Subgroups — C₅₃×#22517 #27230029

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#557	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1998	C₂	1	4	yes	maximal
4	C₅₃×#1997	C₂	1	2	yes	maximal
5	C₅₃×#1801	C₂	1	2	yes	maximal
6	C₅₃×#1801	C₂	1	1	yes	maximal, characteristic
7	C₄×#101×C₅₃	C₂	1	2	yes	maximal
8	C₄×#101×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#557	—	2	2	no	—
10	C₅₃×#557	—	2	4	no	—
11	C₅₃×#557	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#125×C₅₃	C₄	1	4	yes	—
15	C₂×#125×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#446	—	2	4	no	—
21	C₅₃×#446	—	2	2	no	—
22	C₈×Q₈×C₅₃	—	2	2	no	—
23	C₅₃×#423	C₂²	1	2	yes	—
24	C₅₃×#423	C₂²	1	4	yes	—
25	C₂×#101×C₅₃	C₂²	1	2	yes	—
26	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
30	C₅₃×#343	C₂²	1	4	yes	—
31	C₅₃×#343	C₂²	1	4	yes	—
32	C₅₃×#343	C₂²	1	2	yes	—
33	C₅₃×#343	C₂²	1	2	yes	—
34	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
38	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₅₃	—	2	2	no	—
41	C₄⋊Q₈×C₅₃	—	2	2	no	—
42	C₄⋊Q₈×C₅₃	—	2	4	no	—
43	C₄⋊Q₈×C₅₃	—	2	4	no	—
44	#125×C₅₃	—	2	8	no	—
45	#125×C₅₃	—	2	8	no	—
46	#125×C₅₃	—	2	4	no	—
47	#125×C₅₃	—	2	4	no	—
48	C₄×Q₈×C₅₃	—	2	4	no	—
49	C₄×Q₈×C₅₃	—	2	2	no	—
50	C₄×Q₈×C₅₃	—	2	2	no	—
51	C₄×Q₈×C₅₃	—	2	4	no	—
52	C₄×Q₈×C₅₃	—	2	4	no	—
53	C₄×Q₈×C₅₃	—	2	2	no	—
54	C₄×Q₈×C₅₃	—	2	2	no	—
55	C₄×Q₈×C₅₃	D₄	1	2	yes	—
56	C₄×Q₈×C₅₃	D₄	1	2	yes	—
57	C₄×Q₈×C₅₃	D₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
68	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	—	2	4	no	—
72	C₄⋊C₈×C₅₃	—	2	2	no	—
73	C₄⋊C₈×C₅₃	—	2	4	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	#101×C₅₃	—	2	4	no	—
76	#101×C₅₃	—	2	2	no	—
77	#101×C₅₃	Q₈	1	4	yes	complemented
78	#97×C₅₃	—	2	4	no	—
79	#97×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	—	2	4	no	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	—	2	2	no	—
86	C₂×Q₈×C₅₃	—	4	2	no	—
87	C₂×Q₈×C₅₃	—	4	4	no	—
88	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
89	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
91	C₂×Q₈×C₅₃	#41	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₅₃	—	2	2	no	—
101	M₁₆×C₅₃	—	2	4	no	—
102	M₁₆×C₅₃	—	2	2	no	—
103	M₁₆×C₅₃	—	2	4	no	—
104	M₁₆×C₅₃	—	2	2	no	—
105	M₁₆×C₅₃	C₂×Q₈	1	4	yes	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	4	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₄⋊C₄×C₅₃	—	2	4	no	—
110	C₄⋊C₄×C₅₃	—	2	4	no	—
111	C₄⋊C₄×C₅₃	—	2	4	no	—
112	C₄⋊C₄×C₅₃	—	2	8	no	—
113	C₄⋊C₄×C₅₃	—	2	4	no	—
114	C₄⋊C₄×C₅₃	—	2	8	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	2	4	no	—
117	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
118	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
121	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
122	C₄²×C₅₃	—	2	4	no	—
123	C₄²×C₅₃	—	2	4	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	C₂×D₄	1	4	yes	—
127	C₄²×C₅₃	C₂×D₄	1	2	yes	—
128	C₄²×C₅₃	C₂×D₄	1	4	yes	—
129	C₄²×C₅₃	C₂×D₄	1	2	yes	—
130	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	Q₈×C₅₃	—	2	4	no	—
132	Q₈×C₅₃	—	2	4	no	—
133	Q₈×C₅₃	—	2	4	no	—
134	Q₈×C₅₃	—	4	2	no	—
135	Q₈×C₅₃	—	4	4	no	—
136	Q₈×C₅₃	—	4	4	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	4	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	4	no	—
147	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
154	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
155	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
158	C₈×C₅₃	—	2	4	no	—
159	C₈×C₅₃	—	4	4	no	—
160	C₈×C₅₃	—	4	2	no	—
161	#22517	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₂²×C₅₃	#519	1	1	yes	characteristic
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	4	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	4	4	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	#413	1	2	yes	—
176	C₄×C₅₃	#548	1	2	yes	—
177	C₄×C₅₃	D₄×Q₈	1	2	yes	—
178	C₂×#557	C₂×C₅₃	1	1	yes	characteristic
179	Q₈×M₁₆	C₂×C₅₃	1	2	yes	—
180	#1998	C₂×C₅₃	1	4	yes	—
181	#1997	C₂×C₅₃	1	2	yes	—
182	#1801	C₂×C₅₃	1	2	yes	—
183	#1801	C₂×C₅₃	1	1	yes	characteristic
184	C₄×#101	C₂×C₅₃	1	2	yes	—
185	C₄×#101	C₂×C₅₃	1	1	yes	characteristic
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	2	1	no	—
188	C₂×C₅₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₅₃	C₂×#413	1	1	yes	characteristic
190	C₂×C₅₃	#2889	1	1	yes	characteristic
191	#557	—	2	2	no	—
192	#557	—	2	4	no	—
193	#557	—	2	2	no	—
194	C₂×C₄⋊Q₈	C₄×C₅₃	1	2	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×#125	C₄×C₅₃	1	4	yes	—
197	C₂×#125	C₂²×C₅₃	1	2	yes	—
198	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
199	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
200	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
202	#446	—	2	4	no	—
203	#446	—	2	2	no	—
204	C₈×Q₈	—	2	2	no	—
205	#423	C₂²×C₅₃	1	2	yes	—
206	#423	C₂²×C₅₃	1	4	yes	—
207	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
208	C₂×#101	C₂²×C₅₃	1	2	yes	—
209	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
210	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
211	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
212	#343	C₂²×C₅₃	1	4	yes	—
213	#343	C₂²×C₅₃	1	2	yes	—
214	#343	C₂²×C₅₃	1	4	yes	—
215	#343	C₂²×C₅₃	1	2	yes	—
216	C₅₃	#22517	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
218	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
220	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
221	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
222	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
223	C₄⋊Q₈	—	2	4	no	—
224	C₄⋊Q₈	—	2	4	no	—
225	C₄⋊Q₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	#125	—	2	8	no	—
228	#125	—	2	8	no	—
229	#125	—	2	4	no	—
230	#125	—	2	4	no	—
231	C₄×Q₈	—	2	4	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	—	2	4	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	D₄×C₅₃	1	2	yes	—
239	C₄×Q₈	D₄×C₅₃	1	4	yes	—
240	C₄×Q₈	D₄×C₅₃	1	2	yes	—
241	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
242	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
244	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
245	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
250	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
251	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
252	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
253	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
254	C₄⋊C₈	—	2	4	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	4	no	—
257	C₄⋊C₈	—	2	2	no	—
258	#101	—	2	2	no	—
259	#101	—	2	4	no	—
260	#101	Q₈×C₅₃	1	4	yes	complemented
261	#97	—	2	4	no	—
262	#97	—	2	2	no	—
263	C₄×C₈	—	2	4	no	—
264	C₄×C₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
272	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
273	C₂×Q₈	C₂×Q₈×C₅₃	1	2	yes	—
274	C₂×Q₈	#41×C₅₃	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
277	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
278	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
279	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
280	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
282	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	4	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	4	no	—
288	M₁₆	C₂×Q₈×C₅₃	1	4	yes	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
301	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
302	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
303	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
304	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
305	C₄²	—	2	4	no	—
306	C₄²	—	2	4	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	C₂×D₄×C₅₃	1	4	yes	—
310	C₄²	C₂×D₄×C₅₃	1	2	yes	—
311	C₄²	C₂×D₄×C₅₃	1	4	yes	—
312	C₄²	C₂×D₄×C₅₃	1	2	yes	—
313	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
314	Q₈	—	2	4	no	—
315	Q₈	—	2	4	no	—
316	Q₈	—	2	4	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	4	no	—
319	Q₈	—	4	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
331	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
334	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
337	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
338	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
340	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
341	C₈	—	2	4	no	—
342	C₈	—	4	4	no	—
343	C₈	—	4	2	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
347	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
348	C₂²	C₅₃×#519	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	C₅₃×#413	1	2	yes	—
358	C₄	C₅₃×#548	1	2	yes	—
359	C₄	D₄×Q₈×C₅₃	1	2	yes	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	Q₈×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
363	C₂	C₂×C₅₃×#413	1	1	yes	characteristic
364	C₂	C₅₃×#2889	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
77. #101×C₅₃	Q₈	semidirect
161. #22517	C₅₃	direct (N × H)
216. C₅₃	#22517	direct (N × H)
260. #101	Q₈×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×#557	2
Q₈×M₁₆×C₅₃	2
C₅₃×#1998	2
C₅₃×#1997	2
C₅₃×#1801	2
C₄×#101×C₅₃	2
#22517	53

Subgroups — C53×#22517 #27230029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22517 #27230029