Subgroups — C₅₃×#32666 #27240178

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2538	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2094	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2084	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2065	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1972	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1944	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1879	C₂	1	2	yes	maximal
9	C₅₃×#1862	C₂	1	2	yes	maximal, complemented
10	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#396	—	2	2	no	—
24	C₅₃×#396	—	2	2	no	—
25	C₅₃×#390	C₂²	1	2	yes	—
26	C₅₃×#390	C₂²	1	2	yes	—
27	C₅₃×#389	C₂²	1	2	yes	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#384	C₂²	1	2	yes	—
30	C₅₃×#380	—	2	2	no	—
31	C₅₃×#340	C₂²	1	2	yes	—
32	C₅₃×#340	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₅₃	—	2	2	no	—
38	C₂×D₈×C₅₃	—	4	1	no	—
39	C₂×D₈×C₅₃	—	4	1	no	—
40	C₂×D₈×C₅₃	Q₈	1	4	yes	complemented
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²⋊Q₈×C₅₃	—	4	8	no	—
45	C₄×Q₈×C₅₃	—	4	1	no	—
46	C₄×Q₈×C₅₃	—	4	1	no	—
47	C₄×D₄×C₅₃	—	4	2	no	—
48	C₄×D₄×C₅₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	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	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#107×C₅₃	—	4	1	no	—
66	#106×C₅₃	—	2	2	no	—
67	#106×C₅₃	—	4	2	no	—
68	#106×C₅₃	—	4	1	no	—
69	C₄⋊C₈×C₅₃	—	4	1	no	—
70	#102×C₅₃	—	2	4	no	—
71	#102×C₅₃	—	4	2	no	—
72	#102×C₅₃	—	4	2	no	—
73	#97×C₅₃	—	4	1	no	—
74	#95×C₅₃	—	2	2	no	—
75	#95×C₅₃	—	2	2	no	—
76	C₂⁴×C₅₃	—	2	2	no	—
77	C₂×Q₈×C₅₃	—	2	4	no	—
78	C₂×Q₈×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	2	2	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	C₂×Q₈	1	4	yes	—
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	—	4	4	no	—
89	C₂²×C₄×C₅₃	—	4	4	no	—
90	C₂²×C₄×C₅₃	—	4	4	no	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	D₈×C₅₃	—	4	4	no	—
97	C₂×C₈×C₅₃	—	2	1	no	—
98	C₂×C₈×C₅₃	—	2	1	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	—	2	1	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	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₅₃	#41	1	2	yes	—
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	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	4	2	no	—
113	C₄⋊C₄×C₅₃	—	4	1	no	—
114	C₄⋊C₄×C₅₃	—	4	4	no	—
115	C₄⋊C₄×C₅₃	—	4	4	no	—
116	C₄⋊C₄×C₅₃	—	4	4	no	—
117	C₄⋊C₄×C₅₃	—	8	2	no	—
118	C₂²⋊C₄×C₅₃	—	4	4	no	—
119	C₂²⋊C₄×C₅₃	—	4	4	no	—
120	C₂²⋊C₄×C₅₃	—	4	4	no	—
121	C₂²⋊C₄×C₅₃	—	8	2	no	—
122	C₂²⋊C₄×C₅₃	—	8	2	no	—
123	C₄²×C₅₃	—	2	1	no	—
124	C₄²×C₅₃	—	2	1	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	—	2	4	no	—
128	C₂³×C₅₃	—	4	2	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	—	4	2	no	—
131	C₂³×C₅₃	—	4	2	no	—
132	C₂³×C₅₃	—	4	2	no	—
133	C₂³×C₅₃	—	4	2	no	—
134	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	Q₈×C₅₃	—	4	4	no	—
136	D₄×C₅₃	—	2	4	no	—
137	D₄×C₅₃	—	2	4	no	—
138	D₄×C₅₃	—	4	2	no	—
139	D₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	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	—	4	2	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	8	8	no	—
159	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
166	C₈×C₅₃	—	2	2	no	—
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	4	2	no	—
169	#32666	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂²×C₅₃	—	4	4	no	—
171	C₂²×C₅₃	—	4	4	no	—
172	C₂²×C₅₃	—	4	4	no	—
173	C₂²×C₅₃	—	8	2	no	—
174	C₂²×C₅₃	—	8	2	no	—
175	C₂²×C₅₃	—	8	2	no	—
176	C₂²×C₅₃	—	8	2	no	—
177	C₂²×C₅₃	#534	1	1	yes	characteristic
178	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
179	C₂²×C₅₃	#550	1	1	yes	characteristic
180	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
181	C₂²×C₅₃	#577	1	2	yes	—
182	C₄×C₅₃	—	2	4	no	—
183	C₄×C₅₃	—	4	1	no	—
184	C₄×C₅₃	—	4	2	no	—
185	C₄×C₅₃	—	4	1	no	—
186	C₄×C₅₃	—	8	2	no	—
187	C₄×C₅₃	—	8	2	no	—
188	C₄×C₅₃	—	8	2	no	—
189	C₄×C₅₃	#539	1	1	yes	characteristic
190	C₄×C₅₃	#543	1	1	yes	characteristic
191	C₄×C₅₃	#554	1	1	yes	characteristic
192	C₄×C₅₃	#562	1	1	yes	characteristic
193	#2538	C₂×C₅₃	1	2	yes	complemented
194	#2094	C₂×C₅₃	1	2	yes	complemented
195	#2084	C₂×C₅₃	1	2	yes	complemented
196	#2065	C₂×C₅₃	1	2	yes	complemented
197	#1985	C₂×C₅₃	1	1	yes	characteristic, complemented
198	#1972	C₂×C₅₃	1	1	yes	characteristic
199	#1944	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#1879	C₂×C₅₃	1	2	yes	—
201	#1862	C₂×C₅₃	1	2	yes	complemented
202	C₂×C₅₃	—	8	4	no	—
203	C₂×C₅₃	#2732	1	1	yes	characteristic
204	C₂×C₅₃	#3192	1	1	yes	characteristic
205	C₂×C₅₃	#3204	1	1	yes	characteristic
206	C₂×C₅₃	#3384	1	2	yes	—
207	C₂×C₅₃	#3397	1	2	yes	—
208	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
209	C₂×C₂²⋊Q₈	—	2	2	no	—
210	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
212	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
213	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×#106	C₂²×C₅₃	1	2	yes	—
215	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×#102	C₂²×C₅₃	1	2	yes	—
218	C₂×#102	C₂²×C₅₃	1	2	yes	—
219	C₂×#102	C₂²×C₅₃	1	2	yes	—
220	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
221	#396	—	2	2	no	—
222	#396	—	2	2	no	—
223	#390	C₂²×C₅₃	1	2	yes	—
224	#390	C₂²×C₅₃	1	2	yes	—
225	#389	C₂²×C₅₃	1	2	yes	—
226	#389	C₂²×C₅₃	1	2	yes	—
227	#384	C₂²×C₅₃	1	2	yes	—
228	#380	—	2	2	no	—
229	#340	C₂²×C₅₃	1	2	yes	—
230	#340	C₂²×C₅₃	1	2	yes	—
231	#336	C₂²×C₅₃	1	2	yes	—
232	#336	C₂²×C₅₃	1	2	yes	—
233	C₅₃	#32666	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
235	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
236	C₂³×C₄	—	2	2	no	—
237	C₂×D₈	—	4	1	no	—
238	C₂×D₈	—	4	1	no	—
239	C₂×D₈	Q₈×C₅₃	1	4	yes	complemented
240	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
241	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
243	C₂²⋊Q₈	—	4	8	no	—
244	C₄×Q₈	—	4	1	no	—
245	C₄×Q₈	—	4	1	no	—
246	C₄×D₄	—	4	2	no	—
247	C₄×D₄	—	4	2	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
253	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
257	C₂×C₂²⋊C₄	—	2	2	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₂²⋊C₄	—	2	2	no	—
262	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
263	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
264	#107	—	4	1	no	—
265	#106	—	2	2	no	—
266	#106	—	4	2	no	—
267	#106	—	4	1	no	—
268	C₄⋊C₈	—	4	1	no	—
269	#102	—	2	4	no	—
270	#102	—	4	2	no	—
271	#102	—	4	2	no	—
272	#97	—	4	1	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	C₂⁴	—	2	2	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂×Q₈×C₅₃	1	4	yes	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	4	no	—
288	C₂²×C₄	—	4	4	no	—
289	C₂²×C₄	—	4	4	no	—
290	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
292	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
294	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
295	D₈	—	4	4	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
304	C₂×C₈	#41×C₅₃	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	4	no	—
314	C₄⋊C₄	—	4	4	no	—
315	C₄⋊C₄	—	4	4	no	—
316	C₄⋊C₄	—	8	2	no	—
317	C₂²⋊C₄	—	4	4	no	—
318	C₂²⋊C₄	—	4	4	no	—
319	C₂²⋊C₄	—	4	4	no	—
320	C₂²⋊C₄	—	8	2	no	—
321	C₂²⋊C₄	—	8	2	no	—
322	C₄²	—	2	1	no	—
323	C₄²	—	2	1	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₂³	—	2	4	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	2	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
334	Q₈	—	4	4	no	—
335	D₄	—	2	4	no	—
336	D₄	—	2	4	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	8	no	—
358	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
359	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
360	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
361	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
362	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
363	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
364	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	2	2	no	—
367	C₈	—	4	2	no	—
368	C₂²	—	4	4	no	—
369	C₂²	—	4	4	no	—
370	C₂²	—	4	4	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	C₅₃×#534	1	1	yes	characteristic
376	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
377	C₂²	C₅₃×#550	1	1	yes	characteristic
378	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
379	C₂²	C₅₃×#577	1	2	yes	—
380	C₄	—	2	4	no	—
381	C₄	—	4	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	1	no	—
384	C₄	—	8	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	8	2	no	—
387	C₄	C₅₃×#539	1	1	yes	characteristic
388	C₄	C₅₃×#543	1	1	yes	characteristic
389	C₄	C₅₃×#554	1	1	yes	characteristic
390	C₄	C₅₃×#562	1	1	yes	characteristic
391	C₂	—	8	4	no	—
392	C₂	C₅₃×#2732	1	1	yes	characteristic
393	C₂	C₅₃×#3192	1	1	yes	characteristic
394	C₂	C₅₃×#3204	1	1	yes	characteristic
395	C₂	C₅₃×#3384	1	2	yes	—
396	C₂	C₅₃×#3397	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2538	C₂	semidirect
2. C₅₃×#2094	C₂	semidirect
3. C₅₃×#2084	C₂	semidirect
4. C₅₃×#2065	C₂	semidirect
5. C₅₃×#1985	C₂	semidirect
7. C₅₃×#1944	C₂	semidirect
9. C₅₃×#1862	C₂	semidirect
40. C₂×D₈×C₅₃	Q₈	semidirect
169. #32666	C₅₃	direct (N × H)
193. #2538	C₂×C₅₃	semidirect
194. #2094	C₂×C₅₃	semidirect
195. #2084	C₂×C₅₃	semidirect
196. #2065	C₂×C₅₃	semidirect
197. #1985	C₂×C₅₃	semidirect
199. #1944	C₂×C₅₃	semidirect
201. #1862	C₂×C₅₃	semidirect
233. C₅₃	#32666	direct (N × H)
239. C₂×D₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2538	2
C₅₃×#2094	2
C₅₃×#2084	2
C₅₃×#2065	2
C₅₃×#1985	2
C₅₃×#1972	2
C₅₃×#1944	2
C₅₃×#1879	2
C₅₃×#1862	2
#32666	53

Subgroups — C53×#32666 #27240178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#32666 #27240178