Subgroups — C₅₃×#10536 #27218048

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1982	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#331	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₃	—	2	1	no	—
7	C₂²×D₈×C₅₃	—	2	1	no	—
8	C₂×#127×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#493	—	2	4	no	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#102×C₅₃	—	2	2	no	—
14	C₂×#102×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#331	—	2	4	no	—
19	C₂²×D₄×C₅₃	—	2	2	no	—
20	C₂²×D₄×C₅₃	—	2	2	no	—
21	C₂²×D₄×C₅₃	—	2	2	no	—
22	C₂×D₈×C₅₃	—	2	4	no	—
23	C₂×D₈×C₅₃	—	2	4	no	—
24	C₂×D₈×C₅₃	—	4	2	no	—
25	C₂×D₈×C₅₃	—	4	2	no	—
26	C₂×D₈×C₅₃	—	4	2	no	—
27	C₂×D₈×C₅₃	—	4	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	#127×C₅₃	—	2	2	no	—
33	#127×C₅₃	—	2	2	no	—
34	#127×C₅₃	—	2	2	no	—
35	#127×C₅₃	D₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	4	2	no	—
44	#102×C₅₃	—	4	4	no	—
45	#102×C₅₃	—	4	4	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	#95×C₅₃	—	2	1	no	—
48	C₂⁴×C₅₃	—	4	2	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	—	2	4	no	—
51	C₂×D₄×C₅₃	—	2	4	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂×D₄×C₅₃	—	8	2	no	—
64	C₂×D₄×C₅₃	—	8	2	no	—
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	D₈×C₅₃	—	4	8	no	—
71	D₈×C₅₃	—	4	8	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	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	D₈	1	2	yes	—
83	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	D₈	1	2	yes	complemented
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂×D₄	1	2	yes	—
90	C₄²×C₅₃	#41	1	2	yes	—
91	C₂³×C₅₃	—	4	4	no	—
92	C₂³×C₅₃	—	8	2	no	—
93	C₂³×C₅₃	—	8	2	no	—
94	C₂³×C₅₃	—	8	2	no	—
95	C₂³×C₅₃	—	8	2	no	—
96	C₂³×C₅₃	—	8	2	no	—
97	C₂³×C₅₃	—	8	2	no	—
98	C₂³×C₅₃	#121	1	1	yes	characteristic
99	D₄×C₅₃	—	4	4	no	—
100	D₄×C₅₃	—	4	4	no	—
101	D₄×C₅₃	—	4	4	no	—
102	D₄×C₅₃	—	4	4	no	—
103	D₄×C₅₃	—	8	4	no	—
104	D₄×C₅₃	—	8	4	no	—
105	D₄×C₅₃	—	8	2	no	—
106	D₄×C₅₃	—	8	2	no	—
107	D₄×C₅₃	—	8	2	no	—
108	D₄×C₅₃	—	8	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	2	yes	—
119	C₂×C₄×C₅₃	#123	1	2	yes	—
120	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
125	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	—	8	1	no	—
131	C₈×C₅₃	—	8	2	no	—
132	#10536	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	8	4	no	—
134	C₂²×C₅₃	—	8	4	no	—
135	C₂²×C₅₃	—	8	4	no	—
136	C₂²×C₅₃	—	16	2	no	—
137	C₂²×C₅₃	—	16	2	no	—
138	C₂²×C₅₃	—	16	2	no	—
139	C₂²×C₅₃	—	16	2	no	—
140	C₂²×C₅₃	#394	1	1	yes	characteristic
141	C₂²×C₅₃	#459	1	1	yes	characteristic
142	C₂²×C₅₃	#460	1	1	yes	characteristic
143	C₂²×C₅₃	#466	1	1	yes	characteristic
144	C₂²×C₅₃	#469	1	1	yes	characteristic
145	C₂²×C₅₃	#472	1	1	yes	characteristic
146	C₂²×C₅₃	#472	1	1	yes	characteristic
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	4	2	no	—
149	C₄×C₅₃	—	8	1	no	—
150	C₄×C₅₃	#447	1	1	yes	characteristic
151	C₄×C₅₃	#450	1	1	yes	characteristic
152	C₄×C₅₃	#459	1	2	yes	—
153	C₄×C₅₃	#463	1	2	yes	—
154	C₄×C₅₃	#486	1	1	yes	characteristic
155	C₄×C₅₃	#490	1	1	yes	characteristic
156	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
157	#1982	C₂×C₅₃	1	2	yes	complemented
158	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
159	C₂×#331	C₂×C₅₃	1	2	yes	complemented
160	#1277	C₂×C₅₃	1	1	yes	characteristic, complemented
161	C₂×C₅₃	—	16	4	no	—
162	C₂×C₅₃	#1667	1	1	yes	characteristic
163	C₂×C₅₃	#1668	1	1	yes	characteristic
164	C₂×C₅₃	#1687	1	1	yes	characteristic
165	C₂×C₅₃	#1688	1	1	yes	characteristic
166	C₂×C₅₃	#2041	1	1	yes	characteristic
167	C₂×C₅₃	#2045	1	1	yes	characteristic
168	C₂×C₅₃	#2054	1	1	yes	characteristic
169	C₂²×D₈	—	2	1	no	—
170	C₂²×D₈	—	2	1	no	—
171	C₂×#127	C₂²×C₅₃	1	2	yes	—
172	#493	—	2	4	no	—
173	C₂×C₄⋊C₈	—	2	1	no	—
174	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
176	C₂×#102	—	2	2	no	—
177	C₂×#102	—	2	2	no	—
178	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
180	#336	—	2	1	no	—
181	#331	—	2	4	no	—
182	C₅₃	#10536	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂×D₈	—	2	4	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	—	4	2	no	—
189	C₂×D₈	—	4	2	no	—
190	C₂×D₈	—	4	2	no	—
191	C₂×D₈	—	4	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
196	#127	—	2	2	no	—
197	#127	—	2	2	no	—
198	#127	—	2	2	no	—
199	#127	D₄×C₅₃	1	4	yes	—
200	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	4	2	no	—
208	#102	—	4	4	no	—
209	#102	—	4	4	no	—
210	C₄×C₈	—	2	2	no	—
211	#95	—	2	1	no	—
212	C₂⁴	—	4	2	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	8	2	no	—
228	C₂×D₄	—	8	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
234	D₈	—	4	8	no	—
235	D₈	—	4	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	D₈×C₅₃	1	2	yes	—
247	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	D₈×C₅₃	1	2	yes	complemented
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	C₂×D₄×C₅₃	1	2	yes	—
254	C₄²	#41×C₅₃	1	2	yes	—
255	C₂³	—	4	4	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	—	8	2	no	—
262	C₂³	#121×C₅₃	1	1	yes	characteristic
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
282	C₂×C₄	#121×C₅₃	1	2	yes	—
283	C₂×C₄	#123×C₅₃	1	2	yes	—
284	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
285	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
288	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
289	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
290	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
291	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	2	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	16	2	no	—
300	C₂²	—	16	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	C₅₃×#394	1	1	yes	characteristic
304	C₂²	C₅₃×#459	1	1	yes	characteristic
305	C₂²	C₅₃×#460	1	1	yes	characteristic
306	C₂²	C₅₃×#466	1	1	yes	characteristic
307	C₂²	C₅₃×#469	1	1	yes	characteristic
308	C₂²	C₅₃×#472	1	1	yes	characteristic
309	C₂²	C₅₃×#472	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	C₅₃×#447	1	1	yes	characteristic
314	C₄	C₅₃×#450	1	1	yes	characteristic
315	C₄	C₅₃×#459	1	2	yes	—
316	C₄	C₅₃×#463	1	2	yes	—
317	C₄	C₅₃×#486	1	1	yes	characteristic
318	C₄	C₅₃×#490	1	1	yes	characteristic
319	C₂	—	16	4	no	—
320	C₂	C₅₃×#1667	1	1	yes	characteristic
321	C₂	C₅₃×#1668	1	1	yes	characteristic
322	C₂	C₅₃×#1687	1	1	yes	characteristic
323	C₂	C₅₃×#1688	1	1	yes	characteristic
324	C₂	C₅₃×#2041	1	1	yes	characteristic
325	C₂	C₅₃×#2045	1	1	yes	characteristic
326	C₂	C₅₃×#2054	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1982	C₂	semidirect
3. C₅₃×#1939	C₂	semidirect
4. C₂×C₅₃×#331	C₂	semidirect
5. C₅₃×#1277	C₂	semidirect
86. C₄⋊C₄×C₅₃	D₈	semidirect
132. #10536	C₅₃	direct (N × H)
157. #1982	C₂×C₅₃	semidirect
158. #1939	C₂×C₅₃	semidirect
159. C₂×#331	C₂×C₅₃	semidirect
160. #1277	C₂×C₅₃	semidirect
182. C₅₃	#10536	direct (N × H)
250. C₄⋊C₄	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#493	2
C₅₃×#1982	2
C₅₃×#1939	2
C₂×C₅₃×#331	2
C₅₃×#1277	2
#10536	53

Subgroups — C53×#10536 #27218048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10536 #27218048