Subgroups — C₅₃×#11460 #27218972

Order 13568.

374 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	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2163	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1322	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1314	C₂	1	2	yes	maximal
5	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
6	C₅₃×#431	—	2	2	no	—
7	C₅₃×#431	—	2	2	no	—
8	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	2	no	—
15	C₅₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
16	C₅₃×#346	—	2	2	no	—
17	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	—	2	1	no	—
25	C₂×C₄²×C₅₃	—	2	1	no	—
26	C₂×C₄²×C₅₃	—	2	1	no	—
27	C₂×C₄²×C₅₃	—	2	1	no	—
28	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	—	2	4	no	—
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₂²⋊C₈×C₅₃	—	2	2	no	—
36	C₂²⋊C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	1	no	—
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	—	4	1	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	—	4	1	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
67	C₄²×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
76	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
77	C₁₆×C₅₃	—	4	4	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	4	1	no	—
81	C₂³×C₅₃	—	4	1	no	—
82	C₂³×C₅₃	M₃₂	1	2	yes	—
83	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	—	4	1	no	—
112	C₈×C₅₃	—	4	1	no	—
113	#11460	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	—	4	1	no	—
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	4	1	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	#349	1	2	yes	—
132	C₄×C₅₃	#410	1	1	yes	characteristic
133	C₄×C₅₃	#413	1	1	yes	characteristic
134	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
135	#2163	C₂×C₅₃	1	2	yes	complemented
136	#1322	C₂×C₅₃	1	2	yes	complemented
137	#1314	C₂×C₅₃	1	2	yes	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	#1458	1	1	yes	characteristic
142	C₂×C₅₃	#2112	1	1	yes	characteristic
143	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
144	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
145	#431	—	2	2	no	—
146	#431	—	2	2	no	—
147	C₂×C₄⋊C₈	C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
149	C₂×C₂²⋊C₈	C₄×C₅₃	1	1	yes	characteristic
150	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
153	C₄⋊C₁₆	—	2	2	no	—
154	C₂²⋊C₁₆	C₂²×C₅₃	1	4	yes	—
155	#346	—	2	2	no	—
156	C₅₃	#11460	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³×C₄	C₈×C₅₃	1	1	yes	characteristic
158	C₂³×C₄	C₈×C₅₃	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₈×C₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₂×C₁₆	—	2	4	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	D₄×C₅₃	1	2	yes	—
180	C₄×C₈	D₄×C₅₃	1	2	yes	—
181	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
214	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
215	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
216	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
217	C₁₆	—	4	4	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	1	no	—
222	C₂³	M₃₂×C₅₃	1	2	yes	—
223	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
246	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
261	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
262	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	C₅₃×#349	1	2	yes	—
271	C₄	C₅₃×#410	1	1	yes	characteristic
272	C₄	C₅₃×#413	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	4	2	no	—
276	C₂	C₅₃×#1458	1	1	yes	characteristic
277	C₂	C₅₃×#2112	1	1	yes	characteristic
278	C₂	C₂×C₅₃×#349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2163	C₂	semidirect
3. C₅₃×#1322	C₂	semidirect
113. #11460	C₅₃	direct (N × H)
135. #2163	C₂×C₅₃	semidirect
136. #1322	C₂×C₅₃	semidirect
156. C₅₃	#11460	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#431	2
C₅₃×#2163	2
C₅₃×#1322	2
C₅₃×#1314	2
#11460	53

Subgroups — C53×#11460 #27218972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11460 #27218972