Subgroups — C₅₃×#19214 #27226726

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1702	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1682	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#494	—	2	1	no	—
14	C₅₃×#492	—	2	1	no	—
15	C₅₃×#487	—	2	1	no	—
16	C₅₃×#486	—	2	1	no	—
17	C₅₃×#486	—	2	1	no	—
18	C₅₃×#486	—	2	1	no	—
19	C₅₃×#486	C₂²	1	2	yes	—
20	C₅₃×#486	C₂²	1	2	yes	—
21	C₅₃×#462	C₂²	1	4	yes	complemented
22	C₅₃×#460	C₂²	1	4	yes	—
23	C₅₃×#451	—	2	2	no	—
24	C₅₃×#450	—	2	2	no	—
25	C₅₃×#446	—	2	1	no	—
26	C₅₃×#446	C₂²	1	2	yes	complemented
27	C₈×Q₈×C₅₃	—	2	1	no	—
28	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#424	—	2	1	no	—
32	C₅₃×#423	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₃	—	2	1	no	—
34	C₂×#102×C₅₃	—	2	1	no	—
35	C₅₃×#405	—	2	1	no	—
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	2	yes	complemented
38	C₅₃×#334	C₂²	1	2	yes	complemented
39	C₅₃×#332	C₂²	1	4	yes	complemented
40	C₅₃×#331	C₂²	1	4	yes	—
41	C₂²×Q₈×C₅₃	—	2	1	no	—
42	C₂²×D₄×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	—	2	1	no	—
44	C₂²×D₄×C₅₃	—	2	1	no	—
45	C₂×Q₁₆×C₅₃	—	2	4	no	—
46	C₂×SD₁₆×C₅₃	—	2	4	no	—
47	C₂×M₁₆×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
56	#127×C₅₃	—	2	1	no	—
57	#127×C₅₃	—	2	1	no	—
58	#127×C₅₃	—	2	1	no	—
59	#127×C₅₃	C₂³	1	2	yes	—
60	#125×C₅₃	—	2	1	no	—
61	C₂²⋊Q₈×C₅₃	—	2	1	no	—
62	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
63	C₄×Q₈×C₅₃	—	2	1	no	—
64	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
65	#117×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
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	#103×C₅₃	—	2	4	no	—
76	#102×C₅₃	—	2	2	no	—
77	#102×C₅₃	—	2	2	no	—
78	#102×C₅₃	—	2	2	no	—
79	#102×C₅₃	—	2	2	no	—
80	#102×C₅₃	—	2	4	no	—
81	C₂²⋊C₈×C₅₃	—	2	1	no	—
82	C₂²⋊C₈×C₅₃	—	2	1	no	—
83	#97×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	—	2	2	no	—
85	C₄×C₈×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂⁴×C₅₃	—	4	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	4	1	no	—
95	C₂×Q₈×C₅₃	—	8	2	no	—
96	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×D₄×C₅₃	—	2	2	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	4	1	no	—
101	C₂×D₄×C₅₃	—	4	1	no	—
102	C₂×D₄×C₅₃	—	4	1	no	—
103	C₂×D₄×C₅₃	—	4	2	no	—
104	C₂×D₄×C₅₃	—	4	2	no	—
105	C₂×D₄×C₅₃	—	8	2	no	—
106	C₂×D₄×C₅₃	—	8	2	no	—
107	C₂²×C₄×C₅₃	—	4	1	no	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
111	Q₁₆×C₅₃	—	8	4	no	—
112	SD₁₆×C₅₃	—	8	4	no	—
113	M₁₆×C₅₃	—	4	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	4	1	no	—
131	C₄⋊C₄×C₅₃	—	4	1	no	—
132	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
133	C₂²⋊C₄×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₅₃	#41	1	1	yes	characteristic
138	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	8	2	no	—
141	C₂³×C₅₃	—	8	2	no	—
142	C₂³×C₅₃	—	8	2	no	—
143	C₂³×C₅₃	#121	1	1	yes	characteristic
144	Q₈×C₅₃	—	2	4	no	—
145	Q₈×C₅₃	—	4	2	no	—
146	Q₈×C₅₃	—	4	4	no	—
147	Q₈×C₅₃	—	8	2	no	—
148	D₄×C₅₃	—	4	2	no	—
149	D₄×C₅₃	—	4	2	no	—
150	D₄×C₅₃	—	4	4	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	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	8	2	no	—
161	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₅₃	#135	1	2	yes	—
165	C₂×C₄×C₅₃	#137	1	2	yes	—
166	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
169	C₈×C₅₃	—	2	1	no	—
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	2	1	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	#19214	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	8	2	no	—
180	C₂²×C₅₃	—	8	2	no	—
181	C₂²×C₅₃	—	8	2	no	—
182	C₂²×C₅₃	—	16	4	no	—
183	C₂²×C₅₃	#464	1	2	yes	—
184	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	4	2	no	—
188	C₄×C₅₃	—	4	2	no	—
189	C₄×C₅₃	—	8	2	no	—
190	C₄×C₅₃	#464	1	2	yes	—
191	C₄×C₅₃	#471	1	2	yes	—
192	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
193	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
194	C₄×C₅₃	#576	1	2	yes	—
195	#3209	C₂×C₅₃	1	1	yes	characteristic, complemented
196	#3208	C₂×C₅₃	1	1	yes	characteristic
197	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
198	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#1702	C₂×C₅₃	1	4	yes	complemented
200	#1682	C₂×C₅₃	1	4	yes	complemented
201	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
203	#1545	C₂×C₅₃	1	1	yes	characteristic
204	C₂×C₅₃	—	2	2	no	—
205	C₂×C₅₃	—	16	2	no	—
206	C₂×C₅₃	C₂×#464	1	1	yes	characteristic
207	C₂×C₅₃	#3058	1	1	yes	characteristic
208	C₂×C₅₃	C₂×#471	1	1	yes	characteristic
209	#533	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
212	#494	—	2	1	no	—
213	#492	—	2	1	no	—
214	#487	—	2	1	no	—
215	#486	—	2	1	no	—
216	#486	—	2	1	no	—
217	#486	—	2	1	no	—
218	#486	C₂²×C₅₃	1	2	yes	—
219	#486	C₂²×C₅₃	1	2	yes	—
220	#462	C₂²×C₅₃	1	4	yes	complemented
221	#460	C₂²×C₅₃	1	4	yes	—
222	#451	—	2	2	no	—
223	#450	—	2	2	no	—
224	#446	—	2	1	no	—
225	#446	C₂²×C₅₃	1	2	yes	complemented
226	C₈×Q₈	—	2	1	no	—
227	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
228	#431	C₂²×C₅₃	1	1	yes	characteristic
229	#431	C₂²×C₅₃	1	1	yes	characteristic
230	#424	—	2	1	no	—
231	#423	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×#102	—	2	1	no	—
233	C₂×#102	—	2	1	no	—
234	#405	—	2	1	no	—
235	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
236	#335	C₂²×C₅₃	1	2	yes	complemented
237	#334	C₂²×C₅₃	1	2	yes	complemented
238	#332	C₂²×C₅₃	1	4	yes	complemented
239	#331	C₂²×C₅₃	1	4	yes	—
240	C₅₃	#19214	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²×Q₈	—	2	1	no	—
242	C₂²×D₄	—	2	1	no	—
243	C₂²×D₄	—	2	1	no	—
244	C₂²×D₄	—	2	1	no	—
245	C₂×Q₁₆	—	2	4	no	—
246	C₂×SD₁₆	—	2	4	no	—
247	C₂×M₁₆	—	2	1	no	—
248	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
249	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
254	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
255	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
256	#127	—	2	1	no	—
257	#127	—	2	1	no	—
258	#127	—	2	1	no	—
259	#127	C₂³×C₅₃	1	2	yes	—
260	#125	—	2	1	no	—
261	C₂²⋊Q₈	—	2	1	no	—
262	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
265	#117	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
273	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
274	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
275	#103	—	2	4	no	—
276	#102	—	2	2	no	—
277	#102	—	2	2	no	—
278	#102	—	2	2	no	—
279	#102	—	2	2	no	—
280	#102	—	2	4	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	#97	—	2	2	no	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
288	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
290	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
291	C₂⁴	—	4	1	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	8	2	no	—
296	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	8	2	no	—
306	C₂×D₄	—	8	2	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
311	Q₁₆	—	8	4	no	—
312	SD₁₆	—	8	4	no	—
313	M₁₆	—	4	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
321	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₄²	#41×C₅₃	1	1	yes	characteristic
338	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
339	C₂³	—	4	2	no	—
340	C₂³	—	8	2	no	—
341	C₂³	—	8	2	no	—
342	C₂³	—	8	2	no	—
343	C₂³	#121×C₅₃	1	1	yes	characteristic
344	Q₈	—	2	4	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	4	4	no	—
347	Q₈	—	8	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	4	no	—
351	D₄	—	8	2	no	—
352	D₄	—	8	2	no	—
353	D₄	—	8	2	no	—
354	D₄	—	8	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
362	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
363	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
364	C₂×C₄	#135×C₅₃	1	2	yes	—
365	C₂×C₄	#137×C₅₃	1	2	yes	—
366	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	—	16	4	no	—
382	C₂²	C₅₃×#464	1	2	yes	—
383	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	8	2	no	—
389	C₄	C₅₃×#464	1	2	yes	—
390	C₄	C₅₃×#471	1	2	yes	—
391	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
392	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
393	C₄	C₅₃×#576	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	16	2	no	—
396	C₂	C₂×C₅₃×#464	1	1	yes	characteristic
397	C₂	C₅₃×#3058	1	1	yes	characteristic
398	C₂	C₂×C₅₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
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×#3209	C₂	semidirect
4. C₅₃×#2960	C₂	semidirect
5. C₅₃×#1702	C₂	semidirect
6. C₅₃×#1682	C₂	semidirect
7. C₅₃×#1564	C₂	semidirect
8. C₅₃×#1549	C₂	semidirect
21. C₅₃×#462	C₂²	semidirect
26. C₅₃×#446	C₂²	semidirect
28. C₈×Q₈×C₅₃	C₂²	semidirect
37. C₅₃×#335	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
39. C₅₃×#332	C₂²	semidirect
176. #19214	C₅₃	direct (N × H)
195. #3209	C₂×C₅₃	semidirect
198. #2960	C₂×C₅₃	semidirect
199. #1702	C₂×C₅₃	semidirect
200. #1682	C₂×C₅₃	semidirect
201. #1564	C₂×C₅₃	semidirect
202. #1549	C₂×C₅₃	semidirect
220. #462	C₂²×C₅₃	semidirect
225. #446	C₂²×C₅₃	semidirect
227. C₈×Q₈	C₂²×C₅₃	semidirect
236. #335	C₂²×C₅₃	semidirect
237. #334	C₂²×C₅₃	semidirect
238. #332	C₂²×C₅₃	semidirect
240. C₅₃	#19214	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×#3209	2
C₅₃×#3208	2
C₂×C₅₃×#486	2
C₅₃×#2960	2
C₅₃×#1702	2
C₅₃×#1682	2
C₅₃×#1564	2
C₅₃×#1549	2
C₅₃×#1545	2
#19214	53

Subgroups — C53×#19214 #27226726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19214 #27226726