Subgroups — C₅₃×#17593 #27225105

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1521	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1515	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1514	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1504	C₂	1	4	yes	maximal, complemented
8	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
9	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
10	C₅₃×#533	C₂²	1	1	yes	characteristic
11	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₅₃×#529	—	2	1	no	—
15	C₅₃×#529	—	2	1	no	—
16	C₅₃×#529	—	2	1	no	—
17	C₅₃×#529	—	2	1	no	—
18	C₅₃×#529	C₄	1	2	yes	complemented
19	C₅₃×#529	C₂²	1	2	yes	—
20	C₅₃×#525	—	2	1	no	—
21	C₅₃×#525	—	2	1	no	—
22	C₅₃×#525	C₂²	1	2	yes	—
23	C₄×#41×C₅₃	—	2	1	no	—
24	C₄×#41×C₅₃	—	2	1	no	—
25	C₄×#41×C₅₃	C₄	1	2	yes	complemented
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#432	—	2	1	no	—
28	C₅₃×#432	C₂²	1	2	yes	—
29	C₅₃×#431	—	2	1	no	—
30	C₅₃×#431	C₂²	1	2	yes	—
31	C₅₃×#423	—	2	1	no	—
32	C₅₃×#407	C₂²	1	2	yes	—
33	C₄×M₁₆×C₅₃	—	2	1	no	—
34	C₅₃×#383	—	2	1	no	—
35	C₅₃×#383	—	2	1	no	—
36	C₅₃×#383	C₂²	1	2	yes	—
37	C₅₃×#383	C₂²	1	2	yes	—
38	C₂×#95×C₅₃	—	2	1	no	—
39	C₂×#95×C₅₃	—	2	1	no	—
40	C₂×#95×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#328	C₂²	1	8	yes	—
42	C₅₃×#327	C₂²	1	8	yes	—
43	C₂×#41×C₅₃	—	2	1	no	—
44	C₂×#41×C₅₃	—	2	1	no	—
45	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₅₃	—	2	2	no	—
47	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
54	#127×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
55	#126×C₅₃	—	2	2	no	—
56	#126×C₅₃	—	2	2	no	—
57	#125×C₅₃	—	2	1	no	—
58	#125×C₅₃	—	2	1	no	—
59	#125×C₅₃	C₂×C₄	1	2	yes	complemented
60	#124×C₅₃	—	2	1	no	—
61	#124×C₅₃	—	2	1	no	—
62	#124×C₅₃	C₂×C₄	1	2	yes	complemented
63	#123×C₅₃	—	2	2	no	—
64	#123×C₅₃	—	2	2	no	—
65	#123×C₅₃	—	2	2	no	—
66	C₂²⋊Q₈×C₅₃	—	2	1	no	—
67	C₂²⋊Q₈×C₅₃	—	2	1	no	—
68	C₂²⋊Q₈×C₅₃	—	2	1	no	—
69	C₂²⋊Q₈×C₅₃	—	2	1	no	—
70	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
72	#121×C₅₃	—	2	1	no	—
73	#121×C₅₃	—	2	1	no	—
74	#121×C₅₃	—	2	1	no	—
75	#121×C₅₃	—	2	1	no	—
76	#121×C₅₃	C₂×C₄	1	2	yes	—
77	#121×C₅₃	C₂³	1	2	yes	—
78	#120×C₅₃	—	2	2	no	—
79	C₄×Q₈×C₅₃	—	2	1	no	—
80	C₄×Q₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	—	2	1	no	—
82	C₄×Q₈×C₅₃	—	2	1	no	—
83	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
84	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
85	C₄×D₄×C₅₃	—	2	1	no	—
86	C₄×D₄×C₅₃	—	2	1	no	—
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	2	no	—
90	C₄×D₄×C₅₃	—	2	2	no	—
91	C₄×D₄×C₅₃	—	2	2	no	—
92	C₄×D₄×C₅₃	—	2	2	no	—
93	C₄×D₄×C₅₃	—	2	2	no	—
94	C₄×D₄×C₅₃	—	2	2	no	—
95	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
96	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
97	#117×C₅₃	—	2	2	no	—
98	#117×C₅₃	—	2	2	no	—
99	#117×C₅₃	—	2	2	no	—
100	#117×C₅₃	—	2	1	no	—
101	#117×C₅₃	—	2	1	no	—
102	#117×C₅₃	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₅₃	—	2	2	no	—
104	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₅₃	—	2	2	no	—
117	C₄⋊C₈×C₅₃	—	2	2	no	—
118	C₂²⋊C₈×C₅₃	—	2	2	no	—
119	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
120	#97×C₅₃	—	2	2	no	—
121	C₄×C₈×C₅₃	—	2	2	no	—
122	#95×C₅₃	—	2	2	no	—
123	#95×C₅₃	—	2	2	no	—
124	#95×C₅₃	—	2	2	no	—
125	#95×C₅₃	—	2	2	no	—
126	#95×C₅₃	—	2	2	no	—
127	#95×C₅₃	C₂³	1	4	yes	—
128	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
129	#41×C₅₃	—	4	4	no	—
130	#41×C₅₃	—	4	4	no	—
131	#41×C₅₃	—	4	4	no	—
132	C₂×Q₈×C₅₃	—	2	1	no	—
133	C₂×Q₈×C₅₃	—	2	1	no	—
134	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₅₃	—	2	2	no	—
142	C₂²×C₄×C₅₃	—	2	2	no	—
143	C₂²×C₄×C₅₃	—	2	2	no	—
144	C₂²×C₄×C₅₃	—	2	4	no	—
145	C₂²×C₄×C₅₃	—	4	2	no	—
146	C₂²×C₄×C₅₃	—	4	2	no	—
147	C₂²×C₄×C₅₃	—	4	2	no	—
148	C₂²×C₄×C₅₃	—	4	4	no	—
149	C₂²×C₄×C₅₃	—	4	4	no	—
150	C₂²×C₄×C₅₃	—	4	4	no	—
151	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	M₁₆×C₅₃	—	4	4	no	—
170	C₂×C₈×C₅₃	—	2	4	no	—
171	C₄⋊C₄×C₅₃	—	2	1	no	—
172	C₄⋊C₄×C₅₃	—	2	1	no	—
173	C₄⋊C₄×C₅₃	—	2	1	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	2	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	2	no	—
178	C₄⋊C₄×C₅₃	—	2	2	no	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
183	C₂²⋊C₄×C₅₃	—	2	2	no	—
184	C₂²⋊C₄×C₅₃	—	2	2	no	—
185	C₂²⋊C₄×C₅₃	—	2	2	no	—
186	C₂²⋊C₄×C₅₃	—	2	2	no	—
187	C₂²⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	2	no	—
189	C₂²⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₂²⋊C₄×C₅₃	—	2	2	no	—
192	C₄²×C₅₃	—	2	2	no	—
193	C₄²×C₅₃	—	2	2	no	—
194	C₄²×C₅₃	—	2	2	no	—
195	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
196	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
197	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
198	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
211	C₂³×C₅₃	—	2	2	no	—
212	C₂³×C₅₃	—	2	2	no	—
213	C₂³×C₅₃	—	2	2	no	—
214	C₂³×C₅₃	—	2	2	no	—
215	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₅₃	—	4	2	no	—
219	Q₈×C₅₃	—	4	2	no	—
220	Q₈×C₅₃	—	4	2	no	—
221	D₄×C₅₃	—	4	2	no	—
222	D₄×C₅₃	—	4	2	no	—
223	D₄×C₅₃	—	4	2	no	—
224	D₄×C₅₃	—	4	4	no	—
225	D₄×C₅₃	—	4	4	no	—
226	D₄×C₅₃	—	4	4	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
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	—	4	1	no	—
237	C₂×C₄×C₅₃	—	4	1	no	—
238	C₂×C₄×C₅₃	—	4	1	no	—
239	C₂×C₄×C₅₃	—	4	1	no	—
240	C₂×C₄×C₅₃	—	4	1	no	—
241	C₂×C₄×C₅₃	—	4	1	no	—
242	C₂×C₄×C₅₃	—	4	1	no	—
243	C₂×C₄×C₅₃	—	4	1	no	—
244	C₂×C₄×C₅₃	—	4	1	no	—
245	C₂×C₄×C₅₃	—	4	1	no	—
246	C₂×C₄×C₅₃	—	4	1	no	—
247	C₂×C₄×C₅₃	—	4	1	no	—
248	C₂×C₄×C₅₃	—	4	4	no	—
249	C₂×C₄×C₅₃	—	4	4	no	—
250	C₂×C₄×C₅₃	—	4	4	no	—
251	C₂×C₄×C₅₃	—	4	4	no	—
252	C₂×C₄×C₅₃	—	4	4	no	—
253	C₂×C₄×C₅₃	—	4	4	no	—
254	C₂×C₄×C₅₃	—	8	8	no	—
255	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
266	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
267	C₈×C₅₃	—	4	4	no	—
268	#17593	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂²×C₅₃	—	2	8	no	—
270	C₂²×C₅₃	—	2	1	no	—
271	C₂²×C₅₃	—	2	1	no	—
272	C₂²×C₅₃	—	2	1	no	—
273	C₂²×C₅₃	—	2	1	no	—
274	C₂²×C₅₃	—	2	1	no	—
275	C₂²×C₅₃	—	2	1	no	—
276	C₂²×C₅₃	—	4	2	no	—
277	C₂²×C₅₃	—	4	2	no	—
278	C₂²×C₅₃	—	4	2	no	—
279	C₂²×C₅₃	#410	1	2	yes	—
280	C₂²×C₅₃	#421	1	2	yes	—
281	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
282	C₂²×C₅₃	#575	1	2	yes	—
283	C₄×C₅₃	—	2	1	no	—
284	C₄×C₅₃	—	2	1	no	—
285	C₄×C₅₃	—	2	1	no	—
286	C₄×C₅₃	—	2	1	no	—
287	C₄×C₅₃	—	2	1	no	—
288	C₄×C₅₃	—	2	1	no	—
289	C₄×C₅₃	—	2	1	no	—
290	C₄×C₅₃	—	2	1	no	—
291	C₄×C₅₃	—	2	1	no	—
292	C₄×C₅₃	—	2	1	no	—
293	C₄×C₅₃	—	2	1	no	—
294	C₄×C₅₃	—	2	1	no	—
295	C₄×C₅₃	—	4	2	no	—
296	C₄×C₅₃	—	4	2	no	—
297	C₄×C₅₃	—	4	2	no	—
298	C₄×C₅₃	—	8	4	no	—
299	#3444	C₂×C₅₃	1	1	yes	characteristic
300	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
301	C₂×#383	C₂×C₅₃	1	1	yes	characteristic, complemented
302	#1521	C₂×C₅₃	1	4	yes	complemented
303	#1515	C₂×C₅₃	1	2	yes	complemented
304	#1514	C₂×C₅₃	1	2	yes	complemented
305	#1504	C₂×C₅₃	1	4	yes	complemented
306	C₂×C₅₃	—	2	2	no	—
307	C₂×C₅₃	—	2	2	no	—
308	C₂×C₅₃	—	2	2	no	—
309	C₂×C₅₃	—	8	2	no	—
310	C₂×C₅₃	C₂×#410	1	1	yes	characteristic
311	C₂×C₅₃	#2896	1	1	yes	characteristic
312	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
313	#533	C₄×C₅₃	1	1	yes	characteristic, complemented
314	#533	C₄×C₅₃	1	1	yes	characteristic, complemented
315	#533	C₂²×C₅₃	1	1	yes	characteristic
316	#532	C₄×C₅₃	1	1	yes	characteristic, complemented
317	#532	C₄×C₅₃	1	1	yes	characteristic, complemented
318	#532	C₂²×C₅₃	1	1	yes	characteristic
319	#529	—	2	1	no	—
320	#529	—	2	1	no	—
321	#529	—	2	1	no	—
322	#529	—	2	1	no	—
323	#529	C₄×C₅₃	1	2	yes	complemented
324	#529	C₂²×C₅₃	1	2	yes	—
325	#525	—	2	1	no	—
326	#525	—	2	1	no	—
327	#525	C₂²×C₅₃	1	2	yes	—
328	C₄×#41	—	2	1	no	—
329	C₄×#41	—	2	1	no	—
330	C₄×#41	C₄×C₅₃	1	2	yes	complemented
331	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
332	#432	—	2	1	no	—
333	#432	C₂²×C₅₃	1	2	yes	—
334	#431	—	2	1	no	—
335	#431	C₂²×C₅₃	1	2	yes	—
336	#423	—	2	1	no	—
337	#407	C₂²×C₅₃	1	2	yes	—
338	C₄×M₁₆	—	2	1	no	—
339	#383	—	2	1	no	—
340	#383	—	2	1	no	—
341	#383	C₂²×C₅₃	1	2	yes	—
342	#383	C₂²×C₅₃	1	2	yes	—
343	C₂×#95	—	2	1	no	—
344	C₂×#95	—	2	1	no	—
345	C₂×#95	C₂²×C₅₃	1	2	yes	—
346	#328	C₂²×C₅₃	1	8	yes	—
347	#327	C₂²×C₅₃	1	8	yes	—
348	C₅₃	#17593	1	1	yes	characteristic, direct product factor, Sylow
349	C₂×#41	—	2	1	no	—
350	C₂×#41	—	2	1	no	—
351	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
352	C₂³×C₄	—	2	2	no	—
353	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
354	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
356	C₂×M₁₆	—	2	2	no	—
357	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
358	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
359	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
360	#127	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
361	#126	—	2	2	no	—
362	#126	—	2	2	no	—
363	#125	—	2	1	no	—
364	#125	—	2	1	no	—
365	#125	C₂×C₄×C₅₃	1	2	yes	complemented
366	#124	—	2	1	no	—
367	#124	—	2	1	no	—
368	#124	C₂×C₄×C₅₃	1	2	yes	complemented
369	#123	—	2	2	no	—
370	#123	—	2	2	no	—
371	#123	—	2	2	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
377	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
378	#121	—	2	1	no	—
379	#121	—	2	1	no	—
380	#121	—	2	1	no	—
381	#121	—	2	1	no	—
382	#121	C₂×C₄×C₅₃	1	2	yes	—
383	#121	C₂³×C₅₃	1	2	yes	—
384	#120	—	2	2	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
390	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	—	2	1	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	complemented
402	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	complemented
403	#117	—	2	1	no	—
404	#117	—	2	1	no	—
405	#117	—	2	2	no	—
406	#117	—	2	2	no	—
407	#117	—	2	2	no	—
408	#117	C₂×C₄×C₅₃	1	2	yes	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
411	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
412	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
413	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
414	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
415	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
417	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
418	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
419	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
422	C₄⋊C₈	—	2	2	no	—
423	C₄⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
426	#97	—	2	2	no	—
427	C₄×C₈	—	2	2	no	—
428	#95	—	2	2	no	—
429	#95	—	2	2	no	—
430	#95	—	2	2	no	—
431	#95	—	2	2	no	—
432	#95	—	2	2	no	—
433	#95	C₂³×C₅₃	1	4	yes	—
434	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
435	#41	—	4	4	no	—
436	#41	—	4	4	no	—
437	#41	—	4	4	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×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	C₂⁴×C₅₃	1	1	yes	characteristic
475	M₁₆	—	4	4	no	—
476	C₂×C₈	—	2	4	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
488	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	2	no	—
501	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
502	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
503	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
504	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
505	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
506	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
507	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
508	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
509	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
510	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
511	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
512	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
514	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
515	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
516	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
522	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
523	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
524	Q₈	—	4	2	no	—
525	Q₈	—	4	2	no	—
526	Q₈	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	2	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	4	no	—
531	D₄	—	4	4	no	—
532	D₄	—	4	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	8	8	no	—
561	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
562	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
563	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
564	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
565	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
566	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
567	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
569	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
570	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
571	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
573	C₈	—	4	4	no	—
574	C₂²	—	2	8	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	C₅₃×#410	1	2	yes	—
585	C₂²	C₅₃×#421	1	2	yes	—
586	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
587	C₂²	C₅₃×#575	1	2	yes	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	1	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	8	4	no	—
604	C₂	—	2	2	no	—
605	C₂	—	2	2	no	—
606	C₂	—	2	2	no	—
607	C₂	—	8	2	no	—
608	C₂	C₂×C₅₃×#410	1	1	yes	characteristic
609	C₂	C₅₃×#2896	1	1	yes	characteristic
610	C₂	C₂×C₅₃×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2920	C₂	semidirect
3. C₂×C₅₃×#383	C₂	semidirect
4. C₅₃×#1521	C₂	semidirect
5. C₅₃×#1515	C₂	semidirect
6. C₅₃×#1514	C₂	semidirect
7. C₅₃×#1504	C₂	semidirect
8. C₅₃×#533	C₄	semidirect
9. C₅₃×#533	C₄	semidirect
11. C₅₃×#532	C₄	semidirect
12. C₅₃×#532	C₄	semidirect
18. C₅₃×#529	C₄	semidirect
25. C₄×#41×C₅₃	C₄	semidirect
51. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
52. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
53. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
54. #127×C₅₃	C₂×C₄	semidirect
59. #125×C₅₃	C₂×C₄	semidirect
62. #124×C₅₃	C₂×C₄	semidirect
83. C₄×Q₈×C₅₃	C₂×C₄	semidirect
84. C₄×Q₈×C₅₃	C₂×C₄	semidirect
95. C₄×D₄×C₅₃	C₂×C₄	semidirect
96. C₄×D₄×C₅₃	C₂×C₄	semidirect
268. #17593	C₅₃	direct (N × H)
300. #2920	C₂×C₅₃	semidirect
301. C₂×#383	C₂×C₅₃	semidirect
302. #1521	C₂×C₅₃	semidirect
303. #1515	C₂×C₅₃	semidirect
304. #1514	C₂×C₅₃	semidirect
305. #1504	C₂×C₅₃	semidirect
313. #533	C₄×C₅₃	semidirect
314. #533	C₄×C₅₃	semidirect
316. #532	C₄×C₅₃	semidirect
317. #532	C₄×C₅₃	semidirect
323. #529	C₄×C₅₃	semidirect
330. C₄×#41	C₄×C₅₃	semidirect
348. C₅₃	#17593	direct (N × H)
357. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
358. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
359. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
360. #127	C₂×C₄×C₅₃	semidirect
365. #125	C₂×C₄×C₅₃	semidirect
368. #124	C₂×C₄×C₅₃	semidirect
389. C₄×Q₈	C₂×C₄×C₅₃	semidirect
390. C₄×Q₈	C₂×C₄×C₅₃	semidirect
401. C₄×D₄	C₂×C₄×C₅₃	semidirect
402. C₄×D₄	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
C₅₃×#3444	2
C₅₃×#2920	2
C₂×C₅₃×#383	2
C₅₃×#1521	2
C₅₃×#1515	2
C₅₃×#1514	2
C₅₃×#1504	2
#17593	53

Subgroups — C53×#17593 #27225105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17593 #27225105