Subgroups — C₅₃×#7930 #27215442

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1282	C₂	1	4	yes	maximal
5	C₂×#117×C₅₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#432	—	2	1	no	—
8	C₅₃×#432	—	2	1	no	—
9	C₅₃×#432	—	2	1	no	—
10	C₅₃×#432	—	2	1	no	—
11	C₅₃×#432	—	2	1	no	—
12	C₅₃×#432	—	2	1	no	—
13	C₅₃×#432	C₄	1	2	yes	complemented
14	C₅₃×#432	C₄	1	2	yes	complemented
15	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
17	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	—	2	1	no	—
20	C₂×#97×C₅₃	C₄	1	2	yes	complemented
21	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#376	—	2	1	no	—
26	C₄³×C₅₃	—	2	1	no	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	#117×C₅₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₄⋊C₈×C₅₃	—	2	2	no	—
54	C₂²⋊C₈×C₅₃	—	2	2	no	—
55	C₂²⋊C₈×C₅₃	—	2	2	no	—
56	C₂²⋊C₈×C₅₃	—	2	2	no	—
57	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
58	#97×C₅₃	—	2	2	no	—
59	#97×C₅₃	—	2	2	no	—
60	#97×C₅₃	—	2	2	no	—
61	#97×C₅₃	—	2	2	no	—
62	#95×C₅₃	—	2	2	no	—
63	#95×C₅₃	—	2	2	no	—
64	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	4	4	no	—
90	C₂×C₈×C₅₃	—	4	4	no	—
91	C₄⋊C₄×C₅₃	—	2	8	no	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₂²⋊C₄×C₅₃	M₁₆	1	8	yes	—
94	C₄²×C₅₃	—	2	8	no	—
95	C₄²×C₅₃	—	4	4	no	—
96	C₄²×C₅₃	—	4	4	no	—
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₅₃	—	4	1	no	—
114	C₂³×C₅₃	—	4	1	no	—
115	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	8	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	—	4	4	no	—
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₅₃	—	4	4	no	—
153	C₈×C₅₃	—	4	4	no	—
154	#7930	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	4	4	no	—
162	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
163	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
164	C₂²×C₅₃	#407	1	2	yes	—
165	C₂²×C₅₃	#423	1	2	yes	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	4	8	no	—
177	C₄×C₅₃	#344	1	2	yes	—
178	C₄×C₅₃	#410	1	2	yes	—
179	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
180	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
181	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
182	#1282	C₂×C₅₃	1	4	yes	—
183	C₂×C₅₃	—	2	2	no	—
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	—	2	2	no	—
186	C₂×C₅₃	#1728	1	1	yes	characteristic
187	C₂×C₅₃	#1740	1	1	yes	characteristic
188	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
189	C₂×#117	C₄×C₅₃	1	2	yes	—
190	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
191	#432	—	2	1	no	—
192	#432	—	2	1	no	—
193	#432	—	2	1	no	—
194	#432	—	2	1	no	—
195	#432	—	2	1	no	—
196	#432	—	2	1	no	—
197	#432	C₄×C₅₃	1	2	yes	complemented
198	#432	C₄×C₅₃	1	2	yes	complemented
199	C₂×C₄⋊C₈	—	2	1	no	—
200	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	complemented
201	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
202	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
203	C₂×#97	—	2	1	no	—
204	C₂×#97	C₄×C₅₃	1	2	yes	complemented
205	C₄×C₄⋊C₄	—	2	1	no	—
206	C₄×C₄⋊C₄	—	2	1	no	—
207	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
208	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
209	#376	—	2	1	no	—
210	C₄³	—	2	1	no	—
211	C₅₃	#7930	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
213	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
214	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
217	#117	—	2	8	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
234	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₂²⋊C₈	—	2	2	no	—
240	C₂²⋊C₈	—	2	2	no	—
241	C₂²⋊C₈	—	2	2	no	—
242	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
243	#97	—	2	2	no	—
244	#97	—	2	2	no	—
245	#97	—	2	2	no	—
246	#97	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	4	no	—
276	C₄⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	4	no	—
278	C₂²⋊C₄	M₁₆×C₅₃	1	8	yes	—
279	C₄²	—	2	8	no	—
280	C₄²	—	4	4	no	—
281	C₄²	—	4	4	no	—
282	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
283	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
284	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
285	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
286	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
287	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
288	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
289	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
290	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
291	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
292	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
293	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
294	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
295	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
296	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
297	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
298	C₂³	—	4	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
301	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
302	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
303	C₂³	C₂×M₁₆×C₅₃	1	4	yes	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
326	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
327	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
328	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
329	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
330	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
331	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
332	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
337	C₈	—	4	4	no	—
338	C₈	—	4	4	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	4	4	no	—
346	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
347	C₂²	C₄×M₁₆×C₅₃	1	2	yes	—
348	C₂²	C₅₃×#407	1	2	yes	—
349	C₂²	C₅₃×#423	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	8	no	—
361	C₄	C₅₃×#344	1	2	yes	—
362	C₄	C₅₃×#410	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	C₅₃×#1728	1	1	yes	characteristic
367	C₂	C₅₃×#1740	1	1	yes	characteristic
368	C₂	C₂×C₅₃×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ 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
13. C₅₃×#432	C₄	semidirect
14. C₅₃×#432	C₄	semidirect
16. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
20. C₂×#97×C₅₃	C₄	semidirect
154. #7930	C₅₃	direct (N × H)
197. #432	C₄×C₅₃	semidirect
198. #432	C₄×C₅₃	semidirect
200. C₂×C₄⋊C₈	C₄×C₅₃	semidirect
204. C₂×#97	C₄×C₅₃	semidirect
211. C₅₃	#7930	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#432	2
C₄×#117×C₅₃	2
C₅₃×#1282	2
#7930	53

Subgroups — C53×#7930 #27215442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7930 #27215442