Subgroups — C₅₃×#60717 #27268229

Order 13568.

1870 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3512	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3183	C₂	1	4	yes	maximal, complemented
3	C₅₃×#3177	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3176	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3174	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3168	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3142	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3061	C₂	1	2	yes	maximal, complemented
11	C₅₃×#3060	C₂	1	2	yes	maximal, complemented
12	C₅₃×#2952	C₂	1	2	yes	maximal, complemented
13	C₅₃×#2951	C₂	1	2	yes	maximal, complemented
14	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#135×C₅₃	C₂²	1	2	yes	—
16	C₂×#135×C₅₃	C₂²	1	2	yes	—
17	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#553	—	2	4	no	—
20	C₅₃×#553	—	2	8	no	—
21	C₅₃×#552	—	2	4	no	—
22	C₅₃×#541	C₂²	1	4	yes	—
23	C₅₃×#540	C₂²	1	4	yes	—
24	C₅₃×#537	C₂²	1	4	yes	—
25	C₅₃×#532	C₂²	1	2	yes	—
26	C₅₃×#529	C₂²	1	8	yes	—
27	C₂×#125×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#520	C₂²	1	4	yes	—
29	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#499	C₂²	1	2	yes	—
31	C₅₃×#499	C₂²	1	2	yes	—
32	C₅₃×#499	C₂²	1	2	yes	—
33	C₅₃×#499	C₂²	1	2	yes	—
34	C₅₃×#495	C₂²	1	2	yes	—
35	C₅₃×#495	C₂²	1	2	yes	—
36	C₅₃×#494	C₂²	1	1	yes	characteristic
37	C₅₃×#493	C₂²	1	1	yes	characteristic
38	C₅₃×#492	C₂²	1	2	yes	—
39	C₅₃×#491	C₂²	1	4	yes	—
40	C₅₃×#490	C₂²	1	4	yes	—
41	C₅₃×#488	C₂²	1	8	yes	—
42	C₅₃×#470	C₂²	1	4	yes	—
43	C₅₃×#469	C₂²	1	4	yes	—
44	C₅₃×#468	C₂²	1	8	yes	—
45	C₅₃×#467	C₂²	1	4	yes	—
46	C₅₃×#466	C₂²	1	4	yes	—
47	C₅₃×#465	C₂²	1	8	yes	—
48	C₅₃×#442	C₂²	1	4	yes	—
49	C₅₃×#441	C₂²	1	4	yes	—
50	C₅₃×#440	C₂²	1	8	yes	—
51	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
52	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	—
53	C₄×D₈×C₅₃	C₂²	1	4	yes	—
54	C₂×#107×C₅₃	C₂²	1	2	yes	—
55	C₂×#106×C₅₃	C₂²	1	2	yes	—
56	C₅₃×#417	C₂²	1	4	yes	—
57	C₅₃×#417	C₂²	1	4	yes	—
58	C₅₃×#405	C₂²	1	1	yes	characteristic
59	C₅₃×#405	C₂²	1	1	yes	characteristic
60	C₅₃×#405	C₂²	1	2	yes	—
61	C₅₃×#405	C₂²	1	2	yes	—
62	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
63	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
64	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
66	C₂×#41×C₅₃	C₂³	1	4	yes	—
67	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
68	#135×C₅₃	—	2	8	no	—
69	#135×C₅₃	—	2	8	no	—
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
74	C₂×D₈×C₅₃	C₂³	1	2	yes	—
75	C₂×D₈×C₅₃	C₂³	1	2	yes	—
76	#131×C₅₃	—	2	4	no	—
77	#131×C₅₃	—	2	4	no	—
78	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
81	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
82	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
83	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
84	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
87	#127×C₅₃	C₂³	1	2	yes	—
88	#126×C₅₃	—	2	8	no	—
89	#126×C₅₃	—	2	8	no	—
90	#125×C₅₃	—	2	8	no	—
91	#125×C₅₃	—	2	4	no	—
92	#125×C₅₃	C₂³	1	4	yes	—
93	#125×C₅₃	C₂³	1	4	yes	—
94	#124×C₅₃	C₂³	1	4	yes	—
95	#123×C₅₃	—	2	8	no	—
96	#123×C₅₃	—	2	8	no	—
97	C₂²⋊Q₈×C₅₃	—	2	4	no	—
98	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
99	#121×C₅₃	—	2	8	no	—
100	#121×C₅₃	—	2	8	no	—
101	#121×C₅₃	—	2	4	no	—
102	#121×C₅₃	—	2	4	no	—
103	#121×C₅₃	C₂³	1	8	yes	—
104	C₄×Q₈×C₅₃	D₄	1	2	yes	—
105	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
106	C₄×D₄×C₅₃	—	2	8	no	—
107	C₄×D₄×C₅₃	—	2	4	no	—
108	C₄×D₄×C₅₃	—	2	8	no	—
109	C₄×D₄×C₅₃	D₄	1	4	yes	—
110	C₄×D₄×C₅₃	D₄	1	2	yes	—
111	C₄×D₄×C₅₃	C₂³	1	8	yes	—
112	#117×C₅₃	—	2	4	no	—
113	#117×C₅₃	—	2	8	no	—
114	#117×C₅₃	C₂³	1	1	yes	characteristic
115	#117×C₅₃	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
117	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
121	#107×C₅₃	C₂³	1	4	yes	—
122	#107×C₅₃	C₂³	1	4	yes	—
123	#106×C₅₃	C₂³	1	4	yes	—
124	#106×C₅₃	C₂³	1	4	yes	—
125	#103×C₅₃	C₂³	1	8	yes	—
126	#103×C₅₃	C₂³	1	8	yes	—
127	#102×C₅₃	C₂³	1	8	yes	—
128	#102×C₅₃	C₂³	1	8	yes	—
129	#97×C₅₃	C₂³	1	2	yes	—
130	#97×C₅₃	C₂³	1	2	yes	—
131	#97×C₅₃	C₂³	1	2	yes	—
132	C₄×C₈×C₅₃	C₂³	1	2	yes	—
133	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
134	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
135	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₅₃	C₂³	1	2	yes	—
138	C₄×C₈×C₅₃	C₂³	1	2	yes	—
139	#41×C₅₃	—	2	2	no	—
140	#41×C₅₃	—	2	2	no	—
141	#41×C₅₃	—	2	8	no	—
142	#41×C₅₃	—	2	8	no	—
143	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	8	no	—
147	C₂×D₄×C₅₃	—	2	4	no	—
148	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
149	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₅₃	—	2	4	no	—
152	C₂²×C₄×C₅₃	—	2	4	no	—
153	C₂²×C₄×C₅₃	—	2	8	no	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₅₃	—	2	4	no	—
159	Q₁₆×C₅₃	—	2	4	no	—
160	SD₁₆×C₅₃	—	2	8	no	—
161	SD₁₆×C₅₃	—	2	8	no	—
162	D₈×C₅₃	—	2	4	no	—
163	D₈×C₅₃	—	2	4	no	—
164	M₁₆×C₅₃	—	2	2	no	—
165	M₁₆×C₅₃	—	2	2	no	—
166	M₁₆×C₅₃	—	2	4	no	—
167	M₁₆×C₅₃	—	2	4	no	—
168	C₂×C₈×C₅₃	—	2	4	no	—
169	C₂×C₈×C₅₃	—	2	4	no	—
170	C₂×C₈×C₅₃	#41	1	4	yes	—
171	C₂×C₈×C₅₃	#41	1	4	yes	—
172	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
173	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
174	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₅₃	—	2	4	no	—
179	C₄⋊C₄×C₅₃	—	2	8	no	—
180	C₄⋊C₄×C₅₃	—	2	8	no	—
181	C₄⋊C₄×C₅₃	—	2	8	no	—
182	C₄⋊C₄×C₅₃	—	2	8	no	—
183	C₄⋊C₄×C₅₃	—	2	4	no	—
184	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
186	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
187	C₂²⋊C₄×C₅₃	—	2	8	no	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	—	2	8	no	—
190	C₂²⋊C₄×C₅₃	—	2	8	no	—
191	C₂²⋊C₄×C₅₃	—	2	4	no	—
192	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
193	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
194	C₄²×C₅₃	—	2	8	no	—
195	C₄²×C₅₃	C₂×D₄	1	4	yes	—
196	C₄²×C₅₃	C₂×D₄	1	2	yes	—
197	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
198	C₄²×C₅₃	C₂⁴	1	2	yes	—
199	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₂³×C₅₃	—	2	4	no	—
201	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
202	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₅₃	—	2	2	no	—
204	Q₈×C₅₃	—	2	4	no	—
205	Q₈×C₅₃	—	2	4	no	—
206	D₄×C₅₃	—	2	4	no	—
207	D₄×C₅₃	—	2	2	no	—
208	D₄×C₅₃	—	2	4	no	—
209	D₄×C₅₃	—	2	4	no	—
210	D₄×C₅₃	—	4	8	no	—
211	D₄×C₅₃	—	8	4	no	—
212	D₄×C₅₃	—	8	8	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	8	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	2	no	—
219	C₂×C₄×C₅₃	—	4	8	no	—
220	C₂×C₄×C₅₃	—	8	8	no	—
221	C₂×C₄×C₅₃	—	8	4	no	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
228	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
229	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
230	C₈×C₅₃	—	2	2	no	—
231	C₈×C₅₃	—	2	2	no	—
232	C₈×C₅₃	C₂×#41	1	2	yes	—
233	C₈×C₅₃	C₂×#41	1	2	yes	—
234	C₈×C₅₃	C₂×#41	1	2	yes	—
235	C₈×C₅₃	C₂×#41	1	2	yes	—
236	#60717	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂²×C₅₃	—	2	2	no	—
238	C₂²×C₅₃	—	2	4	no	—
239	C₂²×C₅₃	—	2	2	no	—
240	C₂²×C₅₃	—	4	4	no	—
241	C₂²×C₅₃	—	4	4	no	—
242	C₂²×C₅₃	—	4	4	no	—
243	C₂²×C₅₃	#532	1	2	yes	—
244	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
245	C₄×C₅₃	—	2	2	no	—
246	C₄×C₅₃	—	2	4	no	—
247	C₄×C₅₃	—	2	2	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	4	8	no	—
250	C₄×C₅₃	—	4	4	no	—
251	C₄×C₅₃	#532	1	2	yes	—
252	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
253	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
254	#3512	C₂×C₅₃	1	2	yes	complemented
255	#3183	C₂×C₅₃	1	4	yes	complemented
256	#3177	C₂×C₅₃	1	2	yes	complemented
257	#3176	C₂×C₅₃	1	2	yes	complemented
258	#3174	C₂×C₅₃	1	4	yes	complemented
259	#3168	C₂×C₅₃	1	4	yes	complemented
260	C₂×#499	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#3147	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#3142	C₂×C₅₃	1	2	yes	complemented
263	#3061	C₂×C₅₃	1	2	yes	complemented
264	#3060	C₂×C₅₃	1	2	yes	complemented
265	#2952	C₂×C₅₃	1	2	yes	complemented
266	#2951	C₂×C₅₃	1	2	yes	complemented
267	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
268	C₂×C₅₃	—	2	2	no	—
269	C₂×C₅₃	—	4	2	no	—
270	C₂×C₅₃	—	8	4	no	—
271	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
272	C₂×C₅₃	#3584	1	2	yes	—
273	C₂×#135	C₂²×C₅₃	1	2	yes	—
274	C₂×#135	C₂²×C₅₃	1	2	yes	—
275	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
276	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
277	#553	—	2	8	no	—
278	#553	—	2	4	no	—
279	#552	—	2	4	no	—
280	#541	C₂²×C₅₃	1	4	yes	—
281	#540	C₂²×C₅₃	1	4	yes	—
282	#537	C₂²×C₅₃	1	4	yes	—
283	#532	C₂²×C₅₃	1	2	yes	—
284	#529	C₂²×C₅₃	1	8	yes	—
285	C₂×#125	C₂²×C₅₃	1	2	yes	—
286	#520	C₂²×C₅₃	1	4	yes	—
287	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
288	#499	C₂²×C₅₃	1	2	yes	—
289	#499	C₂²×C₅₃	1	2	yes	—
290	#499	C₂²×C₅₃	1	2	yes	—
291	#499	C₂²×C₅₃	1	2	yes	—
292	#495	C₂²×C₅₃	1	2	yes	—
293	#495	C₂²×C₅₃	1	2	yes	—
294	#494	C₂²×C₅₃	1	1	yes	characteristic
295	#493	C₂²×C₅₃	1	1	yes	characteristic
296	#492	C₂²×C₅₃	1	2	yes	—
297	#491	C₂²×C₅₃	1	4	yes	—
298	#490	C₂²×C₅₃	1	4	yes	—
299	#488	C₂²×C₅₃	1	8	yes	—
300	#470	C₂²×C₅₃	1	4	yes	—
301	#469	C₂²×C₅₃	1	4	yes	—
302	#468	C₂²×C₅₃	1	8	yes	—
303	#467	C₂²×C₅₃	1	4	yes	—
304	#466	C₂²×C₅₃	1	4	yes	—
305	#465	C₂²×C₅₃	1	8	yes	—
306	#442	C₂²×C₅₃	1	4	yes	—
307	#441	C₂²×C₅₃	1	4	yes	—
308	#440	C₂²×C₅₃	1	8	yes	—
309	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
310	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	—
311	C₄×D₈	C₂²×C₅₃	1	4	yes	—
312	C₂×#107	C₂²×C₅₃	1	2	yes	—
313	C₂×#106	C₂²×C₅₃	1	2	yes	—
314	#417	C₂²×C₅₃	1	4	yes	—
315	#417	C₂²×C₅₃	1	4	yes	—
316	#405	C₂²×C₅₃	1	2	yes	—
317	#405	C₂²×C₅₃	1	2	yes	—
318	#405	C₂²×C₅₃	1	1	yes	characteristic
319	#405	C₂²×C₅₃	1	1	yes	characteristic
320	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
321	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
322	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
324	C₅₃	#60717	1	1	yes	characteristic, direct product factor, Sylow
325	C₂×#41	C₂³×C₅₃	1	4	yes	—
326	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
327	#135	—	2	8	no	—
328	#135	—	2	8	no	—
329	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
330	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
331	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
332	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
333	C₂×D₈	C₂³×C₅₃	1	2	yes	—
334	C₂×D₈	C₂³×C₅₃	1	2	yes	—
335	#131	—	2	4	no	—
336	#131	—	2	4	no	—
337	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
338	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
339	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
340	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
341	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
342	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
343	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
345	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
346	#127	C₂³×C₅₃	1	2	yes	—
347	#126	—	2	8	no	—
348	#126	—	2	8	no	—
349	#125	—	2	8	no	—
350	#125	—	2	4	no	—
351	#125	C₂³×C₅₃	1	4	yes	—
352	#125	C₂³×C₅₃	1	4	yes	—
353	#124	C₂³×C₅₃	1	4	yes	—
354	#123	—	2	8	no	—
355	#123	—	2	8	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
358	#121	—	2	4	no	—
359	#121	—	2	8	no	—
360	#121	—	2	4	no	—
361	#121	—	2	8	no	—
362	#121	C₂³×C₅₃	1	8	yes	—
363	C₄×Q₈	D₄×C₅₃	1	2	yes	—
364	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	—	2	8	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	D₄×C₅₃	1	4	yes	—
369	C₄×D₄	D₄×C₅₃	1	2	yes	—
370	C₄×D₄	C₂³×C₅₃	1	8	yes	—
371	#117	—	2	4	no	—
372	#117	—	2	8	no	—
373	#117	C₂³×C₅₃	1	1	yes	characteristic
374	#117	C₂³×C₅₃	1	2	yes	—
375	C₂×C₄⋊C₄	—	2	4	no	—
376	C₂×C₄⋊C₄	—	2	4	no	—
377	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
378	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
380	#107	C₂³×C₅₃	1	4	yes	—
381	#107	C₂³×C₅₃	1	4	yes	—
382	#106	C₂³×C₅₃	1	4	yes	—
383	#106	C₂³×C₅₃	1	4	yes	—
384	#103	C₂³×C₅₃	1	8	yes	—
385	#103	C₂³×C₅₃	1	8	yes	—
386	#102	C₂³×C₅₃	1	8	yes	—
387	#102	C₂³×C₅₃	1	8	yes	—
388	#97	C₂³×C₅₃	1	2	yes	—
389	#97	C₂³×C₅₃	1	2	yes	—
390	#97	C₂³×C₅₃	1	2	yes	—
391	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
392	C₄×C₈	C₂³×C₅₃	1	2	yes	—
393	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
394	C₄×C₈	C₂³×C₅₃	1	2	yes	—
395	C₄×C₈	C₂³×C₅₃	1	2	yes	—
396	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
398	#41	—	2	2	no	—
399	#41	—	2	2	no	—
400	#41	—	2	8	no	—
401	#41	—	2	8	no	—
402	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
403	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
408	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
409	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	8	no	—
413	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
415	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	Q₁₆	—	2	4	no	—
418	Q₁₆	—	2	4	no	—
419	SD₁₆	—	2	8	no	—
420	SD₁₆	—	2	8	no	—
421	D₈	—	2	4	no	—
422	D₈	—	2	4	no	—
423	M₁₆	—	2	2	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	2	4	no	—
426	M₁₆	—	2	4	no	—
427	C₂×C₈	—	2	4	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	#41×C₅₃	1	4	yes	—
430	C₂×C₈	#41×C₅₃	1	4	yes	—
431	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
432	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
433	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
434	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
435	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
436	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
444	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
452	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
453	C₄²	—	2	8	no	—
454	C₄²	C₂×D₄×C₅₃	1	4	yes	—
455	C₄²	C₂×D₄×C₅₃	1	2	yes	—
456	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
457	C₄²	C₂⁴×C₅₃	1	2	yes	—
458	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
459	C₂³	—	2	4	no	—
460	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
461	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
462	Q₈	—	2	2	no	—
463	Q₈	—	2	4	no	—
464	Q₈	—	2	4	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	4	8	no	—
470	D₄	—	8	4	no	—
471	D₄	—	8	8	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	8	8	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
482	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
483	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
487	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
488	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	2	no	—
491	C₈	C₂×#41×C₅₃	1	2	yes	—
492	C₈	C₂×#41×C₅₃	1	2	yes	—
493	C₈	C₂×#41×C₅₃	1	2	yes	—
494	C₈	C₂×#41×C₅₃	1	2	yes	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	C₅₃×#532	1	2	yes	—
502	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
503	C₄	—	2	2	no	—
504	C₄	—	2	4	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	4	8	no	—
508	C₄	—	4	4	no	—
509	C₄	C₅₃×#532	1	2	yes	—
510	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
511	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
512	C₂	—	2	2	no	—
513	C₂	—	4	2	no	—
514	C₂	—	8	4	no	—
515	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
516	C₂	C₅₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3512	C₂	semidirect
2. C₅₃×#3183	C₂	semidirect
3. C₅₃×#3177	C₂	semidirect
4. C₅₃×#3176	C₂	semidirect
5. C₅₃×#3174	C₂	semidirect
6. C₅₃×#3168	C₂	semidirect
7. C₂×C₅₃×#499	C₂	semidirect
8. C₅₃×#3147	C₂	semidirect
9. C₅₃×#3142	C₂	semidirect
10. C₅₃×#3061	C₂	semidirect
11. C₅₃×#3060	C₂	semidirect
12. C₅₃×#2952	C₂	semidirect
13. C₅₃×#2951	C₂	semidirect
14. C₄×#131×C₅₃	C₂	semidirect
236. #60717	C₅₃	direct (N × H)
254. #3512	C₂×C₅₃	semidirect
255. #3183	C₂×C₅₃	semidirect
256. #3177	C₂×C₅₃	semidirect
257. #3176	C₂×C₅₃	semidirect
258. #3174	C₂×C₅₃	semidirect
259. #3168	C₂×C₅₃	semidirect
260. C₂×#499	C₂×C₅₃	semidirect
261. #3147	C₂×C₅₃	semidirect
262. #3142	C₂×C₅₃	semidirect
263. #3061	C₂×C₅₃	semidirect
264. #3060	C₂×C₅₃	semidirect
265. #2952	C₂×C₅₃	semidirect
266. #2951	C₂×C₅₃	semidirect
267. C₄×#131	C₂×C₅₃	semidirect
324. C₅₃	#60717	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×#3512	2
C₅₃×#3183	2
C₅₃×#3177	2
C₅₃×#3176	2
C₅₃×#3174	2
C₅₃×#3168	2
C₂×C₅₃×#499	2
C₅₃×#3147	2
C₅₃×#3142	2
C₅₃×#3061	2
C₅₃×#3060	2
C₅₃×#2952	2
C₅₃×#2951	2
C₄×#131×C₅₃	2
#60717	53

Subgroups — C53×#60717 #27268229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60717 #27268229