Subgroups — C₅₃×#22315 #27229827

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	2	yes	maximal
3	C₅₃×#1984	C₂	1	2	yes	maximal
4	C₅₃×#1940	C₂	1	2	yes	maximal
5	C₅₃×#1917	C₂	1	2	yes	maximal
6	C₅₃×#1916	C₂	1	2	yes	maximal
7	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1771	C₂	1	2	yes	maximal
9	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#391	C₄	1	1	yes	characteristic
21	C₅₃×#391	C₄	1	1	yes	characteristic
22	C₅₃×#389	C₄	1	2	yes	—
23	C₅₃×#389	C₄	1	2	yes	—
24	C₅₃×#384	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₄³×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#336	C₂²	1	1	yes	characteristic
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	1	yes	characteristic
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₅₃	—	2	4	no	—
44	C₄×Q₈×C₅₃	—	4	2	no	—
45	C₄×Q₈×C₅₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	—	2	4	no	—
64	C₄⋊C₈×C₅₃	—	4	2	no	—
65	C₄⋊C₈×C₅₃	—	4	2	no	—
66	#97×C₅₃	—	2	2	no	—
67	#97×C₅₃	—	4	2	no	—
68	#97×C₅₃	Q₈	1	4	yes	complemented
69	C₄×C₈×C₅₃	—	4	2	no	—
70	#95×C₅₃	C₂×C₄	1	2	yes	—
71	#95×C₅₃	C₂×C₄	1	2	yes	—
72	#95×C₅₃	C₂×C₄	1	2	yes	—
73	#95×C₅₃	C₂×C₄	1	2	yes	—
74	C₂×Q₈×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	—	4	1	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	C₂×Q₈×C₅₃	—	4	1	no	—
78	C₂×Q₈×C₅₃	—	4	1	no	—
79	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
103	C₄⋊C₄×C₅₃	—	2	4	no	—
104	C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₄⋊C₄×C₅₃	—	2	4	no	—
106	C₄⋊C₄×C₅₃	—	4	2	no	—
107	C₄⋊C₄×C₅₃	—	4	1	no	—
108	C₄⋊C₄×C₅₃	—	4	1	no	—
109	C₄⋊C₄×C₅₃	—	4	2	no	—
110	C₄⋊C₄×C₅₃	—	4	2	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	—	2	2	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	—	2	2	no	—
119	C₄²×C₅₃	—	4	1	no	—
120	C₄²×C₅₃	—	4	1	no	—
121	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
122	C₄²×C₅₃	#41	1	2	yes	—
123	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	Q₈×C₅₃	—	4	4	no	—
125	Q₈×C₅₃	—	4	4	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	2	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	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	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	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
152	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
153	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
155	C₂×C₄×C₅₃	#131	1	2	yes	—
156	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
160	C₈×C₅₃	—	2	4	no	—
161	C₈×C₅₃	—	4	2	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	—	4	2	no	—
164	#22315	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₂²×C₅₃	#518	1	1	yes	characteristic
167	C₂²×C₅₃	#519	1	1	yes	characteristic
168	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
169	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
170	C₂²×C₅₃	#568	1	1	yes	characteristic
171	C₂²×C₅₃	#568	1	1	yes	characteristic
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	—	2	4	no	—
175	C₄×C₅₃	—	2	2	no	—
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	—	4	1	no	—
181	C₄×C₅₃	—	4	1	no	—
182	C₄×C₅₃	#542	1	2	yes	—
183	C₄×C₅₃	#544	1	2	yes	—
184	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
185	#1985	C₂×C₅₃	1	2	yes	—
186	#1984	C₂×C₅₃	1	2	yes	—
187	#1940	C₂×C₅₃	1	2	yes	—
188	#1917	C₂×C₅₃	1	2	yes	—
189	#1916	C₂×C₅₃	1	2	yes	—
190	#1831	C₂×C₅₃	1	1	yes	characteristic
191	#1771	C₂×C₅₃	1	2	yes	—
192	#1745	C₂×C₅₃	1	1	yes	characteristic
193	C₂×C₅₃	#2331	1	1	yes	characteristic
194	C₂×C₅₃	#2962	1	1	yes	characteristic
195	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
196	C₂×C₅₃	#2973	1	1	yes	characteristic
197	C₂×C₅₃	#2981	1	1	yes	characteristic
198	C₂×C₅₃	#2984	1	1	yes	characteristic
199	C₂×C₅₃	#2987	1	1	yes	characteristic
200	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
203	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
204	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
206	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
207	C₂×#97	C₂²×C₅₃	1	2	yes	—
208	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
210	#391	C₄×C₅₃	1	1	yes	characteristic
211	#391	C₄×C₅₃	1	1	yes	characteristic
212	#389	C₄×C₅₃	1	2	yes	—
213	#389	C₄×C₅₃	1	2	yes	—
214	#384	C₄×C₅₃	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
216	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
217	C₄³	C₂²×C₅₃	1	1	yes	characteristic
218	#336	C₂²×C₅₃	1	1	yes	characteristic
219	#336	C₂²×C₅₃	1	1	yes	characteristic
220	#336	C₂²×C₅₃	1	2	yes	—
221	#336	C₂²×C₅₃	1	2	yes	—
222	#336	C₂²×C₅₃	1	2	yes	—
223	#336	C₂²×C₅₃	1	2	yes	—
224	#336	C₂²×C₅₃	1	2	yes	—
225	#336	C₂²×C₅₃	1	2	yes	—
226	#336	C₂²×C₅₃	1	2	yes	—
227	C₅₃	#22315	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
231	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
232	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
233	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
234	C₄⋊Q₈	—	2	4	no	—
235	C₄×Q₈	—	4	2	no	—
236	C₄×Q₈	—	4	2	no	—
237	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
238	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
239	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
240	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
244	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
245	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
251	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
252	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
253	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
254	C₄⋊C₈	—	2	4	no	—
255	C₄⋊C₈	—	4	2	no	—
256	C₄⋊C₈	—	4	2	no	—
257	#97	—	2	2	no	—
258	#97	—	4	2	no	—
259	#97	Q₈×C₅₃	1	4	yes	complemented
260	C₄×C₈	—	4	2	no	—
261	#95	C₂×C₄×C₅₃	1	2	yes	—
262	#95	C₂×C₄×C₅₃	1	2	yes	—
263	#95	C₂×C₄×C₅₃	1	2	yes	—
264	#95	C₂×C₄×C₅₃	1	2	yes	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
271	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
273	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	4	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
313	C₄²	#41×C₅₃	1	2	yes	—
314	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
315	Q₈	—	4	4	no	—
316	Q₈	—	4	4	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	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	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
343	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
344	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
345	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
346	C₂×C₄	#131×C₅₃	1	2	yes	—
347	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
349	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
350	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
351	C₈	—	2	4	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
356	C₂²	C₅₃×#518	1	1	yes	characteristic
357	C₂²	C₅₃×#519	1	1	yes	characteristic
358	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
359	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
360	C₂²	C₅₃×#568	1	1	yes	characteristic
361	C₂²	C₅₃×#568	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	4	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	1	no	—
372	C₄	C₅₃×#542	1	2	yes	—
373	C₄	C₅₃×#544	1	2	yes	—
374	C₂	C₅₃×#2331	1	1	yes	characteristic
375	C₂	C₅₃×#2962	1	1	yes	characteristic
376	C₂	Q₈×M₁₆×C₅₃	1	1	yes	characteristic
377	C₂	C₅₃×#2973	1	1	yes	characteristic
378	C₂	C₅₃×#2981	1	1	yes	characteristic
379	C₂	C₅₃×#2984	1	1	yes	characteristic
380	C₂	C₅₃×#2987	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
68. #97×C₅₃	Q₈	semidirect
164. #22315	C₅₃	direct (N × H)
227. C₅₃	#22315	direct (N × H)
259. #97	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1985	2
C₅₃×#1984	2
C₅₃×#1940	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1831	2
C₅₃×#1771	2
C₅₃×#1745	2
#22315	53

Subgroups — C53×#22315 #27229827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22315 #27229827