Subgroups — C₅₃×#2485 #20271422

Order 6784.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#397	C₂	1	2	yes	maximal, complemented
4	C₅₃×#394	C₂	1	2	yes	maximal
5	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#388	C₂	1	2	yes	maximal, complemented
8	C₅₃×#387	C₂	1	2	yes	maximal, complemented
9	C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₅₃	—	2	4	no	—
16	#117×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	#95×C₅₃	C₂²	1	2	yes	—
33	#95×C₅₃	C₂²	1	2	yes	—
34	#95×C₅₃	C₂²	1	2	yes	—
35	#95×C₅₃	C₂²	1	2	yes	—
36	#95×C₅₃	C₂²	1	2	yes	—
37	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	D₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₂²⋊C₄×C₅₃	—	2	4	no	—
67	C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂²⋊C₄×C₅₃	—	4	2	no	—
69	C₂²⋊C₄×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	2	1	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	C₂×D₄	1	2	yes	—
85	C₂³×C₅₃	#41	1	2	yes	—
86	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
87	D₄×C₅₃	—	2	8	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	8	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	#41	1	2	yes	—
115	C₂×C₄×C₅₃	#41	1	2	yes	—
116	C₂×C₄×C₅₃	#41	1	2	yes	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	—	4	1	no	—
130	C₂²×C₅₃	—	4	1	no	—
131	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
136	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	4	1	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	4	1	no	—
143	C₄×C₅₃	—	4	2	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	#123	1	4	yes	—
147	#2485	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₅₃	—	4	2	no	—
149	C₂×C₅₃	—	4	2	no	—
150	C₂×C₅₃	C₂×#123	1	1	yes	characteristic
151	C₂×C₅₃	#525	1	1	yes	characteristic
152	C₂×C₅₃	#537	1	1	yes	characteristic
153	C₂×C₅₃	#552	1	1	yes	characteristic
154	C₂×C₅₃	#553	1	1	yes	characteristic
155	C₂×C₅₃	#555	1	1	yes	characteristic
156	C₂×C₅₃	#556	1	1	yes	characteristic
157	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
158	C₂×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
159	#397	C₂×C₅₃	1	2	yes	complemented
160	#394	C₂×C₅₃	1	2	yes	—
161	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
162	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
163	#388	C₂×C₅₃	1	2	yes	complemented
164	#387	C₂×C₅₃	1	2	yes	complemented
165	#380	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#380	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#380	C₂×C₅₃	1	1	yes	characteristic, complemented
168	C₅₃	#2485	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
170	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
171	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
172	C₄×D₄	—	2	4	no	—
173	#117	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
180	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
187	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
188	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
189	#95	C₂²×C₅₃	1	2	yes	—
190	#95	C₂²×C₅₃	1	2	yes	—
191	#95	C₂²×C₅₃	1	2	yes	—
192	#95	C₂²×C₅₃	1	2	yes	—
193	#95	C₂²×C₅₃	1	2	yes	—
194	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
195	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	D₄×C₅₃	1	4	yes	—
205	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
207	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
208	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
210	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
211	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	C₂×D₄×C₅₃	1	2	yes	—
242	C₂³	#41×C₅₃	1	2	yes	—
243	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
244	D₄	—	2	8	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
270	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
271	C₂×C₄	#41×C₅₃	1	2	yes	—
272	C₂×C₄	#41×C₅₃	1	2	yes	—
273	C₂×C₄	#41×C₅₃	1	2	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
289	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
290	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
291	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
292	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
293	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
294	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	#123×C₅₃	1	4	yes	—
304	C₂	—	4	2	no	—
305	C₂	—	4	2	no	—
306	C₂	C₂×#123×C₅₃	1	1	yes	characteristic
307	C₂	C₅₃×#525	1	1	yes	characteristic
308	C₂	C₅₃×#537	1	1	yes	characteristic
309	C₂	C₅₃×#552	1	1	yes	characteristic
310	C₂	C₅₃×#553	1	1	yes	characteristic
311	C₂	C₅₃×#555	1	1	yes	characteristic
312	C₂	C₅₃×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
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
2. C₂×#117×C₅₃	C₂	semidirect
3. C₅₃×#397	C₂	semidirect
5. C₅₃×#389	C₂	semidirect
6. C₅₃×#389	C₂	semidirect
7. C₅₃×#388	C₂	semidirect
8. C₅₃×#387	C₂	semidirect
9. C₅₃×#380	C₂	semidirect
10. C₅₃×#380	C₂	semidirect
11. C₅₃×#380	C₂	semidirect
147. #2485	C₅₃	direct (N × H)
158. C₂×#117	C₂×C₅₃	semidirect
159. #397	C₂×C₅₃	semidirect
161. #389	C₂×C₅₃	semidirect
162. #389	C₂×C₅₃	semidirect
163. #388	C₂×C₅₃	semidirect
164. #387	C₂×C₅₃	semidirect
165. #380	C₂×C₅₃	semidirect
166. #380	C₂×C₅₃	semidirect
167. #380	C₂×C₅₃	semidirect
168. C₅₃	#2485	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₅₃	2
C₂×#117×C₅₃	2
C₅₃×#397	2
C₅₃×#394	2
C₅₃×#389	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#380	2
#2485	53

Subgroups — C53×#2485 #20271422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2485 #20271422