Subgroups — C₅₃×#23128 #27230640

Order 13568.

910 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×#467	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2360	C₂	1	2	yes	maximal
3	C₅₃×#1933	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1921	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1878	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1872	C₂	1	2	yes	maximal
7	C₄×#103×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#467	C₄	1	8	yes	complemented
16	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#103×C₅₃	C₂²	1	2	yes	—
18	C₂×#103×C₅₃	C₂²	1	2	yes	—
19	C₂×#103×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#391	C₂²	1	2	yes	—
23	C₅₃×#389	—	2	2	no	—
24	C₅₃×#389	C₂²	1	2	yes	—
25	C₅₃×#387	—	2	2	no	—
26	C₅₃×#385	—	2	2	no	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
29	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#340	C₂²	1	2	yes	—
31	C₅₃×#340	C₂²	1	2	yes	—
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×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₅₃	—	2	2	no	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₅₃	—	4	2	no	—
40	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
41	C₂²×C₈×C₅₃	—	4	2	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²⋊Q₈×C₅₃	—	2	4	no	—
47	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
48	C₄×Q₈×C₅₃	—	2	4	no	—
49	C₄×Q₈×C₅₃	—	4	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
65	#107×C₅₃	C₂×C₄	1	4	yes	complemented
66	#103×C₅₃	—	2	4	no	—
67	#103×C₅₃	—	4	2	no	—
68	#103×C₅₃	C₂×C₄	1	8	yes	complemented
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	C₄×C₈×C₅₃	—	4	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	#95×C₅₃	D₄	1	2	yes	—
76	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×Q₈×C₅₃	—	2	2	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	4	4	no	—
90	C₂²×C₄×C₅₃	—	4	4	no	—
91	C₂²×C₄×C₅₃	—	8	2	no	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₅₃	#41	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₅₃	—	4	4	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	4	4	no	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄⋊C₄×C₅₃	—	2	8	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	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	4	4	no	—
113	C₄⋊C₄×C₅₃	—	4	4	no	—
114	C₄⋊C₄×C₅₃	—	8	2	no	—
115	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
116	C₄⋊C₄×C₅₃	#41	1	4	yes	—
117	C₂²⋊C₄×C₅₃	—	2	8	no	—
118	C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₂²⋊C₄×C₅₃	Q₁₆	1	4	yes	complemented
120	C₄²×C₅₃	—	2	2	no	—
121	C₄²×C₅₃	—	2	2	no	—
122	C₄²×C₅₃	—	4	4	no	—
123	C₄²×C₅₃	—	8	2	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	C₄×D₄	1	2	yes	—
127	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
128	C₂³×C₅₃	#135	1	2	yes	—
129	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	Q₈×C₅₃	—	2	4	no	—
131	Q₈×C₅₃	—	2	4	no	—
132	Q₈×C₅₃	—	4	2	no	—
133	Q₈×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	4	no	—
140	C₂×C₄×C₅₃	—	2	4	no	—
141	C₂×C₄×C₅₃	—	2	4	no	—
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	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	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	8	8	no	—
156	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
157	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
158	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
162	C₈×C₅₃	—	2	2	no	—
163	C₈×C₅₃	—	2	2	no	—
164	C₈×C₅₃	—	4	2	no	—
165	#23128	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₅₃	—	2	2	no	—
167	C₂²×C₅₃	—	2	2	no	—
168	C₂²×C₅₃	—	2	4	no	—
169	C₂²×C₅₃	—	2	4	no	—
170	C₂²×C₅₃	C₄×Q₁₆	1	4	yes	—
171	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₅₃	#546	1	2	yes	—
173	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
175	C₂²×C₅₃	#577	1	2	yes	—
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	2	4	no	—
178	C₄×C₅₃	—	4	4	no	—
179	C₄×C₅₃	—	4	4	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	4	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	#553	1	2	yes	—
186	C₂×#467	C₂×C₅₃	1	1	yes	characteristic
187	#2360	C₂×C₅₃	1	2	yes	—
188	#1933	C₂×C₅₃	1	2	yes	complemented
189	#1921	C₂×C₅₃	1	2	yes	complemented
190	#1878	C₂×C₅₃	1	2	yes	complemented
191	#1872	C₂×C₅₃	1	2	yes	—
192	C₄×#103	C₂×C₅₃	1	2	yes	complemented
193	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
194	C₂×#340	C₂×C₅₃	1	1	yes	characteristic
195	C₂×C₅₃	—	2	4	no	—
196	C₂×C₅₃	#2364	1	1	yes	characteristic
197	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
198	C₂×C₅₃	#2952	1	1	yes	characteristic
199	C₂×C₅₃	#3296	1	2	yes	—
200	C₂×C₅₃	#3299	1	2	yes	—
201	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
202	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
203	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
204	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
205	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
206	#467	C₄×C₅₃	1	8	yes	complemented
207	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
208	C₂×#103	C₂²×C₅₃	1	2	yes	—
209	C₂×#103	C₂²×C₅₃	1	2	yes	—
210	C₂×#103	C₂²×C₅₃	1	2	yes	—
211	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
213	#391	C₂²×C₅₃	1	2	yes	—
214	#389	—	2	2	no	—
215	#389	C₂²×C₅₃	1	2	yes	—
216	#387	—	2	2	no	—
217	#385	—	2	2	no	—
218	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
219	C₄×C₂²⋊C₄	—	2	2	no	—
220	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
221	#340	C₂²×C₅₃	1	2	yes	—
222	#340	C₂²×C₅₃	1	2	yes	—
223	#340	C₂²×C₅₃	1	2	yes	—
224	#340	C₂²×C₅₃	1	1	yes	characteristic
225	#340	C₂²×C₅₃	1	1	yes	characteristic
226	#336	C₂²×C₅₃	1	2	yes	—
227	C₅₃	#23128	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
229	C₂³×C₄	—	2	2	no	—
230	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
231	C₂×Q₁₆	—	4	2	no	—
232	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
233	C₂²×C₈	—	4	2	no	—
234	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
235	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
236	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
237	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
238	C₂²⋊Q₈	—	2	4	no	—
239	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
240	C₄×Q₈	—	2	4	no	—
241	C₄×Q₈	—	4	2	no	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	—	2	2	no	—
247	C₂×C₄⋊C₄	—	4	2	no	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
249	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
250	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
257	#107	C₂×C₄×C₅₃	1	4	yes	complemented
258	#103	—	2	4	no	—
259	#103	—	4	2	no	—
260	#103	C₂×C₄×C₅₃	1	8	yes	complemented
261	C₂²⋊C₈	—	2	2	no	—
262	C₄×C₈	—	4	2	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	#95	—	2	2	no	—
266	#95	—	2	2	no	—
267	#95	D₄×C₅₃	1	2	yes	—
268	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
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	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	4	4	no	—
282	C₂²×C₄	—	4	4	no	—
283	C₂²×C₄	—	8	2	no	—
284	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
286	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
288	C₂²×C₄	#41×C₅₃	1	2	yes	—
289	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
290	Q₁₆	—	4	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	4	no	—
297	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
298	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
299	C₄⋊C₄	—	2	8	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	4	no	—
305	C₄⋊C₄	—	4	4	no	—
306	C₄⋊C₄	—	8	2	no	—
307	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
308	C₄⋊C₄	#41×C₅₃	1	4	yes	—
309	C₂²⋊C₄	—	2	8	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	Q₁₆×C₅₃	1	4	yes	complemented
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	4	4	no	—
315	C₄²	—	8	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	2	2	no	—
318	C₂³	C₄×D₄×C₅₃	1	2	yes	—
319	C₂³	C₂×Q₁₆×C₅₃	1	2	yes	—
320	C₂³	#135×C₅₃	1	2	yes	—
321	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	2	4	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	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	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	8	8	no	—
348	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
349	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
350	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
352	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
353	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
354	C₈	—	2	2	no	—
355	C₈	—	2	2	no	—
356	C₈	—	4	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	4	no	—
359	C₂²	—	2	4	no	—
360	C₂²	—	2	2	no	—
361	C₂²	C₄×Q₁₆×C₅₃	1	4	yes	—
362	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
363	C₂²	C₅₃×#546	1	2	yes	—
364	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
365	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
366	C₂²	C₅₃×#577	1	2	yes	—
367	C₄	—	2	4	no	—
368	C₄	—	2	2	no	—
369	C₄	—	4	4	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	4	no	—
372	C₄	—	4	4	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×#553	1	2	yes	—
377	C₂	—	2	4	no	—
378	C₂	C₅₃×#2364	1	1	yes	characteristic
379	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
380	C₂	C₅₃×#2952	1	1	yes	characteristic
381	C₂	C₅₃×#3296	1	2	yes	—
382	C₂	C₅₃×#3299	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	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₂²⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1933	C₂	semidirect
4. C₅₃×#1921	C₂	semidirect
5. C₅₃×#1878	C₂	semidirect
7. C₄×#103×C₅₃	C₂	semidirect
15. C₅₃×#467	C₄	semidirect
40. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
65. #107×C₅₃	C₂×C₄	semidirect
68. #103×C₅₃	C₂×C₄	semidirect
119. C₂²⋊C₄×C₅₃	Q₁₆	semidirect
165. #23128	C₅₃	direct (N × H)
188. #1933	C₂×C₅₃	semidirect
189. #1921	C₂×C₅₃	semidirect
190. #1878	C₂×C₅₃	semidirect
192. C₄×#103	C₂×C₅₃	semidirect
206. #467	C₄×C₅₃	semidirect
227. C₅₃	#23128	direct (N × H)
232. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
257. #107	C₂×C₄×C₅₃	semidirect
260. #103	C₂×C₄×C₅₃	semidirect
311. C₂²⋊C₄	Q₁₆×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×#467	2
C₅₃×#2360	2
C₅₃×#1933	2
C₅₃×#1921	2
C₅₃×#1878	2
C₅₃×#1872	2
C₄×#103×C₅₃	2
C₈×C₂²⋊C₄×C₅₃	2
C₂×C₅₃×#340	2
#23128	53

Subgroups — C53×#23128 #27230640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23128 #27230640