Subgroups — C₅₃×#21939 #27229451

Order 13568.

1122 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2280	C₂	1	2	yes	maximal
3	C₅₃×#1854	C₂	1	8	yes	maximal, complemented
4	C₅₃×#1808	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1807	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	4	yes	—
10	C₂×#107×C₅₃	C₂²	1	4	yes	complemented
11	C₂×#106×C₅₃	C₂²	1	4	yes	complemented
12	C₂×C₂²⋊C₈×C₅₃	C₄	1	8	yes	complemented
13	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
14	C₅₃×#381	—	2	2	no	—
15	C₅₃×#380	C₂²	1	4	yes	—
16	C₅₃×#380	C₂²	1	4	yes	—
17	C₅₃×#379	—	2	2	no	—
18	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
20	C₅₃×#340	C₂²	1	4	yes	complemented
21	C₅₃×#340	C₂²	1	4	yes	complemented
22	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	—	4	2	no	—
24	C₂³×C₄×C₅₃	—	4	2	no	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	D₄	1	2	yes	—
28	C₂³×C₄×C₅₃	Q₈	1	4	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	—	4	2	no	—
32	C₂²×C₈×C₅₃	—	4	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	#117×C₅₃	—	4	4	no	—
38	#117×C₅₃	—	4	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
43	C₂×C₄²×C₅₃	—	2	4	no	—
44	#107×C₅₃	—	4	4	no	—
45	#106×C₅₃	—	4	4	no	—
46	C₂²⋊C₈×C₅₃	C₂×C₄	1	16	yes	—
47	#95×C₅₃	—	2	4	no	—
48	#95×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	—	2	4	no	—
50	C₂⁴×C₅₃	—	4	2	no	—
51	C₂⁴×C₅₃	—	4	2	no	—
52	C₂⁴×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
54	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	8	no	—
60	C₂²×C₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	—	4	8	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	4	4	no	—
77	C₂×C₈×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
80	C₄⋊C₄×C₅₃	—	2	8	no	—
81	C₄⋊C₄×C₅₃	—	2	8	no	—
82	C₂²⋊C₄×C₅₃	—	4	8	no	—
83	C₂²⋊C₄×C₅₃	—	4	8	no	—
84	C₂²⋊C₄×C₅₃	—	4	8	no	—
85	C₄²×C₅₃	—	4	8	no	—
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	—	2	2	no	—
88	C₂³×C₅₃	—	2	4	no	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	2	4	no	—
91	C₂³×C₅₃	—	2	8	no	—
92	C₂³×C₅₃	—	4	4	no	—
93	C₂³×C₅₃	—	4	4	no	—
94	C₂³×C₅₃	—	4	16	no	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₅₃	#135	1	4	yes	—
98	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	8	no	—
109	C₂×C₄×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	—	4	8	no	—
111	C₂×C₄×C₅₃	—	4	8	no	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
117	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
118	C₈×C₅₃	—	2	4	no	—
119	C₈×C₅₃	—	2	4	no	—
120	#21939	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	2	8	no	—
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	4	16	no	—
128	C₂²×C₅₃	—	4	8	no	—
129	C₂²×C₅₃	#427	1	4	yes	—
130	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₅₃	#534	1	2	yes	—
132	C₂²×C₅₃	C₂×#135	1	2	yes	—
133	C₂²×C₅₃	#575	1	2	yes	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	—	8	8	no	—
137	C₄×C₅₃	#518	1	1	yes	characteristic
138	C₄×C₅₃	#518	1	1	yes	characteristic
139	C₄×C₅₃	#543	1	2	yes	—
140	C₂²×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic
141	#2280	C₂×C₅₃	1	2	yes	—
142	#1854	C₂×C₅₃	1	8	yes	complemented
143	#1808	C₂×C₅₃	1	2	yes	complemented
144	#1807	C₂×C₅₃	1	2	yes	complemented
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	—	4	4	no	—
147	C₂×C₅₃	#2315	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
149	C₂×C₅₃	#2911	1	1	yes	characteristic
150	C₂×C₅₃	#3191	1	4	yes	—
151	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
152	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×#117	C₂²×C₅₃	1	4	yes	—
155	C₂×#107	C₂²×C₅₃	1	4	yes	complemented
156	C₂×#106	C₂²×C₅₃	1	4	yes	complemented
157	C₂×C₂²⋊C₈	C₄×C₅₃	1	8	yes	complemented
158	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
159	#381	—	2	2	no	—
160	#380	C₂²×C₅₃	1	4	yes	—
161	#380	C₂²×C₅₃	1	4	yes	—
162	#379	—	2	2	no	—
163	C₄×C₂²⋊C₄	—	2	2	no	—
164	C₄×C₂²⋊C₄	—	2	2	no	—
165	#340	C₂²×C₅₃	1	4	yes	complemented
166	#340	C₂²×C₅₃	1	4	yes	complemented
167	C₅₃	#21939	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁵	D₄×C₅₃	1	1	yes	characteristic
169	C₂³×C₄	—	4	2	no	—
170	C₂³×C₄	—	4	2	no	—
171	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
172	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₅₃	1	2	yes	—
174	C₂³×C₄	Q₈×C₅₃	1	4	yes	—
175	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
176	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
177	C₂²×C₈	—	4	2	no	—
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
180	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
181	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
183	#117	—	4	4	no	—
184	#117	—	4	4	no	—
185	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
186	C₂×C₂²⋊C₄	—	2	4	no	—
187	C₂×C₂²⋊C₄	—	2	4	no	—
188	C₂×C₂²⋊C₄	—	2	4	no	—
189	C₂×C₄²	—	2	4	no	—
190	#107	—	4	4	no	—
191	#106	—	4	4	no	—
192	C₂²⋊C₈	C₂×C₄×C₅₃	1	16	yes	—
193	#95	—	2	4	no	—
194	#95	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	4	2	no	—
197	C₂⁴	—	4	2	no	—
198	C₂⁴	C₄⋊C₄×C₅₃	1	4	yes	—
199	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
200	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
212	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
217	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
218	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
225	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	4	8	no	—
229	C₂²⋊C₄	—	4	8	no	—
230	C₂²⋊C₄	—	4	8	no	—
231	C₄²	—	4	8	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	16	no	—
241	C₂³	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
242	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
243	C₂³	#135×C₅₃	1	4	yes	—
244	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
262	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
263	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	2	4	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	8	no	—
274	C₂²	C₅₃×#427	1	4	yes	—
275	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
276	C₂²	C₅₃×#534	1	2	yes	—
277	C₂²	C₂×#135×C₅₃	1	2	yes	—
278	C₂²	C₅₃×#575	1	2	yes	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	8	no	—
282	C₄	C₅₃×#518	1	1	yes	characteristic
283	C₄	C₅₃×#518	1	1	yes	characteristic
284	C₄	C₅₃×#543	1	2	yes	—
285	C₂	—	2	4	no	—
286	C₂	—	4	4	no	—
287	C₂	C₅₃×#2315	1	1	yes	characteristic
288	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
289	C₂	C₅₃×#2911	1	1	yes	characteristic
290	C₂	C₅₃×#3191	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁵×C₅₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1854	C₂	semidirect
4. C₅₃×#1808	C₂	semidirect
5. C₅₃×#1807	C₂	semidirect
10. C₂×#107×C₅₃	C₂²	semidirect
11. C₂×#106×C₅₃	C₂²	semidirect
12. C₂×C₂²⋊C₈×C₅₃	C₄	semidirect
20. C₅₃×#340	C₂²	semidirect
21. C₅₃×#340	C₂²	semidirect
120. #21939	C₅₃	direct (N × H)
142. #1854	C₂×C₅₃	semidirect
143. #1808	C₂×C₅₃	semidirect
144. #1807	C₂×C₅₃	semidirect
155. C₂×#107	C₂²×C₅₃	semidirect
156. C₂×#106	C₂²×C₅₃	semidirect
157. C₂×C₂²⋊C₈	C₄×C₅₃	semidirect
165. #340	C₂²×C₅₃	semidirect
166. #340	C₂²×C₅₃	semidirect
167. C₅₃	#21939	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×C₂²⋊C₈×C₅₃	2
C₅₃×#2280	2
C₅₃×#1854	2
C₅₃×#1808	2
C₅₃×#1807	2
#21939	53

Subgroups — C53×#21939 #27229451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21939 #27229451