Subgroups — C₅₃×#9910 #27217422

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2851	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1294	C₂	1	2	yes	maximal
5	C₅₃×#1294	C₂	1	2	yes	maximal
6	C₂×#125×C₅₃	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₄²×C₅₃	—	2	1	no	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#389	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#389	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#387	—	2	1	no	—
14	C₅₃×#387	—	2	1	no	—
15	C₅₃×#385	—	2	1	no	—
16	C₅₃×#385	—	2	1	no	—
17	C₅₃×#382	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#380	C₂²	1	1	yes	characteristic
19	C₅₃×#380	C₂²	1	1	yes	characteristic
20	C₂×#95×C₅₃	—	2	1	no	—
21	C₂×#95×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#336	—	2	1	no	—
23	C₂³×C₄×C₅₃	—	2	2	no	—
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	#125×C₅₃	—	4	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	—	2	4	no	—
40	C₂×C₄²×C₅₃	—	4	2	no	—
41	C₂×C₄²×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	—	4	2	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	4	2	no	—
48	C₂²⋊C₈×C₅₃	—	2	4	no	—
49	#95×C₅₃	—	2	4	no	—
50	#95×C₅₃	—	2	1	no	—
51	#95×C₅₃	—	2	1	no	—
52	#95×C₅₃	—	2	1	no	—
53	#95×C₅₃	—	2	1	no	—
54	#95×C₅₃	—	2	1	no	—
55	#95×C₅₃	—	2	1	no	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	Q₈	1	4	yes	—
58	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	1	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	4	1	no	—
73	C₂²×C₄×C₅₃	—	4	1	no	—
74	C₂²×C₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	—	4	2	no	—
78	C₂²×C₄×C₅₃	—	4	2	no	—
79	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²⋊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×Q₈	1	2	yes	—
86	C₂²×C₄×C₅₃	#41	1	2	yes	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₂×C₈×C₅₃	—	4	2	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	1	no	—
92	C₄⋊C₄×C₅₃	—	4	1	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₂²⋊C₄×C₅₃	—	4	1	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	4	1	no	—
104	C₄²×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	—	4	8	no	—
106	C₂³×C₅₃	—	2	1	no	—
107	C₂³×C₅₃	—	2	1	no	—
108	C₂³×C₅₃	—	2	1	no	—
109	C₂³×C₅₃	—	2	1	no	—
110	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
112	C₂³×C₅₃	#135	1	2	yes	—
113	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂³×C₅₃	#137	1	1	yes	characteristic
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	8	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	4	no	—
134	C₂×C₄×C₅₃	—	4	4	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	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	—	4	1	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	—	4	1	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	#99	1	2	yes	—
153	C₂×C₄×C₅₃	#104	1	2	yes	—
154	C₂×C₄×C₅₃	#104	1	2	yes	—
155	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
158	C₈×C₅₃	—	8	2	no	—
159	#9910	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	#391	1	1	yes	characteristic
169	C₂²×C₅₃	C₂×#99	1	1	yes	characteristic
170	C₂²×C₅₃	#410	1	1	yes	characteristic
171	C₂²×C₅₃	#416	1	1	yes	characteristic
172	C₂²×C₅₃	#417	1	1	yes	characteristic
173	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
174	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
175	C₂²×C₅₃	#478	1	2	yes	—
176	C₂²×C₅₃	#479	1	2	yes	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	2	2	no	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	4	8	no	—
184	C₄×C₅₃	—	4	1	no	—
185	C₄×C₅₃	—	8	1	no	—
186	C₄×C₅₃	—	8	1	no	—
187	C₄×C₅₃	—	8	1	no	—
188	C₄×C₅₃	—	8	1	no	—
189	#2851	C₂×C₅₃	1	1	yes	characteristic
190	#1797	C₂×C₅₃	1	1	yes	characteristic
191	C₄×#95	C₂×C₅₃	1	1	yes	characteristic
192	#1294	C₂×C₅₃	1	2	yes	—
193	#1294	C₂×C₅₃	1	2	yes	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	#1500	1	1	yes	characteristic
197	C₂×C₅₃	#1501	1	1	yes	characteristic
198	C₂×C₅₃	#1504	1	1	yes	characteristic
199	C₂×C₅₃	#1509	1	1	yes	characteristic
200	C₂×C₅₃	#1989	1	1	yes	characteristic
201	C₂×C₅₃	#1994	1	1	yes	characteristic
202	C₂×C₅₃	#1996	1	1	yes	characteristic
203	C₂×#125	C₄×C₅₃	1	1	yes	characteristic, complemented
204	C₂²×C₄²	—	2	1	no	—
205	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×C₄⋊C₈	—	2	1	no	—
207	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
208	#389	C₄×C₅₃	1	1	yes	characteristic, complemented
209	#389	C₄×C₅₃	1	1	yes	characteristic, complemented
210	#387	—	2	1	no	—
211	#387	—	2	1	no	—
212	#385	—	2	1	no	—
213	#385	—	2	1	no	—
214	#382	C₄×C₅₃	1	1	yes	characteristic, complemented
215	#380	C₂²×C₅₃	1	1	yes	characteristic
216	#380	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×#95	—	2	1	no	—
218	C₂×#95	C₂²×C₅₃	1	2	yes	—
219	#336	—	2	1	no	—
220	C₅₃	#9910	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³×C₄	—	2	2	no	—
222	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
223	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
224	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
225	C₂²×C₈	—	2	2	no	—
226	#125	—	4	2	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	—	2	1	no	—
236	C₂×C₂²⋊C₄	—	2	1	no	—
237	C₂×C₄²	—	2	4	no	—
238	C₂×C₄²	—	4	2	no	—
239	C₂×C₄²	—	4	2	no	—
240	C₂×C₄²	—	4	2	no	—
241	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
244	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
245	C₄⋊C₈	—	4	2	no	—
246	C₂²⋊C₈	—	2	4	no	—
247	#95	—	2	4	no	—
248	#95	—	2	1	no	—
249	#95	—	2	1	no	—
250	#95	—	2	2	no	—
251	#95	—	2	1	no	—
252	#95	—	2	1	no	—
253	#95	—	2	1	no	—
254	#95	—	2	1	no	—
255	#95	Q₈×C₅₃	1	4	yes	—
256	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
284	C₂²×C₄	#41×C₅₃	1	2	yes	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	8	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
309	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
310	C₂³	#135×C₅₃	1	2	yes	—
311	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
312	C₂³	#137×C₅₃	1	1	yes	characteristic
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	#99×C₅₃	1	2	yes	—
351	C₂×C₄	#104×C₅₃	1	2	yes	—
352	C₂×C₄	#104×C₅₃	1	2	yes	—
353	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
354	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
356	C₈	—	8	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	C₅₃×#391	1	1	yes	characteristic
366	C₂²	C₂×#99×C₅₃	1	1	yes	characteristic
367	C₂²	C₅₃×#410	1	1	yes	characteristic
368	C₂²	C₅₃×#416	1	1	yes	characteristic
369	C₂²	C₅₃×#417	1	1	yes	characteristic
370	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
371	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
372	C₂²	C₅₃×#478	1	2	yes	—
373	C₂²	C₅₃×#479	1	2	yes	—
374	C₄	—	2	1	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	2	no	—
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	4	8	no	—
381	C₄	—	4	1	no	—
382	C₄	—	8	1	no	—
383	C₄	—	8	1	no	—
384	C₄	—	8	1	no	—
385	C₄	—	8	1	no	—
386	C₂	—	2	2	no	—
387	C₂	—	2	2	no	—
388	C₂	C₅₃×#1500	1	1	yes	characteristic
389	C₂	C₅₃×#1501	1	1	yes	characteristic
390	C₂	C₅₃×#1504	1	1	yes	characteristic
391	C₂	C₅₃×#1509	1	1	yes	characteristic
392	C₂	C₅₃×#1989	1	1	yes	characteristic
393	C₂	C₅₃×#1994	1	1	yes	characteristic
394	C₂	C₅₃×#1996	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×#125×C₅₃	C₄	semidirect
11. C₅₃×#389	C₄	semidirect
12. C₅₃×#389	C₄	semidirect
17. C₅₃×#382	C₄	semidirect
159. #9910	C₅₃	direct (N × H)
203. C₂×#125	C₄×C₅₃	semidirect
208. #389	C₄×C₅₃	semidirect
209. #389	C₄×C₅₃	semidirect
214. #382	C₄×C₅₃	semidirect
220. C₅₃	#9910	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×#2851	2
C₅₃×#1797	2
C₄×#95×C₅₃	2
C₅₃×#1294	2
#9910	53

Subgroups — C53×#9910 #27217422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9910 #27217422