Subgroups — C₅₃×#36625 #27244137

Order 13568.

3110 conjugacy classes of subgroups.

The subgroup lattice has 446 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, complemented
2	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2857	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2855	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2851	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2848	C₂	1	4	yes	maximal
7	C₄×C₂²⋊Q₈×C₅₃	C₂	1	8	yes	maximal, complemented
8	C₅₃×#2296	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2280	C₂	1	2	yes	maximal
11	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#533	—	2	8	no	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
15	C₂×#125×C₅₃	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	—
17	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	complemented
19	C₂×#117×C₅₃	C₂²	1	2	yes	—
20	C₂×#117×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅₃	—	2	8	no	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#391	C₂²	1	8	yes	complemented
29	C₅₃×#389	C₂²	1	8	yes	complemented
30	C₅₃×#389	C₂²	1	8	yes	complemented
31	C₅₃×#387	C₂²	1	16	yes	complemented
32	C₅₃×#385	C₂²	1	8	yes	—
33	C₅₃×#385	C₂²	1	16	yes	—
34	C₅₃×#384	C₂²	1	2	yes	complemented
35	C₅₃×#382	C₂²	1	2	yes	complemented
36	C₅₃×#381	C₂²	1	4	yes	—
37	C₅₃×#380	C₂²	1	8	yes	—
38	C₅₃×#380	C₂²	1	4	yes	—
39	C₅₃×#380	C₂²	1	8	yes	—
40	C₅₃×#379	C₂²	1	4	yes	—
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
43	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
44	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
46	C₂³×C₄×C₅₃	—	2	8	no	—
47	C₂³×C₄×C₅₃	—	2	4	no	—
48	C₂³×C₄×C₅₃	—	2	16	no	—
49	C₂³×C₄×C₅₃	—	2	4	no	—
50	C₂³×C₄×C₅₃	Q₈	1	8	yes	complemented
51	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₅₃	—	2	8	no	—
60	#125×C₅₃	—	2	8	no	—
61	C₂²⋊Q₈×C₅₃	—	2	16	no	—
62	C₂²⋊Q₈×C₅₃	—	2	32	no	—
63	C₄×Q₈×C₅₃	—	2	16	no	—
64	#117×C₅₃	—	2	8	no	—
65	#117×C₅₃	—	2	8	no	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
72	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
75	C₂×C₄²×C₅₃	—	2	16	no	—
76	C₂×C₄²×C₅₃	—	2	8	no	—
77	C₂×C₄²×C₅₃	—	2	8	no	—
78	C₂×C₄²×C₅₃	—	2	8	no	—
79	C₂×C₄²×C₅₃	—	2	4	no	—
80	C₂×C₄²×C₅₃	D₄	1	8	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
87	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
90	#95×C₅₃	C₂³	1	16	yes	—
91	#95×C₅₃	C₂³	1	8	yes	—
92	#95×C₅₃	C₂³	1	16	yes	—
93	C₂⁴×C₅₃	—	2	2	no	—
94	C₂⁴×C₅₃	—	2	8	no	—
95	C₂⁴×C₅₃	—	2	2	no	—
96	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
99	C₂×Q₈×C₅₃	—	2	8	no	—
100	C₂×Q₈×C₅₃	—	2	16	no	—
101	C₂²×C₄×C₅₃	—	2	64	no	—
102	C₂²×C₄×C₅₃	—	2	4	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	8	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	32	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	4	no	—
110	C₂²×C₄×C₅₃	—	2	8	no	—
111	C₂²×C₄×C₅₃	—	2	8	no	—
112	C₂²×C₄×C₅₃	—	2	8	no	—
113	C₂²×C₄×C₅₃	—	2	4	no	—
114	C₂²×C₄×C₅₃	—	2	4	no	—
115	C₂²×C₄×C₅₃	—	2	8	no	—
116	C₂²×C₄×C₅₃	—	2	8	no	—
117	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
119	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
120	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
121	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
122	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
123	C₂²×C₄×C₅₃	#41	1	16	yes	—
124	C₂²×C₄×C₅₃	#41	1	4	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₄⋊C₄×C₅₃	—	2	8	no	—
134	C₄⋊C₄×C₅₃	—	2	16	no	—
135	C₄⋊C₄×C₅₃	—	2	8	no	—
136	C₄⋊C₄×C₅₃	—	2	8	no	—
137	C₄⋊C₄×C₅₃	—	2	32	no	—
138	C₄⋊C₄×C₅₃	—	2	16	no	—
139	C₂²⋊C₄×C₅₃	—	2	8	no	—
140	C₂²⋊C₄×C₅₃	—	2	8	no	—
141	C₂²⋊C₄×C₅₃	—	2	32	no	—
142	C₄²×C₅₃	—	2	8	no	—
143	C₄²×C₅₃	—	2	4	no	—
144	C₄²×C₅₃	—	2	4	no	—
145	C₄²×C₅₃	—	2	8	no	—
146	C₄²×C₅₃	—	2	8	no	—
147	C₄²×C₅₃	—	2	8	no	—
148	C₄²×C₅₃	C₂×D₄	1	8	yes	—
149	C₄²×C₅₃	C₂×D₄	1	4	yes	—
150	C₄²×C₅₃	C₂×D₄	1	4	yes	—
151	C₂³×C₅₃	—	2	16	no	—
152	C₂³×C₅₃	—	2	16	no	—
153	C₂³×C₅₃	—	2	8	no	—
154	C₂³×C₅₃	—	2	16	no	—
155	C₂³×C₅₃	—	2	2	no	—
156	C₂³×C₅₃	—	2	2	no	—
157	C₂³×C₅₃	—	2	4	no	—
158	C₂³×C₅₃	—	2	4	no	—
159	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
160	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
161	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
162	C₂³×C₅₃	C₂×#41	1	2	yes	—
163	C₂³×C₅₃	C₂×#41	1	8	yes	—
164	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
165	Q₈×C₅₃	—	4	16	no	—
166	C₂×C₄×C₅₃	—	2	32	no	—
167	C₂×C₄×C₅₃	—	2	16	no	—
168	C₂×C₄×C₅₃	—	2	8	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	4	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	2	16	no	—
175	C₂×C₄×C₅₃	—	2	16	no	—
176	C₂×C₄×C₅₃	—	2	16	no	—
177	C₂×C₄×C₅₃	—	2	32	no	—
178	C₂×C₄×C₅₃	—	2	8	no	—
179	C₂×C₄×C₅₃	—	2	16	no	—
180	C₂×C₄×C₅₃	—	2	8	no	—
181	C₂×C₄×C₅₃	—	2	16	no	—
182	C₂×C₄×C₅₃	C₂²⋊Q₈	1	16	yes	complemented
183	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
185	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
187	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
188	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
189	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
190	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
191	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
192	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
193	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
194	#36625	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂²×C₅₃	—	2	32	no	—
196	C₂²×C₅₃	—	2	4	no	—
197	C₂²×C₅₃	—	2	4	no	—
198	C₂²×C₅₃	—	2	8	no	—
199	C₂²×C₅₃	—	2	4	no	—
200	C₂²×C₅₃	—	2	4	no	—
201	C₂²×C₅₃	—	2	4	no	—
202	C₂²×C₅₃	—	2	8	no	—
203	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
204	C₂²×C₅₃	#533	1	8	yes	complemented
205	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
206	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
207	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
208	C₂²×C₅₃	C₂²×#41	1	4	yes	—
209	C₄×C₅₃	—	2	4	no	—
210	C₄×C₅₃	—	2	4	no	—
211	C₄×C₅₃	—	2	8	no	—
212	C₄×C₅₃	—	4	8	no	—
213	C₄×C₅₃	—	4	8	no	—
214	C₄×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
215	C₄×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
216	C₄×C₅₃	#525	1	8	yes	—
217	C₂×#533	C₂×C₅₃	1	1	yes	characteristic, complemented
218	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic
219	#2857	C₂×C₅₃	1	4	yes	complemented
220	#2855	C₂×C₅₃	1	4	yes	complemented
221	#2851	C₂×C₅₃	1	4	yes	complemented
222	#2848	C₂×C₅₃	1	4	yes	—
223	C₄×C₂²⋊Q₈	C₂×C₅₃	1	8	yes	complemented
224	#2296	C₂×C₅₃	1	2	yes	complemented
225	#2285	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#2280	C₂×C₅₃	1	2	yes	—
227	C₂×C₅₃	—	2	4	no	—
228	C₂×C₅₃	—	2	8	no	—
229	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₅₃	C₂×#525	1	2	yes	—
231	C₂×C₅₃	C₂×#533	1	2	yes	—
232	C₂×C₅₃	#3444	1	2	yes	—
233	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
234	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
235	#533	—	2	8	no	—
236	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
237	C₂×#125	C₂²×C₅₃	1	2	yes	complemented
238	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
239	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	—
240	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	complemented
241	C₂×#117	C₂²×C₅₃	1	2	yes	—
242	C₂×#117	C₂²×C₅₃	1	2	yes	—
243	C₂²×C₄²	—	2	8	no	—
244	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
245	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
246	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
247	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
248	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
249	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
250	#391	C₂²×C₅₃	1	8	yes	complemented
251	#389	C₂²×C₅₃	1	8	yes	complemented
252	#389	C₂²×C₅₃	1	8	yes	complemented
253	#387	C₂²×C₅₃	1	16	yes	complemented
254	#385	C₂²×C₅₃	1	16	yes	—
255	#385	C₂²×C₅₃	1	8	yes	—
256	#384	C₂²×C₅₃	1	2	yes	complemented
257	#382	C₂²×C₅₃	1	2	yes	complemented
258	#381	C₂²×C₅₃	1	4	yes	—
259	#380	C₂²×C₅₃	1	4	yes	—
260	#380	C₂²×C₅₃	1	8	yes	—
261	#380	C₂²×C₅₃	1	8	yes	—
262	#379	C₂²×C₅₃	1	4	yes	—
263	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
264	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
265	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
266	C₅₃	#36625	1	1	yes	characteristic, direct product factor, Sylow
267	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
268	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
269	C₂³×C₄	—	2	8	no	—
270	C₂³×C₄	—	2	4	no	—
271	C₂³×C₄	—	2	16	no	—
272	C₂³×C₄	—	2	4	no	—
273	C₂³×C₄	Q₈×C₅₃	1	8	yes	complemented
274	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
275	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
276	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
277	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
278	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
279	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
281	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
282	C₄⋊Q₈	—	2	8	no	—
283	#125	—	2	8	no	—
284	C₂²⋊Q₈	—	2	16	no	—
285	C₂²⋊Q₈	—	2	32	no	—
286	C₄×Q₈	—	2	16	no	—
287	#117	—	2	8	no	—
288	#117	—	2	8	no	—
289	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
295	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
296	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
298	C₂×C₄²	—	2	8	no	—
299	C₂×C₄²	—	2	4	no	—
300	C₂×C₄²	—	2	16	no	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	—	2	8	no	—
303	C₂×C₄²	D₄×C₅₃	1	8	yes	—
304	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
305	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
306	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
307	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
308	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
309	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
310	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
311	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
313	#95	C₂³×C₅₃	1	16	yes	—
314	#95	C₂³×C₅₃	1	8	yes	—
315	#95	C₂³×C₅₃	1	16	yes	—
316	C₂⁴	—	2	2	no	—
317	C₂⁴	—	2	8	no	—
318	C₂⁴	—	2	2	no	—
319	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
320	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
321	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
322	C₂×Q₈	—	2	8	no	—
323	C₂×Q₈	—	2	16	no	—
324	C₂²×C₄	—	2	64	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	32	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	2	8	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	8	no	—
340	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
341	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
342	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
343	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
344	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
345	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
346	C₂²×C₄	#41×C₅₃	1	16	yes	—
347	C₂²×C₄	#41×C₅₃	1	4	yes	—
348	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
349	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
350	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
351	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
353	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
354	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
356	C₄⋊C₄	—	2	8	no	—
357	C₄⋊C₄	—	2	16	no	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	8	no	—
360	C₄⋊C₄	—	2	16	no	—
361	C₄⋊C₄	—	2	32	no	—
362	C₂²⋊C₄	—	2	8	no	—
363	C₂²⋊C₄	—	2	8	no	—
364	C₂²⋊C₄	—	2	32	no	—
365	C₄²	—	2	8	no	—
366	C₄²	—	2	4	no	—
367	C₄²	—	2	4	no	—
368	C₄²	—	2	8	no	—
369	C₄²	—	2	8	no	—
370	C₄²	—	2	8	no	—
371	C₄²	C₂×D₄×C₅₃	1	8	yes	—
372	C₄²	C₂×D₄×C₅₃	1	4	yes	—
373	C₄²	C₂×D₄×C₅₃	1	4	yes	—
374	C₂³	—	2	16	no	—
375	C₂³	—	2	16	no	—
376	C₂³	—	2	8	no	—
377	C₂³	—	2	16	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	2	4	no	—
382	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
383	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
384	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
385	C₂³	C₂×#41×C₅₃	1	2	yes	—
386	C₂³	C₂×#41×C₅₃	1	8	yes	—
387	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
388	Q₈	—	4	16	no	—
389	C₂×C₄	—	2	32	no	—
390	C₂×C₄	—	2	16	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	16	no	—
398	C₂×C₄	—	2	16	no	—
399	C₂×C₄	—	2	16	no	—
400	C₂×C₄	—	2	32	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	16	no	—
404	C₂×C₄	—	2	16	no	—
405	C₂×C₄	C₂²⋊Q₈×C₅₃	1	16	yes	complemented
406	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
407	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
408	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
409	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
410	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
411	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
412	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
413	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
414	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
415	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
416	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
417	C₂²	—	2	32	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	2	4	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	4	no	—
424	C₂²	—	2	8	no	—
425	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
426	C₂²	C₅₃×#533	1	8	yes	complemented
427	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
428	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
429	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
430	C₂²	C₂²×#41×C₅₃	1	4	yes	—
431	C₄	—	2	4	no	—
432	C₄	—	2	4	no	—
433	C₄	—	2	8	no	—
434	C₄	—	4	8	no	—
435	C₄	—	4	8	no	—
436	C₄	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
437	C₄	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
438	C₄	C₅₃×#525	1	8	yes	—
439	C₂	—	2	4	no	—
440	C₂	—	2	8	no	—
441	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
442	C₂	C₂×C₅₃×#525	1	2	yes	—
443	C₂	C₂×C₅₃×#533	1	2	yes	—
444	C₂	C₅₃×#3444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁵×C₅₃	C₂³
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×#533	C₂	semidirect
3. C₅₃×#2857	C₂	semidirect
4. C₅₃×#2855	C₂	semidirect
5. C₅₃×#2851	C₂	semidirect
7. C₄×C₂²⋊Q₈×C₅₃	C₂	semidirect
8. C₅₃×#2296	C₂	semidirect
9. C₅₃×#2285	C₂	semidirect
14. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
15. C₂×#125×C₅₃	C₂²	semidirect
18. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
28. C₅₃×#391	C₂²	semidirect
29. C₅₃×#389	C₂²	semidirect
30. C₅₃×#389	C₂²	semidirect
31. C₅₃×#387	C₂²	semidirect
34. C₅₃×#384	C₂²	semidirect
35. C₅₃×#382	C₂²	semidirect
41. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
42. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
50. C₂³×C₄×C₅₃	Q₈	semidirect
182. C₂×C₄×C₅₃	C₂²⋊Q₈	semidirect
194. #36625	C₅₃	direct (N × H)
204. C₂²×C₅₃	#533	semidirect
217. C₂×#533	C₂×C₅₃	semidirect
219. #2857	C₂×C₅₃	semidirect
220. #2855	C₂×C₅₃	semidirect
221. #2851	C₂×C₅₃	semidirect
223. C₄×C₂²⋊Q₈	C₂×C₅₃	semidirect
224. #2296	C₂×C₅₃	semidirect
225. #2285	C₂×C₅₃	semidirect
236. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
237. C₂×#125	C₂²×C₅₃	semidirect
240. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
250. #391	C₂²×C₅₃	semidirect
251. #389	C₂²×C₅₃	semidirect
252. #389	C₂²×C₅₃	semidirect
253. #387	C₂²×C₅₃	semidirect
256. #384	C₂²×C₅₃	semidirect
257. #382	C₂²×C₅₃	semidirect
263. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
264. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
266. C₅₃	#36625	direct (N × H)
273. C₂³×C₄	Q₈×C₅₃	semidirect
405. C₂×C₄	C₂²⋊Q₈×C₅₃	semidirect
426. C₂²	C₅₃×#533	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
C₂³×C₄²×C₅₃	2
C₅₃×#2857	2
C₅₃×#2855	2
C₅₃×#2851	2
C₅₃×#2848	2
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#2296	2
C₅₃×#2285	2
C₅₃×#2280	2
#36625	53

Subgroups — C53×#36625 #27244137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36625 #27244137