Subgroups — C₅₃×#2322 #20271259

Order 6784.

632 conjugacy classes of subgroups.

The subgroup lattice has 290 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
2	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#389	C₂	1	2	yes	maximal, complemented
5	C₅₃×#388	C₂	1	2	yes	maximal, complemented
6	C₅₃×#387	C₂	1	2	yes	maximal, complemented
7	C₅₃×#386	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#385	C₂	1	2	yes	maximal, complemented
9	C₅₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₅₃	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	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	#123×C₅₃	C₄	1	8	yes	complemented
18	C₄×D₄×C₅₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊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	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×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	#95×C₅₃	C₂²	1	2	yes	—
38	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	—	2	2	no	—
42	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
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	—	4	2	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₅₃	D₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
66	C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂²⋊C₄×C₅₃	—	2	2	no	—
69	C₂²⋊C₄×C₅₃	—	4	2	no	—
70	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
71	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	complemented
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂³×C₅₃	C₂×D₄	1	2	yes	—
83	C₂³×C₅₃	#41	1	2	yes	—
84	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
85	D₄×C₅₃	—	2	8	no	—
86	C₂×C₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	2	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	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
109	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	2	4	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	4	no	—
119	C₂²×C₅₃	—	4	1	no	—
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	—	4	2	no	—
122	C₂²×C₅₃	C₄×D₄	1	4	yes	—
123	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
127	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
128	C₄×C₅₃	—	2	4	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	2	4	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	—	4	2	no	—
135	C₄×C₅₃	—	4	1	no	—
136	C₄×C₅₃	—	4	2	no	—
137	#2322	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₅₃	—	2	4	no	—
139	C₂×C₅₃	—	4	2	no	—
140	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₅₃	#520	1	2	yes	—
142	C₂×C₅₃	#534	1	1	yes	characteristic
143	C₂×C₅₃	#536	1	1	yes	characteristic
144	C₂×C₅₃	#539	1	2	yes	—
145	C₂×#123	C₂×C₅₃	1	1	yes	characteristic
146	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
147	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#389	C₂×C₅₃	1	2	yes	complemented
149	#388	C₂×C₅₃	1	2	yes	complemented
150	#387	C₂×C₅₃	1	2	yes	complemented
151	#386	C₂×C₅₃	1	1	yes	characteristic
152	#385	C₂×C₅₃	1	2	yes	complemented
153	#385	C₂×C₅₃	1	1	yes	characteristic, complemented
154	#381	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
156	C₅₃	#2322	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
160	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
161	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
162	#123	C₄×C₅₃	1	8	yes	complemented
163	C₄×D₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
176	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
178	#95	C₂²×C₅₃	1	2	yes	—
179	#95	C₂²×C₅₃	1	2	yes	—
180	#95	C₂²×C₅₃	1	2	yes	—
181	#95	C₂²×C₅₃	1	2	yes	—
182	#95	C₂²×C₅₃	1	2	yes	—
183	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
184	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₂×C₄×C₅₃	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂×C₄×C₅₃	1	4	yes	complemented
196	C₂²×C₄	D₄×C₅₃	1	4	yes	—
197	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
202	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	complemented
216	C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
217	C₄²	—	2	2	no	—
218	C₄²	—	4	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
226	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
227	C₂³	C₂×D₄×C₅₃	1	2	yes	—
228	C₂³	#41×C₅₃	1	2	yes	—
229	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
230	D₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
254	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
255	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
256	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	C₄×D₄×C₅₃	1	4	yes	—
268	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
269	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
270	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
271	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
272	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	2	no	—
282	C₂	—	2	4	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
285	C₂	C₅₃×#520	1	2	yes	—
286	C₂	C₅₃×#534	1	1	yes	characteristic
287	C₂	C₅₃×#536	1	1	yes	characteristic
288	C₂	C₅₃×#539	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊C₄×C₅₃	C₂	semidirect
4. C₅₃×#389	C₂	semidirect
5. C₅₃×#388	C₂	semidirect
6. C₅₃×#387	C₂	semidirect
8. C₅₃×#385	C₂	semidirect
9. C₅₃×#385	C₂	semidirect
10. C₅₃×#381	C₂	semidirect
11. C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
17. #123×C₅₃	C₄	semidirect
50. C₂²×C₄×C₅₃	C₂×C₄	semidirect
65. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
70. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
71. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
137. #2322	C₅₃	direct (N × H)
147. C₂²×C₄⋊C₄	C₂×C₅₃	semidirect
148. #389	C₂×C₅₃	semidirect
149. #388	C₂×C₅₃	semidirect
150. #387	C₂×C₅₃	semidirect
152. #385	C₂×C₅₃	semidirect
153. #385	C₂×C₅₃	semidirect
154. #381	C₂×C₅₃	semidirect
155. C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
156. C₅₃	#2322	direct (N × H)
162. #123	C₄×C₅₃	semidirect
195. C₂²×C₄	C₂×C₄×C₅₃	semidirect
210. C₄⋊C₄	C₂×C₄×C₅₃	semidirect
215. C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
216. C₂²⋊C₄	C₂×C₄×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×C₄×D₄×C₅₃	2
C₂²×C₄⋊C₄×C₅₃	2
C₅₃×#389	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#386	2
C₅₃×#385	2
C₅₃×#381	2
C₄×C₂²⋊C₄×C₅₃	2
#2322	53

Subgroups — C53×#2322 #20271259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2322 #20271259