Subgroups — C₅₃×#7944 #27215456

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1284	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#432	—	2	1	no	—
7	C₅₃×#432	C₄	1	2	yes	complemented
8	C₅₃×#431	—	2	1	no	—
9	C₅₃×#431	C₄	1	2	yes	complemented
10	C₅₃×#423	—	2	1	no	—
11	C₅₃×#407	C₂²	1	2	yes	—
12	C₄×M₁₆×C₅₃	—	2	1	no	—
13	C₅₃×#383	—	2	1	no	—
14	C₅₃×#383	—	2	1	no	—
15	C₅₃×#382	—	2	1	no	—
16	C₅₃×#382	C₄	1	2	yes	complemented
17	C₅₃×#381	—	2	1	no	—
18	C₅₃×#381	C₂²	1	2	yes	—
19	C₅₃×#380	—	2	1	no	—
20	C₅₃×#380	—	2	1	no	—
21	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
23	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
24	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#376	C₄	1	2	yes	complemented
26	C₅₃×#376	C₄	1	2	yes	complemented
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	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	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
53	#97×C₅₃	—	2	2	no	—
54	C₄×C₈×C₅₃	—	2	2	no	—
55	#95×C₅₃	—	2	2	no	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	2	no	—
58	#95×C₅₃	—	2	2	no	—
59	#95×C₅₃	—	2	2	no	—
60	#95×C₅₃	—	2	2	no	—
61	#95×C₅₃	—	2	2	no	—
62	#95×C₅₃	C₂×C₄	1	4	yes	—
63	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₂²×C₄	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	M₁₆×C₅₃	—	4	4	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	4	4	no	—
90	C₄⋊C₄×C₅₃	—	4	4	no	—
91	C₂²⋊C₄×C₅₃	—	4	4	no	—
92	C₂²⋊C₄×C₅₃	—	4	4	no	—
93	C₄²×C₅₃	—	4	4	no	—
94	C₄²×C₅₃	—	4	4	no	—
95	C₄²×C₅₃	C₄²	1	1	yes	characteristic
96	C₄²×C₅₃	C₄²	1	1	yes	characteristic
97	C₄²×C₅₃	C₄²	1	1	yes	characteristic
98	C₄²×C₅₃	C₄²	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	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₄⋊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	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	—	4	1	no	—
114	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
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	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	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	4	no	—
136	C₂×C₄×C₅₃	—	4	4	no	—
137	C₂×C₄×C₅₃	—	4	4	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	—	4	4	no	—
140	C₂×C₄×C₅₃	—	4	4	no	—
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	#95	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₅₃	—	4	4	no	—
154	#7944	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	—	2	1	no	—
162	C₂²×C₅₃	—	4	4	no	—
163	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
164	C₂²×C₅₃	#410	1	2	yes	—
165	C₂²×C₅₃	#421	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	—	2	1	no	—
177	C₄×C₅₃	—	8	4	no	—
178	C₄×C₅₃	—	8	4	no	—
179	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
180	#2920	C₂×C₅₃	1	1	yes	characteristic
181	#2283	C₂×C₅₃	1	1	yes	characteristic
182	#2283	C₂×C₅₃	1	1	yes	characteristic
183	#1284	C₂×C₅₃	1	4	yes	—
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	—	2	2	no	—
186	C₂×C₅₃	—	4	1	no	—
187	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
188	C₂×C₅₃	#1733	1	1	yes	characteristic
189	C₂×C₅₃	#1740	1	1	yes	characteristic
190	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
191	#432	—	2	1	no	—
192	#432	C₄×C₅₃	1	2	yes	complemented
193	#431	—	2	1	no	—
194	#431	C₄×C₅₃	1	2	yes	complemented
195	#423	—	2	1	no	—
196	#407	C₂²×C₅₃	1	2	yes	—
197	C₄×M₁₆	—	2	1	no	—
198	#383	—	2	1	no	—
199	#383	—	2	1	no	—
200	#382	—	2	1	no	—
201	#382	C₄×C₅₃	1	2	yes	complemented
202	#381	—	2	1	no	—
203	#381	C₂²×C₅₃	1	2	yes	—
204	#380	—	2	1	no	—
205	#380	—	2	1	no	—
206	C₄×C₄⋊C₄	—	2	1	no	—
207	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	complemented
208	C₄×C₂²⋊C₄	—	2	1	no	—
209	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
210	#376	C₄×C₅₃	1	2	yes	complemented
211	#376	C₄×C₅₃	1	2	yes	complemented
212	C₅₃	#7944	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
214	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
215	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
216	C₂×M₁₆	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
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	C₂×C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄²	C₂×C₄×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	C₂²⋊C₈	—	2	2	no	—
238	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
239	#97	—	2	2	no	—
240	C₄×C₈	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	C₂×C₄×C₅₃	1	4	yes	—
249	C₂⁴	C₂×D₄×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	—	4	2	no	—
255	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊C₄×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	C₂²⋊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	C₄⋊C₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
273	M₁₆	—	4	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₄²	—	4	4	no	—
280	C₄²	—	4	4	no	—
281	C₄²	C₄²×C₅₃	1	1	yes	characteristic
282	C₄²	C₄²×C₅₃	1	1	yes	characteristic
283	C₄²	C₄²×C₅₃	1	1	yes	characteristic
284	C₄²	C₄²×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	C₄⋊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	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	C₂×C₂²⋊C₄×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	C₂×C₄	—	2	2	no	—
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	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	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	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	4	no	—
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	#95×C₅₃	1	1	yes	characteristic
334	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
339	C₈	—	4	4	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	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
349	C₂²	C₅₃×#410	1	2	yes	—
350	C₂²	C₅₃×#421	1	2	yes	—
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	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	4	no	—
363	C₄	—	8	4	no	—
364	C₄	C₅₃×H(3,ℤ/4)	1	2	yes	—
365	C₂	—	2	2	no	—
366	C₂	—	2	2	no	—
367	C₂	—	4	1	no	—
368	C₂	C₂×C₅₃×H(3,ℤ/4)	1	1	yes	characteristic
369	C₂	C₅₃×#1733	1	1	yes	characteristic
370	C₂	C₅₃×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#95
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
7. C₅₃×#432	C₄	semidirect
9. C₅₃×#431	C₄	semidirect
16. C₅₃×#382	C₄	semidirect
22. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
25. C₅₃×#376	C₄	semidirect
26. C₅₃×#376	C₄	semidirect
154. #7944	C₅₃	direct (N × H)
192. #432	C₄×C₅₃	semidirect
194. #431	C₄×C₅₃	semidirect
201. #382	C₄×C₅₃	semidirect
207. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
210. #376	C₄×C₅₃	semidirect
211. #376	C₄×C₅₃	semidirect
212. C₅₃	#7944	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×#2920	2
C₅₃×#2283	2
C₅₃×#1284	2
#7944	53

Subgroups — C53×#7944 #27215456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7944 #27215456