Subgroups — C₅₃×#19636 #27227148

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3229	C₂	1	2	yes	maximal
2	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3139	C₂	1	2	yes	maximal
5	C₅₃×#1718	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1710	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1707	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#533	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#500	—	2	2	no	—
12	C₅₃×#500	C₂²	1	2	yes	—
13	C₅₃×#500	C₂²	1	1	yes	characteristic
14	C₅₃×#500	C₂²	1	2	yes	—
15	C₅₃×#498	C₂²	1	4	yes	—
16	C₅₃×#494	C₂²	1	1	yes	characteristic
17	C₅₃×#491	—	2	4	no	—
18	C₅₃×#487	C₂²	1	4	yes	—
19	C₅₃×#487	C₂²	1	4	yes	—
20	C₅₃×#487	C₂²	1	4	yes	—
21	C₅₃×#487	C₂²	1	2	yes	—
22	C₅₃×#483	—	2	4	no	—
23	C₅₃×#475	—	2	4	no	—
24	C₅₃×#467	—	2	2	no	—
25	C₄×Q₁₆×C₅₃	—	2	2	no	—
26	C₅₃×#431	C₂²	1	1	yes	characteristic
27	C₂×#107×C₅₃	—	2	2	no	—
28	C₅₃×#419	—	2	4	no	—
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₂²×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₄⋊Q₈×C₅₃	—	2	2	no	—
38	C₄⋊Q₈×C₅₃	—	2	2	no	—
39	C₄⋊Q₈×C₅₃	—	2	2	no	—
40	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
42	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
43	#125×C₅₃	—	2	2	no	—
44	C₂²⋊Q₈×C₅₃	—	2	2	no	—
45	C₂²⋊Q₈×C₅₃	—	2	2	no	—
46	C₄×Q₈×C₅₃	—	2	2	no	—
47	C₄×Q₈×C₅₃	—	2	2	no	—
48	#117×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	—	2	4	no	—
53	#107×C₅₃	—	2	4	no	—
54	#107×C₅₃	—	2	2	no	—
55	#106×C₅₃	—	2	8	no	—
56	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
57	#103×C₅₃	—	2	8	no	—
58	#103×C₅₃	—	2	8	no	—
59	#103×C₅₃	—	2	4	no	—
60	#103×C₅₃	—	2	8	no	—
61	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
62	#97×C₅₃	D₄	1	2	yes	—
63	C₄×C₈×C₅₃	D₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	—	2	4	no	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×Q₈×C₅₃	—	4	2	no	—
72	C₂×Q₈×C₅₃	—	4	8	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₅₃	—	4	4	no	—
78	M₁₆×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	—	4	2	no	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂³×C₅₃	#127	1	1	yes	characteristic
101	Q₈×C₅₃	—	4	8	no	—
102	Q₈×C₅₃	—	4	4	no	—
103	Q₈×C₅₃	—	4	4	no	—
104	Q₈×C₅₃	—	8	4	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	8	4	no	—
111	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#135	1	4	yes	—
115	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₅₃	—	2	4	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	#137	1	4	yes	—
122	#19636	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	#495	1	2	yes	—
126	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	8	4	no	—
130	C₄×C₅₃	—	8	2	no	—
131	C₄×C₅₃	—	8	2	no	—
132	C₄×C₅₃	#495	1	2	yes	—
133	C₄×C₅₃	C₂×#135	1	2	yes	—
134	C₄×C₅₃	C₂×#137	1	2	yes	—
135	C₄×C₅₃	#578	1	2	yes	—
136	#3229	C₂×C₅₃	1	2	yes	—
137	#3170	C₂×C₅₃	1	1	yes	characteristic
138	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
139	#3139	C₂×C₅₃	1	2	yes	—
140	#1718	C₂×C₅₃	1	4	yes	complemented
141	#1710	C₂×C₅₃	1	2	yes	complemented
142	#1707	C₂×C₅₃	1	2	yes	complemented
143	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
146	C₂×C₅₃	#3152	1	1	yes	characteristic
147	C₂×C₅₃	#3156	1	1	yes	characteristic
148	#533	C₂²×C₅₃	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
150	#500	—	2	2	no	—
151	#500	C₂²×C₅₃	1	2	yes	—
152	#500	C₂²×C₅₃	1	2	yes	—
153	#500	C₂²×C₅₃	1	1	yes	characteristic
154	#498	C₂²×C₅₃	1	4	yes	—
155	#494	C₂²×C₅₃	1	1	yes	characteristic
156	#491	—	2	4	no	—
157	#487	C₂²×C₅₃	1	4	yes	—
158	#487	C₂²×C₅₃	1	4	yes	—
159	#487	C₂²×C₅₃	1	2	yes	—
160	#487	C₂²×C₅₃	1	4	yes	—
161	#483	—	2	4	no	—
162	#475	—	2	4	no	—
163	#467	—	2	2	no	—
164	C₄×Q₁₆	—	2	2	no	—
165	#431	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×#107	—	2	2	no	—
167	#419	—	2	4	no	—
168	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
170	#322	C₂²×C₅₃	1	2	yes	—
171	C₈²	C₂²×C₅₃	1	2	yes	—
172	C₅₃	#19636	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×Q₈	—	2	2	no	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
176	C₂²×C₈	D₄×C₅₃	1	2	yes	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
181	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
182	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	#117	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	#107	—	2	2	no	—
195	#106	—	2	8	no	—
196	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
197	#103	—	2	8	no	—
198	#103	—	2	8	no	—
199	#103	—	2	4	no	—
200	#103	—	2	8	no	—
201	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
202	#97	D₄×C₅₃	1	2	yes	—
203	C₄×C₈	D₄×C₅₃	1	2	yes	—
204	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₅₃	1	2	yes	—
206	C₄×C₈	C₂³×C₅₃	1	2	yes	—
207	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	8	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	Q₁₆	—	4	4	no	—
218	M₁₆	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
222	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
223	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
239	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
240	C₂³	#127×C₅₃	1	1	yes	characteristic
241	Q₈	—	4	8	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
253	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
254	C₂×C₄	#135×C₅₃	1	4	yes	—
255	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
258	C₈	—	2	4	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	4	no	—
261	C₈	#137×C₅₃	1	4	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	C₅₃×#495	1	2	yes	—
265	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	2	no	—
270	C₄	—	8	2	no	—
271	C₄	C₅₃×#495	1	2	yes	—
272	C₄	C₂×#135×C₅₃	1	2	yes	—
273	C₄	C₂×#137×C₅₃	1	2	yes	—
274	C₄	C₅₃×#578	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	C₂×C₅₃×#495	1	1	yes	characteristic
277	C₂	C₅₃×#3152	1	1	yes	characteristic
278	C₂	C₅₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂³×C₅₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1718	C₂	semidirect
6. C₅₃×#1710	C₂	semidirect
7. C₅₃×#1707	C₂	semidirect
122. #19636	C₅₃	direct (N × H)
140. #1718	C₂×C₅₃	semidirect
141. #1710	C₂×C₅₃	semidirect
142. #1707	C₂×C₅₃	semidirect
172. C₅₃	#19636	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3229	2
C₅₃×#3170	2
C₂×C₅₃×#500	2
C₅₃×#3139	2
C₅₃×#1718	2
C₅₃×#1710	2
C₅₃×#1707	2
C₈×M₁₆×C₅₃	2
#19636	53

Subgroups — C53×#19636 #27227148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19636 #27227148