Subgroups — C₅₃×#2678 #20271615

Order 6784.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#120×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#120×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#396	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#392	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#392	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#388	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#386	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	#123×C₅₃	—	2	4	no	—
26	#121×C₅₃	—	2	4	no	—
27	#120×C₅₃	—	2	4	no	—
28	#120×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
52	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
53	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
54	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₂⁴×C₅₃	—	2	2	no	—
56	C₂⁴×C₅₃	—	2	2	no	—
57	C₂⁴×C₅₃	—	2	2	no	—
58	C₂⁴×C₅₃	—	4	1	no	—
59	C₂⁴×C₅₃	—	4	1	no	—
60	C₂⁴×C₅₃	—	4	1	no	—
61	C₂⁴×C₅₃	—	4	1	no	—
62	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
63	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
64	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
65	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
66	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	2	2	no	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	4	1	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	4	1	no	—
80	C₂×D₄×C₅₃	—	4	1	no	—
81	C₂×D₄×C₅₃	—	4	1	no	—
82	C₂×D₄×C₅₃	—	4	1	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	—	4	1	no	—
85	C₂×D₄×C₅₃	—	4	1	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×D₄×C₅₃	D₄	1	4	yes	complemented
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	4	1	no	—
91	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₂²⋊C₄×C₅₃	—	2	2	no	—
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	—	4	1	no	—
115	C₂²⋊C₄×C₅₃	—	4	1	no	—
116	C₂²⋊C₄×C₅₃	—	4	1	no	—
117	C₂²⋊C₄×C₅₃	—	4	1	no	—
118	C₂²⋊C₄×C₅₃	—	4	1	no	—
119	C₂²⋊C₄×C₅₃	—	4	1	no	—
120	C₂²⋊C₄×C₅₃	—	4	1	no	—
121	C₂²⋊C₄×C₅₃	D₄	1	4	yes	complemented
122	C₄²×C₅₃	—	4	1	no	—
123	C₂³×C₅₃	—	2	1	no	—
124	C₂³×C₅₃	—	2	1	no	—
125	C₂³×C₅₃	—	2	1	no	—
126	C₂³×C₅₃	—	2	1	no	—
127	C₂³×C₅₃	—	2	1	no	—
128	C₂³×C₅₃	—	2	1	no	—
129	C₂³×C₅₃	—	2	1	no	—
130	C₂³×C₅₃	—	2	1	no	—
131	C₂³×C₅₃	—	2	1	no	—
132	C₂³×C₅₃	—	2	1	no	—
133	C₂³×C₅₃	—	2	1	no	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	—	2	1	no	—
136	C₂³×C₅₃	—	2	1	no	—
137	C₂³×C₅₃	—	2	1	no	—
138	C₂³×C₅₃	—	2	1	no	—
139	C₂³×C₅₃	—	2	1	no	—
140	C₂³×C₅₃	—	2	1	no	—
141	C₂³×C₅₃	—	2	1	no	—
142	C₂³×C₅₃	—	2	1	no	—
143	C₂³×C₅₃	—	2	1	no	—
144	C₂³×C₅₃	—	2	1	no	—
145	C₂³×C₅₃	—	2	1	no	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	—	2	1	no	—
148	C₂³×C₅₃	—	2	1	no	—
149	C₂³×C₅₃	—	2	1	no	—
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	—	2	1	no	—
152	C₂³×C₅₃	—	2	1	no	—
153	C₂³×C₅₃	—	2	1	no	—
154	C₂³×C₅₃	—	4	4	no	—
155	C₂³×C₅₃	—	4	4	no	—
156	C₂³×C₅₃	—	4	4	no	—
157	C₂³×C₅₃	—	4	4	no	—
158	C₂³×C₅₃	—	4	4	no	—
159	C₂³×C₅₃	—	4	4	no	—
160	C₂³×C₅₃	—	4	4	no	—
161	C₂³×C₅₃	C₂×D₄	1	2	yes	—
162	C₂³×C₅₃	C₂×D₄	1	2	yes	—
163	C₂³×C₅₃	C₂×D₄	1	2	yes	—
164	C₂³×C₅₃	#41	1	2	yes	—
165	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
166	D₄×C₅₃	—	4	4	no	—
167	D₄×C₅₃	—	4	4	no	—
168	D₄×C₅₃	—	4	4	no	—
169	D₄×C₅₃	—	4	4	no	—
170	D₄×C₅₃	—	4	4	no	—
171	D₄×C₅₃	—	4	4	no	—
172	D₄×C₅₃	—	4	4	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	1	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	4	4	no	—
201	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
202	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
203	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	2	2	no	—
209	C₂²×C₅₃	—	2	2	no	—
210	C₂²×C₅₃	—	2	2	no	—
211	C₂²×C₅₃	—	2	2	no	—
212	C₂²×C₅₃	—	2	2	no	—
213	C₂²×C₅₃	—	2	2	no	—
214	C₂²×C₅₃	—	2	2	no	—
215	C₂²×C₅₃	—	2	2	no	—
216	C₂²×C₅₃	—	4	16	no	—
217	C₂²×C₅₃	—	4	1	no	—
218	C₂²×C₅₃	—	4	1	no	—
219	C₂²×C₅₃	—	4	1	no	—
220	C₂²×C₅₃	—	4	1	no	—
221	C₂²×C₅₃	—	4	1	no	—
222	C₂²×C₅₃	—	4	1	no	—
223	C₂²×C₅₃	—	4	1	no	—
224	C₂²×C₅₃	—	4	1	no	—
225	C₂²×C₅₃	—	4	1	no	—
226	C₂²×C₅₃	—	4	1	no	—
227	C₂²×C₅₃	—	4	1	no	—
228	C₂²×C₅₃	—	4	1	no	—
229	C₂²×C₅₃	—	4	1	no	—
230	C₂²×C₅₃	—	4	1	no	—
231	C₂²×C₅₃	—	4	1	no	—
232	C₂²×C₅₃	—	4	1	no	—
233	C₂²×C₅₃	—	4	1	no	—
234	C₂²×C₅₃	—	4	1	no	—
235	C₂²×C₅₃	—	4	1	no	—
236	C₂²×C₅₃	—	4	1	no	—
237	C₂²×C₅₃	—	4	1	no	—
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	—	4	1	no	—
240	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
241	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
242	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
243	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
245	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
246	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
247	C₄×C₅₃	—	2	2	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	2	2	no	—
250	C₄×C₅₃	—	4	1	no	—
251	C₄×C₅₃	—	4	1	no	—
252	C₄×C₅₃	—	4	1	no	—
253	C₄×C₅₃	—	4	1	no	—
254	C₄×C₅₃	—	4	1	no	—
255	C₄×C₅₃	—	4	1	no	—
256	C₄×C₅₃	—	4	1	no	—
257	#2678	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂×C₅₃	—	4	2	no	—
259	C₂×C₅₃	—	4	2	no	—
260	C₂×C₅₃	—	4	2	no	—
261	C₂×C₅₃	—	4	2	no	—
262	C₂×C₅₃	—	8	1	no	—
263	C₂×C₅₃	#534	1	1	yes	characteristic
264	C₂×C₅₃	#535	1	1	yes	characteristic
265	C₂×C₅₃	#538	1	1	yes	characteristic
266	C₂×C₅₃	D₄²	1	1	yes	characteristic
267	C₂×C₅₃	#546	1	1	yes	characteristic
268	C₂×C₅₃	#546	1	1	yes	characteristic
269	C₂×C₅₃	#560	1	1	yes	characteristic
270	C₂×#123	C₂×C₅₃	1	1	yes	characteristic, complemented
271	C₂×#121	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂×#120	C₂×C₅₃	1	1	yes	characteristic, complemented
273	C₂×#120	C₂×C₅₃	1	1	yes	characteristic, complemented
274	#397	C₂×C₅₃	1	1	yes	characteristic, complemented
275	#396	C₂×C₅₃	1	1	yes	characteristic, complemented
276	#394	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#394	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#392	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#392	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#390	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#390	C₂×C₅₃	1	1	yes	characteristic, complemented
282	#388	C₂×C₅₃	1	1	yes	characteristic, complemented
283	#386	C₂×C₅₃	1	1	yes	characteristic, complemented
284	#379	C₂×C₅₃	1	1	yes	characteristic, complemented
285	C₅₃	#2678	1	1	yes	characteristic, direct product factor, Sylow
286	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
287	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
288	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
289	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
290	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
291	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
292	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
293	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
294	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
295	#123	—	2	4	no	—
296	#121	—	2	4	no	—
297	#120	—	2	4	no	—
298	#120	—	2	4	no	—
299	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
300	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
301	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
302	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
303	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
310	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
316	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
317	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
318	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
319	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
320	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
321	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
322	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
323	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
324	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
325	C₂⁴	—	2	2	no	—
326	C₂⁴	—	2	2	no	—
327	C₂⁴	—	2	2	no	—
328	C₂⁴	—	4	1	no	—
329	C₂⁴	—	4	1	no	—
330	C₂⁴	—	4	1	no	—
331	C₂⁴	—	4	1	no	—
332	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
333	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
334	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
335	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
336	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂×D₄	—	4	1	no	—
349	C₂×D₄	—	4	1	no	—
350	C₂×D₄	—	4	1	no	—
351	C₂×D₄	—	4	1	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	1	no	—
354	C₂×D₄	—	4	1	no	—
355	C₂×D₄	—	4	1	no	—
356	C₂×D₄	—	4	1	no	—
357	C₂×D₄	D₄×C₅₃	1	4	yes	complemented
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
362	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
363	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
370	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₂²⋊C₄	—	2	2	no	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	2	2	no	—
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	—	4	1	no	—
385	C₂²⋊C₄	—	4	1	no	—
386	C₂²⋊C₄	—	4	1	no	—
387	C₂²⋊C₄	—	4	1	no	—
388	C₂²⋊C₄	—	4	1	no	—
389	C₂²⋊C₄	—	4	1	no	—
390	C₂²⋊C₄	—	4	1	no	—
391	C₂²⋊C₄	D₄×C₅₃	1	4	yes	complemented
392	C₄²	—	4	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	2	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	1	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	4	4	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	4	no	—
428	C₂³	—	4	4	no	—
429	C₂³	—	4	4	no	—
430	C₂³	—	4	4	no	—
431	C₂³	C₂×D₄×C₅₃	1	2	yes	—
432	C₂³	C₂×D₄×C₅₃	1	2	yes	—
433	C₂³	C₂×D₄×C₅₃	1	2	yes	—
434	C₂³	#41×C₅₃	1	2	yes	—
435	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
436	D₄	—	4	4	no	—
437	D₄	—	4	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	4	no	—
442	D₄	—	4	4	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
472	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
473	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	2	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	4	16	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	4	1	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	4	1	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	—	4	1	no	—
510	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
511	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
512	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
513	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
514	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
515	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
516	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	2	no	—
520	C₄	—	4	1	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	1	no	—
523	C₄	—	4	1	no	—
524	C₄	—	4	1	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	1	no	—
527	C₂	—	4	2	no	—
528	C₂	—	4	2	no	—
529	C₂	—	4	2	no	—
530	C₂	—	4	2	no	—
531	C₂	—	8	1	no	—
532	C₂	C₅₃×#534	1	1	yes	characteristic
533	C₂	C₅₃×#535	1	1	yes	characteristic
534	C₂	C₅₃×#538	1	1	yes	characteristic
535	C₂	D₄²×C₅₃	1	1	yes	characteristic
536	C₂	C₅₃×#546	1	1	yes	characteristic
537	C₂	C₅₃×#546	1	1	yes	characteristic
538	C₂	C₅₃×#560	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#123×C₅₃	C₂	semidirect
2. C₂×#121×C₅₃	C₂	semidirect
3. C₂×#120×C₅₃	C₂	semidirect
4. C₂×#120×C₅₃	C₂	semidirect
5. C₅₃×#397	C₂	semidirect
6. C₅₃×#396	C₂	semidirect
7. C₅₃×#394	C₂	semidirect
8. C₅₃×#394	C₂	semidirect
9. C₅₃×#392	C₂	semidirect
10. C₅₃×#392	C₂	semidirect
11. C₅₃×#390	C₂	semidirect
12. C₅₃×#390	C₂	semidirect
13. C₅₃×#388	C₂	semidirect
14. C₅₃×#386	C₂	semidirect
15. C₅₃×#379	C₂	semidirect
23. C₂²×D₄×C₅₃	C₂²	semidirect
24. C₂³×C₄×C₅₃	C₂²	semidirect
29. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
30. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
31. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
32. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
45. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
46. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
47. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
50. C₂×C₄²×C₅₃	C₂²	semidirect
51. #95×C₅₃	C₂²	semidirect
52. #95×C₅₃	C₂²	semidirect
53. #95×C₅₃	C₂²	semidirect
54. #95×C₅₃	C₂²	semidirect
87. C₂×D₄×C₅₃	D₄	semidirect
121. C₂²⋊C₄×C₅₃	D₄	semidirect
257. #2678	C₅₃	direct (N × H)
270. C₂×#123	C₂×C₅₃	semidirect
271. C₂×#121	C₂×C₅₃	semidirect
272. C₂×#120	C₂×C₅₃	semidirect
273. C₂×#120	C₂×C₅₃	semidirect
274. #397	C₂×C₅₃	semidirect
275. #396	C₂×C₅₃	semidirect
276. #394	C₂×C₅₃	semidirect
277. #394	C₂×C₅₃	semidirect
278. #392	C₂×C₅₃	semidirect
279. #392	C₂×C₅₃	semidirect
280. #390	C₂×C₅₃	semidirect
281. #390	C₂×C₅₃	semidirect
282. #388	C₂×C₅₃	semidirect
283. #386	C₂×C₅₃	semidirect
284. #379	C₂×C₅₃	semidirect
285. C₅₃	#2678	direct (N × H)
293. C₂²×D₄	C₂²×C₅₃	semidirect
294. C₂³×C₄	C₂²×C₅₃	semidirect
299. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
300. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
301. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
302. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
315. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
316. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
317. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
318. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
319. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
320. C₂×C₄²	C₂²×C₅₃	semidirect
321. #95	C₂²×C₅₃	semidirect
322. #95	C₂²×C₅₃	semidirect
323. #95	C₂²×C₅₃	semidirect
324. #95	C₂²×C₅₃	semidirect
357. C₂×D₄	D₄×C₅₃	semidirect
391. C₂²⋊C₄	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₅₃	2
C₂×#121×C₅₃	2
C₂×#120×C₅₃	2
C₅₃×#397	2
C₅₃×#396	2
C₅₃×#394	2
C₅₃×#392	2
C₅₃×#390	2
C₅₃×#388	2
C₅₃×#386	2
C₅₃×#379	2
#2678	53

Subgroups — C53×#2678 #20271615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2678 #20271615