Subgroups — C₅₃×#34199 #27241711

Order 13568.

3190 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#518	C₂	1	1	yes	maximal, characteristic
3	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2359	C₂	1	8	yes	maximal, complemented
5	C₅₃×#2344	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2342	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2322	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2308	C₂	1	1	yes	maximal, characteristic
10	C₄×#123×C₅₃	C₂	1	4	yes	maximal, complemented
11	C₂×C₅₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2273	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#123×C₅₃	C₄	1	16	yes	complemented
16	C₂×#123×C₅₃	C₂²	1	2	yes	—
17	C₂×#123×C₅₃	C₂²	1	2	yes	—
18	C₂×#123×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#518	—	2	8	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
22	C₂×#117×C₅₃	—	2	8	no	—
23	C₂×#117×C₅₃	C₂²	1	4	yes	—
24	C₂×#117×C₅₃	C₂²	1	2	yes	—
25	C₂×#117×C₅₃	C₂²	1	4	yes	complemented
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
36	C₅₃×#389	C₂²	1	8	yes	complemented
37	C₅₃×#388	C₂²	1	8	yes	complemented
38	C₅₃×#388	C₂²	1	8	yes	complemented
39	C₅₃×#387	C₂²	1	8	yes	complemented
40	C₅₃×#387	C₂²	1	8	yes	complemented
41	C₅₃×#386	C₂²	1	4	yes	—
42	C₅₃×#386	C₂²	1	4	yes	—
43	C₅₃×#385	C₂²	1	4	yes	complemented
44	C₅₃×#385	C₂²	1	4	yes	complemented
45	C₅₃×#385	C₂²	1	8	yes	complemented
46	C₅₃×#381	C₂²	1	4	yes	—
47	C₅₃×#381	C₂²	1	2	yes	—
48	C₅₃×#381	C₂²	1	2	yes	complemented
49	C₅₃×#381	C₂²	1	2	yes	complemented
50	C₅₃×#380	C₂²	1	4	yes	—
51	C₅₃×#379	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
53	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
55	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
56	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
57	C₂×#95×C₅₃	C₂²	1	4	yes	complemented
58	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
59	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₅₃	—	2	4	no	—
61	C₂²×D₄×C₅₃	—	2	4	no	—
62	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
63	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
64	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	—	2	4	no	—
66	C₂³×C₄×C₅₃	—	2	4	no	—
67	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
68	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
76	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
78	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
79	#123×C₅₃	C₂×C₄	1	16	yes	complemented
80	#123×C₅₃	C₂×C₄	1	16	yes	complemented
81	#123×C₅₃	C₂×C₄	1	32	yes	complemented
82	C₄×D₄×C₅₃	—	2	16	no	—
83	C₄×D₄×C₅₃	—	2	16	no	—
84	#117×C₅₃	—	2	16	no	—
85	#117×C₅₃	—	2	16	no	—
86	#117×C₅₃	—	2	8	no	—
87	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
88	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	complemented
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
95	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	complemented
99	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
100	C₂×C₂²⋊C₄×C₅₃	D₄	1	8	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₅₃	—	2	8	no	—
116	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
118	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
119	#95×C₅₃	C₂³	1	8	yes	—
120	#95×C₅₃	C₂³	1	8	yes	—
121	#95×C₅₃	C₂³	1	8	yes	—
122	#95×C₅₃	C₂³	1	8	yes	—
123	#95×C₅₃	C₂³	1	8	yes	—
124	C₂⁴×C₅₃	—	2	2	no	—
125	C₂⁴×C₅₃	—	2	2	no	—
126	C₂⁴×C₅₃	—	4	4	no	—
127	C₂⁴×C₅₃	—	4	4	no	—
128	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
129	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
130	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
131	C₂⁴×C₅₃	#41	1	4	yes	—
132	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
133	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂×D₄×C₅₃	—	2	32	no	—
137	C₂×D₄×C₅₃	—	2	4	no	—
138	C₂×D₄×C₅₃	—	2	4	no	—
139	C₂×D₄×C₅₃	—	2	4	no	—
140	C₂×D₄×C₅₃	—	2	4	no	—
141	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
143	C₂²×C₄×C₅₃	—	2	32	no	—
144	C₂²×C₄×C₅₃	—	2	2	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	4	no	—
147	C₂²×C₄×C₅₃	—	2	4	no	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	4	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	2	2	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	4	no	—
155	C₂²×C₄×C₅₃	—	2	4	no	—
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	8	no	—
158	C₂²×C₄×C₅₃	—	4	4	no	—
159	C₂²×C₄×C₅₃	—	4	4	no	—
160	C₂²×C₄×C₅₃	—	4	4	no	—
161	C₂²×C₄×C₅₃	—	4	8	no	—
162	C₂²×C₄×C₅₃	—	4	4	no	—
163	C₂²×C₄×C₅₃	—	4	4	no	—
164	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
165	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₅₃	—	2	16	no	—
180	C₄⋊C₄×C₅₃	—	2	8	no	—
181	C₄⋊C₄×C₅₃	—	4	8	no	—
182	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
183	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
184	C₂²⋊C₄×C₅₃	—	2	4	no	—
185	C₂²⋊C₄×C₅₃	—	2	4	no	—
186	C₂²⋊C₄×C₅₃	—	2	4	no	—
187	C₂²⋊C₄×C₅₃	—	2	4	no	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	—	2	4	no	—
190	C₂²⋊C₄×C₅₃	—	2	8	no	—
191	C₂²⋊C₄×C₅₃	—	4	8	no	—
192	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
193	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
194	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
195	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
196	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
197	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
198	C₄²×C₅₃	—	2	16	no	—
199	C₄²×C₅₃	—	2	8	no	—
200	C₄²×C₅₃	—	4	8	no	—
201	C₂³×C₅₃	—	2	4	no	—
202	C₂³×C₅₃	—	2	4	no	—
203	C₂³×C₅₃	—	2	4	no	—
204	C₂³×C₅₃	—	2	2	no	—
205	C₂³×C₅₃	—	2	2	no	—
206	C₂³×C₅₃	—	2	8	no	—
207	C₂³×C₅₃	—	2	8	no	—
208	C₂³×C₅₃	—	4	8	no	—
209	C₂³×C₅₃	—	4	8	no	—
210	C₂³×C₅₃	—	4	8	no	—
211	C₂³×C₅₃	—	4	8	no	—
212	C₂³×C₅₃	—	8	4	no	—
213	C₂³×C₅₃	—	8	4	no	—
214	C₂³×C₅₃	C₄×D₄	1	16	yes	—
215	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
216	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
217	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
218	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
219	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
220	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
221	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
222	C₂³×C₅₃	C₂×#41	1	2	yes	—
223	C₂³×C₅₃	C₂×#41	1	2	yes	—
224	C₂³×C₅₃	C₂×#41	1	2	yes	—
225	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
226	D₄×C₅₃	—	2	16	no	—
227	D₄×C₅₃	—	2	16	no	—
228	C₂×C₄×C₅₃	—	2	16	no	—
229	C₂×C₄×C₅₃	—	2	8	no	—
230	C₂×C₄×C₅₃	—	2	8	no	—
231	C₂×C₄×C₅₃	—	2	16	no	—
232	C₂×C₄×C₅₃	—	2	8	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	8	no	—
240	C₂×C₄×C₅₃	—	2	8	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	2	4	no	—
243	C₂×C₄×C₅₃	—	2	4	no	—
244	C₂×C₄×C₅₃	—	2	8	no	—
245	C₂×C₄×C₅₃	—	2	8	no	—
246	C₂×C₄×C₅₃	—	2	8	no	—
247	C₂×C₄×C₅₃	—	4	8	no	—
248	C₂×C₄×C₅₃	—	4	8	no	—
249	C₂×C₄×C₅₃	—	4	8	no	—
250	C₂×C₄×C₅₃	—	8	8	no	—
251	C₂×C₄×C₅₃	—	8	4	no	—
252	C₂×C₄×C₅₃	—	8	4	no	—
253	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
254	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
255	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
256	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
257	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
258	#34199	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂²×C₅₃	—	2	4	no	—
260	C₂²×C₅₃	—	2	4	no	—
261	C₂²×C₅₃	—	2	4	no	—
262	C₂²×C₅₃	—	2	4	no	—
263	C₂²×C₅₃	—	2	4	no	—
264	C₂²×C₅₃	—	2	4	no	—
265	C₂²×C₅₃	—	2	4	no	—
266	C₂²×C₅₃	—	2	4	no	—
267	C₂²×C₅₃	—	2	4	no	—
268	C₂²×C₅₃	—	2	4	no	—
269	C₂²×C₅₃	—	2	8	no	—
270	C₂²×C₅₃	—	4	16	no	—
271	C₂²×C₅₃	—	4	4	no	—
272	C₂²×C₅₃	—	4	4	no	—
273	C₂²×C₅₃	—	8	8	no	—
274	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
275	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
276	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
277	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
278	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
279	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
280	C₂²×C₅₃	#585	1	4	yes	—
281	C₄×C₅₃	—	2	4	no	—
282	C₄×C₅₃	—	2	4	no	—
283	C₄×C₅₃	—	2	4	no	—
284	C₄×C₅₃	—	2	8	no	—
285	C₄×C₅₃	—	2	8	no	—
286	C₄×C₅₃	—	4	4	no	—
287	C₄×C₅₃	—	4	4	no	—
288	C₄×C₅₃	—	4	4	no	—
289	C₂²×#123	C₂×C₅₃	1	1	yes	characteristic
290	C₂×#518	C₂×C₅₃	1	1	yes	characteristic
291	C₂²×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#2359	C₂×C₅₃	1	8	yes	complemented
293	#2344	C₂×C₅₃	1	4	yes	complemented
294	#2342	C₂×C₅₃	1	4	yes	complemented
295	#2322	C₂×C₅₃	1	4	yes	complemented
296	#2315	C₂×C₅₃	1	1	yes	characteristic, complemented
297	#2308	C₂×C₅₃	1	1	yes	characteristic
298	C₄×#123	C₂×C₅₃	1	4	yes	complemented
299	C₂×#381	C₂×C₅₃	1	1	yes	characteristic, complemented
300	#2273	C₂×C₅₃	1	1	yes	characteristic, complemented
301	C₂×C₅₃	—	2	4	no	—
302	C₂×C₅₃	—	2	4	no	—
303	C₂×C₅₃	—	2	4	no	—
304	C₂×C₅₃	—	4	4	no	—
305	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
306	C₂×C₅₃	#3428	1	2	yes	—
307	C₂×C₅₃	#3485	1	2	yes	—
308	C₂×C₅₃	#3491	1	2	yes	—
309	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
310	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
311	C₂×#123	C₄×C₅₃	1	16	yes	complemented
312	C₂×#123	C₂²×C₅₃	1	2	yes	—
313	C₂×#123	C₂²×C₅₃	1	4	yes	—
314	C₂×#123	C₂²×C₅₃	1	2	yes	—
315	#518	—	2	8	no	—
316	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
317	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
318	C₂×#117	—	2	8	no	—
319	C₂×#117	C₂²×C₅₃	1	4	yes	—
320	C₂×#117	C₂²×C₅₃	1	2	yes	—
321	C₂×#117	C₂²×C₅₃	1	4	yes	complemented
322	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
323	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
324	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
325	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
326	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
327	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
328	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
329	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
330	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
331	C₂²×C₄²	C₂²×C₅₃	1	2	yes	complemented
332	#389	C₂²×C₅₃	1	8	yes	complemented
333	#388	C₂²×C₅₃	1	8	yes	complemented
334	#388	C₂²×C₅₃	1	8	yes	complemented
335	#387	C₂²×C₅₃	1	8	yes	complemented
336	#387	C₂²×C₅₃	1	8	yes	complemented
337	#386	C₂²×C₅₃	1	4	yes	—
338	#386	C₂²×C₅₃	1	4	yes	—
339	#385	C₂²×C₅₃	1	4	yes	complemented
340	#385	C₂²×C₅₃	1	4	yes	complemented
341	#385	C₂²×C₅₃	1	8	yes	complemented
342	#381	C₂²×C₅₃	1	2	yes	—
343	#381	C₂²×C₅₃	1	4	yes	—
344	#381	C₂²×C₅₃	1	2	yes	complemented
345	#381	C₂²×C₅₃	1	2	yes	complemented
346	#380	C₂²×C₅₃	1	4	yes	—
347	#379	C₂²×C₅₃	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
349	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
350	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
351	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
352	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
353	C₂×#95	C₂²×C₅₃	1	4	yes	complemented
354	C₅₃	#34199	1	1	yes	characteristic, direct product factor, Sylow
355	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
356	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	—	2	4	no	—
359	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
360	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
361	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
362	C₂³×C₄	—	2	4	no	—
363	C₂³×C₄	—	2	4	no	—
364	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	complemented
365	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
366	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
369	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
370	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
371	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
372	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
373	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
374	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
375	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
376	#123	C₂×C₄×C₅₃	1	16	yes	complemented
377	#123	C₂×C₄×C₅₃	1	32	yes	complemented
378	#123	C₂×C₄×C₅₃	1	16	yes	complemented
379	C₄×D₄	—	2	16	no	—
380	C₄×D₄	—	2	16	no	—
381	#117	—	2	16	no	—
382	#117	—	2	16	no	—
383	#117	—	2	8	no	—
384	C₂×C₄⋊C₄	—	2	8	no	—
385	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	complemented
386	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
387	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
389	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
391	C₂×C₂²⋊C₄	—	2	4	no	—
392	C₂×C₂²⋊C₄	—	2	4	no	—
393	C₂×C₂²⋊C₄	—	2	4	no	—
394	C₂×C₂²⋊C₄	—	2	8	no	—
395	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	16	yes	complemented
396	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
397	C₂×C₂²⋊C₄	D₄×C₅₃	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
412	C₂×C₄²	—	2	8	no	—
413	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
414	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
415	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
416	#95	C₂³×C₅₃	1	8	yes	—
417	#95	C₂³×C₅₃	1	8	yes	—
418	#95	C₂³×C₅₃	1	8	yes	—
419	#95	C₂³×C₅₃	1	8	yes	—
420	#95	C₂³×C₅₃	1	8	yes	—
421	C₂⁴	—	2	2	no	—
422	C₂⁴	—	2	2	no	—
423	C₂⁴	—	4	4	no	—
424	C₂⁴	—	4	4	no	—
425	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
426	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
427	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
428	C₂⁴	#41×C₅₃	1	4	yes	—
429	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
430	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
431	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
432	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
433	C₂×D₄	—	2	32	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
439	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
440	C₂²×C₄	—	2	32	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	8	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	8	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
462	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
463	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
464	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
465	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
466	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
467	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
468	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
469	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
470	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
476	C₄⋊C₄	—	2	16	no	—
477	C₄⋊C₄	—	2	8	no	—
478	C₄⋊C₄	—	4	8	no	—
479	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
480	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	4	8	no	—
489	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
490	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
491	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
493	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
494	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
495	C₄²	—	2	16	no	—
496	C₄²	—	2	8	no	—
497	C₄²	—	4	8	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	8	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	4	8	no	—
506	C₂³	—	4	8	no	—
507	C₂³	—	4	8	no	—
508	C₂³	—	4	8	no	—
509	C₂³	—	8	4	no	—
510	C₂³	—	8	4	no	—
511	C₂³	C₄×D₄×C₅₃	1	16	yes	—
512	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
513	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
514	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
515	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
516	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
517	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
518	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
519	C₂³	C₂×#41×C₅₃	1	2	yes	—
520	C₂³	C₂×#41×C₅₃	1	2	yes	—
521	C₂³	C₂×#41×C₅₃	1	2	yes	—
522	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
523	D₄	—	2	16	no	—
524	D₄	—	2	16	no	—
525	C₂×C₄	—	2	16	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	2	16	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	8	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	8	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	—	4	8	no	—
547	C₂×C₄	—	8	8	no	—
548	C₂×C₄	—	8	4	no	—
549	C₂×C₄	—	8	4	no	—
550	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
551	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
552	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
553	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
554	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
555	C₂²	—	2	4	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	4	16	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	8	8	no	—
570	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
571	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
572	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
573	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
574	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
575	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
576	C₂²	C₅₃×#585	1	4	yes	—
577	C₄	—	2	8	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	4	no	—
580	C₄	—	2	4	no	—
581	C₄	—	2	8	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	4	no	—
585	C₂	—	2	4	no	—
586	C₂	—	2	4	no	—
587	C₂	—	2	4	no	—
588	C₂	—	4	4	no	—
589	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
590	C₂	C₅₃×#3428	1	2	yes	—
591	C₂	C₅₃×#3485	1	2	yes	—
592	C₂	C₅₃×#3491	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×#117×C₅₃	C₂	semidirect
4. C₅₃×#2359	C₂	semidirect
5. C₅₃×#2344	C₂	semidirect
6. C₅₃×#2342	C₂	semidirect
7. C₅₃×#2322	C₂	semidirect
8. C₅₃×#2315	C₂	semidirect
10. C₄×#123×C₅₃	C₂	semidirect
11. C₂×C₅₃×#381	C₂	semidirect
12. C₅₃×#2273	C₂	semidirect
15. C₂×#123×C₅₃	C₄	semidirect
25. C₂×#117×C₅₃	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
35. C₂²×C₄²×C₅₃	C₂²	semidirect
36. C₅₃×#389	C₂²	semidirect
37. C₅₃×#388	C₂²	semidirect
38. C₅₃×#388	C₂²	semidirect
39. C₅₃×#387	C₂²	semidirect
40. C₅₃×#387	C₂²	semidirect
43. C₅₃×#385	C₂²	semidirect
44. C₅₃×#385	C₂²	semidirect
45. C₅₃×#385	C₂²	semidirect
48. C₅₃×#381	C₂²	semidirect
49. C₅₃×#381	C₂²	semidirect
52. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
55. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
57. C₂×#95×C₅₃	C₂²	semidirect
67. C₂³×C₄×C₅₃	C₂×C₄	semidirect
79. #123×C₅₃	C₂×C₄	semidirect
80. #123×C₅₃	C₂×C₄	semidirect
81. #123×C₅₃	C₂×C₄	semidirect
88. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
98. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
99. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
258. #34199	C₅₃	direct (N × H)
291. C₂²×#117	C₂×C₅₃	semidirect
292. #2359	C₂×C₅₃	semidirect
293. #2344	C₂×C₅₃	semidirect
294. #2342	C₂×C₅₃	semidirect
295. #2322	C₂×C₅₃	semidirect
296. #2315	C₂×C₅₃	semidirect
298. C₄×#123	C₂×C₅₃	semidirect
299. C₂×#381	C₂×C₅₃	semidirect
300. #2273	C₂×C₅₃	semidirect
311. C₂×#123	C₄×C₅₃	semidirect
321. C₂×#117	C₂²×C₅₃	semidirect
323. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
331. C₂²×C₄²	C₂²×C₅₃	semidirect
332. #389	C₂²×C₅₃	semidirect
333. #388	C₂²×C₅₃	semidirect
334. #388	C₂²×C₅₃	semidirect
335. #387	C₂²×C₅₃	semidirect
336. #387	C₂²×C₅₃	semidirect
339. #385	C₂²×C₅₃	semidirect
340. #385	C₂²×C₅₃	semidirect
341. #385	C₂²×C₅₃	semidirect
344. #381	C₂²×C₅₃	semidirect
345. #381	C₂²×C₅₃	semidirect
348. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
350. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
351. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
352. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
353. C₂×#95	C₂²×C₅₃	semidirect
354. C₅₃	#34199	direct (N × H)
364. C₂³×C₄	C₂×C₄×C₅₃	semidirect
376. #123	C₂×C₄×C₅₃	semidirect
377. #123	C₂×C₄×C₅₃	semidirect
378. #123	C₂×C₄×C₅₃	semidirect
385. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
395. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
396. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#123×C₅₃	2
C₂×C₅₃×#518	2
C₂²×#117×C₅₃	2
C₅₃×#2359	2
C₅₃×#2344	2
C₅₃×#2342	2
C₅₃×#2322	2
C₅₃×#2315	2
C₅₃×#2308	2
C₄×#123×C₅₃	2
C₂×C₅₃×#381	2
C₅₃×#2273	2
#34199	53

Subgroups — C53×#34199 #27241711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34199 #27241711