Subgroups — C₅₃×#2437 #20271374

Order 6784.

780 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#396	C₂	1	2	yes	maximal, complemented
5	C₅₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#385	C₂	1	2	yes	maximal, complemented
7	C₅₃×#385	C₂	1	2	yes	maximal, complemented
8	C₅₃×#380	C₂	1	2	yes	maximal, complemented
9	C₅₃×#379	C₂	1	1	yes	maximal, characteristic
10	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂²⋊Q₈×C₅₃	—	2	8	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
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	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	#95×C₅₃	C₂²	1	2	yes	complemented
31	#95×C₅₃	C₂²	1	2	yes	complemented
32	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	C₂⁴×C₅₃	—	2	4	no	—
34	C₂⁴×C₅₃	—	2	4	no	—
35	C₂⁴×C₅₃	—	4	1	no	—
36	C₂⁴×C₅₃	—	4	1	no	—
37	C₂⁴×C₅₃	Q₈	1	4	yes	complemented
38	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₅₃	C₂³	1	2	yes	—
40	C₂×Q₈×C₅₃	—	2	4	no	—
41	C₂×Q₈×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
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	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	1	no	—
60	C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₂²⋊C₄×C₅₃	—	2	4	no	—
62	C₂²⋊C₄×C₅₃	—	2	4	no	—
63	C₂²⋊C₄×C₅₃	—	2	4	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₂²⋊C₄×C₅₃	—	4	1	no	—
66	C₂²⋊C₄×C₅₃	—	4	1	no	—
67	C₂²⋊C₄×C₅₃	D₄	1	8	yes	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	—	2	16	no	—
73	C₂³×C₅₃	—	2	2	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	4	4	no	—
78	C₂³×C₅₃	—	4	4	no	—
79	C₂³×C₅₃	—	4	8	no	—
80	C₂³×C₅₃	—	4	4	no	—
81	C₂³×C₅₃	C₂×D₄	1	4	yes	—
82	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
84	C₂³×C₅₃	#41	1	4	yes	—
85	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	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	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	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	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	4	8	no	—
105	C₂×C₄×C₅₃	—	4	8	no	—
106	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
108	C₂²×C₅₃	—	2	4	no	—
109	C₂²×C₅₃	—	2	4	no	—
110	C₂²×C₅₃	—	2	4	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	16	no	—
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	—	4	1	no	—
120	C₂²×C₅₃	C₂²⋊Q₈	1	8	yes	complemented
121	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
122	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×#41	1	2	yes	—
124	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	4	2	no	—
129	C₄×C₅₃	—	4	1	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	—	4	1	no	—
133	#2437	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₅₃	—	2	8	no	—
135	C₂×C₅₃	—	4	2	no	—
136	C₂×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₂×C₅₃	#543	1	1	yes	characteristic
138	C₂×C₅₃	D₄²	1	1	yes	characteristic
139	C₂×C₅₃	#546	1	2	yes	—
140	C₂×C₅₃	#551	1	1	yes	characteristic
141	C₂×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
142	C₂²×C₂²⋊C₄	C₂×C₅₃	1	2	yes	—
143	#399	C₂×C₅₃	1	1	yes	characteristic, complemented
144	#396	C₂×C₅₃	1	2	yes	complemented
145	#393	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#385	C₂×C₅₃	1	2	yes	complemented
147	#385	C₂×C₅₃	1	2	yes	complemented
148	#380	C₂×C₅₃	1	2	yes	complemented
149	#379	C₂×C₅₃	1	1	yes	characteristic
150	C₅₃	#2437	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
152	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
153	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
154	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
155	C₂²⋊Q₈	—	2	8	no	—
156	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
157	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
158	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
159	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
160	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
161	C₂×C₂²⋊C₄	—	2	8	no	—
162	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
170	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
171	#95	C₂²×C₅₃	1	2	yes	complemented
172	#95	C₂²×C₅₃	1	2	yes	complemented
173	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
174	C₂⁴	—	2	4	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	—	4	1	no	—
177	C₂⁴	—	4	1	no	—
178	C₂⁴	Q₈×C₅₃	1	4	yes	complemented
179	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
180	C₂⁴	C₂³×C₅₃	1	2	yes	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	4	1	no	—
207	C₂²⋊C₄	—	4	1	no	—
208	C₂²⋊C₄	D₄×C₅₃	1	8	yes	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	16	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	4	8	no	—
221	C₂³	—	4	4	no	—
222	C₂³	C₂×D₄×C₅₃	1	4	yes	—
223	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
224	C₂³	C₂×Q₈×C₅₃	1	4	yes	—
225	C₂³	#41×C₅₃	1	4	yes	—
226	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	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	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
248	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	16	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	C₂²⋊Q₈×C₅₃	1	8	yes	complemented
262	C₂²	C₂²×D₄×C₅₃	1	2	yes	—
263	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
264	C₂²	C₂×#41×C₅₃	1	2	yes	—
265	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
266	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₂	—	2	8	no	—
275	C₂	—	4	2	no	—
276	C₂	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
277	C₂	C₅₃×#543	1	1	yes	characteristic
278	C₂	D₄²×C₅₃	1	1	yes	characteristic
279	C₂	C₅₃×#546	1	2	yes	—
280	C₂	C₅₃×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁵×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
1. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#399	C₂	semidirect
4. C₅₃×#396	C₂	semidirect
5. C₅₃×#393	C₂	semidirect
6. C₅₃×#385	C₂	semidirect
7. C₅₃×#385	C₂	semidirect
8. C₅₃×#380	C₂	semidirect
11. C₂²×Q₈×C₅₃	C₂²	semidirect
15. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
16. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
17. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
18. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
19. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
29. #95×C₅₃	C₂²	semidirect
30. #95×C₅₃	C₂²	semidirect
31. #95×C₅₃	C₂²	semidirect
32. #95×C₅₃	C₂²	semidirect
37. C₂⁴×C₅₃	Q₈	semidirect
120. C₂²×C₅₃	C₂²⋊Q₈	semidirect
133. #2437	C₅₃	direct (N × H)
141. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
143. #399	C₂×C₅₃	semidirect
144. #396	C₂×C₅₃	semidirect
145. #393	C₂×C₅₃	semidirect
146. #385	C₂×C₅₃	semidirect
147. #385	C₂×C₅₃	semidirect
148. #380	C₂×C₅₃	semidirect
150. C₅₃	#2437	direct (N × H)
152. C₂²×Q₈	C₂²×C₅₃	semidirect
156. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
157. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
158. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
159. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
160. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
170. #95	C₂²×C₅₃	semidirect
171. #95	C₂²×C₅₃	semidirect
172. #95	C₂²×C₅₃	semidirect
173. #95	C₂²×C₅₃	semidirect
178. C₂⁴	Q₈×C₅₃	semidirect
261. C₂²	C₂²⋊Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₅₃	2
C₂²×C₂²⋊C₄×C₅₃	2
C₅₃×#399	2
C₅₃×#396	2
C₅₃×#393	2
C₅₃×#385	2
C₅₃×#380	2
C₅₃×#379	2
#2437	53

Subgroups — C53×#2437 #20271374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2437 #20271374