Subgroups — C₅₃×#22575 #27230087

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#103×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2345	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1933	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1928	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1863	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1824	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1787	C₂	1	1	yes	maximal, characteristic
10	C₄×#103×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#103×C₅₃	C₄	1	8	yes	complemented
18	C₂×#103×C₅₃	C₂²	1	2	yes	—
19	C₂×#103×C₅₃	C₂²	1	2	yes	—
20	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#391	—	2	2	no	—
27	C₅₃×#391	C₂²	1	2	yes	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#387	—	2	2	no	—
30	C₅₃×#380	—	2	1	no	—
31	C₅₃×#380	—	2	1	no	—
32	C₅₃×#380	C₂²	1	1	yes	characteristic
33	C₅₃×#380	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
37	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×Q₈×C₅₃	—	2	2	no	—
44	C₂²×Q₈×C₅₃	—	2	4	no	—
45	C₂²×Q₈×C₅₃	—	4	2	no	—
46	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	—	2	1	no	—
50	C₂³×C₄×C₅₃	—	2	1	no	—
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	—	4	2	no	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₅₃	—	2	4	no	—
58	C₄×Q₈×C₅₃	—	4	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	—	2	2	no	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	#107×C₅₃	—	4	2	no	—
76	#103×C₅₃	—	2	4	no	—
77	#103×C₅₃	—	4	2	no	—
78	#103×C₅₃	C₂×C₄	1	4	yes	—
79	#103×C₅₃	C₂×C₄	1	4	yes	—
80	#103×C₅₃	C₂×C₄	1	4	yes	—
81	#103×C₅₃	C₂×C₄	1	4	yes	—
82	C₂²⋊C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	4	2	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	—	2	2	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	1	no	—
89	#95×C₅₃	—	2	1	no	—
90	#95×C₅₃	D₄	1	2	yes	—
91	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₅₃	—	2	4	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	2	2	no	—
96	C₂×Q₈×C₅₃	—	2	8	no	—
97	C₂×Q₈×C₅₃	—	2	2	no	—
98	C₂×Q₈×C₅₃	—	2	2	no	—
99	C₂×Q₈×C₅₃	—	4	4	no	—
100	C₂×Q₈×C₅₃	—	4	4	no	—
101	C₂×Q₈×C₅₃	—	4	8	no	—
102	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂×Q₈×C₅₃	#41	1	4	yes	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	4	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	4	4	no	—
116	C₂²×C₄×C₅₃	—	8	2	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₅₃	#41	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	4	4	no	—
129	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	4	4	no	—
139	C₄⋊C₄×C₅₃	—	8	2	no	—
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₂²⋊C₄×C₅₃	—	4	2	no	—
144	C₂²⋊C₄×C₅₃	—	4	2	no	—
145	C₂²⋊C₄×C₅₃	Q₁₆	1	4	yes	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	2	no	—
148	C₄²×C₅₃	—	4	4	no	—
149	C₄²×C₅₃	—	8	2	no	—
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	C₄×D₄	1	2	yes	—
153	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
154	C₂³×C₅₃	#135	1	2	yes	—
155	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₅₃	—	2	2	no	—
157	Q₈×C₅₃	—	2	4	no	—
158	Q₈×C₅₃	—	2	2	no	—
159	Q₈×C₅₃	—	2	2	no	—
160	Q₈×C₅₃	—	4	8	no	—
161	Q₈×C₅₃	—	4	2	no	—
162	Q₈×C₅₃	#117	1	4	yes	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	4	8	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	4	2	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	8	4	no	—
188	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
189	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
190	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
194	C₈×C₅₃	—	2	2	no	—
195	C₈×C₅₃	—	2	2	no	—
196	C₈×C₅₃	—	4	2	no	—
197	#22575	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂²×C₅₃	—	2	2	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	2	4	no	—
201	C₂²×C₅₃	—	2	4	no	—
202	C₂²×C₅₃	C₄×Q₁₆	1	4	yes	—
203	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₅₃	#546	1	2	yes	—
205	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
206	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
207	C₂²×C₅₃	#577	1	2	yes	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	4	no	—
210	C₄×C₅₃	—	2	4	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	2	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	8	2	no	—
216	C₄×C₅₃	—	8	2	no	—
217	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
218	C₄×C₅₃	#518	1	1	yes	characteristic
219	C₄×C₅₃	#551	1	2	yes	—
220	C₂²×#103	C₂×C₅₃	1	1	yes	characteristic
221	#2345	C₂×C₅₃	1	1	yes	characteristic
222	Q₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
223	#1933	C₂×C₅₃	1	2	yes	complemented
224	#1928	C₂×C₅₃	1	2	yes	complemented
225	#1863	C₂×C₅₃	1	2	yes	complemented
226	#1824	C₂×C₅₃	1	1	yes	characteristic
227	#1808	C₂×C₅₃	1	1	yes	characteristic
228	#1787	C₂×C₅₃	1	1	yes	characteristic
229	C₄×#103	C₂×C₅₃	1	2	yes	complemented
230	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
231	C₂×C₅₃	—	2	4	no	—
232	C₂×C₅₃	#2342	1	1	yes	characteristic
233	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
234	C₂×C₅₃	#2949	1	1	yes	characteristic
235	C₂×C₅₃	#3299	1	2	yes	—
236	C₂×C₅₃	#3301	1	2	yes	—
237	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
238	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
240	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×#107	C₂²×C₅₃	1	2	yes	—
242	C₂×#103	C₄×C₅₃	1	8	yes	complemented
243	C₂×#103	C₂²×C₅₃	1	2	yes	—
244	C₂×#103	C₂²×C₅₃	1	2	yes	—
245	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
251	#391	—	2	2	no	—
252	#391	C₂²×C₅₃	1	2	yes	—
253	#389	C₂²×C₅₃	1	2	yes	—
254	#387	—	2	2	no	—
255	#380	—	2	1	no	—
256	#380	—	2	1	no	—
257	#380	C₂²×C₅₃	1	1	yes	characteristic
258	#380	C₂²×C₅₃	1	1	yes	characteristic
259	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
260	C₄×C₂²⋊C₄	—	2	1	no	—
261	C₄×C₂²⋊C₄	—	2	1	no	—
262	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	#340	C₂²×C₅₃	1	2	yes	—
264	#340	C₂²×C₅₃	1	2	yes	—
265	#340	C₂²×C₅₃	1	2	yes	—
266	#336	C₂²×C₅₃	1	2	yes	—
267	C₅₃	#22575	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²×Q₈	—	2	1	no	—
269	C₂²×Q₈	—	2	2	no	—
270	C₂²×Q₈	—	2	4	no	—
271	C₂²×Q₈	—	4	2	no	—
272	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
273	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
274	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₂³×C₄	—	2	1	no	—
276	C₂³×C₄	—	2	1	no	—
277	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
278	C₂²×C₈	—	4	2	no	—
279	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
280	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
281	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
283	C₄×Q₈	—	2	4	no	—
284	C₄×Q₈	—	4	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
301	#107	—	4	2	no	—
302	#103	—	2	4	no	—
303	#103	—	4	2	no	—
304	#103	C₂×C₄×C₅₃	1	4	yes	—
305	#103	C₂×C₄×C₅₃	1	4	yes	—
306	#103	C₂×C₄×C₅₃	1	4	yes	—
307	#103	C₂×C₄×C₅₃	1	4	yes	—
308	C₂²⋊C₈	—	2	2	no	—
309	C₄×C₈	—	4	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	—	2	1	no	—
315	#95	—	2	1	no	—
316	#95	D₄×C₅₃	1	2	yes	—
317	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂×Q₈	—	2	4	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	8	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	—	4	4	no	—
326	C₂×Q₈	—	4	4	no	—
327	C₂×Q₈	—	4	8	no	—
328	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
329	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
330	C₂×Q₈	#41×C₅₃	1	4	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	4	4	no	—
342	C₂²×C₄	—	8	2	no	—
343	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
347	C₂²×C₄	#41×C₅₃	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	4	4	no	—
355	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
356	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
357	C₄⋊C₄	—	2	4	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	4	4	no	—
365	C₄⋊C₄	—	8	2	no	—
366	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
367	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
368	C₂²⋊C₄	—	2	2	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₂²⋊C₄	Q₁₆×C₅₃	1	4	yes	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	4	4	no	—
375	C₄²	—	8	2	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	2	no	—
378	C₂³	C₄×D₄×C₅₃	1	2	yes	—
379	C₂³	C₂×Q₁₆×C₅₃	1	2	yes	—
380	C₂³	#135×C₅₃	1	2	yes	—
381	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	4	no	—
384	Q₈	—	2	2	no	—
385	Q₈	—	2	2	no	—
386	Q₈	—	4	8	no	—
387	Q₈	—	4	2	no	—
388	Q₈	#117×C₅₃	1	4	yes	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	8	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	8	4	no	—
414	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
415	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
416	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
418	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
419	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
420	C₈	—	2	2	no	—
421	C₈	—	2	2	no	—
422	C₈	—	4	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	4	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	2	2	no	—
427	C₂²	C₄×Q₁₆×C₅₃	1	4	yes	—
428	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
429	C₂²	C₅₃×#546	1	2	yes	—
430	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
431	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
432	C₂²	C₅₃×#577	1	2	yes	—
433	C₄	—	2	4	no	—
434	C₄	—	2	2	no	—
435	C₄	—	2	4	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	2	no	—
442	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
443	C₄	C₅₃×#518	1	1	yes	characteristic
444	C₄	C₅₃×#551	1	2	yes	—
445	C₂	—	2	4	no	—
446	C₂	C₅₃×#2342	1	1	yes	characteristic
447	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
448	C₂	C₅₃×#2949	1	1	yes	characteristic
449	C₂	C₅₃×#3299	1	2	yes	—
450	C₂	C₅₃×#3301	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1933	C₂	semidirect
5. C₅₃×#1928	C₂	semidirect
6. C₅₃×#1863	C₂	semidirect
10. C₄×#103×C₅₃	C₂	semidirect
17. C₂×#103×C₅₃	C₄	semidirect
197. #22575	C₅₃	direct (N × H)
223. #1933	C₂×C₅₃	semidirect
224. #1928	C₂×C₅₃	semidirect
225. #1863	C₂×C₅₃	semidirect
229. C₄×#103	C₂×C₅₃	semidirect
242. C₂×#103	C₄×C₅₃	semidirect
267. C₅₃	#22575	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#103×C₅₃	2
C₅₃×#2345	2
Q₈×C₂²⋊C₄×C₅₃	2
C₅₃×#1933	2
C₅₃×#1928	2
C₅₃×#1863	2
C₅₃×#1824	2
C₅₃×#1808	2
C₅₃×#1787	2
C₄×#103×C₅₃	2
C₈×C₂²⋊C₄×C₅₃	2
#22575	53

Subgroups — C53×#22575 #27230087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22575 #27230087