Subgroups — C₅₃×#22725 #27230237

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2355	C₂	1	2	yes	maximal
3	C₅₃×#1949	C₂	1	4	yes	maximal
4	C₅₃×#1947	C₂	1	2	yes	maximal
5	C₅₃×#1846	C₂	1	2	yes	maximal
6	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×#106×C₅₃	C₂	1	2	yes	maximal
8	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₅₃	C₄	1	8	yes	complemented
11	C₂×#107×C₅₃	C₂²	1	4	yes	—
12	C₂×#106×C₅₃	C₂²	1	2	yes	—
13	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#391	—	2	2	no	—
21	C₅₃×#387	—	2	4	no	—
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₅₃×#382	C₂²	1	4	yes	—
24	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#340	C₂²	1	4	yes	—
28	C₅₃×#340	C₂²	1	2	yes	—
29	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	4	no	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#107×C₅₃	—	4	4	no	—
47	#106×C₅₃	—	2	4	no	—
48	#106×C₅₃	—	2	2	no	—
49	#106×C₅₃	—	4	2	no	—
50	#106×C₅₃	Q₈	1	4	yes	complemented
51	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	4	no	—
58	#95×C₅₃	—	2	4	no	—
59	C₂×Q₈×C₅₃	—	2	4	no	—
60	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	4	4	no	—
84	C₄⋊C₄×C₅₃	—	8	2	no	—
85	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
86	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
87	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
88	C₄²×C₅₃	—	4	4	no	—
89	C₄²×C₅₃	—	8	4	no	—
90	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
91	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
93	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
94	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₅₃	—	2	8	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	4	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	C₄×Q₈	1	4	yes	—
122	#22725	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
125	C₂²×C₅₃	#550	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#135	1	2	yes	—
127	C₂²×C₅₃	#577	1	2	yes	—
128	C₄×C₅₃	—	2	4	no	—
129	C₄×C₅₃	—	2	4	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	—	8	2	no	—
133	C₄×C₅₃	—	8	4	no	—
134	C₄×C₅₃	#427	1	4	yes	—
135	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
136	C₄×C₅₃	#520	1	1	yes	characteristic
137	C₄×C₅₃	#557	1	1	yes	characteristic
138	C₄×C₅₃	Q₈²	1	1	yes	characteristic
139	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
140	#2355	C₂×C₅₃	1	2	yes	—
141	#1949	C₂×C₅₃	1	4	yes	—
142	#1947	C₂×C₅₃	1	2	yes	—
143	#1846	C₂×C₅₃	1	2	yes	—
144	#1844	C₂×C₅₃	1	1	yes	characteristic
145	C₄×#106	C₂×C₅₃	1	2	yes	—
146	C₄×#106	C₂×C₅₃	1	1	yes	characteristic
147	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
149	C₂×C₅₃	#2912	1	1	yes	characteristic
150	C₂×C₅₃	#3381	1	2	yes	—
151	C₂×C₅₃	#3394	1	2	yes	—
152	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
153	C₈×Q₈	C₄×C₅₃	1	8	yes	complemented
154	C₂×#107	C₂²×C₅₃	1	4	yes	—
155	C₂×#106	C₂²×C₅₃	1	2	yes	—
156	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×#106	C₂²×C₅₃	1	2	yes	—
158	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
163	#391	—	2	2	no	—
164	#387	—	2	4	no	—
165	#384	C₂²×C₅₃	1	2	yes	—
166	#382	C₂²×C₅₃	1	4	yes	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
170	#340	C₂²×C₅₃	1	4	yes	—
171	#340	C₂²×C₅₃	1	2	yes	—
172	C₅₃	#22725	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
176	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
177	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
178	C₄×Q₈	Q₈×C₅₃	1	4	yes	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	4	no	—
188	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
189	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
190	#107	—	4	4	no	—
191	#106	—	2	4	no	—
192	#106	—	2	2	no	—
193	#106	—	4	2	no	—
194	#106	Q₈×C₅₃	1	4	yes	complemented
195	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
196	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
197	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
198	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
199	#95	—	2	4	no	—
200	#95	—	2	2	no	—
201	#95	—	2	4	no	—
202	#95	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
219	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
220	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
221	C₂×C₈	#41×C₅₃	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	8	2	no	—
229	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
230	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
231	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
232	C₄²	—	4	4	no	—
233	C₄²	—	8	4	no	—
234	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
235	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
236	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
237	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
238	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
256	C₂×C₄	#135×C₅₃	1	4	yes	—
257	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
261	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
262	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	C₄×Q₈×C₅₃	1	4	yes	—
266	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
267	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
268	C₂²	C₅₃×#550	1	1	yes	characteristic
269	C₂²	C₂×#135×C₅₃	1	2	yes	—
270	C₂²	C₅₃×#577	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₅₃×#427	1	4	yes	—
278	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
279	C₄	C₅₃×#520	1	1	yes	characteristic
280	C₄	C₅₃×#557	1	1	yes	characteristic
281	C₄	Q₈²×C₅₃	1	1	yes	characteristic
282	C₂	Q₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
283	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
284	C₂	C₅₃×#2912	1	1	yes	characteristic
285	C₂	C₅₃×#3381	1	2	yes	—
286	C₂	C₅₃×#3394	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₅₃	C₄	semidirect
50. #106×C₅₃	Q₈	semidirect
122. #22725	C₅₃	direct (N × H)
153. C₈×Q₈	C₄×C₅₃	semidirect
172. C₅₃	#22725	direct (N × H)
194. #106	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×C₈×Q₈×C₅₃	2
C₅₃×#2355	2
C₅₃×#1949	2
C₅₃×#1947	2
C₅₃×#1846	2
C₅₃×#1844	2
C₄×#106×C₅₃	2
#22725	53

Subgroups — C53×#22725 #27230237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22725 #27230237