Subgroups — C₅₃×#21311 #27228823

Order 13568.

1194 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1989	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1988	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1987	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1735	C₂	1	4	yes	maximal
8	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#533	—	2	4	no	—
10	C₅₃×#533	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#125×C₅₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
16	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
17	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
18	C₂×#117×C₅₃	C₂²	1	2	yes	—
19	C₂×#117×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₅₃	—	2	4	no	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
27	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
28	C₅₃×#376	—	2	1	no	—
29	C₄³×C₅₃	—	2	1	no	—
30	C₅₃×#340	C₂²	1	8	yes	—
31	C₅₃×#340	C₂²	1	8	yes	—
32	C₂²×Q₈×C₅₃	—	2	1	no	—
33	C₂²×Q₈×C₅₃	—	2	1	no	—
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	—	2	4	no	—
38	C₂²×C₈×C₅₃	—	2	8	no	—
39	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₅₃	—	2	2	no	—
41	C₄⋊Q₈×C₅₃	—	2	2	no	—
42	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
44	#125×C₅₃	—	2	4	no	—
45	#125×C₅₃	C₂×C₄	1	8	yes	—
46	C₂²⋊Q₈×C₅₃	—	2	8	no	—
47	C₂²⋊Q₈×C₅₃	—	2	8	no	—
48	C₄×Q₈×C₅₃	—	2	8	no	—
49	C₄×Q₈×C₅₃	—	2	8	no	—
50	#117×C₅₃	—	2	4	no	—
51	#117×C₅₃	—	2	4	no	—
52	#117×C₅₃	C₂×C₄	1	8	yes	—
53	#117×C₅₃	Q₈	1	8	yes	complemented
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	—	2	2	no	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
68	C₂×C₄²×C₅₃	D₄	1	4	yes	—
69	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₃	—	2	4	no	—
74	C₄⋊C₈×C₅₃	—	2	4	no	—
75	#95×C₅₃	—	2	2	no	—
76	#95×C₅₃	—	2	2	no	—
77	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×Q₈×C₅₃	—	2	4	no	—
79	C₂×Q₈×C₅₃	—	2	4	no	—
80	C₂×Q₈×C₅₃	—	4	4	no	—
81	C₂×Q₈×C₅₃	—	4	4	no	—
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	—	2	4	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
88	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
97	C₂²×C₄×C₅₃	#41	1	4	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	—	2	8	no	—
101	C₂×C₈×C₅₃	—	4	8	no	—
102	C₄⋊C₄×C₅₃	—	2	8	no	—
103	C₄⋊C₄×C₅₃	—	2	8	no	—
104	C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₄⋊C₄×C₅₃	—	2	4	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
110	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
111	C₄⋊C₄×C₅₃	C₂×Q₈	1	8	yes	—
112	C₂²⋊C₄×C₅₃	—	2	8	no	—
113	C₂²⋊C₄×C₅₃	—	2	8	no	—
114	C₄²×C₅₃	—	2	8	no	—
115	C₄²×C₅₃	—	2	8	no	—
116	C₄²×C₅₃	—	4	4	no	—
117	C₄²×C₅₃	—	4	4	no	—
118	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
119	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
120	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
121	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄²×C₅₃	C₂×D₄	1	2	yes	—
123	C₄²×C₅₃	C₂×D₄	1	2	yes	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂³×C₅₃	C₂²⋊Q₈	1	4	yes	—
128	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₅₃	—	4	8	no	—
130	Q₈×C₅₃	—	4	8	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	8	no	—
135	C₂×C₄×C₅₃	—	2	4	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	8	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	—	4	4	no	—
140	C₂×C₄×C₅₃	—	4	4	no	—
141	C₂×C₄×C₅₃	—	4	4	no	—
142	C₂×C₄×C₅₃	—	4	4	no	—
143	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
146	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
147	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
149	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
150	C₂×C₄×C₅₃	#124	1	4	yes	—
151	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
152	C₂×C₄×C₅₃	#135	1	4	yes	—
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
157	C₈×C₅₃	—	4	4	no	—
158	#21311	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂²×C₅₃	—	2	8	no	—
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	#391	1	4	yes	—
163	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
165	C₂²×C₅₃	C₂×#135	1	2	yes	—
166	C₂²×C₅₃	#575	1	2	yes	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	4	8	no	—
171	C₄×C₅₃	—	4	4	no	—
172	C₄×C₅₃	—	4	4	no	—
173	C₄×C₅₃	#391	1	4	yes	—
174	C₄×C₅₃	#416	1	4	yes	—
175	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
176	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
177	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
178	C₂×#533	C₂×C₅₃	1	1	yes	characteristic
179	C₂²×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
180	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
181	#1989	C₂×C₅₃	1	4	yes	complemented
182	#1988	C₂×C₅₃	1	2	yes	complemented
183	#1987	C₂×C₅₃	1	2	yes	complemented
184	#1735	C₂×C₅₃	1	4	yes	—
185	C₂×C₅₃	—	2	4	no	—
186	C₂×C₅₃	C₂×#391	1	1	yes	characteristic
187	C₂×C₅₃	C₂×#416	1	1	yes	characteristic
188	C₂×C₅₃	#2896	1	1	yes	characteristic
189	C₂×C₅₃	#3076	1	4	yes	—
190	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
191	#533	—	2	4	no	—
192	#533	C₄×C₅₃	1	8	yes	complemented
193	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
195	C₂×#125	C₂²×C₅₃	1	2	yes	—
196	C₂×C₂²⋊Q₈	—	2	1	no	—
197	C₂×C₂²⋊Q₈	—	2	1	no	—
198	C₂×C₄×Q₈	—	2	1	no	—
199	C₂×C₄×Q₈	—	2	1	no	—
200	C₂×#117	C₂²×C₅₃	1	2	yes	—
201	C₂×#117	C₂²×C₅₃	1	2	yes	—
202	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
203	C₂×C₄⋊C₈	—	2	4	no	—
204	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
206	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
207	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
208	C₄×C₂²⋊C₄	—	2	1	no	—
209	C₄×C₂²⋊C₄	—	2	1	no	—
210	#376	—	2	1	no	—
211	C₄³	—	2	1	no	—
212	#340	C₂²×C₅₃	1	8	yes	—
213	#340	C₂²×C₅₃	1	8	yes	—
214	C₅₃	#21311	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×Q₈	—	2	1	no	—
217	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
218	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
219	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	2	8	no	—
222	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	2	2	no	—
225	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
226	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
227	#125	—	2	4	no	—
228	#125	C₂×C₄×C₅₃	1	8	yes	—
229	C₂²⋊Q₈	—	2	8	no	—
230	C₂²⋊Q₈	—	2	8	no	—
231	C₄×Q₈	—	2	8	no	—
232	C₄×Q₈	—	2	8	no	—
233	#117	—	2	4	no	—
234	#117	—	2	4	no	—
235	#117	C₂×C₄×C₅₃	1	8	yes	—
236	#117	Q₈×C₅₃	1	8	yes	complemented
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
242	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
244	C₂×C₂²⋊C₄	—	2	2	no	—
245	C₂×C₂²⋊C₄	—	2	2	no	—
246	C₂×C₄²	—	2	2	no	—
247	C₂×C₄²	—	2	2	no	—
248	C₂×C₄²	—	2	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
251	C₂×C₄²	D₄×C₅₃	1	4	yes	—
252	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
256	C₄⋊C₈	—	2	4	no	—
257	C₄⋊C₈	—	2	4	no	—
258	#95	—	2	2	no	—
259	#95	—	2	2	no	—
260	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
261	C₂×Q₈	—	2	4	no	—
262	C₂×Q₈	—	2	4	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
271	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
272	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
279	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
280	C₂²×C₄	#41×C₅₃	1	4	yes	—
281	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	8	no	—
284	C₂×C₈	—	4	8	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
293	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
294	C₄⋊C₄	C₂×Q₈×C₅₃	1	8	yes	—
295	C₂²⋊C₄	—	2	8	no	—
296	C₂²⋊C₄	—	2	8	no	—
297	C₄²	—	2	8	no	—
298	C₄²	—	2	8	no	—
299	C₄²	—	4	4	no	—
300	C₄²	—	4	4	no	—
301	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
302	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
303	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
304	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
305	C₄²	C₂×D₄×C₅₃	1	2	yes	—
306	C₄²	C₂×D₄×C₅₃	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
310	C₂³	C₂²⋊Q₈×C₅₃	1	4	yes	—
311	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
312	Q₈	—	4	8	no	—
313	Q₈	—	4	8	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
327	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
328	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
329	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
331	C₂×C₄	C₄×Q₈×C₅₃	1	8	yes	—
332	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
333	C₂×C₄	#124×C₅₃	1	4	yes	—
334	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
335	C₂×C₄	#135×C₅₃	1	4	yes	—
336	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
339	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
340	C₈	—	4	4	no	—
341	C₂²	—	2	8	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	C₅₃×#391	1	4	yes	—
345	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
346	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
347	C₂²	C₂×#135×C₅₃	1	2	yes	—
348	C₂²	C₅₃×#575	1	2	yes	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	4	8	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	4	no	—
355	C₄	C₅₃×#391	1	4	yes	—
356	C₄	C₅₃×#416	1	4	yes	—
357	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
358	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
359	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
360	C₂	—	2	4	no	—
361	C₂	C₂×C₅₃×#391	1	1	yes	characteristic
362	C₂	C₂×C₅₃×#416	1	1	yes	characteristic
363	C₂	C₅₃×#2896	1	1	yes	characteristic
364	C₂	C₅₃×#3076	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1989	C₂	semidirect
5. C₅₃×#1988	C₂	semidirect
6. C₅₃×#1987	C₂	semidirect
10. C₅₃×#533	C₄	semidirect
53. #117×C₅₃	Q₈	semidirect
158. #21311	C₅₃	direct (N × H)
181. #1989	C₂×C₅₃	semidirect
182. #1988	C₂×C₅₃	semidirect
183. #1987	C₂×C₅₃	semidirect
192. #533	C₄×C₅₃	semidirect
214. C₅₃	#21311	direct (N × H)
236. #117	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×#533	2
C₂²×C₄⋊C₈×C₅₃	2
C₄×#117×C₅₃	2
C₅₃×#1989	2
C₅₃×#1988	2
C₅₃×#1987	2
C₅₃×#1735	2
#21311	53

Subgroups — C53×#21311 #27228823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21311 #27228823