Subgroups — C₂×C₅₃×#2487 #27242276

Order 13568.

3374 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2487	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#400	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#399	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#389	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#386	C₂	1	2	yes	maximal
8	C₂×C₅₃×#385	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#380	C₂	1	2	yes	maximal, complemented
11	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₅₃	—	2	8	no	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
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	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#400	C₂²	1	16	yes	complemented
30	C₅₃×#399	C₂²	1	16	yes	complemented
31	C₅₃×#389	C₂²	1	16	yes	complemented
32	C₅₃×#386	C₂²	1	16	yes	—
33	C₅₃×#385	C₂²	1	16	yes	complemented
34	C₅₃×#380	C₂²	1	16	yes	complemented
35	C₅₃×#380	C₂²	1	8	yes	complemented
36	C₂×#95×C₅₃	C₂²	1	4	yes	—
37	C₂×#95×C₅₃	C₂²	1	2	yes	—
38	C₂×#95×C₅₃	C₂²	1	4	yes	—
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₅₃	—	2	4	no	—
42	C₂²×D₄×C₅₃	—	2	4	no	—
43	C₂²×D₄×C₅₃	Q₈	1	8	yes	—
44	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₅₃	—	2	8	no	—
46	C₂³×C₄×C₅₃	—	2	8	no	—
47	C₂³×C₄×C₅₃	—	2	4	no	—
48	C₂³×C₄×C₅₃	—	2	8	no	—
49	C₂³×C₄×C₅₃	—	4	8	no	—
50	C₂³×C₄×C₅₃	D₄	1	8	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂³	1	4	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	8	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×D₄×C₅₃	—	2	32	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	32	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
73	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
77	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
81	C₂×C₄²×C₅₃	—	2	4	no	—
82	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
83	#95×C₅₃	C₂³	1	16	yes	—
84	#95×C₅₃	C₂³	1	8	yes	—
85	#95×C₅₃	C₂³	1	16	yes	—
86	C₂⁴×C₅₃	—	2	4	no	—
87	C₂⁴×C₅₃	—	2	4	no	—
88	C₂⁴×C₅₃	—	2	2	no	—
89	C₂⁴×C₅₃	—	2	4	no	—
90	C₂⁴×C₅₃	—	2	2	no	—
91	C₂⁴×C₅₃	—	2	4	no	—
92	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
93	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
94	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
95	C₂⁴×C₅₃	#41	1	4	yes	—
96	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
98	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂×D₄×C₅₃	—	2	32	no	—
100	C₂×D₄×C₅₃	—	2	8	no	—
101	C₂×D₄×C₅₃	—	2	8	no	—
102	C₂×D₄×C₅₃	C₂×Q₈	1	16	yes	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	32	no	—
107	C₂²×C₄×C₅₃	—	2	16	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	2	16	no	—
111	C₂²×C₄×C₅₃	—	2	8	no	—
112	C₂²×C₄×C₅₃	—	2	8	no	—
113	C₂²×C₄×C₅₃	—	2	8	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	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	8	no	—
119	C₂²×C₄×C₅₃	—	2	8	no	—
120	C₂²×C₄×C₅₃	—	2	16	no	—
121	C₂²×C₄×C₅₃	—	4	16	no	—
122	C₂²×C₄×C₅₃	—	4	32	no	—
123	C₂²×C₄×C₅₃	—	4	16	no	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
128	C₂²×C₄×C₅₃	#41	1	8	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
136	C₄⋊C₄×C₅₃	—	2	16	no	—
137	C₄⋊C₄×C₅₃	—	2	16	no	—
138	C₄⋊C₄×C₅₃	—	2	8	no	—
139	C₄⋊C₄×C₅₃	—	2	16	no	—
140	C₄⋊C₄×C₅₃	—	4	16	no	—
141	C₂²⋊C₄×C₅₃	—	2	8	no	—
142	C₂²⋊C₄×C₅₃	—	2	16	no	—
143	C₂²⋊C₄×C₅₃	—	2	8	no	—
144	C₂²⋊C₄×C₅₃	—	4	16	no	—
145	C₄²×C₅₃	—	2	8	no	—
146	C₂³×C₅₃	—	2	8	no	—
147	C₂³×C₅₃	—	2	8	no	—
148	C₂³×C₅₃	—	2	16	no	—
149	C₂³×C₅₃	—	2	16	no	—
150	C₂³×C₅₃	—	2	4	no	—
151	C₂³×C₅₃	—	2	4	no	—
152	C₂³×C₅₃	—	2	8	no	—
153	C₂³×C₅₃	—	2	16	no	—
154	C₂³×C₅₃	—	2	2	no	—
155	C₂³×C₅₃	—	2	4	no	—
156	C₂³×C₅₃	—	2	2	no	—
157	C₂³×C₅₃	—	2	4	no	—
158	C₂³×C₅₃	—	4	4	no	—
159	C₂³×C₅₃	—	4	4	no	—
160	C₂³×C₅₃	—	4	8	no	—
161	C₂³×C₅₃	C₂²⋊Q₈	1	16	yes	—
162	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
163	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
164	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
165	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
166	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
167	C₂³×C₅₃	C₂×#41	1	4	yes	—
168	C₂³×C₅₃	C₂×#41	1	2	yes	—
169	C₂³×C₅₃	C₂×#41	1	4	yes	—
170	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
171	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
172	D₄×C₅₃	—	2	32	no	—
173	C₂×C₄×C₅₃	—	2	8	no	—
174	C₂×C₄×C₅₃	—	2	8	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	4	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	32	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	8	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	4	no	—
183	C₂×C₄×C₅₃	—	2	8	no	—
184	C₂×C₄×C₅₃	—	2	8	no	—
185	C₂×C₄×C₅₃	—	2	8	no	—
186	C₂×C₄×C₅₃	—	2	8	no	—
187	C₂×C₄×C₅₃	—	2	8	no	—
188	C₂×C₄×C₅₃	—	2	4	no	—
189	C₂×C₄×C₅₃	—	4	32	no	—
190	C₂×C₄×C₅₃	—	4	4	no	—
191	C₂×C₄×C₅₃	—	4	4	no	—
192	C₂×C₄×C₅₃	—	4	16	no	—
193	C₂×C₄×C₅₃	—	4	16	no	—
194	C₂×C₄×C₅₃	—	4	16	no	—
195	C₂×C₄×C₅₃	#123	1	16	yes	—
196	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
197	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
199	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
200	C₂×#2487	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂²×C₅₃	—	2	4	no	—
202	C₂²×C₅₃	—	2	4	no	—
203	C₂²×C₅₃	—	2	8	no	—
204	C₂²×C₅₃	—	2	8	no	—
205	C₂²×C₅₃	—	2	32	no	—
206	C₂²×C₅₃	—	2	4	no	—
207	C₂²×C₅₃	—	2	8	no	—
208	C₂²×C₅₃	—	4	2	no	—
209	C₂²×C₅₃	—	4	2	no	—
210	C₂²×C₅₃	—	4	4	no	—
211	C₂²×C₅₃	—	4	16	no	—
212	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	8	yes	—
213	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
214	C₂²×C₅₃	C₂×#123	1	4	yes	—
215	C₂²×C₅₃	#535	1	4	yes	—
216	C₂²×C₅₃	#553	1	8	yes	—
217	C₂²×C₅₃	#554	1	8	yes	—
218	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
219	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
220	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
221	C₂²×C₅₃	C₂²×#41	1	2	yes	—
222	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
223	C₄×C₅₃	—	2	4	no	—
224	C₄×C₅₃	—	2	4	no	—
225	C₄×C₅₃	—	4	2	no	—
226	C₄×C₅₃	—	4	2	no	—
227	C₄×C₅₃	—	4	8	no	—
228	C₄×C₅₃	—	4	8	no	—
229	C₄×C₅₃	C₂×#123	1	8	yes	—
230	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
231	C₂³×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#2487	C₂×C₅₃	1	16	yes	direct product factor
233	C₂×#400	C₂×C₅₃	1	2	yes	complemented
234	C₂×#399	C₂×C₅₃	1	2	yes	complemented
235	C₂×#389	C₂×C₅₃	1	2	yes	complemented
236	C₂×#386	C₂×C₅₃	1	2	yes	—
237	C₂×#385	C₂×C₅₃	1	2	yes	complemented
238	C₂×#380	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×#380	C₂×C₅₃	1	2	yes	complemented
240	C₂×C₅₃	—	2	8	no	—
241	C₂×C₅₃	—	4	4	no	—
242	C₂×C₅₃	#2487	1	8	yes	direct product factor
243	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₅₃	C₂²×#123	1	1	yes	characteristic
245	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
246	C₂×C₅₃	C₂×#553	1	2	yes	—
247	C₂×C₅₃	C₂×#554	1	2	yes	—
248	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
249	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
250	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
251	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
252	C₂×C₄×D₄	—	2	8	no	—
253	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
254	C₂²×C₄⋊C₄	—	2	8	no	—
255	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
256	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
258	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
259	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
260	C₂²×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
261	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
262	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
264	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
265	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
266	#400	C₂²×C₅₃	1	16	yes	complemented
267	#399	C₂²×C₅₃	1	16	yes	complemented
268	#389	C₂²×C₅₃	1	16	yes	complemented
269	#386	C₂²×C₅₃	1	16	yes	—
270	#385	C₂²×C₅₃	1	16	yes	complemented
271	#380	C₂²×C₅₃	1	16	yes	complemented
272	#380	C₂²×C₅₃	1	8	yes	complemented
273	C₂×#95	C₂²×C₅₃	1	2	yes	—
274	C₂×#95	C₂²×C₅₃	1	4	yes	—
275	C₂×#95	C₂²×C₅₃	1	4	yes	—
276	C₅₃	C₂×#2487	1	1	yes	characteristic, direct product factor, Sylow
277	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
278	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
279	C₂²×D₄	—	2	4	no	—
280	C₂²×D₄	—	2	4	no	—
281	C₂²×D₄	Q₈×C₅₃	1	8	yes	—
282	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
283	C₂³×C₄	—	2	8	no	—
284	C₂³×C₄	—	2	8	no	—
285	C₂³×C₄	—	2	4	no	—
286	C₂³×C₄	—	2	8	no	—
287	C₂³×C₄	—	4	8	no	—
288	C₂³×C₄	D₄×C₅₃	1	8	yes	—
289	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
290	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
291	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
292	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
294	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
295	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
296	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
298	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
299	C₄×D₄	—	2	32	no	—
300	C₂×C₄⋊C₄	—	2	8	no	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	—	2	32	no	—
303	C₂×C₄⋊C₄	—	2	4	no	—
304	C₂×C₄⋊C₄	—	2	8	no	—
305	C₂×C₄⋊C₄	—	4	8	no	—
306	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
307	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
309	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
310	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
311	C₂×C₂²⋊C₄	—	2	4	no	—
312	C₂×C₂²⋊C₄	—	2	8	no	—
313	C₂×C₂²⋊C₄	—	2	4	no	—
314	C₂×C₂²⋊C₄	—	4	8	no	—
315	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
316	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
317	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
318	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
319	C₂×C₄²	—	2	4	no	—
320	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
321	#95	C₂³×C₅₃	1	16	yes	—
322	#95	C₂³×C₅₃	1	8	yes	—
323	#95	C₂³×C₅₃	1	16	yes	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	2	no	—
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	2	2	no	—
329	C₂⁴	—	2	4	no	—
330	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
331	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
332	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
333	C₂⁴	#41×C₅₃	1	4	yes	—
334	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
335	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
336	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
337	C₂×D₄	—	2	32	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	8	no	—
340	C₂×D₄	C₂×Q₈×C₅₃	1	16	yes	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	32	no	—
345	C₂²×C₄	—	2	16	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	16	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	2	8	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	8	no	—
357	C₂²×C₄	—	2	8	no	—
358	C₂²×C₄	—	2	16	no	—
359	C₂²×C₄	—	4	16	no	—
360	C₂²×C₄	—	4	32	no	—
361	C₂²×C₄	—	4	16	no	—
362	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
363	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
364	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
365	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
366	C₂²×C₄	#41×C₅₃	1	8	yes	—
367	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
369	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
370	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
372	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
373	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
374	C₄⋊C₄	—	2	16	no	—
375	C₄⋊C₄	—	2	16	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	16	no	—
378	C₄⋊C₄	—	4	16	no	—
379	C₂²⋊C₄	—	2	8	no	—
380	C₂²⋊C₄	—	2	16	no	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	4	16	no	—
383	C₄²	—	2	8	no	—
384	C₂³	—	2	8	no	—
385	C₂³	—	2	8	no	—
386	C₂³	—	2	16	no	—
387	C₂³	—	2	16	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	8	no	—
391	C₂³	—	2	16	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	4	4	no	—
398	C₂³	—	4	8	no	—
399	C₂³	C₂²⋊Q₈×C₅₃	1	16	yes	—
400	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
401	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
402	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
403	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
404	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
405	C₂³	C₂×#41×C₅₃	1	4	yes	—
406	C₂³	C₂×#41×C₅₃	1	2	yes	—
407	C₂³	C₂×#41×C₅₃	1	4	yes	—
408	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
409	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
410	D₄	—	2	32	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	32	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	8	no	—
421	C₂×C₄	—	2	8	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	8	no	—
424	C₂×C₄	—	2	8	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	4	32	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	16	no	—
431	C₂×C₄	—	4	16	no	—
432	C₂×C₄	—	4	16	no	—
433	C₂×C₄	#123×C₅₃	1	16	yes	—
434	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
435	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
436	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
437	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
438	C₂²	—	2	8	no	—
439	C₂²	—	2	8	no	—
440	C₂²	—	2	32	no	—
441	C₂²	—	2	4	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	8	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	4	4	no	—
446	C₂²	—	4	16	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	8	yes	—
450	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
451	C₂²	C₂×#123×C₅₃	1	4	yes	—
452	C₂²	C₅₃×#535	1	4	yes	—
453	C₂²	C₅₃×#553	1	8	yes	—
454	C₂²	C₅₃×#554	1	8	yes	—
455	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
456	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
457	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
458	C₂²	C₂²×#41×C₅₃	1	2	yes	—
459	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
460	C₄	—	2	4	no	—
461	C₄	—	2	4	no	—
462	C₄	—	4	8	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	8	no	—
465	C₄	—	4	2	no	—
466	C₄	C₂×#123×C₅₃	1	8	yes	—
467	C₂	—	2	8	no	—
468	C₂	—	4	4	no	—
469	C₂	C₅₃×#2487	1	8	yes	direct product factor
470	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
471	C₂	C₂²×#123×C₅₃	1	1	yes	characteristic
472	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
473	C₂	C₂×C₅₃×#553	1	2	yes	—
474	C₂	C₂×C₅₃×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
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
2. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
3. C₅₃×#2487	C₂	direct (N × H)
4. C₂×C₅₃×#400	C₂	semidirect
5. C₂×C₅₃×#399	C₂	semidirect
6. C₂×C₅₃×#389	C₂	semidirect
8. C₂×C₅₃×#385	C₂	semidirect
9. C₂×C₅₃×#380	C₂	semidirect
10. C₂×C₅₃×#380	C₂	semidirect
23. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₅₃×#400	C₂²	semidirect
30. C₅₃×#399	C₂²	semidirect
31. C₅₃×#389	C₂²	semidirect
33. C₅₃×#385	C₂²	semidirect
34. C₅₃×#380	C₂²	semidirect
35. C₅₃×#380	C₂²	semidirect
200. C₂×#2487	C₅₃	direct (N × H)
231. C₂³×C₄⋊C₄	C₂×C₅₃	semidirect
232. #2487	C₂×C₅₃	direct (N × H)
233. C₂×#400	C₂×C₅₃	semidirect
234. C₂×#399	C₂×C₅₃	semidirect
235. C₂×#389	C₂×C₅₃	semidirect
237. C₂×#385	C₂×C₅₃	semidirect
238. C₂×#380	C₂×C₅₃	semidirect
239. C₂×#380	C₂×C₅₃	semidirect
242. C₂×C₅₃	#2487	direct (N × H)
260. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
266. #400	C₂²×C₅₃	semidirect
267. #399	C₂²×C₅₃	semidirect
268. #389	C₂²×C₅₃	semidirect
270. #385	C₂²×C₅₃	semidirect
271. #380	C₂²×C₅₃	semidirect
272. #380	C₂²×C₅₃	semidirect
276. C₅₃	C₂×#2487	direct (N × H)
469. C₂	C₅₃×#2487	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×C₄×D₄×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₅₃×#2487	2
C₂×C₅₃×#400	2
C₂×C₅₃×#399	2
C₂×C₅₃×#389	2
C₂×C₅₃×#386	2
C₂×C₅₃×#385	2
C₂×C₅₃×#380	2
C₂×#2487	53

Subgroups — C2×C53×#2487 #27242276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2487 #27242276