Subgroups — C₅₃×#36624 #27244136

Order 13568.

3902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#525	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2857	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2856	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2854	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2850	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2847	C₂	1	4	yes	maximal, complemented
9	C₄×#121×C₅₃	C₂	1	8	yes	maximal, complemented
10	C₅₃×#2297	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₅₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#532	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
18	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
19	C₅₃×#525	—	2	16	no	—
20	C₂×#123×C₅₃	C₂²	1	4	yes	—
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
22	C₂×#121×C₅₃	C₂²	1	8	yes	—
23	C₂×#121×C₅₃	C₂²	1	4	yes	—
24	C₂×#121×C₅₃	C₂²	1	4	yes	—
25	C₂×#120×C₅₃	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
27	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
28	C₂×#117×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅₃	—	2	8	no	—
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
32	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
33	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#391	C₂²	1	4	yes	complemented
37	C₅₃×#390	C₂²	1	4	yes	complemented
38	C₅₃×#390	C₂²	1	8	yes	complemented
39	C₅₃×#389	C₂²	1	8	yes	complemented
40	C₅₃×#388	C₂²	1	16	yes	complemented
41	C₅₃×#386	C₂²	1	8	yes	—
42	C₅₃×#386	C₂²	1	16	yes	—
43	C₅₃×#384	C₂²	1	2	yes	complemented
44	C₅₃×#382	C₂²	1	2	yes	complemented
45	C₅₃×#380	C₂²	1	4	yes	—
46	C₅₃×#380	C₂²	1	8	yes	—
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
49	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×#41×C₅₃	—	2	8	no	—
51	C₂×#41×C₅₃	—	2	8	no	—
52	C₂×#41×C₅₃	—	2	8	no	—
53	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
55	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
58	C₂³×C₄×C₅₃	—	2	8	no	—
59	C₂³×C₄×C₅₃	—	2	4	no	—
60	C₂³×C₄×C₅₃	—	2	4	no	—
61	C₂³×C₄×C₅₃	—	2	16	no	—
62	C₂³×C₄×C₅₃	D₄	1	8	yes	complemented
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
68	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
71	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₅₃	—	2	4	no	—
73	#127×C₅₃	—	2	4	no	—
74	#124×C₅₃	—	2	8	no	—
75	#123×C₅₃	—	2	16	no	—
76	C₂²⋊Q₈×C₅₃	—	2	16	no	—
77	#121×C₅₃	—	2	16	no	—
78	#121×C₅₃	—	2	16	no	—
79	#121×C₅₃	—	2	32	no	—
80	#120×C₅₃	—	2	16	no	—
81	C₄×D₄×C₅₃	—	2	16	no	—
82	C₄×D₄×C₅₃	—	2	32	no	—
83	#117×C₅₃	—	2	8	no	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
93	C₂×C₄²×C₅₃	—	2	16	no	—
94	C₂×C₄²×C₅₃	—	2	8	no	—
95	C₂×C₄²×C₅₃	—	2	8	no	—
96	C₂×C₄²×C₅₃	—	2	8	no	—
97	C₂×C₄²×C₅₃	—	2	4	no	—
98	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
99	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
105	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
106	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
107	#95×C₅₃	C₂³	1	16	yes	—
108	#95×C₅₃	C₂³	1	8	yes	—
109	C₂⁴×C₅₃	—	2	2	no	—
110	C₂⁴×C₅₃	—	2	8	no	—
111	C₂⁴×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
113	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
114	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
116	#41×C₅₃	—	4	32	no	—
117	C₂×Q₈×C₅₃	—	2	4	no	—
118	C₂×Q₈×C₅₃	—	2	4	no	—
119	C₂×Q₈×C₅₃	—	2	4	no	—
120	C₂×D₄×C₅₃	—	2	16	no	—
121	C₂×D₄×C₅₃	—	2	8	no	—
122	C₂×D₄×C₅₃	—	2	8	no	—
123	C₂×D₄×C₅₃	—	2	32	no	—
124	C₂×D₄×C₅₃	—	2	8	no	—
125	C₂×D₄×C₅₃	—	2	8	no	—
126	C₂×D₄×C₅₃	—	2	4	no	—
127	C₂×D₄×C₅₃	—	2	4	no	—
128	C₂×D₄×C₅₃	—	2	16	no	—
129	C₂×D₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	64	no	—
131	C₂²×C₄×C₅₃	—	2	4	no	—
132	C₂²×C₄×C₅₃	—	2	2	no	—
133	C₂²×C₄×C₅₃	—	2	8	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	32	no	—
136	C₂²×C₄×C₅₃	—	2	8	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	2	4	no	—
139	C₂²×C₄×C₅₃	—	2	8	no	—
140	C₂²×C₄×C₅₃	—	2	8	no	—
141	C₂²×C₄×C₅₃	—	2	8	no	—
142	C₂²×C₄×C₅₃	—	2	4	no	—
143	C₂²×C₄×C₅₃	—	2	4	no	—
144	C₂²×C₄×C₅₃	—	2	8	no	—
145	C₂²×C₄×C₅₃	—	2	8	no	—
146	C₂²×C₄×C₅₃	—	2	8	no	—
147	C₂²×C₄×C₅₃	—	2	8	no	—
148	C₂²×C₄×C₅₃	—	2	8	no	—
149	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
150	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
151	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
152	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
155	C₂²×C₄×C₅₃	#41	1	4	yes	—
156	C₂²×C₄×C₅₃	#41	1	16	yes	complemented
157	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	—	2	8	no	—
166	C₄⋊C₄×C₅₃	—	2	8	no	—
167	C₄⋊C₄×C₅₃	—	2	16	no	—
168	C₄⋊C₄×C₅₃	—	2	16	no	—
169	C₂²⋊C₄×C₅₃	—	2	8	no	—
170	C₂²⋊C₄×C₅₃	—	2	16	no	—
171	C₂²⋊C₄×C₅₃	—	2	16	no	—
172	C₂²⋊C₄×C₅₃	—	2	16	no	—
173	C₂²⋊C₄×C₅₃	—	2	32	no	—
174	C₄²×C₅₃	—	2	8	no	—
175	C₄²×C₅₃	—	2	4	no	—
176	C₄²×C₅₃	—	2	4	no	—
177	C₄²×C₅₃	—	2	8	no	—
178	C₄²×C₅₃	—	2	8	no	—
179	C₄²×C₅₃	C₂×D₄	1	8	yes	—
180	C₄²×C₅₃	C₂×D₄	1	4	yes	—
181	C₄²×C₅₃	C₂×D₄	1	4	yes	—
182	C₂³×C₅₃	—	2	16	no	—
183	C₂³×C₅₃	—	2	16	no	—
184	C₂³×C₅₃	—	2	8	no	—
185	C₂³×C₅₃	—	2	16	no	—
186	C₂³×C₅₃	—	2	2	no	—
187	C₂³×C₅₃	—	2	2	no	—
188	C₂³×C₅₃	—	2	4	no	—
189	C₂³×C₅₃	—	2	4	no	—
190	C₂³×C₅₃	—	2	4	no	—
191	C₂³×C₅₃	—	2	4	no	—
192	C₂³×C₅₃	—	2	4	no	—
193	C₂³×C₅₃	—	2	16	no	—
194	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
195	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
196	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
197	C₂³×C₅₃	C₂×#41	1	2	yes	—
198	C₂³×C₅₃	C₂×#41	1	8	yes	—
199	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
200	Q₈×C₅₃	—	4	8	no	—
201	D₄×C₅₃	—	4	16	no	—
202	D₄×C₅₃	—	4	32	no	—
203	D₄×C₅₃	—	4	8	no	—
204	D₄×C₅₃	—	4	16	no	—
205	C₂×C₄×C₅₃	—	2	32	no	—
206	C₂×C₄×C₅₃	—	2	16	no	—
207	C₂×C₄×C₅₃	—	2	8	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	16	no	—
214	C₂×C₄×C₅₃	—	2	16	no	—
215	C₂×C₄×C₅₃	—	2	16	no	—
216	C₂×C₄×C₅₃	—	2	32	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	2	8	no	—
220	C₂×C₄×C₅₃	—	2	4	no	—
221	C₂×C₄×C₅₃	—	2	16	no	—
222	C₂×C₄×C₅₃	—	4	16	no	—
223	C₂×C₄×C₅₃	#121	1	16	yes	complemented
224	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
232	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
233	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
234	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
235	#36624	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂²×C₅₃	—	2	32	no	—
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	2	4	no	—
239	C₂²×C₅₃	—	2	8	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	4	no	—
242	C₂²×C₅₃	—	2	4	no	—
243	C₂²×C₅₃	—	2	8	no	—
244	C₂²×C₅₃	—	4	4	no	—
245	C₂²×C₅₃	—	4	8	no	—
246	C₂²×C₅₃	—	4	8	no	—
247	C₂²×C₅₃	—	4	8	no	—
248	C₂²×C₅₃	C₂×#121	1	4	yes	—
249	C₂²×C₅₃	#532	1	8	yes	complemented
250	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
251	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
252	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
253	C₂²×C₅₃	C₂²×#41	1	4	yes	—
254	C₄×C₅₃	—	2	4	no	—
255	C₄×C₅₃	—	2	4	no	—
256	C₄×C₅₃	—	2	8	no	—
257	C₄×C₅₃	—	4	4	no	—
258	C₄×C₅₃	—	4	8	no	—
259	C₄×C₅₃	C₂×#121	1	4	yes	—
260	C₄×C₅₃	C₂×#121	1	4	yes	—
261	C₄×C₅₃	#525	1	8	yes	complemented
262	C₂×#532	C₂×C₅₃	1	1	yes	characteristic, complemented
263	C₂×#525	C₂×C₅₃	1	2	yes	complemented
264	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#2857	C₂×C₅₃	1	2	yes	complemented
266	#2856	C₂×C₅₃	1	2	yes	complemented
267	#2854	C₂×C₅₃	1	4	yes	complemented
268	#2850	C₂×C₅₃	1	4	yes	complemented
269	#2847	C₂×C₅₃	1	4	yes	complemented
270	C₄×#121	C₂×C₅₃	1	8	yes	complemented
271	#2297	C₂×C₅₃	1	2	yes	complemented
272	#2285	C₂×C₅₃	1	1	yes	characteristic, complemented
273	C₂×C₅₃	—	2	4	no	—
274	C₂×C₅₃	—	2	8	no	—
275	C₂×C₅₃	—	8	4	no	—
276	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
277	C₂×C₅₃	C₂×#525	1	2	yes	—
278	C₂×C₅₃	C₂×#532	1	2	yes	—
279	C₂×C₅₃	#3444	1	2	yes	—
280	C₂²×#41	C₂²×C₅₃	1	2	yes	—
281	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
282	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
283	#532	—	2	8	no	—
284	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
285	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
286	C₂×#124	C₂²×C₅₃	1	2	yes	complemented
287	#525	—	2	16	no	—
288	C₂×#123	C₂²×C₅₃	1	4	yes	—
289	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
290	C₂×#121	C₂²×C₅₃	1	4	yes	—
291	C₂×#121	C₂²×C₅₃	1	8	yes	—
292	C₂×#121	C₂²×C₅₃	1	4	yes	—
293	C₂×#120	C₂²×C₅₃	1	4	yes	—
294	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
295	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
296	C₂×#117	C₂²×C₅₃	1	2	yes	—
297	C₂²×C₄²	—	2	8	no	—
298	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
299	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
300	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
301	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
302	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
303	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
304	#391	C₂²×C₅₃	1	4	yes	complemented
305	#390	C₂²×C₅₃	1	4	yes	complemented
306	#390	C₂²×C₅₃	1	8	yes	complemented
307	#389	C₂²×C₅₃	1	8	yes	complemented
308	#388	C₂²×C₅₃	1	16	yes	complemented
309	#386	C₂²×C₅₃	1	16	yes	—
310	#386	C₂²×C₅₃	1	8	yes	—
311	#384	C₂²×C₅₃	1	2	yes	complemented
312	#382	C₂²×C₅₃	1	2	yes	complemented
313	#380	C₂²×C₅₃	1	4	yes	—
314	#380	C₂²×C₅₃	1	8	yes	—
315	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
316	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
317	C₅₃	#36624	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
319	C₂×#41	—	2	8	no	—
320	C₂×#41	—	2	8	no	—
321	C₂×#41	—	2	8	no	—
322	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
323	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
324	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
325	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
326	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
327	C₂³×C₄	—	2	8	no	—
328	C₂³×C₄	—	2	4	no	—
329	C₂³×C₄	—	2	4	no	—
330	C₂³×C₄	—	2	16	no	—
331	C₂³×C₄	D₄×C₅₃	1	8	yes	complemented
332	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
333	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
334	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
335	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
336	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
337	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
338	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
340	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
341	C₄⋊Q₈	—	2	4	no	—
342	#127	—	2	4	no	—
343	#124	—	2	8	no	—
344	#123	—	2	16	no	—
345	C₂²⋊Q₈	—	2	16	no	—
346	#121	—	2	16	no	—
347	#121	—	2	16	no	—
348	#121	—	2	32	no	—
349	#120	—	2	16	no	—
350	C₄×D₄	—	2	32	no	—
351	C₄×D₄	—	2	16	no	—
352	#117	—	2	8	no	—
353	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
355	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
362	C₂×C₄²	—	2	8	no	—
363	C₂×C₄²	—	2	4	no	—
364	C₂×C₄²	—	2	16	no	—
365	C₂×C₄²	—	2	8	no	—
366	C₂×C₄²	—	2	8	no	—
367	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
368	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
369	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
370	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
371	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
372	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
373	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
374	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
376	#95	C₂³×C₅₃	1	16	yes	—
377	#95	C₂³×C₅₃	1	8	yes	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	2	8	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
382	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
383	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
384	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
385	#41	—	4	32	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×D₄	—	2	16	no	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	32	no	—
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	8	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	16	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂²×C₄	—	2	64	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	8	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	32	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	2	8	no	—
418	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
419	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
420	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
421	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
423	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
424	C₂²×C₄	#41×C₅₃	1	4	yes	—
425	C₂²×C₄	#41×C₅₃	1	16	yes	complemented
426	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
427	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	16	no	—
437	C₄⋊C₄	—	2	16	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	16	no	—
440	C₂²⋊C₄	—	2	16	no	—
441	C₂²⋊C₄	—	2	16	no	—
442	C₂²⋊C₄	—	2	32	no	—
443	C₄²	—	2	8	no	—
444	C₄²	—	2	4	no	—
445	C₄²	—	2	4	no	—
446	C₄²	—	2	8	no	—
447	C₄²	—	2	8	no	—
448	C₄²	C₂×D₄×C₅₃	1	8	yes	—
449	C₄²	C₂×D₄×C₅₃	1	4	yes	—
450	C₄²	C₂×D₄×C₅₃	1	4	yes	—
451	C₂³	—	2	16	no	—
452	C₂³	—	2	16	no	—
453	C₂³	—	2	8	no	—
454	C₂³	—	2	16	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	16	no	—
462	C₂³	—	2	4	no	—
463	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
464	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
465	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
466	C₂³	C₂×#41×C₅₃	1	2	yes	—
467	C₂³	C₂×#41×C₅₃	1	8	yes	—
468	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
469	Q₈	—	4	8	no	—
470	D₄	—	4	16	no	—
471	D₄	—	4	32	no	—
472	D₄	—	4	8	no	—
473	D₄	—	4	16	no	—
474	C₂×C₄	—	2	32	no	—
475	C₂×C₄	—	2	16	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	16	no	—
483	C₂×C₄	—	2	16	no	—
484	C₂×C₄	—	2	16	no	—
485	C₂×C₄	—	2	32	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	16	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	4	16	no	—
492	C₂×C₄	#121×C₅₃	1	16	yes	complemented
493	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
495	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
496	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
498	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
500	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
501	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
502	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
503	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
504	C₂²	—	2	32	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	8	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	8	no	—
516	C₂²	C₂×#121×C₅₃	1	4	yes	—
517	C₂²	C₅₃×#532	1	8	yes	complemented
518	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
519	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
520	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
521	C₂²	C₂²×#41×C₅₃	1	4	yes	—
522	C₄	—	2	4	no	—
523	C₄	—	2	4	no	—
524	C₄	—	2	8	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	8	no	—
527	C₄	C₂×#121×C₅₃	1	4	yes	—
528	C₄	C₂×#121×C₅₃	1	4	yes	—
529	C₄	C₅₃×#525	1	8	yes	complemented
530	C₂	—	2	4	no	—
531	C₂	—	2	8	no	—
532	C₂	—	8	4	no	—
533	C₂	C₂²×#121×C₅₃	1	1	yes	characteristic
534	C₂	C₂×C₅₃×#525	1	2	yes	—
535	C₂	C₂×C₅₃×#532	1	2	yes	—
536	C₂	C₅₃×#3444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
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
1. C₂×C₅₃×#532	C₂	semidirect
2. C₂×C₅₃×#525	C₂	semidirect
3. C₂³×C₄²×C₅₃	C₂	semidirect
4. C₅₃×#2857	C₂	semidirect
5. C₅₃×#2856	C₂	semidirect
6. C₅₃×#2854	C₂	semidirect
7. C₅₃×#2850	C₂	semidirect
8. C₅₃×#2847	C₂	semidirect
9. C₄×#121×C₅₃	C₂	semidirect
10. C₅₃×#2297	C₂	semidirect
11. C₅₃×#2285	C₂	semidirect
16. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
17. C₂×#127×C₅₃	C₂²	semidirect
18. C₂×#124×C₅₃	C₂²	semidirect
27. C₂×C₄×D₄×C₅₃	C₂²	semidirect
36. C₅₃×#391	C₂²	semidirect
37. C₅₃×#390	C₂²	semidirect
38. C₅₃×#390	C₂²	semidirect
39. C₅₃×#389	C₂²	semidirect
40. C₅₃×#388	C₂²	semidirect
43. C₅₃×#384	C₂²	semidirect
44. C₅₃×#382	C₂²	semidirect
47. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
62. C₂³×C₄×C₅₃	D₄	semidirect
98. C₂×C₄²×C₅₃	D₄	semidirect
156. C₂²×C₄×C₅₃	#41	semidirect
223. C₂×C₄×C₅₃	#121	semidirect
235. #36624	C₅₃	direct (N × H)
249. C₂²×C₅₃	#532	semidirect
261. C₄×C₅₃	#525	semidirect
262. C₂×#532	C₂×C₅₃	semidirect
263. C₂×#525	C₂×C₅₃	semidirect
264. C₂³×C₄²	C₂×C₅₃	semidirect
265. #2857	C₂×C₅₃	semidirect
266. #2856	C₂×C₅₃	semidirect
267. #2854	C₂×C₅₃	semidirect
268. #2850	C₂×C₅₃	semidirect
269. #2847	C₂×C₅₃	semidirect
270. C₄×#121	C₂×C₅₃	semidirect
271. #2297	C₂×C₅₃	semidirect
272. #2285	C₂×C₅₃	semidirect
284. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
285. C₂×#127	C₂²×C₅₃	semidirect
286. C₂×#124	C₂²×C₅₃	semidirect
295. C₂×C₄×D₄	C₂²×C₅₃	semidirect
304. #391	C₂²×C₅₃	semidirect
305. #390	C₂²×C₅₃	semidirect
306. #390	C₂²×C₅₃	semidirect
307. #389	C₂²×C₅₃	semidirect
308. #388	C₂²×C₅₃	semidirect
311. #384	C₂²×C₅₃	semidirect
312. #382	C₂²×C₅₃	semidirect
315. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
316. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
317. C₅₃	#36624	direct (N × H)
331. C₂³×C₄	D₄×C₅₃	semidirect
367. C₂×C₄²	D₄×C₅₃	semidirect
425. C₂²×C₄	#41×C₅₃	semidirect
492. C₂×C₄	#121×C₅₃	semidirect
517. C₂²	C₅₃×#532	semidirect
529. C₄	C₅₃×#525	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#532	2
C₂×C₅₃×#525	2
C₂³×C₄²×C₅₃	2
C₅₃×#2857	2
C₅₃×#2856	2
C₅₃×#2854	2
C₅₃×#2850	2
C₅₃×#2847	2
C₄×#121×C₅₃	2
C₅₃×#2297	2
C₅₃×#2285	2
#36624	53

Subgroups — C53×#36624 #27244136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36624 #27244136