Subgroups — C₅₃×#9588 #27217100

Order 13568.

304 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	C₅₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1484	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#532	C₂²	1	1	yes	characteristic
9	C₅₃×#431	C₂²	1	1	yes	characteristic
10	C₅₃×#431	C₂²	1	1	yes	characteristic
11	C₅₃×#431	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#335	—	2	1	no	—
16	C₅₃×#334	—	2	1	no	—
17	C₅₃×#333	C₄	1	1	yes	characteristic
18	C₅₃×#333	C₄	1	1	yes	characteristic
19	C₅₃×#331	C₄	1	1	yes	characteristic
20	C₅₃×#331	C₄	1	1	yes	characteristic
21	C₅₃×D₄⋊C₈	—	2	1	no	—
22	C₅₃×D₄⋊C₈	—	2	1	no	—
23	C₅₃×D₄⋊C₈	—	2	2	no	—
24	C₅₃×#322	—	2	2	no	—
25	C₂×#41×C₅₃	—	2	1	no	—
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂×M₁₆×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#124×C₅₃	—	2	1	no	—
32	#121×C₅₃	—	2	1	no	—
33	#121×C₅₃	—	2	1	no	—
34	C₄×D₄×C₅₃	—	2	1	no	—
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²⋊C₈×C₅₃	—	2	1	no	—
46	C₂²⋊C₈×C₅₃	—	2	1	no	—
47	C₂²⋊C₈×C₅₃	—	2	1	no	—
48	#97×C₅₃	—	2	1	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	—	2	1	no	—
51	C₄×C₈×C₅₃	—	2	1	no	—
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	D₄	1	2	yes	—
55	#41×C₅₃	—	8	1	no	—
56	#41×C₅₃	—	8	1	no	—
57	C₂×Q₈×C₅₃	—	2	1	no	—
58	C₂×D₄×C₅₃	—	2	1	no	—
59	C₂×D₄×C₅₃	—	2	1	no	—
60	C₂×D₄×C₅₃	—	2	1	no	—
61	C₂×D₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	M₁₆×C₅₃	—	2	2	no	—
67	M₁₆×C₅₃	—	4	1	no	—
68	M₁₆×C₅₃	—	8	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂²⋊C₄×C₅₃	—	4	1	no	—
83	C₂²⋊C₄×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	C₂³×C₅₃	—	4	1	no	—
89	C₂³×C₅₃	#126	1	1	yes	characteristic
90	Q₈×C₅₃	—	8	1	no	—
91	D₄×C₅₃	—	4	1	no	—
92	D₄×C₅₃	—	4	1	no	—
93	D₄×C₅₃	—	4	1	no	—
94	D₄×C₅₃	—	4	1	no	—
95	D₄×C₅₃	—	8	1	no	—
96	D₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	8	1	no	—
105	C₂×C₄×C₅₃	—	8	1	no	—
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
112	C₈×C₅₃	—	4	2	no	—
113	C₈×C₅₃	—	4	2	no	—
114	C₈×C₅₃	—	4	1	no	—
115	C₈×C₅₃	—	4	1	no	—
116	C₈×C₅₃	—	4	1	no	—
117	C₈×C₅₃	—	4	1	no	—
118	C₈×C₅₃	—	4	1	no	—
119	C₈×C₅₃	—	4	1	no	—
120	C₈×C₅₃	—	8	1	no	—
121	#9588	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	8	1	no	—
126	C₂²×C₅₃	—	8	1	no	—
127	C₂²×C₅₃	—	8	1	no	—
128	C₂²×C₅₃	#388	1	1	yes	characteristic
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	8	1	no	—
133	C₄×C₅₃	#443	1	1	yes	characteristic
134	C₄×C₅₃	#444	1	1	yes	characteristic
135	C₄×C₅₃	#444	1	1	yes	characteristic
136	C₄×C₅₃	#444	1	1	yes	characteristic
137	C₄×C₅₃	#471	1	1	yes	characteristic
138	C₄×C₅₃	#472	1	1	yes	characteristic
139	#1574	C₂×C₅₃	1	1	yes	characteristic, complemented
140	#1543	C₂×C₅₃	1	1	yes	characteristic
141	#1492	C₂×C₅₃	1	1	yes	characteristic
142	#1484	C₂×C₅₃	1	1	yes	characteristic
143	#1452	C₂×C₅₃	1	1	yes	characteristic, complemented
144	#1299	C₂×C₅₃	1	1	yes	characteristic, complemented
145	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
146	C₂×C₅₃	—	4	1	no	—
147	C₂×C₅₃	—	16	1	no	—
148	C₂×C₅₃	#1929	1	1	yes	characteristic
149	C₂×C₅₃	#1931	1	1	yes	characteristic
150	C₂×C₅₃	#1938	1	1	yes	characteristic
151	#532	C₂²×C₅₃	1	1	yes	characteristic
152	#431	C₂²×C₅₃	1	1	yes	characteristic
153	#431	C₂²×C₅₃	1	1	yes	characteristic
154	#431	C₂²×C₅₃	1	1	yes	characteristic
155	#423	C₂²×C₅₃	1	1	yes	characteristic
156	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
158	#335	—	2	1	no	—
159	#334	—	2	1	no	—
160	#333	C₄×C₅₃	1	1	yes	characteristic
161	#333	C₄×C₅₃	1	1	yes	characteristic
162	#331	C₄×C₅₃	1	1	yes	characteristic
163	#331	C₄×C₅₃	1	1	yes	characteristic
164	D₄⋊C₈	—	2	1	no	—
165	D₄⋊C₈	—	2	1	no	—
166	D₄⋊C₈	—	2	2	no	—
167	#322	—	2	2	no	—
168	C₅₃	#9588	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×#41	—	2	1	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
174	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
175	#124	—	2	1	no	—
176	#121	—	2	1	no	—
177	#121	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
187	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
188	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	D₄×C₅₃	1	2	yes	—
199	#41	—	8	1	no	—
200	#41	—	8	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	8	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₂²⋊C₄	—	4	1	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
229	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
231	C₄²	#41×C₅₃	1	1	yes	characteristic
232	C₂³	—	4	1	no	—
233	C₂³	#126×C₅₃	1	1	yes	characteristic
234	Q₈	—	8	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	8	1	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
254	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
255	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	8	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	8	1	no	—
271	C₂²	C₅₃×#388	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	1	no	—
276	C₄	C₅₃×#443	1	1	yes	characteristic
277	C₄	C₅₃×#444	1	1	yes	characteristic
278	C₄	C₅₃×#444	1	1	yes	characteristic
279	C₄	C₅₃×#444	1	1	yes	characteristic
280	C₄	C₅₃×#471	1	1	yes	characteristic
281	C₄	C₅₃×#472	1	1	yes	characteristic
282	C₂	—	4	1	no	—
283	C₂	—	16	1	no	—
284	C₂	C₅₃×#1929	1	1	yes	characteristic
285	C₂	C₅₃×#1931	1	1	yes	characteristic
286	C₂	C₅₃×#1938	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#388
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#1574	C₂	semidirect
5. C₅₃×#1452	C₂	semidirect
6. C₅₃×#1299	C₂	semidirect
7. C₅₃×#1278	C₂	semidirect
121. #9588	C₅₃	direct (N × H)
139. #1574	C₂×C₅₃	semidirect
143. #1452	C₂×C₅₃	semidirect
144. #1299	C₂×C₅₃	semidirect
145. #1278	C₂×C₅₃	semidirect
168. C₅₃	#9588	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×#1574	2
C₅₃×#1543	2
C₅₃×#1492	2
C₅₃×#1484	2
C₅₃×#1452	2
C₅₃×#1299	2
C₅₃×#1278	2
#9588	53

Subgroups — C53×#9588 #27217100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9588 #27217100