Subgroups — C₅₃×#21117 #27228629

Order 13568.

978 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1927	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1920	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1877	C₂	1	2	yes	maximal
7	C₅₃×#1864	C₂	1	2	yes	maximal
8	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#103×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	—	2	1	no	—
17	C₄×SD₁₆×C₅₃	C₄	1	8	yes	complemented
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#391	C₂²	1	2	yes	—
27	C₅₃×#389	C₂²	1	2	yes	—
28	C₅₃×#387	—	2	2	no	—
29	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
34	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
35	C₅₃×#376	—	2	1	no	—
36	C₄³×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	2	no	—
45	C₂³×C₄×C₅₃	—	2	1	no	—
46	C₂×SD₁₆×C₅₃	—	2	4	no	—
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₄×D₄×C₅₃	—	2	4	no	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₄×D₄×C₅₃	—	2	4	no	—
56	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄²×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	—	2	1	no	—
67	C₂×C₄²×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	—	2	1	no	—
69	C₂×C₄²×C₅₃	—	4	4	no	—
70	C₂×C₄²×C₅₃	D₄	1	2	yes	—
71	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	#106×C₅₃	—	2	4	no	—
76	#106×C₅₃	C₂×C₄	1	4	yes	complemented
77	#103×C₅₃	—	2	4	no	—
78	#103×C₅₃	C₂×C₄	1	4	yes	complemented
79	#102×C₅₃	—	2	4	no	—
80	#102×C₅₃	C₂×C₄	1	4	yes	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	—	2	2	no	—
86	#95×C₅₃	—	2	1	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	1	no	—
89	C₂⁴×C₅₃	—	2	1	no	—
90	C₂×Q₈×C₅₃	—	2	1	no	—
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	1	no	—
96	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₅₃	#41	1	4	yes	—
98	C₂²×C₄×C₅₃	—	2	8	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	1	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	4	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	4	4	no	—
106	C₂²×C₄×C₅₃	—	4	4	no	—
107	C₂²×C₄×C₅₃	—	4	2	no	—
108	C₂²×C₄×C₅₃	—	4	2	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	SD₁₆×C₅₃	—	2	8	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	—	2	2	no	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₅₃	—	2	4	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	2	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	—	4	4	no	—
129	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
130	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
131	C₄⋊C₄×C₅₃	#41	1	4	yes	—
132	C₂²⋊C₄×C₅₃	—	2	8	no	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₄²×C₅₃	—	2	4	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	4	2	no	—
139	C₄²×C₅₃	—	4	4	no	—
140	C₄²×C₅₃	—	4	2	no	—
141	C₄²×C₅₃	—	4	2	no	—
142	C₄²×C₅₃	SD₁₆	1	4	yes	complemented
143	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
144	C₄²×C₅₃	#41	1	2	yes	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	—	2	4	no	—
147	C₂³×C₅₃	—	4	1	no	—
148	C₂³×C₅₃	—	4	1	no	—
149	C₂³×C₅₃	—	4	2	no	—
150	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₅₃	—	2	2	no	—
152	Q₈×C₅₃	—	2	2	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	D₄×C₅₃	—	2	2	no	—
155	D₄×C₅₃	—	2	4	no	—
156	D₄×C₅₃	#117	1	4	yes	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	4	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	4	16	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	8	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
180	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
181	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
182	C₂×C₄×C₅₃	#135	1	2	yes	—
183	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
185	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
186	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₈×C₅₃	—	2	2	no	—
188	C₈×C₅₃	—	2	2	no	—
189	C₈×C₅₃	—	4	2	no	—
190	#21117	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂²×C₅₃	—	2	4	no	—
192	C₂²×C₅₃	—	4	4	no	—
193	C₂²×C₅₃	—	4	4	no	—
194	C₂²×C₅₃	—	4	2	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₅₃	#525	1	2	yes	—
198	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
199	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
200	C₂²×C₅₃	#575	1	2	yes	—
201	C₄×C₅₃	—	2	4	no	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	4	4	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	4	2	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	8	2	no	—
209	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
210	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
211	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
212	C₄×C₅₃	#529	1	2	yes	—
213	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
214	#2295	C₂×C₅₃	1	1	yes	characteristic, complemented
215	C₄²×D₄	C₂×C₅₃	1	1	yes	characteristic
216	#1927	C₂×C₅₃	1	2	yes	complemented
217	#1920	C₂×C₅₃	1	2	yes	complemented
218	#1877	C₂×C₅₃	1	2	yes	—
219	#1864	C₂×C₅₃	1	2	yes	—
220	C₄×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
221	C₄×#103	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
223	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×C₅₃	—	4	4	no	—
225	C₂×C₅₃	#2290	1	1	yes	characteristic
226	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
227	C₂×C₅₃	#2943	1	1	yes	characteristic
228	C₂×C₅₃	#3101	1	2	yes	—
229	C₂×C₅₃	#3102	1	2	yes	—
230	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
233	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
234	C₂²×C₄²	—	2	1	no	—
235	C₄×SD₁₆	C₄×C₅₃	1	8	yes	complemented
236	C₂×#106	C₂²×C₅₃	1	2	yes	—
237	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
238	C₂×#103	C₂²×C₅₃	1	2	yes	—
239	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×#102	C₂²×C₅₃	1	2	yes	—
241	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
244	#391	C₂²×C₅₃	1	2	yes	—
245	#389	C₂²×C₅₃	1	2	yes	—
246	#387	—	2	2	no	—
247	C₄×C₄⋊C₄	—	2	1	no	—
248	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
250	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
251	C₄×C₂²⋊C₄	—	2	1	no	—
252	C₄×C₂²⋊C₄	—	2	2	no	—
253	#376	—	2	1	no	—
254	C₄³	C₂²×C₅₃	1	1	yes	characteristic
255	#340	C₂²×C₅₃	1	2	yes	—
256	#340	C₂²×C₅₃	1	2	yes	—
257	#340	C₂²×C₅₃	1	2	yes	—
258	#336	C₂²×C₅₃	1	2	yes	—
259	#336	C₂²×C₅₃	1	2	yes	—
260	C₅₃	#21117	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
262	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
263	C₂³×C₄	—	2	2	no	—
264	C₂³×C₄	—	2	1	no	—
265	C₂×SD₁₆	—	2	4	no	—
266	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
267	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
268	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
270	C₄×Q₈	—	2	2	no	—
271	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
272	C₄×D₄	—	2	4	no	—
273	C₄×D₄	—	2	4	no	—
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	1	no	—
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	—	4	4	no	—
289	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
290	C₂×C₄²	D₄×C₅₃	1	2	yes	—
291	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
293	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
294	#106	—	2	4	no	—
295	#106	C₂×C₄×C₅₃	1	4	yes	complemented
296	#103	—	2	4	no	—
297	#103	C₂×C₄×C₅₃	1	4	yes	complemented
298	#102	—	2	4	no	—
299	#102	C₂×C₄×C₅₃	1	4	yes	—
300	C₄×C₈	—	2	2	no	—
301	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	1	no	—
306	#95	—	2	2	no	—
307	#95	—	2	1	no	—
308	C₂⁴	—	2	1	no	—
309	C₂×Q₈	—	2	1	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
316	C₂×D₄	#41×C₅₃	1	4	yes	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	4	4	no	—
325	C₂²×C₄	—	4	4	no	—
326	C₂²×C₄	—	4	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
329	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
333	SD₁₆	—	2	8	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
340	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
341	C₄⋊C₄	—	2	4	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
349	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
350	C₄⋊C₄	#41×C₅₃	1	4	yes	—
351	C₂²⋊C₄	—	2	8	no	—
352	C₂²⋊C₄	—	2	4	no	—
353	C₄²	—	2	4	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	4	2	no	—
358	C₄²	—	4	4	no	—
359	C₄²	—	4	2	no	—
360	C₄²	—	4	2	no	—
361	C₄²	SD₁₆×C₅₃	1	4	yes	complemented
362	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
363	C₄²	#41×C₅₃	1	2	yes	—
364	C₂³	—	2	2	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	2	no	—
369	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
370	Q₈	—	2	2	no	—
371	Q₈	—	2	2	no	—
372	Q₈	—	4	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	2	4	no	—
375	D₄	#117×C₅₃	1	4	yes	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	16	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	8	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
398	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
399	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
400	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
401	C₂×C₄	#135×C₅₃	1	2	yes	—
402	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
404	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
405	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	4	no	—
413	C₂²	—	4	4	no	—
414	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
415	C₂²	C₅₃×#525	1	2	yes	—
416	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
417	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
418	C₂²	C₅₃×#575	1	2	yes	—
419	C₄	—	2	2	no	—
420	C₄	—	2	4	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	2	no	—
424	C₄	—	4	4	no	—
425	C₄	—	4	2	no	—
426	C₄	—	8	2	no	—
427	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
428	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
429	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
430	C₄	C₅₃×#529	1	2	yes	—
431	C₂	—	4	4	no	—
432	C₂	C₅₃×#2290	1	1	yes	characteristic
433	C₂	C₂×C₄×SD₁₆×C₅₃	1	1	yes	characteristic
434	C₂	C₅₃×#2943	1	1	yes	characteristic
435	C₂	C₅₃×#3101	1	2	yes	—
436	C₂	C₅₃×#3102	1	2	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	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2295	C₂	semidirect
4. C₅₃×#1927	C₂	semidirect
5. C₅₃×#1920	C₂	semidirect
8. C₄×#106×C₅₃	C₂	semidirect
9. C₄×#103×C₅₃	C₂	semidirect
11. C₅₃×#1744	C₂	semidirect
17. C₄×SD₁₆×C₅₃	C₄	semidirect
52. C₄×Q₈×C₅₃	C₂×C₄	semidirect
76. #106×C₅₃	C₂×C₄	semidirect
78. #103×C₅₃	C₂×C₄	semidirect
82. C₄×C₈×C₅₃	C₂×C₄	semidirect
142. C₄²×C₅₃	SD₁₆	semidirect
190. #21117	C₅₃	direct (N × H)
214. #2295	C₂×C₅₃	semidirect
216. #1927	C₂×C₅₃	semidirect
217. #1920	C₂×C₅₃	semidirect
220. C₄×#106	C₂×C₅₃	semidirect
221. C₄×#103	C₂×C₅₃	semidirect
223. #1744	C₂×C₅₃	semidirect
235. C₄×SD₁₆	C₄×C₅₃	semidirect
260. C₅₃	#21117	direct (N × H)
271. C₄×Q₈	C₂×C₄×C₅₃	semidirect
295. #106	C₂×C₄×C₅₃	semidirect
297. #103	C₂×C₄×C₅₃	semidirect
301. C₄×C₈	C₂×C₄×C₅₃	semidirect
361. C₄²	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₅₃	2
C₅₃×#2295	2
C₄²×D₄×C₅₃	2
C₅₃×#1927	2
C₅₃×#1920	2
C₅₃×#1877	2
C₅₃×#1864	2
C₄×#106×C₅₃	2
C₄×#103×C₅₃	2
C₄×#102×C₅₃	2
C₅₃×#1744	2
#21117	53

Subgroups — C53×#21117 #27228629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21117 #27228629