Subgroups — C₅₃×#17601 #27225113

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3523	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2320	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1521	C₂	1	2	yes	maximal
5	C₅₃×#1518	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1515	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1512	C₂	1	2	yes	maximal
8	C₅₃×#1509	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1506	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1503	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1500	C₂	1	1	yes	maximal, characteristic
12	C₅₃×#557	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#557	C₄	1	1	yes	characteristic, complemented
14	C₅₃×#557	C₄	1	2	yes	complemented
15	C₅₃×#556	C₄	1	1	yes	characteristic, complemented
16	C₅₃×#556	C₄	1	2	yes	complemented
17	C₅₃×#552	—	2	2	no	—
18	C₅₃×#552	—	2	1	no	—
19	D₄×Q₈×C₅₃	—	2	1	no	—
20	D₄×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#541	—	2	1	no	—
22	C₅₃×#541	—	2	2	no	—
23	C₅₃×#533	C₂²	1	1	yes	characteristic
24	C₅₃×#533	C₂²	1	2	yes	—
25	C₅₃×#529	C₂²	1	1	yes	characteristic
26	C₅₃×#529	C₂²	1	2	yes	—
27	C₅₃×#519	—	2	1	no	—
28	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#446	—	2	1	no	—
30	C₅₃×#446	—	2	2	no	—
31	C₈×Q₈×C₅₃	—	2	1	no	—
32	C₅₃×#432	C₂²	1	1	yes	characteristic
33	C₅₃×#432	C₂²	1	2	yes	—
34	C₅₃×#431	C₂²	1	1	yes	characteristic
35	C₅₃×#431	C₂²	1	2	yes	—
36	C₅₃×#391	—	2	1	no	—
37	C₅₃×#389	—	2	1	no	—
38	C₅₃×#387	—	2	2	no	—
39	C₅₃×#383	C₂²	1	2	yes	—
40	C₅₃×#382	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#376	C₂²	1	1	yes	characteristic
43	C₅₃×#328	C₂²	1	1	yes	characteristic
44	C₅₃×#328	C₂²	1	1	yes	characteristic
45	C₅₃×#328	C₂²	1	2	yes	—
46	C₅₃×#328	C₂²	1	2	yes	—
47	C₅₃×#328	C₂²	1	2	yes	—
48	C₅₃×#328	C₂²	1	2	yes	—
49	C₅₃×#328	C₂²	1	2	yes	—
50	C₅₃×#327	C₂²	1	1	yes	characteristic
51	C₅₃×#327	C₂²	1	1	yes	characteristic
52	C₅₃×#327	C₂²	1	2	yes	—
53	C₂²×Q₈×C₅₃	—	2	1	no	—
54	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
61	#126×C₅₃	—	2	2	no	—
62	#126×C₅₃	—	2	2	no	—
63	#126×C₅₃	—	2	2	no	—
64	#126×C₅₃	—	2	2	no	—
65	#126×C₅₃	—	2	1	no	—
66	#125×C₅₃	—	2	2	no	—
67	#125×C₅₃	—	2	2	no	—
68	#125×C₅₃	—	2	2	no	—
69	#125×C₅₃	—	2	2	no	—
70	#125×C₅₃	—	2	1	no	—
71	#125×C₅₃	—	2	2	no	—
72	#125×C₅₃	—	2	1	no	—
73	#125×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#125×C₅₃	C₂×C₄	1	2	yes	—
75	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
76	#124×C₅₃	C₂×C₄	1	2	yes	—
77	#123×C₅₃	—	2	1	no	—
78	#123×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	C₂²⋊Q₈×C₅₃	—	2	1	no	—
81	C₂²⋊Q₈×C₅₃	—	2	1	no	—
82	C₂²⋊Q₈×C₅₃	—	2	2	no	—
83	C₂²⋊Q₈×C₅₃	—	2	1	no	—
84	C₂²⋊Q₈×C₅₃	—	2	2	no	—
85	C₂²⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
87	#121×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₅₃	—	2	2	no	—
89	C₄×Q₈×C₅₃	—	2	1	no	—
90	C₄×Q₈×C₅₃	—	2	1	no	—
91	C₄×Q₈×C₅₃	—	2	2	no	—
92	C₄×Q₈×C₅₃	—	2	1	no	—
93	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
96	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
99	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
100	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
101	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
102	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
103	C₄×Q₈×C₅₃	D₄	1	2	yes	—
104	C₄×D₄×C₅₃	—	2	1	no	—
105	C₄×D₄×C₅₃	—	2	2	no	—
106	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
108	#117×C₅₃	—	2	2	no	—
109	#117×C₅₃	—	2	1	no	—
110	#117×C₅₃	—	2	1	no	—
111	#117×C₅₃	—	2	2	no	—
112	#117×C₅₃	—	2	1	no	—
113	#117×C₅₃	—	2	2	no	—
114	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
115	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
117	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	—	2	1	no	—
119	C₂×C₄²×C₅₃	—	2	1	no	—
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
122	C₄⋊C₈×C₅₃	—	2	2	no	—
123	C₄⋊C₈×C₅₃	—	2	1	no	—
124	C₄⋊C₈×C₅₃	—	2	1	no	—
125	C₄⋊C₈×C₅₃	—	2	2	no	—
126	C₂²⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₂²⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
129	#97×C₅₃	—	2	2	no	—
130	#97×C₅₃	—	2	1	no	—
131	C₄×C₈×C₅₃	—	2	1	no	—
132	C₄×C₈×C₅₃	—	2	2	no	—
133	#95×C₅₃	—	2	1	no	—
134	#95×C₅₃	—	2	2	no	—
135	#95×C₅₃	—	2	2	no	—
136	#95×C₅₃	—	2	2	no	—
137	#95×C₅₃	C₂³	1	1	yes	characteristic
138	#95×C₅₃	C₂³	1	1	yes	characteristic
139	#95×C₅₃	C₂³	1	2	yes	—
140	C₂×Q₈×C₅₃	—	2	2	no	—
141	C₂×Q₈×C₅₃	—	2	1	no	—
142	C₂×Q₈×C₅₃	—	4	2	no	—
143	C₂×Q₈×C₅₃	—	4	4	no	—
144	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
147	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₅₃	—	2	1	no	—
150	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	—	2	2	no	—
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	—	2	2	no	—
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	2	no	—
165	C₄⋊C₄×C₅₃	—	2	1	no	—
166	C₄⋊C₄×C₅₃	—	2	1	no	—
167	C₄⋊C₄×C₅₃	—	2	2	no	—
168	C₄⋊C₄×C₅₃	—	2	1	no	—
169	C₄⋊C₄×C₅₃	—	2	2	no	—
170	C₄⋊C₄×C₅₃	—	2	1	no	—
171	C₄⋊C₄×C₅₃	—	2	2	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	1	no	—
178	C₄⋊C₄×C₅₃	—	2	2	no	—
179	C₄⋊C₄×C₅₃	—	2	1	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	—	4	2	no	—
182	C₄⋊C₄×C₅₃	—	8	2	no	—
183	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
184	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
185	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
186	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
187	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
188	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
189	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
190	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
192	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₅₃	—	2	2	no	—
194	C₂²⋊C₄×C₅₃	—	2	1	no	—
195	C₂²⋊C₄×C₅₃	—	2	1	no	—
196	C₂²⋊C₄×C₅₃	—	2	2	no	—
197	C₂²⋊C₄×C₅₃	—	2	1	no	—
198	C₂²⋊C₄×C₅₃	—	2	2	no	—
199	C₂²⋊C₄×C₅₃	—	2	1	no	—
200	C₂²⋊C₄×C₅₃	—	2	2	no	—
201	C₂²⋊C₄×C₅₃	—	2	1	no	—
202	C₄²×C₅₃	—	2	2	no	—
203	C₄²×C₅₃	—	2	2	no	—
204	C₄²×C₅₃	—	2	2	no	—
205	C₄²×C₅₃	—	2	2	no	—
206	C₄²×C₅₃	—	2	1	no	—
207	C₄²×C₅₃	—	4	2	no	—
208	C₄²×C₅₃	—	8	1	no	—
209	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
210	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
213	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₅₃	C₂×D₄	1	2	yes	—
215	C₄²×C₅₃	C₂×D₄	1	2	yes	—
216	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
217	C₂³×C₅₃	—	2	1	no	—
218	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₅₃	—	2	2	no	—
220	Q₈×C₅₃	—	2	2	no	—
221	Q₈×C₅₃	—	4	2	no	—
222	Q₈×C₅₃	—	4	4	no	—
223	Q₈×C₅₃	—	4	1	no	—
224	Q₈×C₅₃	—	4	2	no	—
225	Q₈×C₅₃	—	4	2	no	—
226	D₄×C₅₃	—	4	1	no	—
227	D₄×C₅₃	—	4	1	no	—
228	D₄×C₅₃	—	4	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	1	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	1	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	4	2	no	—
240	C₂×C₄×C₅₃	—	4	1	no	—
241	C₂×C₄×C₅₃	—	4	4	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	1	no	—
244	C₂×C₄×C₅₃	—	4	1	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	—	4	1	no	—
247	C₂×C₄×C₅₃	—	4	1	no	—
248	C₂×C₄×C₅₃	—	4	2	no	—
249	C₂×C₄×C₅₃	—	4	2	no	—
250	C₂×C₄×C₅₃	—	4	1	no	—
251	C₂×C₄×C₅₃	—	4	1	no	—
252	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
255	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
259	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
262	C₈×C₅₃	—	4	1	no	—
263	C₈×C₅₃	—	4	1	no	—
264	C₈×C₅₃	—	4	2	no	—
265	#17601	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂²×C₅₃	—	2	1	no	—
267	C₂²×C₅₃	—	2	1	no	—
268	C₂²×C₅₃	—	4	1	no	—
269	C₂²×C₅₃	—	4	1	no	—
270	C₂²×C₅₃	—	4	1	no	—
271	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
272	C₂²×C₅₃	#577	1	1	yes	characteristic
273	C₂²×C₅₃	#578	1	1	yes	characteristic
274	C₄×C₅₃	—	2	1	no	—
275	C₄×C₅₃	—	2	1	no	—
276	C₄×C₅₃	—	2	1	no	—
277	C₄×C₅₃	—	2	1	no	—
278	C₄×C₅₃	—	2	2	no	—
279	C₄×C₅₃	—	2	2	no	—
280	C₄×C₅₃	—	2	1	no	—
281	C₄×C₅₃	—	2	2	no	—
282	C₄×C₅₃	—	2	2	no	—
283	C₄×C₅₃	—	4	2	no	—
284	C₄×C₅₃	—	4	1	no	—
285	C₄×C₅₃	—	4	1	no	—
286	C₄×C₅₃	—	4	2	no	—
287	C₄×C₅₃	—	4	1	no	—
288	C₄×C₅₃	—	8	1	no	—
289	C₄×C₅₃	—	8	2	no	—
290	C₄×C₅₃	—	8	1	no	—
291	C₄×C₅₃	#421	1	2	yes	—
292	#3523	C₂×C₅₃	1	1	yes	characteristic
293	#2984	C₂×C₅₃	1	1	yes	characteristic, complemented
294	#2320	C₂×C₅₃	1	1	yes	characteristic, complemented
295	#1521	C₂×C₅₃	1	2	yes	—
296	#1518	C₂×C₅₃	1	2	yes	complemented
297	#1515	C₂×C₅₃	1	2	yes	complemented
298	#1512	C₂×C₅₃	1	2	yes	—
299	#1509	C₂×C₅₃	1	1	yes	characteristic
300	#1506	C₂×C₅₃	1	1	yes	characteristic, complemented
301	#1503	C₂×C₅₃	1	1	yes	characteristic, complemented
302	#1500	C₂×C₅₃	1	1	yes	characteristic
303	C₂×C₅₃	—	2	1	no	—
304	C₂×C₅₃	—	2	1	no	—
305	C₂×C₅₃	—	8	1	no	—
306	C₂×C₅₃	#2884	1	1	yes	characteristic
307	C₂×C₅₃	#2898	1	1	yes	characteristic
308	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
309	#557	C₄×C₅₃	1	1	yes	characteristic, complemented
310	#557	C₄×C₅₃	1	1	yes	characteristic, complemented
311	#557	C₄×C₅₃	1	2	yes	complemented
312	#556	C₄×C₅₃	1	1	yes	characteristic, complemented
313	#556	C₄×C₅₃	1	2	yes	complemented
314	#552	—	2	2	no	—
315	#552	—	2	1	no	—
316	D₄×Q₈	—	2	1	no	—
317	D₄×Q₈	C₄×C₅₃	1	1	yes	characteristic, complemented
318	#541	—	2	1	no	—
319	#541	—	2	2	no	—
320	#533	C₂²×C₅₃	1	1	yes	characteristic
321	#533	C₂²×C₅₃	1	2	yes	—
322	#529	C₂²×C₅₃	1	1	yes	characteristic
323	#529	C₂²×C₅₃	1	2	yes	—
324	#519	—	2	1	no	—
325	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
326	#446	—	2	1	no	—
327	#446	—	2	2	no	—
328	C₈×Q₈	—	2	1	no	—
329	#432	C₂²×C₅₃	1	1	yes	characteristic
330	#432	C₂²×C₅₃	1	2	yes	—
331	#431	C₂²×C₅₃	1	1	yes	characteristic
332	#431	C₂²×C₅₃	1	2	yes	—
333	#391	—	2	1	no	—
334	#389	—	2	1	no	—
335	#387	—	2	2	no	—
336	#383	C₂²×C₅₃	1	2	yes	—
337	#382	C₂²×C₅₃	1	2	yes	—
338	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
339	#376	C₂²×C₅₃	1	1	yes	characteristic
340	#328	C₂²×C₅₃	1	1	yes	characteristic
341	#328	C₂²×C₅₃	1	1	yes	characteristic
342	#328	C₂²×C₅₃	1	2	yes	—
343	#328	C₂²×C₅₃	1	2	yes	—
344	#328	C₂²×C₅₃	1	2	yes	—
345	#328	C₂²×C₅₃	1	2	yes	—
346	#328	C₂²×C₅₃	1	2	yes	—
347	#327	C₂²×C₅₃	1	1	yes	characteristic
348	#327	C₂²×C₅₃	1	1	yes	characteristic
349	#327	C₂²×C₅₃	1	2	yes	—
350	C₅₃	#17601	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²×Q₈	—	2	1	no	—
352	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
353	C₄⋊Q₈	—	2	1	no	—
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
357	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
358	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
359	#126	—	2	2	no	—
360	#126	—	2	2	no	—
361	#126	—	2	2	no	—
362	#126	—	2	2	no	—
363	#126	—	2	1	no	—
364	#125	—	2	2	no	—
365	#125	—	2	2	no	—
366	#125	—	2	2	no	—
367	#125	—	2	2	no	—
368	#125	—	2	1	no	—
369	#125	—	2	2	no	—
370	#125	—	2	1	no	—
371	#125	C₂×C₄×C₅₃	1	1	yes	characteristic
372	#125	C₂×C₄×C₅₃	1	2	yes	—
373	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
374	#124	C₂×C₄×C₅₃	1	2	yes	—
375	#123	—	2	1	no	—
376	#123	—	2	2	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	—	2	1	no	—
383	C₂²⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
384	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
385	#121	C₂³×C₅₃	1	1	yes	characteristic
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
392	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
393	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
394	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
395	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
396	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
397	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
398	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
399	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
400	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
401	C₄×Q₈	D₄×C₅₃	1	2	yes	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
405	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
406	#117	—	2	1	no	—
407	#117	—	2	2	no	—
408	#117	—	2	2	no	—
409	#117	—	2	1	no	—
410	#117	—	2	1	no	—
411	#117	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
415	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
416	C₂×C₄²	—	2	1	no	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
420	C₄⋊C₈	—	2	2	no	—
421	C₄⋊C₈	—	2	1	no	—
422	C₄⋊C₈	—	2	1	no	—
423	C₄⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
425	C₂²⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
426	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
427	#97	—	2	2	no	—
428	#97	—	2	1	no	—
429	C₄×C₈	—	2	1	no	—
430	C₄×C₈	—	2	2	no	—
431	#95	—	2	1	no	—
432	#95	—	2	2	no	—
433	#95	—	2	2	no	—
434	#95	—	2	2	no	—
435	#95	C₂³×C₅₃	1	1	yes	characteristic
436	#95	C₂³×C₅₃	1	1	yes	characteristic
437	#95	C₂³×C₅₃	1	2	yes	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	4	4	no	—
442	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
443	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
444	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
445	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
446	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
456	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	4	2	no	—
480	C₄⋊C₄	—	8	2	no	—
481	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
482	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
483	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
484	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
485	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
486	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
487	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
488	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
489	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
490	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	2	2	no	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	4	2	no	—
506	C₄²	—	8	1	no	—
507	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
508	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
509	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
510	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
511	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₅₃	1	2	yes	—
513	C₄²	C₂×D₄×C₅₃	1	2	yes	—
514	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
515	C₂³	—	2	1	no	—
516	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	2	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	4	2	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	4	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
551	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
552	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
553	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
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	2	yes	—
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	C₈	—	4	1	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	2	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	4	1	no	—
568	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
569	C₂²	C₅₃×#577	1	1	yes	characteristic
570	C₂²	C₅₃×#578	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	1	no	—
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	1	no	—
585	C₄	—	8	1	no	—
586	C₄	—	8	2	no	—
587	C₄	—	8	1	no	—
588	C₄	C₅₃×#421	1	2	yes	—
589	C₂	—	2	1	no	—
590	C₂	—	2	1	no	—
591	C₂	—	8	1	no	—
592	C₂	C₅₃×#2884	1	1	yes	characteristic
593	C₂	C₅₃×#2898	1	1	yes	characteristic
594	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	#121×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2984	C₂	semidirect
3. C₅₃×#2320	C₂	semidirect
5. C₅₃×#1518	C₂	semidirect
6. C₅₃×#1515	C₂	semidirect
9. C₅₃×#1506	C₂	semidirect
10. C₅₃×#1503	C₂	semidirect
12. C₅₃×#557	C₄	semidirect
13. C₅₃×#557	C₄	semidirect
14. C₅₃×#557	C₄	semidirect
15. C₅₃×#556	C₄	semidirect
16. C₅₃×#556	C₄	semidirect
20. D₄×Q₈×C₅₃	C₄	semidirect
265. #17601	C₅₃	direct (N × H)
293. #2984	C₂×C₅₃	semidirect
294. #2320	C₂×C₅₃	semidirect
296. #1518	C₂×C₅₃	semidirect
297. #1515	C₂×C₅₃	semidirect
300. #1506	C₂×C₅₃	semidirect
301. #1503	C₂×C₅₃	semidirect
309. #557	C₄×C₅₃	semidirect
310. #557	C₄×C₅₃	semidirect
311. #557	C₄×C₅₃	semidirect
312. #556	C₄×C₅₃	semidirect
313. #556	C₄×C₅₃	semidirect
317. D₄×Q₈	C₄×C₅₃	semidirect
350. C₅₃	#17601	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×#3523	2
C₅₃×#2984	2
C₅₃×#2320	2
C₅₃×#1521	2
C₅₃×#1518	2
C₅₃×#1515	2
C₅₃×#1512	2
C₅₃×#1509	2
C₅₃×#1506	2
C₅₃×#1503	2
C₅₃×#1500	2
#17601	53

Subgroups — C53×#17601 #27225113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17601 #27225113