no	—
264	C₄×C₅₃	—	4	1	no	—
265	C₄×C₅₃	—	4	1	no	—
266	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
267	C₄×C₅₃	#568	1	1	yes	characteristic
268	C₄×C₅₃	#578	1	1	yes	characteristic
269	C₄×C₅₃	#578	1	1	yes	characteristic
270	#3104	C₂×C₅₃	1	1	yes	characteristic
271	#2986	C₂×C₅₃	1	1	yes	characteristic
272	#2984	C₂×C₅₃	1	1	yes	characteristic
273	#1593	C₂×C₅₃	1	1	yes	characteristic, complemented
274	#1591	C₂×C₅₃	1	1	yes	characteristic, complemented
275	#1587	C₂×C₅₃	1	1	yes	characteristic, complemented
276	#1587	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#1584	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#1584	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#1579	C₂×C₅₃	1	1	yes	characteristic, complemented
280	C₈×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#1572	C₂×C₅₃	1	1	yes	characteristic
282	#1497	C₂×C₅₃	1	1	yes	characteristic
283	#1479	C₂×C₅₃	1	1	yes	characteristic
284	#1454	C₂×C₅₃	1	1	yes	characteristic
285	C₂×C₅₃	—	4	1	no	—
286	C₂×C₅₃	#2933	1	1	yes	characteristic
287	C₂×C₅₃	#2950	1	1	yes	characteristic
288	C₂×C₅₃	#2955	1	1	yes	characteristic
289	#533	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
291	#484	C₄×C₅₃	1	1	yes	characteristic
292	#484	C₄×C₅₃	1	1	yes	characteristic
293	#477	C₄×C₅₃	1	1	yes	characteristic
294	#477	C₄×C₅₃	1	1	yes	characteristic
295	#462	C₄×C₅₃	1	1	yes	characteristic
296	#462	C₄×C₅₃	1	1	yes	characteristic
297	#451	C₄×C₅₃	1	1	yes	characteristic
298	#451	C₄×C₅₃	1	1	yes	characteristic
299	#446	—	2	1	no	—
300	#446	—	2	1	no	—
301	#446	—	2	1	no	—
302	#446	C₂²×C₅₃	1	1	yes	characteristic
303	#446	C₂²×C₅₃	1	1	yes	characteristic
304	#446	C₂²×C₅₃	1	1	yes	characteristic
305	#446	C₂²×C₅₃	1	1	yes	characteristic
306	C₈×Q₈	—	2	1	no	—
307	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
308	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
309	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
310	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
311	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
312	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
313	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
314	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
315	#433	—	2	1	no	—
316	#433	—	2	1	no	—
317	#432	—	2	1	no	—
318	#432	C₂²×C₅₃	1	1	yes	characteristic
319	#431	—	2	1	no	—
320	#431	C₂²×C₅₃	1	1	yes	characteristic
321	#431	C₂²×C₅₃	1	1	yes	characteristic
322	#431	C₂²×C₅₃	1	1	yes	characteristic
323	#431	C₂²×C₅₃	1	1	yes	characteristic
324	#335	C₂²×C₅₃	1	1	yes	characteristic
325	#335	C₂²×C₅₃	1	1	yes	characteristic
326	#334	C₂²×C₅₃	1	1	yes	characteristic
327	#334	C₂²×C₅₃	1	1	yes	characteristic
328	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
329	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
330	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
331	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
332	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
335	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
336	#322	C₂²×C₅₃	1	1	yes	characteristic
337	#322	C₂²×C₅₃	1	1	yes	characteristic
338	#322	C₂²×C₅₃	1	1	yes	characteristic
339	C₈²	C₂²×C₅₃	1	1	yes	characteristic
340	C₅₃	#18032	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
342	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
343	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
344	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
345	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
346	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
347	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
348	#125	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
351	C₂²⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
356	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
357	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
358	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
359	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
360	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
361	#117	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
366	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
367	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
368	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
369	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	—	2	1	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
380	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
381	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
382	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
383	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
384	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
385	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
386	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
387	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
388	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
389	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
390	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
391	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
392	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
393	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
394	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
395	C₂²⋊C₈	—	2	1	no	—
396	C₂²⋊C₈	—	2	1	no	—
397	C₂²⋊C₈	—	2	1	no	—
398	C₂²⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
399	C₂²⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
400	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
401	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
402	#97	—	2	1	no	—
403	#97	—	2	1	no	—
404	#97	—	2	1	no	—
405	#97	—	2	1	no	—
406	#97	—	2	1	no	—
407	C₄×C₈	—	2	1	no	—
408	C₄×C₈	—	2	1	no	—
409	C₄×C₈	—	2	1	no	—
410	C₄×C₈	—	2	1	no	—
411	C₄×C₈	—	2	1	no	—
412	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
414	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
415	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
417	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	4	1	no	—
422	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
423	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
424	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
425	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
431	Q₁₆	—	2	2	no	—
432	Q₁₆	—	2	2	no	—
433	Q₁₆	—	4	1	no	—
434	Q₁₆	—	4	1	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	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
442	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
443	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
444	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
445	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
446	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
448	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
462	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
463	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
464	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
475	C₄²	#41×C₅₃	1	1	yes	characteristic
476	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
477	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
478	Q₈	—	2	1	no	—
479	Q₈	—	2	1	no	—
480	Q₈	—	2	1	no	—
481	Q₈	—	2	1	no	—
482	Q₈	—	2	1	no	—
483	Q₈	—	2	1	no	—
484	Q₈	—	2	1	no	—
485	Q₈	—	4	2	no	—
486	Q₈	—	4	2	no	—
487	Q₈	#131×C₅₃	1	2	yes	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
504	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
505	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
506	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
508	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
509	C₈	—	2	1	no	—
510	C₈	—	2	1	no	—
511	C₈	—	2	1	no	—
512	C₈	—	2	1	no	—
513	C₈	—	2	1	no	—
514	C₈	—	2	1	no	—
515	C₈	—	2	1	no	—
516	C₈	—	2	1	no	—
517	C₈	—	4	1	no	—
518	C₈	—	4	1	no	—
519	C₈	—	4	1	no	—
520	C₈	—	4	1	no	—
521	C₈	—	4	1	no	—
522	C₈	—	4	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
527	C₄	—	2	1	no	—
528	C₄	—	2	1	no	—
529	C₄	—	2	1	no	—
530	C₄	—	2	1	no	—
531	C₄	—	2	2	no	—
532	C₄	—	4	1	no	—
533	C₄	—	4	1	no	—
534	C₄	—	4	1	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	1	no	—
538	C₄	—	4	1	no	—
539	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
540	C₄	C₅₃×#568	1	1	yes	characteristic
541	C₄	C₅₃×#578	1	1	yes	characteristic
542	C₄	C₅₃×#578	1	1	yes	characteristic
543	C₂	—	4	1	no	—
544	C₂	C₅₃×#2933	1	1	yes	characteristic
545	C₂	C₅₃×#2950	1	1	yes	characteristic
546	C₂	C₅₃×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1593	C₂	semidirect
5. C₅₃×#1591	C₂	semidirect
6. C₅₃×#1587	C₂	semidirect
7. C₅₃×#1587	C₂	semidirect
8. C₅₃×#1584	C₂	semidirect
9. C₅₃×#1584	C₂	semidirect
10. C₅₃×#1579	C₂	semidirect
11. C₈×Q₁₆×C₅₃	C₂	semidirect
249. #18032	C₅₃	direct (N × H)
273. #1593	C₂×C₅₃	semidirect
274. #1591	C₂×C₅₃	semidirect
275. #1587	C₂×C₅₃	semidirect
276. #1587	C₂×C₅₃	semidirect
277. #1584	C₂×C₅₃	semidirect
278. #1584	C₂×C₅₃	semidirect
279. #1579	C₂×C₅₃	semidirect
280. C₈×Q₁₆	C₂×C₅₃	semidirect
340. C₅₃	#18032	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3104	2
C₅₃×#2986	2
C₅₃×#2984	2
C₅₃×#1593	2
C₅₃×#1591	2
C₅₃×#1587	2
C₅₃×#1584	2
C₅₃×#1579	2
C₈×Q₁₆×C₅₃	2
C₅₃×#1572	2
C₅₃×#1497	2
C₅₃×#1479	2
C₅₃×#1454	2
#18032	53

Subgroups — C53×#18032 #27225544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18032 #27225544