Subgroups — C₅₃×#22714 #27230226

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2352	C₂	1	2	yes	maximal
3	C₅₃×#1949	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1854	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1853	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1809	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1736	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#436	C₄	1	8	yes	complemented
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#106×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#388	—	2	2	no	—
22	C₅₃×#387	—	2	2	no	—
23	C₅₃×#386	—	2	2	no	—
24	C₅₃×#382	C₂²	1	2	yes	—
25	C₅₃×#380	C₂²	1	2	yes	—
26	C₅₃×#380	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
29	C₅₃×#340	C₂²	1	2	yes	—
30	C₅₃×#340	C₂²	1	2	yes	—
31	C₅₃×#340	C₂²	1	1	yes	characteristic
32	C₅₃×#340	C₂²	1	2	yes	—
33	C₅₃×#340	C₂²	1	1	yes	characteristic
34	C₅₃×#340	C₂²	1	1	yes	characteristic
35	C₅₃×#340	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	—	2	2	no	—
38	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₅₃	—	4	2	no	—
40	C₂×M₁₆×C₅₃	—	4	2	no	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₅₃	Q₈	1	4	yes	—
47	#117×C₅₃	—	2	4	no	—
48	#117×C₅₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
61	C₂×C₄²×C₅₃	—	2	2	no	—
62	C₂×C₄²×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	4	2	no	—
65	#106×C₅₃	—	2	4	no	—
66	#106×C₅₃	—	4	2	no	—
67	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
68	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	complemented
69	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
70	#95×C₅₃	—	2	2	no	—
71	#95×C₅₃	—	2	2	no	—
72	#95×C₅₃	—	2	2	no	—
73	#95×C₅₃	—	2	2	no	—
74	#95×C₅₃	—	2	2	no	—
75	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₅₃	—	4	1	no	—
77	C₂×D₄×C₅₃	—	4	1	no	—
78	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	—	4	4	no	—
88	C₂²×C₄×C₅₃	—	4	4	no	—
89	C₂²×C₄×C₅₃	—	8	2	no	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	M₁₆×C₅₃	—	4	8	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	4	4	no	—
112	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
113	C₄⋊C₄×C₅₃	#41	1	4	yes	—
114	C₂²⋊C₄×C₅₃	—	4	4	no	—
115	C₂²⋊C₄×C₅₃	—	4	4	no	—
116	C₂²⋊C₄×C₅₃	—	8	2	no	—
117	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
118	C₄²×C₅₃	—	4	4	no	—
119	C₄²×C₅₃	—	8	2	no	—
120	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
121	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	—	2	2	no	—
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	D₄×C₅₃	—	4	4	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	2	4	no	—
139	C₂×C₄×C₅₃	—	2	4	no	—
140	C₂×C₄×C₅₃	—	2	4	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	4	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	8	8	no	—
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
159	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	—	2	4	no	—
162	C₈×C₅₃	—	2	2	no	—
163	#22714	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂²×C₅₃	—	2	4	no	—
165	C₂²×C₅₃	—	2	4	no	—
166	C₂²×C₅₃	—	2	4	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	4	2	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₅₃	#546	1	2	yes	—
173	C₂²×C₅₃	#575	1	1	yes	characteristic
174	C₂²×C₅₃	#575	1	1	yes	characteristic
175	C₂²×C₅₃	#577	1	2	yes	—
176	C₄×C₅₃	—	2	4	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	4	4	no	—
180	C₄×C₅₃	—	8	2	no	—
181	C₄×C₅₃	—	8	2	no	—
182	C₄×C₅₃	—	8	2	no	—
183	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
184	C₄×C₅₃	#518	1	1	yes	characteristic
185	C₄×C₅₃	#554	1	2	yes	—
186	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
187	#2352	C₂×C₅₃	1	2	yes	—
188	#1949	C₂×C₅₃	1	2	yes	complemented
189	#1854	C₂×C₅₃	1	2	yes	complemented
190	#1853	C₂×C₅₃	1	2	yes	complemented
191	#1835	C₂×C₅₃	1	1	yes	characteristic, complemented
192	#1809	C₂×C₅₃	1	2	yes	complemented
193	C₄×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
194	#1736	C₂×C₅₃	1	2	yes	complemented
195	C₂×C₅₃	—	4	4	no	—
196	C₂×C₅₃	#2349	1	1	yes	characteristic
197	C₂×C₅₃	#2911	1	1	yes	characteristic
198	C₂×C₅₃	#2912	1	1	yes	characteristic
199	C₂×C₅₃	#3323	1	2	yes	—
200	C₂×C₅₃	#3326	1	2	yes	—
201	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
202	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
203	C₂×#117	C₂²×C₅₃	1	2	yes	—
204	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
205	#436	C₄×C₅₃	1	8	yes	complemented
206	C₂×#107	C₂²×C₅₃	1	2	yes	—
207	C₂×#106	C₂²×C₅₃	1	2	yes	—
208	C₂×#106	C₂²×C₅₃	1	2	yes	—
209	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
211	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
212	#388	—	2	2	no	—
213	#387	—	2	2	no	—
214	#386	—	2	2	no	—
215	#382	C₂²×C₅₃	1	2	yes	—
216	#380	C₂²×C₅₃	1	2	yes	—
217	#380	C₂²×C₅₃	1	2	yes	—
218	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
219	C₄×C₂²⋊C₄	—	2	2	no	—
220	#340	C₂²×C₅₃	1	1	yes	characteristic
221	#340	C₂²×C₅₃	1	2	yes	—
222	#340	C₂²×C₅₃	1	1	yes	characteristic
223	#340	C₂²×C₅₃	1	1	yes	characteristic
224	#340	C₂²×C₅₃	1	2	yes	—
225	#340	C₂²×C₅₃	1	2	yes	—
226	#340	C₂²×C₅₃	1	1	yes	characteristic
227	C₅₃	#22714	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
229	C₂³×C₄	—	2	2	no	—
230	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
231	C₂×M₁₆	—	4	2	no	—
232	C₂×M₁₆	—	4	2	no	—
233	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
234	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
235	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
236	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
237	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
238	C₄×D₄	Q₈×C₅₃	1	4	yes	—
239	#117	—	2	4	no	—
240	#117	—	4	2	no	—
241	C₂×C₄⋊C₄	—	2	4	no	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	4	2	no	—
244	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
247	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
249	C₂×C₂²⋊C₄	—	2	2	no	—
250	C₂×C₂²⋊C₄	—	2	2	no	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
256	#107	—	4	2	no	—
257	#106	—	2	4	no	—
258	#106	—	4	2	no	—
259	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
260	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	complemented
261	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
262	#95	—	2	2	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	#95	—	2	2	no	—
266	#95	—	2	2	no	—
267	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	4	4	no	—
280	C₂²×C₄	—	4	4	no	—
281	C₂²×C₄	—	8	2	no	—
282	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
283	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
286	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
287	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
288	M₁₆	—	4	8	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
294	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
295	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	4	4	no	—
304	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
305	C₄⋊C₄	#41×C₅₃	1	4	yes	—
306	C₂²⋊C₄	—	4	4	no	—
307	C₂²⋊C₄	—	4	4	no	—
308	C₂²⋊C₄	—	8	2	no	—
309	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
310	C₄²	—	4	4	no	—
311	C₄²	—	8	2	no	—
312	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
313	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	2	2	no	—
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
321	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
322	D₄	—	4	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	8	no	—
345	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
346	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
347	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
348	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
351	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
352	C₈	—	2	2	no	—
353	C₈	—	2	4	no	—
354	C₈	—	2	2	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	2	4	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
363	C₂²	C₅₃×#546	1	2	yes	—
364	C₂²	C₅₃×#575	1	1	yes	characteristic
365	C₂²	C₅₃×#575	1	1	yes	characteristic
366	C₂²	C₅₃×#577	1	2	yes	—
367	C₄	—	2	4	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	4	4	no	—
371	C₄	—	8	2	no	—
372	C₄	—	8	2	no	—
373	C₄	—	8	2	no	—
374	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
375	C₄	C₅₃×#518	1	1	yes	characteristic
376	C₄	C₅₃×#554	1	2	yes	—
377	C₂	—	4	4	no	—
378	C₂	C₅₃×#2349	1	1	yes	characteristic
379	C₂	C₅₃×#2911	1	1	yes	characteristic
380	C₂	C₅₃×#2912	1	1	yes	characteristic
381	C₂	C₅₃×#3323	1	2	yes	—
382	C₂	C₅₃×#3326	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1949	C₂	semidirect
4. C₅₃×#1854	C₂	semidirect
5. C₅₃×#1853	C₂	semidirect
6. C₅₃×#1835	C₂	semidirect
7. C₅₃×#1809	C₂	semidirect
8. C₄×#106×C₅₃	C₂	semidirect
9. C₅₃×#1736	C₂	semidirect
14. C₅₃×#436	C₄	semidirect
41. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
67. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
68. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
69. C₄×C₈×C₅₃	C₂×C₄	semidirect
163. #22714	C₅₃	direct (N × H)
188. #1949	C₂×C₅₃	semidirect
189. #1854	C₂×C₅₃	semidirect
190. #1853	C₂×C₅₃	semidirect
191. #1835	C₂×C₅₃	semidirect
192. #1809	C₂×C₅₃	semidirect
193. C₄×#106	C₂×C₅₃	semidirect
194. #1736	C₂×C₅₃	semidirect
205. #436	C₄×C₅₃	semidirect
227. C₅₃	#22714	direct (N × H)
233. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
259. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
260. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
261. C₄×C₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₅₃×#2352	2
C₅₃×#1949	2
C₅₃×#1854	2
C₅₃×#1853	2
C₅₃×#1835	2
C₅₃×#1809	2
C₄×#106×C₅₃	2
C₅₃×#1736	2
#22714	53

Subgroups — C53×#22714 #27230226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22714 #27230226