Subgroups — C₅₃×#17662 #27225174

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 600 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×#432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1537	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1536	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1535	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1524	C₂	1	4	yes	maximal, complemented
8	C₅₃×#533	C₄	1	1	yes	characteristic
9	C₅₃×#533	C₄	1	1	yes	characteristic
10	C₅₃×#533	C₂²	1	1	yes	characteristic
11	C₅₃×#532	C₄	1	1	yes	characteristic
12	C₅₃×#532	C₄	1	1	yes	characteristic
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	C₂²	1	2	yes	—
19	C₅₃×#529	C₂²	1	2	yes	—
20	C₅₃×#525	—	2	1	no	—
21	C₅₃×#525	—	2	1	no	—
22	C₅₃×#525	C₄	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	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#432	—	2	1	no	—
28	C₅₃×#432	—	2	1	no	—
29	C₅₃×#432	C₂²	1	2	yes	—
30	C₅₃×#432	C₂²	1	2	yes	—
31	C₅₃×#432	C₂²	1	2	yes	—
32	C₅₃×#432	C₂²	1	2	yes	—
33	C₅₃×#431	—	2	1	no	—
34	C₅₃×#431	—	2	1	no	—
35	C₅₃×#423	—	2	1	no	—
36	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
37	C₅₃×#407	—	2	1	no	—
38	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
39	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
40	C₂×#97×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#330	C₂²	1	8	yes	complemented
42	C₅₃×#329	C₂²	1	8	yes	complemented
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	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	2	yes	—
60	#124×C₅₃	—	2	1	no	—
61	#124×C₅₃	—	2	1	no	—
62	#124×C₅₃	C₂³	1	2	yes	—
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	C₂×C₄	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	C₂×C₄	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	—
84	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
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	—
96	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
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	D₄	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	C₂³	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	C₄⋊C₈×C₅₃	—	2	2	no	—
119	C₄⋊C₈×C₅₃	—	2	2	no	—
120	C₂²⋊C₈×C₅₃	—	2	2	no	—
121	C₂²⋊C₈×C₅₃	—	2	2	no	—
122	C₂²⋊C₈×C₅₃	—	2	2	no	—
123	C₂²⋊C₈×C₅₃	—	2	2	no	—
124	#97×C₅₃	—	2	2	no	—
125	#97×C₅₃	C₂³	1	4	yes	—
126	#97×C₅₃	C₂³	1	4	yes	—
127	C₄×C₈×C₅₃	—	2	2	no	—
128	C₂⁴×C₅₃	C₂²⋊C₄	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	—	4	2	no	—
145	C₂²×C₄×C₅₃	—	4	2	no	—
146	C₂²×C₄×C₅₃	—	4	2	no	—
147	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂×D₄	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	M₁₆×C₅₃	—	4	4	no	—
166	C₂×C₈×C₅₃	—	2	4	no	—
167	C₂×C₈×C₅₃	—	2	4	no	—
168	C₂×C₈×C₅₃	—	4	4	no	—
169	C₄⋊C₄×C₅₃	—	2	1	no	—
170	C₄⋊C₄×C₅₃	—	2	1	no	—
171	C₄⋊C₄×C₅₃	—	2	1	no	—
172	C₄⋊C₄×C₅₃	—	2	1	no	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	—	2	2	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	C₂²×C₄	1	2	yes	—
180	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
181	C₂²⋊C₄×C₅₃	—	2	2	no	—
182	C₂²⋊C₄×C₅₃	—	2	2	no	—
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	C₄²×C₅₃	—	2	2	no	—
191	C₄²×C₅₃	—	2	2	no	—
192	C₄²×C₅₃	—	2	2	no	—
193	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
194	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
209	C₂³×C₅₃	—	2	2	no	—
210	C₂³×C₅₃	—	2	2	no	—
211	C₂³×C₅₃	—	4	1	no	—
212	C₂³×C₅₃	—	4	1	no	—
213	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	Q₈×C₅₃	—	4	2	no	—
217	Q₈×C₅₃	—	4	2	no	—
218	Q₈×C₅₃	—	4	2	no	—
219	D₄×C₅₃	—	4	2	no	—
220	D₄×C₅₃	—	4	2	no	—
221	D₄×C₅₃	—	4	2	no	—
222	D₄×C₅₃	—	4	4	no	—
223	D₄×C₅₃	—	4	4	no	—
224	D₄×C₅₃	—	4	4	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	2	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	4	no	—
247	C₂×C₄×C₅₃	—	4	4	no	—
248	C₂×C₄×C₅₃	—	4	4	no	—
249	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
261	C₈×C₅₃	—	4	4	no	—
262	C₈×C₅₃	—	4	4	no	—
263	#17662	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₅₃	—	2	1	no	—
265	C₂²×C₅₃	—	2	1	no	—
266	C₂²×C₅₃	—	2	1	no	—
267	C₂²×C₅₃	—	2	1	no	—
268	C₂²×C₅₃	—	2	1	no	—
269	C₂²×C₅₃	—	2	1	no	—
270	C₂²×C₅₃	—	2	1	no	—
271	C₂²×C₅₃	—	2	1	no	—
272	C₂²×C₅₃	—	4	4	no	—
273	C₂²×C₅₃	—	4	2	no	—
274	C₂²×C₅₃	—	4	2	no	—
275	C₂²×C₅₃	—	4	2	no	—
276	C₂²×C₅₃	#421	1	2	yes	—
277	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
278	C₄×C₅₃	—	2	1	no	—
279	C₄×C₅₃	—	2	1	no	—
280	C₄×C₅₃	—	2	1	no	—
281	C₄×C₅₃	—	2	1	no	—
282	C₄×C₅₃	—	2	1	no	—
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	—	4	2	no	—
289	C₄×C₅₃	—	4	2	no	—
290	C₄×C₅₃	—	4	2	no	—
291	C₄×C₅₃	#413	1	2	yes	—
292	C₄×C₅₃	#421	1	2	yes	—
293	#3444	C₂×C₅₃	1	1	yes	characteristic
294	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
295	C₂×#432	C₂×C₅₃	1	1	yes	characteristic, complemented
296	#1537	C₂×C₅₃	1	2	yes	complemented
297	#1536	C₂×C₅₃	1	2	yes	complemented
298	#1535	C₂×C₅₃	1	4	yes	complemented
299	#1524	C₂×C₅₃	1	4	yes	complemented
300	C₂×C₅₃	—	2	2	no	—
301	C₂×C₅₃	—	4	1	no	—
302	C₂×C₅₃	—	4	1	no	—
303	C₂×C₅₃	—	8	2	no	—
304	C₂×C₅₃	C₂×#413	1	1	yes	characteristic
305	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
306	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
307	#533	C₄×C₅₃	1	1	yes	characteristic
308	#533	C₄×C₅₃	1	1	yes	characteristic
309	#533	C₂²×C₅₃	1	1	yes	characteristic
310	#532	C₄×C₅₃	1	1	yes	characteristic
311	#532	C₄×C₅₃	1	1	yes	characteristic
312	#532	C₂²×C₅₃	1	1	yes	characteristic
313	#529	—	2	1	no	—
314	#529	—	2	1	no	—
315	#529	—	2	1	no	—
316	#529	—	2	1	no	—
317	#529	C₂²×C₅₃	1	2	yes	—
318	#529	C₂²×C₅₃	1	2	yes	—
319	#525	—	2	1	no	—
320	#525	—	2	1	no	—
321	#525	C₄×C₅₃	1	2	yes	—
322	C₄×#41	—	2	1	no	—
323	C₄×#41	—	2	1	no	—
324	C₄×#41	C₄×C₅₃	1	2	yes	—
325	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
326	#432	—	2	1	no	—
327	#432	—	2	1	no	—
328	#432	C₂²×C₅₃	1	2	yes	—
329	#432	C₂²×C₅₃	1	2	yes	—
330	#432	C₂²×C₅₃	1	2	yes	—
331	#432	C₂²×C₅₃	1	2	yes	—
332	#431	—	2	1	no	—
333	#431	—	2	1	no	—
334	#423	—	2	1	no	—
335	C₂×C₄⋊C₈	—	2	1	no	—
336	#407	—	2	1	no	—
337	C₂×C₂²⋊C₈	—	2	1	no	—
338	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
339	C₂×#97	C₂²×C₅₃	1	2	yes	—
340	#330	C₂²×C₅₃	1	8	yes	complemented
341	#329	C₂²×C₅₃	1	8	yes	complemented
342	C₅₃	#17662	1	1	yes	characteristic, direct product factor, Sylow
343	C₂×#41	—	2	1	no	—
344	C₂×#41	—	2	1	no	—
345	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
346	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
347	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
348	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
349	C₂×M₁₆	—	2	2	no	—
350	C₂²×C₈	—	2	2	no	—
351	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
352	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
353	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
354	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
355	#126	—	2	2	no	—
356	#126	—	2	2	no	—
357	#125	—	2	1	no	—
358	#125	—	2	1	no	—
359	#125	C₂³×C₅₃	1	2	yes	—
360	#124	—	2	1	no	—
361	#124	—	2	1	no	—
362	#124	C₂³×C₅₃	1	2	yes	—
363	#123	—	2	2	no	—
364	#123	—	2	2	no	—
365	#123	—	2	2	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	C₂²⋊Q₈	—	2	1	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
371	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
372	#121	—	2	1	no	—
373	#121	—	2	1	no	—
374	#121	—	2	1	no	—
375	#121	—	2	1	no	—
376	#121	C₂×C₄×C₅₃	1	2	yes	—
377	#121	C₂×C₄×C₅₃	1	2	yes	—
378	#120	—	2	2	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	—	2	1	no	—
383	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
384	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	—	2	1	no	—
395	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
396	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	—	2	2	no	—
400	#117	—	2	2	no	—
401	#117	—	2	2	no	—
402	#117	C₂×C₄×C₅₃	1	2	yes	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
405	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
406	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
407	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
408	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
409	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
410	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
411	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
412	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
416	C₄⋊C₈	—	2	2	no	—
417	C₄⋊C₈	—	2	2	no	—
418	C₄⋊C₈	—	2	2	no	—
419	C₄⋊C₈	—	2	2	no	—
420	C₂²⋊C₈	—	2	2	no	—
421	C₂²⋊C₈	—	2	2	no	—
422	C₂²⋊C₈	—	2	2	no	—
423	C₂²⋊C₈	—	2	2	no	—
424	#97	—	2	2	no	—
425	#97	C₂³×C₅₃	1	4	yes	—
426	#97	C₂³×C₅₃	1	4	yes	—
427	C₄×C₈	—	2	2	no	—
428	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
429	#41	—	4	4	no	—
430	#41	—	4	4	no	—
431	#41	—	4	4	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
448	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
449	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
450	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
451	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
465	M₁₆	—	4	4	no	—
466	C₂×C₈	—	2	4	no	—
467	C₂×C₈	—	2	4	no	—
468	C₂×C₈	—	4	4	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
480	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	2	2	no	—
493	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
494	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
495	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
496	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
497	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
498	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
499	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
500	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
501	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
502	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
503	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
504	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
505	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
506	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
507	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
508	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
514	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
515	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
516	Q₈	—	4	2	no	—
517	Q₈	—	4	2	no	—
518	Q₈	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	2	no	—
522	D₄	—	4	4	no	—
523	D₄	—	4	4	no	—
524	D₄	—	4	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	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	4	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
550	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
551	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
552	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
553	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
554	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
555	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
556	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
557	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
558	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
560	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
561	C₈	—	4	4	no	—
562	C₈	—	4	4	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	2	no	—
575	C₂²	C₅₃×#421	1	2	yes	—
576	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
577	C₄	—	2	1	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	1	no	—
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	C₅₃×#413	1	2	yes	—
591	C₄	C₅₃×#421	1	2	yes	—
592	C₂	—	2	2	no	—
593	C₂	—	4	1	no	—
594	C₂	—	4	1	no	—
595	C₂	—	8	2	no	—
596	C₂	C₂×C₅₃×#413	1	1	yes	characteristic
597	C₂	C₂×C₅₃×#421	1	1	yes	characteristic
598	C₂	C₂×C₅₃×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₅₃×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#432	C₂	semidirect
4. C₅₃×#1537	C₂	semidirect
5. C₅₃×#1536	C₂	semidirect
6. C₅₃×#1535	C₂	semidirect
7. C₅₃×#1524	C₂	semidirect
41. C₅₃×#330	C₂²	semidirect
42. C₅₃×#329	C₂²	semidirect
263. #17662	C₅₃	direct (N × H)
294. #2920	C₂×C₅₃	semidirect
295. C₂×#432	C₂×C₅₃	semidirect
296. #1537	C₂×C₅₃	semidirect
297. #1536	C₂×C₅₃	semidirect
298. #1535	C₂×C₅₃	semidirect
299. #1524	C₂×C₅₃	semidirect
340. #330	C₂²×C₅₃	semidirect
341. #329	C₂²×C₅₃	semidirect
342. C₅₃	#17662	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×#3444	2
C₅₃×#2920	2
C₂×C₅₃×#432	2
C₅₃×#1537	2
C₅₃×#1536	2
C₅₃×#1535	2
C₅₃×#1524	2
#17662	53

Subgroups — C53×#17662 #27225174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17662 #27225174