Subgroups — C₂×C₅₃×#2486 #27242267

Order 13568.

3678 conjugacy classes of subgroups.

The subgroup lattice has 484 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	2	yes	maximal
2	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2486	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#398	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#397	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#389	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#381	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	2	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#123×C₅₃	—	2	16	no	—
16	C₂×#123×C₅₃	C₂²	1	16	yes	—
17	C₂×C₄×D₄×C₅₃	—	2	8	no	—
18	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#398	C₂²	1	16	yes	complemented
32	C₅₃×#397	C₂²	1	16	yes	complemented
33	C₅₃×#389	C₂²	1	16	yes	complemented
34	C₅₃×#381	C₂²	1	16	yes	complemented
35	C₅₃×#380	C₂²	1	16	yes	complemented
36	C₅₃×#380	C₂²	1	8	yes	complemented
37	C₂×#95×C₅₃	C₂²	1	4	yes	—
38	C₂×#95×C₅₃	C₂²	1	2	yes	—
39	C₂×#95×C₅₃	C₂²	1	4	yes	—
40	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
41	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×D₄×C₅₃	—	2	4	no	—
43	C₂²×D₄×C₅₃	—	2	4	no	—
44	C₂²×D₄×C₅₃	D₄	1	8	yes	—
45	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
46	C₂³×C₄×C₅₃	—	2	8	no	—
47	C₂³×C₄×C₅₃	—	2	8	no	—
48	C₂³×C₄×C₅₃	—	2	8	no	—
49	C₂³×C₄×C₅₃	—	2	8	no	—
50	C₂³×C₄×C₅₃	—	2	4	no	—
51	C₂³×C₄×C₅₃	—	2	8	no	—
52	C₂³×C₄×C₅₃	D₄	1	8	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	8	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	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	#123×C₅₃	—	2	64	no	—
64	C₄×D₄×C₅₃	—	2	32	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	32	no	—
68	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
77	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
84	C₂×C₄²×C₅₃	—	2	4	no	—
85	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
86	#95×C₅₃	C₂³	1	16	yes	—
87	#95×C₅₃	C₂³	1	8	yes	—
88	#95×C₅₃	C₂³	1	16	yes	—
89	C₂⁴×C₅₃	—	2	4	no	—
90	C₂⁴×C₅₃	—	2	4	no	—
91	C₂⁴×C₅₃	—	2	2	no	—
92	C₂⁴×C₅₃	—	2	4	no	—
93	C₂⁴×C₅₃	—	2	2	no	—
94	C₂⁴×C₅₃	—	2	4	no	—
95	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
96	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
97	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
98	C₂⁴×C₅₃	#41	1	4	yes	—
99	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
100	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
101	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂×D₄×C₅₃	—	2	32	no	—
103	C₂×D₄×C₅₃	—	2	8	no	—
104	C₂×D₄×C₅₃	—	2	8	no	—
105	C₂×D₄×C₅₃	C₂×D₄	1	16	yes	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	4	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	32	no	—
110	C₂²×C₄×C₅₃	—	2	16	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	8	no	—
113	C₂²×C₄×C₅₃	—	2	8	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	16	no	—
116	C₂²×C₄×C₅₃	—	2	8	no	—
117	C₂²×C₄×C₅₃	—	2	16	no	—
118	C₂²×C₄×C₅₃	—	2	8	no	—
119	C₂²×C₄×C₅₃	—	2	16	no	—
120	C₂²×C₄×C₅₃	—	2	8	no	—
121	C₂²×C₄×C₅₃	—	2	4	no	—
122	C₂²×C₄×C₅₃	—	2	4	no	—
123	C₂²×C₄×C₅₃	—	2	8	no	—
124	C₂²×C₄×C₅₃	—	2	8	no	—
125	C₂²×C₄×C₅₃	—	2	16	no	—
126	C₂²×C₄×C₅₃	—	4	32	no	—
127	C₂²×C₄×C₅₃	—	4	16	no	—
128	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₅₃	#41	1	8	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
136	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
140	C₄⋊C₄×C₅₃	—	2	16	no	—
141	C₄⋊C₄×C₅₃	—	2	16	no	—
142	C₄⋊C₄×C₅₃	—	2	8	no	—
143	C₄⋊C₄×C₅₃	—	2	32	no	—
144	C₄⋊C₄×C₅₃	—	2	16	no	—
145	C₂²⋊C₄×C₅₃	—	2	8	no	—
146	C₂²⋊C₄×C₅₃	—	2	32	no	—
147	C₂²⋊C₄×C₅₃	—	2	16	no	—
148	C₂²⋊C₄×C₅₃	—	2	8	no	—
149	C₄²×C₅₃	—	2	8	no	—
150	C₂³×C₅₃	—	2	8	no	—
151	C₂³×C₅₃	—	2	8	no	—
152	C₂³×C₅₃	—	2	16	no	—
153	C₂³×C₅₃	—	2	16	no	—
154	C₂³×C₅₃	—	2	4	no	—
155	C₂³×C₅₃	—	2	4	no	—
156	C₂³×C₅₃	—	2	8	no	—
157	C₂³×C₅₃	—	2	16	no	—
158	C₂³×C₅₃	—	2	2	no	—
159	C₂³×C₅₃	—	2	4	no	—
160	C₂³×C₅₃	—	2	2	no	—
161	C₂³×C₅₃	—	2	4	no	—
162	C₂³×C₅₃	—	4	4	no	—
163	C₂³×C₅₃	—	4	4	no	—
164	C₂³×C₅₃	—	4	8	no	—
165	C₂³×C₅₃	#121	1	16	yes	—
166	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
167	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
168	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
169	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
170	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
171	C₂³×C₅₃	C₂×#41	1	4	yes	—
172	C₂³×C₅₃	C₂×#41	1	2	yes	—
173	C₂³×C₅₃	C₂×#41	1	4	yes	—
174	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
175	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
176	D₄×C₅₃	—	2	32	no	—
177	C₂×C₄×C₅₃	—	2	8	no	—
178	C₂×C₄×C₅₃	—	2	8	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	4	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	32	no	—
183	C₂×C₄×C₅₃	—	2	8	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	8	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	8	no	—
188	C₂×C₄×C₅₃	—	2	8	no	—
189	C₂×C₄×C₅₃	—	2	4	no	—
190	C₂×C₄×C₅₃	—	2	8	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	8	no	—
193	C₂×C₄×C₅₃	—	4	32	no	—
194	C₂×C₄×C₅₃	—	4	16	no	—
195	C₂×C₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	4	4	no	—
197	C₂×C₄×C₅₃	—	4	16	no	—
198	C₂×C₄×C₅₃	—	4	16	no	—
199	C₂×C₄×C₅₃	#123	1	16	yes	complemented
200	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
201	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
202	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
203	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
204	C₂×#2486	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂²×C₅₃	—	2	4	no	—
206	C₂²×C₅₃	—	2	4	no	—
207	C₂²×C₅₃	—	2	8	no	—
208	C₂²×C₅₃	—	2	8	no	—
209	C₂²×C₅₃	—	2	32	no	—
210	C₂²×C₅₃	—	2	4	no	—
211	C₂²×C₅₃	—	2	8	no	—
212	C₂²×C₅₃	—	4	2	no	—
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	4	4	no	—
215	C₂²×C₅₃	—	4	16	no	—
216	C₂²×C₅₃	C₂×#121	1	8	yes	—
217	C₂²×C₅₃	C₂×#121	1	4	yes	—
218	C₂²×C₅₃	C₂×#123	1	4	yes	—
219	C₂²×C₅₃	#536	1	4	yes	—
220	C₂²×C₅₃	#547	1	8	yes	—
221	C₂²×C₅₃	#552	1	8	yes	—
222	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
223	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
224	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
225	C₂²×C₅₃	C₂²×#41	1	2	yes	—
226	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
227	C₄×C₅₃	—	2	4	no	—
228	C₄×C₅₃	—	2	4	no	—
229	C₄×C₅₃	—	4	8	no	—
230	C₄×C₅₃	—	4	2	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	4	8	no	—
233	C₄×C₅₃	C₂×#123	1	8	yes	complemented
234	C₂²×#123	C₂×C₅₃	1	2	yes	—
235	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
236	C₂³×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#2486	C₂×C₅₃	1	16	yes	direct product factor
238	C₂×#398	C₂×C₅₃	1	2	yes	complemented
239	C₂×#397	C₂×C₅₃	1	2	yes	complemented
240	C₂×#389	C₂×C₅₃	1	2	yes	complemented
241	C₂×#381	C₂×C₅₃	1	2	yes	complemented
242	C₂×#380	C₂×C₅₃	1	2	yes	complemented
243	C₂×#380	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₂×C₅₃	—	2	8	no	—
245	C₂×C₅₃	—	4	4	no	—
246	C₂×C₅₃	#2486	1	8	yes	direct product factor
247	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
248	C₂×C₅₃	C₂²×#123	1	1	yes	characteristic
249	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
250	C₂×C₅₃	C₂×#547	1	2	yes	—
251	C₂×C₅₃	C₂×#552	1	2	yes	—
252	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
253	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
254	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
255	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×#123	—	2	16	no	—
257	C₂×#123	C₂²×C₅₃	1	16	yes	—
258	C₂×C₄×D₄	—	2	8	no	—
259	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
260	C₂²×C₄⋊C₄	—	2	8	no	—
261	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
262	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
264	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
265	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
266	C₂²×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
267	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
268	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
269	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
270	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
271	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
272	#398	C₂²×C₅₃	1	16	yes	complemented
273	#397	C₂²×C₅₃	1	16	yes	complemented
274	#389	C₂²×C₅₃	1	16	yes	complemented
275	#381	C₂²×C₅₃	1	16	yes	complemented
276	#380	C₂²×C₅₃	1	16	yes	complemented
277	#380	C₂²×C₅₃	1	8	yes	complemented
278	C₂×#95	C₂²×C₅₃	1	2	yes	—
279	C₂×#95	C₂²×C₅₃	1	4	yes	—
280	C₂×#95	C₂²×C₅₃	1	4	yes	—
281	C₅₃	C₂×#2486	1	1	yes	characteristic, direct product factor, Sylow
282	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
283	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
284	C₂²×D₄	—	2	4	no	—
285	C₂²×D₄	—	2	4	no	—
286	C₂²×D₄	D₄×C₅₃	1	8	yes	—
287	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
288	C₂³×C₄	—	2	8	no	—
289	C₂³×C₄	—	2	8	no	—
290	C₂³×C₄	—	2	8	no	—
291	C₂³×C₄	—	2	8	no	—
292	C₂³×C₄	—	2	4	no	—
293	C₂³×C₄	—	2	8	no	—
294	C₂³×C₄	D₄×C₅₃	1	8	yes	—
295	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
296	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
298	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
299	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
300	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
301	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
304	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
305	#123	—	2	64	no	—
306	C₄×D₄	—	2	32	no	—
307	C₂×C₄⋊C₄	—	2	8	no	—
308	C₂×C₄⋊C₄	—	2	8	no	—
309	C₂×C₄⋊C₄	—	2	32	no	—
310	C₂×C₄⋊C₄	—	2	4	no	—
311	C₂×C₄⋊C₄	—	2	16	no	—
312	C₂×C₄⋊C₄	—	2	8	no	—
313	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
314	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
315	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
316	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
317	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
318	C₂×C₂²⋊C₄	—	2	4	no	—
319	C₂×C₂²⋊C₄	—	2	8	no	—
320	C₂×C₂²⋊C₄	—	2	16	no	—
321	C₂×C₂²⋊C₄	—	2	4	no	—
322	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
326	C₂×C₄²	—	2	4	no	—
327	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
328	#95	C₂³×C₅₃	1	16	yes	—
329	#95	C₂³×C₅₃	1	8	yes	—
330	#95	C₂³×C₅₃	1	16	yes	—
331	C₂⁴	—	2	4	no	—
332	C₂⁴	—	2	4	no	—
333	C₂⁴	—	2	2	no	—
334	C₂⁴	—	2	4	no	—
335	C₂⁴	—	2	2	no	—
336	C₂⁴	—	2	4	no	—
337	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
338	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
339	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
340	C₂⁴	#41×C₅₃	1	4	yes	—
341	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
342	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
343	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
344	C₂×D₄	—	2	32	no	—
345	C₂×D₄	—	2	8	no	—
346	C₂×D₄	—	2	8	no	—
347	C₂×D₄	C₂×D₄×C₅₃	1	16	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	32	no	—
352	C₂²×C₄	—	2	16	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	8	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	16	no	—
358	C₂²×C₄	—	2	8	no	—
359	C₂²×C₄	—	2	16	no	—
360	C₂²×C₄	—	2	8	no	—
361	C₂²×C₄	—	2	16	no	—
362	C₂²×C₄	—	2	8	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	—	2	8	no	—
366	C₂²×C₄	—	2	8	no	—
367	C₂²×C₄	—	2	16	no	—
368	C₂²×C₄	—	4	32	no	—
369	C₂²×C₄	—	4	16	no	—
370	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
371	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
372	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
373	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
374	C₂²×C₄	#41×C₅₃	1	8	yes	—
375	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
376	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
378	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
379	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
380	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
381	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
382	C₄⋊C₄	—	2	16	no	—
383	C₄⋊C₄	—	2	16	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	32	no	—
386	C₄⋊C₄	—	2	16	no	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	16	no	—
389	C₂²⋊C₄	—	2	32	no	—
390	C₂²⋊C₄	—	2	8	no	—
391	C₄²	—	2	8	no	—
392	C₂³	—	2	8	no	—
393	C₂³	—	2	8	no	—
394	C₂³	—	2	16	no	—
395	C₂³	—	2	16	no	—
396	C₂³	—	2	4	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	2	8	no	—
399	C₂³	—	2	16	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	4	no	—
406	C₂³	—	4	8	no	—
407	C₂³	#121×C₅₃	1	16	yes	—
408	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
409	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
410	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
411	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
412	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
413	C₂³	C₂×#41×C₅₃	1	4	yes	—
414	C₂³	C₂×#41×C₅₃	1	2	yes	—
415	C₂³	C₂×#41×C₅₃	1	4	yes	—
416	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
417	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
418	D₄	—	2	32	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	8	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	32	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	8	no	—
430	C₂×C₄	—	2	8	no	—
431	C₂×C₄	—	2	8	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	8	no	—
435	C₂×C₄	—	4	32	no	—
436	C₂×C₄	—	4	16	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	16	no	—
440	C₂×C₄	—	4	16	no	—
441	C₂×C₄	#123×C₅₃	1	16	yes	complemented
442	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
443	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
444	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
445	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	8	no	—
448	C₂²	—	2	8	no	—
449	C₂²	—	2	32	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	2	8	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	4	16	no	—
456	C₂²	—	4	2	no	—
457	C₂²	C₂×#121×C₅₃	1	8	yes	—
458	C₂²	C₂×#121×C₅₃	1	4	yes	—
459	C₂²	C₂×#123×C₅₃	1	4	yes	—
460	C₂²	C₅₃×#536	1	4	yes	—
461	C₂²	C₅₃×#547	1	8	yes	—
462	C₂²	C₅₃×#552	1	8	yes	—
463	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
464	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
465	C₂²	C₂²×#41×C₅₃	1	2	yes	—
466	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
467	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	8	no	—
472	C₄	—	4	8	no	—
473	C₄	—	4	2	no	—
474	C₄	C₂×#123×C₅₃	1	8	yes	complemented
475	C₂	—	2	8	no	—
476	C₂	—	4	4	no	—
477	C₂	C₅₃×#2486	1	8	yes	direct product factor
478	C₂	C₂²×#121×C₅₃	1	1	yes	characteristic
479	C₂	C₂²×#123×C₅₃	1	1	yes	characteristic
480	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
481	C₂	C₂×C₅₃×#547	1	2	yes	—
482	C₂	C₂×C₅₃×#552	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
3. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
4. C₅₃×#2486	C₂	direct (N × H)
5. C₂×C₅₃×#398	C₂	semidirect
6. C₂×C₅₃×#397	C₂	semidirect
7. C₂×C₅₃×#389	C₂	semidirect
8. C₂×C₅₃×#381	C₂	semidirect
9. C₂×C₅₃×#380	C₂	semidirect
10. C₂×C₅₃×#380	C₂	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
31. C₅₃×#398	C₂²	semidirect
32. C₅₃×#397	C₂²	semidirect
33. C₅₃×#389	C₂²	semidirect
34. C₅₃×#381	C₂²	semidirect
35. C₅₃×#380	C₂²	semidirect
36. C₅₃×#380	C₂²	semidirect
199. C₂×C₄×C₅₃	#123	semidirect
204. C₂×#2486	C₅₃	direct (N × H)
233. C₄×C₅₃	C₂×#123	semidirect
236. C₂³×C₄⋊C₄	C₂×C₅₃	semidirect
237. #2486	C₂×C₅₃	direct (N × H)
238. C₂×#398	C₂×C₅₃	semidirect
239. C₂×#397	C₂×C₅₃	semidirect
240. C₂×#389	C₂×C₅₃	semidirect
241. C₂×#381	C₂×C₅₃	semidirect
242. C₂×#380	C₂×C₅₃	semidirect
243. C₂×#380	C₂×C₅₃	semidirect
246. C₂×C₅₃	#2486	direct (N × H)
266. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
272. #398	C₂²×C₅₃	semidirect
273. #397	C₂²×C₅₃	semidirect
274. #389	C₂²×C₅₃	semidirect
275. #381	C₂²×C₅₃	semidirect
276. #380	C₂²×C₅₃	semidirect
277. #380	C₂²×C₅₃	semidirect
281. C₅₃	C₂×#2486	direct (N × H)
441. C₂×C₄	#123×C₅₃	semidirect
474. C₄	C₂×#123×C₅₃	semidirect
477. C₂	C₅₃×#2486	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×#123×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₅₃×#2486	2
C₂×C₅₃×#398	2
C₂×C₅₃×#397	2
C₂×C₅₃×#389	2
C₂×C₅₃×#381	2
C₂×C₅₃×#380	2
C₂×#2486	53

Subgroups — C2×C53×#2486 #27242267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2486 #27242267