Subgroups — C₅₃×#19420 #27226932

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#459	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1697	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1685	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₅₃	—	2	1	no	—
11	C₅₃×#533	C₂²	1	1	yes	characteristic
12	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#492	C₂²	1	2	yes	complemented
15	C₅₃×#486	C₂²	1	2	yes	complemented
16	C₅₃×#478	—	2	2	no	—
17	C₅₃×#476	C₂²	1	2	yes	complemented
18	C₅₃×#474	C₂²	1	2	yes	complemented
19	C₅₃×#460	—	2	2	no	—
20	C₅₃×#459	—	2	2	no	—
21	C₅₃×#459	C₂²	1	4	yes	—
22	C₅₃×#450	—	2	2	no	—
23	C₅₃×#436	—	2	2	no	—
24	C₅₃×#436	C₂²	1	4	yes	complemented
25	C₅₃×#431	C₂²	1	1	yes	characteristic
26	C₅₃×#431	C₂²	1	1	yes	characteristic
27	C₅₃×#428	—	2	1	no	—
28	C₅₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#418	—	2	1	no	—
31	C₂×#102×C₅₃	—	2	1	no	—
32	C₅₃×#407	—	2	2	no	—
33	C₅₃×#335	C₂²	1	4	yes	complemented
34	C₅₃×#333	C₂²	1	2	yes	complemented
35	C₅₃×#332	C₂²	1	2	yes	complemented
36	C₅₃×#331	C₂²	1	4	yes	complemented
37	C₂²×D₄×C₅₃	—	2	1	no	—
38	C₂²×D₄×C₅₃	—	2	1	no	—
39	C₂²×D₄×C₅₃	—	2	1	no	—
40	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	—	2	1	no	—
42	C₂×SD₁₆×C₅₃	—	2	4	no	—
43	C₂×D₈×C₅₃	—	2	4	no	—
44	C₂×D₈×C₅₃	—	4	4	no	—
45	C₂×M₁₆×C₅₃	—	2	4	no	—
46	C₂×M₁₆×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
50	#127×C₅₃	—	2	1	no	—
51	#127×C₅₃	—	2	1	no	—
52	#127×C₅₃	—	2	1	no	—
53	#127×C₅₃	C₂³	1	2	yes	—
54	#125×C₅₃	—	2	1	no	—
55	C₂²⋊Q₈×C₅₃	—	2	1	no	—
56	C₂²⋊Q₈×C₅₃	—	2	1	no	—
57	C₄×Q₈×C₅₃	—	2	1	no	—
58	C₄×Q₈×C₅₃	—	2	1	no	—
59	C₄×D₄×C₅₃	—	2	1	no	—
60	C₄×D₄×C₅₃	—	2	1	no	—
61	C₄×D₄×C₅₃	D₄	1	2	yes	—
62	C₄×D₄×C₅₃	C₂³	1	2	yes	—
63	#117×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	#107×C₅₃	—	2	2	no	—
68	#106×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	—	2	1	no	—
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
74	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
75	#102×C₅₃	—	2	4	no	—
76	#102×C₅₃	—	2	4	no	—
77	#102×C₅₃	—	2	2	no	—
78	#102×C₅₃	—	2	2	no	—
79	C₂²⋊C₈×C₅₃	—	2	4	no	—
80	C₂²⋊C₈×C₅₃	—	2	1	no	—
81	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
82	C₄×C₈×C₅₃	—	2	1	no	—
83	C₄×C₈×C₅₃	C₂³	1	2	yes	—
84	C₂⁴×C₅₃	—	2	1	no	—
85	C₂⁴×C₅₃	—	4	1	no	—
86	C₂×Q₈×C₅₃	—	2	1	no	—
87	C₂×Q₈×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	—	2	1	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	—	4	1	no	—
95	C₂×D₄×C₅₃	—	4	1	no	—
96	C₂×D₄×C₅₃	—	4	1	no	—
97	C₂×D₄×C₅₃	—	4	2	no	—
98	C₂×D₄×C₅₃	—	4	2	no	—
99	C₂×D₄×C₅₃	—	8	2	no	—
100	C₂×D₄×C₅₃	—	8	2	no	—
101	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	8	2	no	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
109	SD₁₆×C₅₃	—	8	4	no	—
110	D₈×C₅₃	—	4	8	no	—
111	M₁₆×C₅₃	—	4	2	no	—
112	M₁₆×C₅₃	—	8	4	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	4	2	no	—
117	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	4	1	no	—
124	C₄⋊C₄×C₅₃	—	4	1	no	—
125	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₂²⋊C₄×C₅₃	—	4	1	no	—
129	C₄²×C₅₃	—	4	1	no	—
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	#41	1	1	yes	characteristic
133	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂³×C₅₃	—	2	2	no	—
135	C₂³×C₅₃	—	4	2	no	—
136	C₂³×C₅₃	—	4	2	no	—
137	C₂³×C₅₃	—	8	1	no	—
138	C₂³×C₅₃	—	8	1	no	—
139	C₂³×C₅₃	—	8	2	no	—
140	C₂³×C₅₃	—	8	2	no	—
141	C₂³×C₅₃	—	8	2	no	—
142	C₂³×C₅₃	#121	1	1	yes	characteristic
143	Q₈×C₅₃	—	4	2	no	—
144	Q₈×C₅₃	—	8	1	no	—
145	D₄×C₅₃	—	2	2	no	—
146	D₄×C₅₃	—	2	2	no	—
147	D₄×C₅₃	—	4	2	no	—
148	D₄×C₅₃	—	4	2	no	—
149	D₄×C₅₃	—	4	4	no	—
150	D₄×C₅₃	—	4	2	no	—
151	D₄×C₅₃	—	8	2	no	—
152	D₄×C₅₃	—	8	2	no	—
153	D₄×C₅₃	—	8	2	no	—
154	D₄×C₅₃	—	8	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	4	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	8	2	no	—
162	C₂×C₄×C₅₃	—	8	1	no	—
163	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
166	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₅₃	#137	1	2	yes	—
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
171	C₈×C₅₃	—	2	4	no	—
172	C₈×C₅₃	—	4	2	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	—	4	2	no	—
175	#19420	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	4	2	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	8	4	no	—
182	C₂²×C₅₃	—	8	2	no	—
183	C₂²×C₅₃	—	8	2	no	—
184	C₂²×C₅₃	—	8	2	no	—
185	C₂²×C₅₃	—	16	4	no	—
186	C₂²×C₅₃	#468	1	2	yes	—
187	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	—	8	1	no	—
193	C₄×C₅₃	—	8	1	no	—
194	C₄×C₅₃	#468	1	2	yes	—
195	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
197	C₄×C₅₃	#576	1	2	yes	—
198	#3208	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#3077	C₂×C₅₃	1	1	yes	characteristic, complemented
200	C₂×#459	C₂×C₅₃	1	1	yes	characteristic
201	#2975	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#1697	C₂×C₅₃	1	4	yes	complemented
203	#1685	C₂×C₅₃	1	4	yes	complemented
204	#1571	C₂×C₅₃	1	1	yes	characteristic, complemented
205	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#1542	C₂×C₅₃	1	1	yes	characteristic, complemented
207	C₂×C₅₃	—	2	2	no	—
208	C₂×C₅₃	—	8	2	no	—
209	C₂×C₅₃	—	16	2	no	—
210	C₂×C₅₃	#3041	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#468	1	1	yes	characteristic
212	C₂×C₅₃	#3068	1	1	yes	characteristic
213	C₂²×D₈	—	2	1	no	—
214	#533	C₂²×C₅₃	1	1	yes	characteristic
215	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
217	#492	C₂²×C₅₃	1	2	yes	complemented
218	#486	C₂²×C₅₃	1	2	yes	complemented
219	#478	—	2	2	no	—
220	#476	C₂²×C₅₃	1	2	yes	complemented
221	#474	C₂²×C₅₃	1	2	yes	complemented
222	#460	—	2	2	no	—
223	#459	—	2	2	no	—
224	#459	C₂²×C₅₃	1	4	yes	—
225	#450	—	2	2	no	—
226	#436	—	2	2	no	—
227	#436	C₂²×C₅₃	1	4	yes	complemented
228	#431	C₂²×C₅₃	1	1	yes	characteristic
229	#431	C₂²×C₅₃	1	1	yes	characteristic
230	#428	—	2	1	no	—
231	#423	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
233	#418	—	2	1	no	—
234	C₂×#102	—	2	1	no	—
235	#407	—	2	2	no	—
236	#335	C₂²×C₅₃	1	4	yes	complemented
237	#333	C₂²×C₅₃	1	2	yes	complemented
238	#332	C₂²×C₅₃	1	2	yes	complemented
239	#331	C₂²×C₅₃	1	4	yes	complemented
240	C₅₃	#19420	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²×D₄	—	2	1	no	—
242	C₂²×D₄	—	2	1	no	—
243	C₂²×D₄	—	2	1	no	—
244	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
245	C₂³×C₄	—	2	1	no	—
246	C₂×SD₁₆	—	2	4	no	—
247	C₂×D₈	—	2	4	no	—
248	C₂×D₈	—	4	4	no	—
249	C₂×M₁₆	—	2	1	no	—
250	C₂×M₁₆	—	2	4	no	—
251	C₂²×C₈	—	2	1	no	—
252	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
253	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
254	#127	—	2	1	no	—
255	#127	—	2	1	no	—
256	#127	—	2	1	no	—
257	#127	C₂³×C₅₃	1	2	yes	—
258	#125	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	—	2	1	no	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×Q₈	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	D₄×C₅₃	1	2	yes	—
266	C₄×D₄	C₂³×C₅₃	1	2	yes	—
267	#117	—	2	1	no	—
268	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
269	C₂×C₂²⋊C₄	—	2	1	no	—
270	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
271	#107	—	2	2	no	—
272	#106	—	2	2	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
276	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
277	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
278	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
279	#102	—	2	2	no	—
280	#102	—	2	2	no	—
281	#102	—	2	4	no	—
282	#102	—	2	4	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	2	4	no	—
285	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	C₂³×C₅₃	1	2	yes	—
288	C₂⁴	—	2	1	no	—
289	C₂⁴	—	4	1	no	—
290	C₂×Q₈	—	2	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	2	no	—
302	C₂×D₄	—	4	2	no	—
303	C₂×D₄	—	8	2	no	—
304	C₂×D₄	—	8	2	no	—
305	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	8	2	no	—
310	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
313	SD₁₆	—	8	4	no	—
314	D₈	—	4	8	no	—
315	M₁₆	—	4	2	no	—
316	M₁₆	—	8	4	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₂²⋊C₄	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₄²	#41×C₅₃	1	1	yes	characteristic
337	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	8	1	no	—
342	C₂³	—	8	1	no	—
343	C₂³	—	8	2	no	—
344	C₂³	—	8	2	no	—
345	C₂³	—	8	2	no	—
346	C₂³	#121×C₅₃	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	Q₈	—	8	1	no	—
349	D₄	—	2	2	no	—
350	D₄	—	2	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	4	no	—
354	D₄	—	4	2	no	—
355	D₄	—	8	2	no	—
356	D₄	—	8	2	no	—
357	D₄	—	8	2	no	—
358	D₄	—	8	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	8	2	no	—
366	C₂×C₄	—	8	1	no	—
367	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
368	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
369	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
370	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
371	C₂×C₄	#137×C₅₃	1	2	yes	—
372	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
375	C₈	—	2	4	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	8	4	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	16	4	no	—
389	C₂²	C₅₃×#468	1	2	yes	—
390	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	8	1	no	—
396	C₄	—	8	1	no	—
397	C₄	C₅₃×#468	1	2	yes	—
398	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
399	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
400	C₄	C₅₃×#576	1	2	yes	—
401	C₂	—	2	2	no	—
402	C₂	—	8	2	no	—
403	C₂	—	16	2	no	—
404	C₂	C₅₃×#3041	1	1	yes	characteristic
405	C₂	C₂×C₅₃×#468	1	1	yes	characteristic
406	C₂	C₅₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3208	C₂	semidirect
2. C₅₃×#3077	C₂	semidirect
4. C₅₃×#2975	C₂	semidirect
5. C₅₃×#1697	C₂	semidirect
6. C₅₃×#1685	C₂	semidirect
7. C₅₃×#1571	C₂	semidirect
8. C₅₃×#1545	C₂	semidirect
9. C₅₃×#1542	C₂	semidirect
14. C₅₃×#492	C₂²	semidirect
15. C₅₃×#486	C₂²	semidirect
17. C₅₃×#476	C₂²	semidirect
18. C₅₃×#474	C₂²	semidirect
24. C₅₃×#436	C₂²	semidirect
33. C₅₃×#335	C₂²	semidirect
34. C₅₃×#333	C₂²	semidirect
35. C₅₃×#332	C₂²	semidirect
36. C₅₃×#331	C₂²	semidirect
175. #19420	C₅₃	direct (N × H)
198. #3208	C₂×C₅₃	semidirect
199. #3077	C₂×C₅₃	semidirect
201. #2975	C₂×C₅₃	semidirect
202. #1697	C₂×C₅₃	semidirect
203. #1685	C₂×C₅₃	semidirect
204. #1571	C₂×C₅₃	semidirect
205. #1545	C₂×C₅₃	semidirect
206. #1542	C₂×C₅₃	semidirect
217. #492	C₂²×C₅₃	semidirect
218. #486	C₂²×C₅₃	semidirect
220. #476	C₂²×C₅₃	semidirect
221. #474	C₂²×C₅₃	semidirect
227. #436	C₂²×C₅₃	semidirect
236. #335	C₂²×C₅₃	semidirect
237. #333	C₂²×C₅₃	semidirect
238. #332	C₂²×C₅₃	semidirect
239. #331	C₂²×C₅₃	semidirect
240. C₅₃	#19420	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×#3208	2
C₅₃×#3077	2
C₂×C₅₃×#459	2
C₅₃×#2975	2
C₅₃×#1697	2
C₅₃×#1685	2
C₅₃×#1571	2
C₅₃×#1545	2
C₅₃×#1542	2
#19420	53

Subgroups — C53×#19420 #27226932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19420 #27226932