Subgroups — C₂×C₅₃×#2497 #27242281

Order 13568.

2270 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2497	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅₃×#400	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#396	C₂	1	2	yes	maximal
4	C₂×C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#388	C₂	1	2	yes	maximal
7	C₂×C₅₃×#387	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#400	C₂²	1	16	yes	complemented
30	C₅₃×#396	C₂²	1	16	yes	—
31	C₅₃×#389	C₂²	1	8	yes	complemented
32	C₅₃×#389	C₂²	1	8	yes	complemented
33	C₅₃×#388	C₂²	1	16	yes	—
34	C₅₃×#387	C₂²	1	16	yes	complemented
35	C₅₃×#384	C₂²	1	8	yes	complemented
36	C₅₃×#380	C₂²	1	8	yes	—
37	C₅₃×#380	C₂²	1	8	yes	—
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
39	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
40	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#95×C₅₃	C₂²	1	2	yes	—
43	C₂×#95×C₅₃	C₂²	1	2	yes	—
44	C₂×#95×C₅₃	C₂²	1	2	yes	—
45	C₂×#95×C₅₃	C₂²	1	2	yes	—
46	C₂×#95×C₅₃	C₂²	1	2	yes	—
47	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	4	no	—
49	C₂³×C₄×C₅₃	—	2	4	no	—
50	C₂³×C₄×C₅₃	—	2	4	no	—
51	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
68	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
69	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
80	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
81	C₂×C₂²⋊C₄×C₅₃	Q₈	1	8	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄²×C₅₃	—	2	4	no	—
87	C₂×C₄²×C₅₃	—	2	4	no	—
88	C₂×C₄²×C₅₃	—	4	4	no	—
89	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
92	#95×C₅₃	C₂³	1	4	yes	—
93	#95×C₅₃	C₂³	1	4	yes	—
94	#95×C₅₃	C₂³	1	8	yes	—
95	#95×C₅₃	C₂³	1	8	yes	—
96	#95×C₅₃	C₂³	1	8	yes	—
97	#95×C₅₃	C₂³	1	8	yes	—
98	#95×C₅₃	C₂³	1	8	yes	—
99	C₂⁴×C₅₃	—	2	2	no	—
100	C₂⁴×C₅₃	—	2	2	no	—
101	C₂⁴×C₅₃	—	2	2	no	—
102	C₂⁴×C₅₃	—	2	2	no	—
103	C₂⁴×C₅₃	—	2	2	no	—
104	C₂⁴×C₅₃	—	2	2	no	—
105	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
106	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
107	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	4	no	—
117	C₂²×C₄×C₅₃	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	4	no	—
119	C₂²×C₄×C₅₃	—	2	4	no	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	4	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	4	no	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	4	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	4	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	2	8	no	—
133	C₂²×C₄×C₅₃	—	2	8	no	—
134	C₂²×C₄×C₅₃	—	2	8	no	—
135	C₂²×C₄×C₅₃	—	4	16	no	—
136	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
137	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
138	C₂²×C₄×C₅₃	#41	1	4	yes	—
139	C₂²×C₄×C₅₃	#41	1	4	yes	—
140	C₂²×C₄×C₅₃	#41	1	4	yes	—
141	C₂²×C₄×C₅₃	#41	1	4	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
152	C₄⋊C₄×C₅₃	—	2	16	no	—
153	C₄⋊C₄×C₅₃	—	2	8	no	—
154	C₄⋊C₄×C₅₃	—	2	8	no	—
155	C₄⋊C₄×C₅₃	—	2	8	no	—
156	C₄⋊C₄×C₅₃	—	2	8	no	—
157	C₄⋊C₄×C₅₃	—	4	8	no	—
158	C₄⋊C₄×C₅₃	—	4	8	no	—
159	C₄⋊C₄×C₅₃	—	4	8	no	—
160	C₂²⋊C₄×C₅₃	—	2	8	no	—
161	C₂²⋊C₄×C₅₃	—	4	8	no	—
162	C₂²⋊C₄×C₅₃	—	4	8	no	—
163	C₂²⋊C₄×C₅₃	C₂×Q₈	1	16	yes	—
164	C₄²×C₅₃	—	2	8	no	—
165	C₄²×C₅₃	—	2	8	no	—
166	C₄²×C₅₃	—	4	8	no	—
167	C₂³×C₅₃	—	2	8	no	—
168	C₂³×C₅₃	—	2	8	no	—
169	C₂³×C₅₃	—	2	8	no	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	2	2	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	2	no	—
176	C₂³×C₅₃	—	4	4	no	—
177	C₂³×C₅₃	—	4	4	no	—
178	C₂³×C₅₃	—	4	4	no	—
179	C₂³×C₅₃	—	4	4	no	—
180	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
181	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
182	C₂³×C₅₃	C₂×#41	1	2	yes	—
183	C₂³×C₅₃	C₂×#41	1	2	yes	—
184	C₂³×C₅₃	C₂×#41	1	2	yes	—
185	C₂³×C₅₃	C₂×#41	1	2	yes	—
186	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
187	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
188	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
189	C₂×C₄×C₅₃	—	2	8	no	—
190	C₂×C₄×C₅₃	—	2	8	no	—
191	C₂×C₄×C₅₃	—	2	8	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	4	no	—
208	C₂×C₄×C₅₃	—	2	4	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	4	4	no	—
217	C₂×C₄×C₅₃	—	4	8	no	—
218	C₂×C₄×C₅₃	—	4	8	no	—
219	C₂×C₄×C₅₃	—	4	8	no	—
220	C₂×C₄×C₅₃	—	4	4	no	—
221	C₂×C₄×C₅₃	—	4	8	no	—
222	C₂×C₄×C₅₃	—	4	16	no	—
223	C₂×C₄×C₅₃	#126	1	16	yes	—
224	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
225	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
226	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
227	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
228	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
229	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
230	C₂×#2497	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	2	4	no	—
233	C₂²×C₅₃	—	2	4	no	—
234	C₂²×C₅₃	—	4	2	no	—
235	C₂²×C₅₃	—	4	2	no	—
236	C₂²×C₅₃	—	4	2	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	—	4	16	no	—
239	C₂²×C₅₃	C₂×#126	1	4	yes	—
240	C₂²×C₅₃	#533	1	4	yes	—
241	C₂²×C₅₃	#540	1	4	yes	—
242	C₂²×C₅₃	#553	1	4	yes	—
243	C₂²×C₅₃	#554	1	4	yes	—
244	C₂²×C₅₃	#554	1	4	yes	—
245	C₂²×C₅₃	#555	1	4	yes	—
246	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
247	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
248	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
249	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
250	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
251	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
253	C₄×C₅₃	—	2	4	no	—
254	C₄×C₅₃	—	2	4	no	—
255	C₄×C₅₃	—	2	4	no	—
256	C₄×C₅₃	—	4	2	no	—
257	C₄×C₅₃	—	4	4	no	—
258	C₄×C₅₃	—	4	4	no	—
259	C₄×C₅₃	—	4	4	no	—
260	C₄×C₅₃	—	4	2	no	—
261	C₄×C₅₃	—	4	4	no	—
262	C₄×C₅₃	C₂×#126	1	8	yes	—
263	#2497	C₂×C₅₃	1	16	yes	direct product factor
264	C₂×#400	C₂×C₅₃	1	2	yes	complemented
265	C₂×#396	C₂×C₅₃	1	2	yes	—
266	C₂×#389	C₂×C₅₃	1	1	yes	characteristic, complemented
267	C₂×#389	C₂×C₅₃	1	1	yes	characteristic, complemented
268	C₂×#388	C₂×C₅₃	1	2	yes	—
269	C₂×#387	C₂×C₅₃	1	2	yes	complemented
270	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
271	C₂×#380	C₂×C₅₃	1	1	yes	characteristic
272	C₂×#380	C₂×C₅₃	1	1	yes	characteristic
273	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
274	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
275	C₂×C₅₃	—	4	4	no	—
276	C₂×C₅₃	#2497	1	8	yes	direct product factor
277	C₂×C₅₃	C₂²×#126	1	1	yes	characteristic
278	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
279	C₂×C₅₃	C₂×#540	1	1	yes	characteristic
280	C₂×C₅₃	C₂×#553	1	1	yes	characteristic
281	C₂×C₅₃	C₂×#554	1	1	yes	characteristic
282	C₂×C₅₃	C₂×#554	1	1	yes	characteristic
283	C₂×C₅₃	C₂×#555	1	1	yes	characteristic
284	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
285	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
286	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
287	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
288	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
289	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
290	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
291	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
292	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
293	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
294	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
295	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
296	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
297	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
298	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
299	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
300	#400	C₂²×C₅₃	1	16	yes	complemented
301	#396	C₂²×C₅₃	1	16	yes	—
302	#389	C₂²×C₅₃	1	8	yes	complemented
303	#389	C₂²×C₅₃	1	8	yes	complemented
304	#388	C₂²×C₅₃	1	16	yes	—
305	#387	C₂²×C₅₃	1	16	yes	complemented
306	#384	C₂²×C₅₃	1	8	yes	complemented
307	#380	C₂²×C₅₃	1	8	yes	—
308	#380	C₂²×C₅₃	1	8	yes	—
309	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
310	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
311	C₂×#95	C₂²×C₅₃	1	2	yes	—
312	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×#95	C₂²×C₅₃	1	2	yes	—
314	C₂×#95	C₂²×C₅₃	1	2	yes	—
315	C₂×#95	C₂²×C₅₃	1	2	yes	—
316	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
317	C₂×#95	C₂²×C₅₃	1	2	yes	—
318	C₅₃	C₂×#2497	1	1	yes	characteristic, direct product factor, Sylow
319	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
320	C₂³×C₄	—	2	4	no	—
321	C₂³×C₄	—	2	4	no	—
322	C₂³×C₄	—	2	4	no	—
323	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
324	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
325	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
326	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
328	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
330	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
334	C₂×C₄⋊C₄	—	2	8	no	—
335	C₂×C₄⋊C₄	—	2	4	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	—	2	4	no	—
338	C₂×C₄⋊C₄	—	2	4	no	—
339	C₂×C₄⋊C₄	—	4	4	no	—
340	C₂×C₄⋊C₄	—	4	4	no	—
341	C₂×C₄⋊C₄	—	4	4	no	—
342	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
348	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
349	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
350	C₂×C₂²⋊C₄	—	2	4	no	—
351	C₂×C₂²⋊C₄	—	4	4	no	—
352	C₂×C₂²⋊C₄	—	4	4	no	—
353	C₂×C₂²⋊C₄	Q₈×C₅₃	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₂×C₄²	—	2	4	no	—
359	C₂×C₄²	—	2	4	no	—
360	C₂×C₄²	—	4	4	no	—
361	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
362	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
363	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
364	#95	C₂³×C₅₃	1	4	yes	—
365	#95	C₂³×C₅₃	1	4	yes	—
366	#95	C₂³×C₅₃	1	8	yes	—
367	#95	C₂³×C₅₃	1	8	yes	—
368	#95	C₂³×C₅₃	1	8	yes	—
369	#95	C₂³×C₅₃	1	8	yes	—
370	#95	C₂³×C₅₃	1	8	yes	—
371	C₂⁴	—	2	2	no	—
372	C₂⁴	—	2	2	no	—
373	C₂⁴	—	2	2	no	—
374	C₂⁴	—	2	2	no	—
375	C₂⁴	—	2	2	no	—
376	C₂⁴	—	2	2	no	—
377	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
378	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
379	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	8	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	4	16	no	—
408	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
409	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
410	C₂²×C₄	#41×C₅₃	1	4	yes	—
411	C₂²×C₄	#41×C₅₃	1	4	yes	—
412	C₂²×C₄	#41×C₅₃	1	4	yes	—
413	C₂²×C₄	#41×C₅₃	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
418	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
420	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
421	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
424	C₄⋊C₄	—	2	16	no	—
425	C₄⋊C₄	—	2	8	no	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	8	no	—
429	C₄⋊C₄	—	4	8	no	—
430	C₄⋊C₄	—	4	8	no	—
431	C₄⋊C₄	—	4	8	no	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₂²⋊C₄	—	4	8	no	—
434	C₂²⋊C₄	—	4	8	no	—
435	C₂²⋊C₄	C₂×Q₈×C₅₃	1	16	yes	—
436	C₄²	—	2	8	no	—
437	C₄²	—	2	8	no	—
438	C₄²	—	4	8	no	—
439	C₂³	—	2	8	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	8	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	4	no	—
449	C₂³	—	4	4	no	—
450	C₂³	—	4	4	no	—
451	C₂³	—	4	4	no	—
452	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
453	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
454	C₂³	C₂×#41×C₅₃	1	2	yes	—
455	C₂³	C₂×#41×C₅₃	1	2	yes	—
456	C₂³	C₂×#41×C₅₃	1	2	yes	—
457	C₂³	C₂×#41×C₅₃	1	2	yes	—
458	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
459	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
460	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	8	no	—
494	C₂×C₄	—	4	16	no	—
495	C₂×C₄	#126×C₅₃	1	16	yes	—
496	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
497	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
498	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
499	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
500	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
501	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	16	no	—
509	C₂²	—	4	2	no	—
510	C₂²	C₂×#126×C₅₃	1	4	yes	—
511	C₂²	C₅₃×#533	1	4	yes	—
512	C₂²	C₅₃×#540	1	4	yes	—
513	C₂²	C₅₃×#553	1	4	yes	—
514	C₂²	C₅₃×#554	1	4	yes	—
515	C₂²	C₅₃×#554	1	4	yes	—
516	C₂²	C₅₃×#555	1	4	yes	—
517	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
518	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
519	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
520	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
521	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
522	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
523	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
524	C₄	—	2	4	no	—
525	C₄	—	2	4	no	—
526	C₄	—	2	4	no	—
527	C₄	—	4	4	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	4	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	2	no	—
533	C₄	C₂×#126×C₅₃	1	8	yes	—
534	C₂	—	4	4	no	—
535	C₂	C₅₃×#2497	1	8	yes	direct product factor
536	C₂	C₂²×#126×C₅₃	1	1	yes	characteristic
537	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
538	C₂	C₂×C₅₃×#540	1	1	yes	characteristic
539	C₂	C₂×C₅₃×#553	1	1	yes	characteristic
540	C₂	C₂×C₅₃×#554	1	1	yes	characteristic
541	C₂	C₂×C₅₃×#554	1	1	yes	characteristic
542	C₂	C₂×C₅₃×#555	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2497	C₂	direct (N × H)
2. C₂×C₅₃×#400	C₂	semidirect
4. C₂×C₅₃×#389	C₂	semidirect
5. C₂×C₅₃×#389	C₂	semidirect
7. C₂×C₅₃×#387	C₂	semidirect
8. C₂×C₅₃×#384	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
29. C₅₃×#400	C₂²	semidirect
31. C₅₃×#389	C₂²	semidirect
32. C₅₃×#389	C₂²	semidirect
34. C₅₃×#387	C₂²	semidirect
35. C₅₃×#384	C₂²	semidirect
38. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
230. C₂×#2497	C₅₃	direct (N × H)
263. #2497	C₂×C₅₃	direct (N × H)
264. C₂×#400	C₂×C₅₃	semidirect
266. C₂×#389	C₂×C₅₃	semidirect
267. C₂×#389	C₂×C₅₃	semidirect
269. C₂×#387	C₂×C₅₃	semidirect
270. C₂×#384	C₂×C₅₃	semidirect
273. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
276. C₂×C₅₃	#2497	direct (N × H)
300. #400	C₂²×C₅₃	semidirect
302. #389	C₂²×C₅₃	semidirect
303. #389	C₂²×C₅₃	semidirect
305. #387	C₂²×C₅₃	semidirect
306. #384	C₂²×C₅₃	semidirect
309. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
318. C₅₃	C₂×#2497	direct (N × H)
535. C₂	C₅₃×#2497	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2497	2
C₂×C₅₃×#400	2
C₂×C₅₃×#396	2
C₂×C₅₃×#389	2
C₂×C₅₃×#388	2
C₂×C₅₃×#387	2
C₂×C₅₃×#384	2
C₂×C₅₃×#380	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×#2497	53

Subgroups — C2×C53×#2497 #27242281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2497 #27242281