Subgroups — C₅₃×#10168 #27217680

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₅₃	—	2	2	no	—
9	C₅₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#497	—	2	2	no	—
13	C₅₃×#494	—	2	1	no	—
14	C₅₃×#493	—	2	1	no	—
15	C₅₃×#492	—	2	1	no	—
16	C₅₃×#486	—	2	1	no	—
17	C₅₃×#486	—	2	2	no	—
18	C₅₃×#486	—	2	2	no	—
19	C₅₃×#459	—	2	2	no	—
20	C₅₃×#447	—	2	2	no	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₅₃×#423	C₂²	1	1	yes	characteristic
23	C₂×#102×C₅₃	—	2	2	no	—
24	C₂×#100×C₅₃	—	2	2	no	—
25	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#332	—	2	2	no	—
28	C₅₃×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₅₃	—	2	1	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	—	2	1	no	—
33	C₂²×D₄×C₅₃	D₄	1	2	yes	—
34	#137×C₅₃	—	4	4	no	—
35	#137×C₅₃	—	4	4	no	—
36	C₂×Q₁₆×C₅₃	—	2	4	no	—
37	C₂×SD₁₆×C₅₃	—	2	4	no	—
38	C₂×D₈×C₅₃	—	2	4	no	—
39	C₂×M₁₆×C₅₃	—	2	1	no	—
40	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	—	2	1	no	—
43	C₄⋊Q₈×C₅₃	—	2	1	no	—
44	C₄⋊Q₈×C₅₃	—	2	1	no	—
45	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
47	#127×C₅₃	—	2	1	no	—
48	#127×C₅₃	—	2	1	no	—
49	#127×C₅₃	—	2	1	no	—
50	#127×C₅₃	—	2	1	no	—
51	#127×C₅₃	D₄	1	1	yes	characteristic
52	#124×C₅₃	—	2	1	no	—
53	#121×C₅₃	—	2	1	no	—
54	#121×C₅₃	—	2	1	no	—
55	C₄×D₄×C₅₃	—	2	1	no	—
56	C₄×D₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
63	#102×C₅₃	—	2	8	no	—
64	#102×C₅₃	—	2	4	no	—
65	#100×C₅₃	—	4	4	no	—
66	C₂²⋊C₈×C₅₃	—	2	1	no	—
67	#97×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	C₄×C₈×C₅₃	—	2	1	no	—
70	C₄×C₈×C₅₃	—	2	1	no	—
71	C₄×C₈×C₅₃	D₄	1	2	yes	—
72	C₂⁴×C₅₃	—	2	2	no	—
73	#41×C₅₃	—	4	2	no	—
74	#41×C₅₃	—	4	2	no	—
75	C₂×Q₈×C₅₃	—	2	1	no	—
76	C₂×Q₈×C₅₃	—	2	2	no	—
77	C₂×Q₈×C₅₃	—	4	1	no	—
78	C₂×Q₈×C₅₃	—	4	2	no	—
79	C₂×Q₈×C₅₃	—	4	2	no	—
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	1	no	—
84	C₂×D₄×C₅₃	—	2	1	no	—
85	C₂×D₄×C₅₃	—	4	4	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×D₄×C₅₃	—	4	1	no	—
88	C₂×D₄×C₅₃	—	4	1	no	—
89	C₂×D₄×C₅₃	—	4	1	no	—
90	C₂×D₄×C₅₃	—	4	1	no	—
91	C₂×D₄×C₅₃	—	4	2	no	—
92	C₂×D₄×C₅₃	—	4	2	no	—
93	C₂×D₄×C₅₃	—	8	2	no	—
94	C₂²×C₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	—	4	1	no	—
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	4	8	no	—
100	SD₁₆×C₅₃	—	4	8	no	—
101	D₈×C₅₃	—	8	4	no	—
102	M₁₆×C₅₃	—	2	2	no	—
103	M₁₆×C₅₃	—	4	2	no	—
104	M₁₆×C₅₃	D₈	1	4	yes	complemented
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	4	2	no	—
109	C₂×C₈×C₅₃	—	4	1	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	1	no	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	4	1	no	—
115	C₄⋊C₄×C₅₃	—	4	1	no	—
116	C₂²⋊C₄×C₅₃	—	4	1	no	—
117	C₂²⋊C₄×C₅₃	—	4	1	no	—
118	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂³×C₅₃	—	4	1	no	—
123	C₂³×C₅₃	—	4	2	no	—
124	C₂³×C₅₃	—	4	4	no	—
125	C₂³×C₅₃	—	8	2	no	—
126	C₂³×C₅₃	—	8	2	no	—
127	C₂³×C₅₃	#121	1	1	yes	characteristic
128	Q₈×C₅₃	—	4	4	no	—
129	Q₈×C₅₃	—	4	2	no	—
130	Q₈×C₅₃	—	8	1	no	—
131	Q₈×C₅₃	—	8	1	no	—
132	D₄×C₅₃	—	4	4	no	—
133	D₄×C₅₃	—	4	2	no	—
134	D₄×C₅₃	—	4	4	no	—
135	D₄×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	8	2	no	—
138	D₄×C₅₃	—	8	2	no	—
139	D₄×C₅₃	—	8	2	no	—
140	D₄×C₅₃	—	8	2	no	—
141	D₄×C₅₃	—	8	1	no	—
142	D₄×C₅₃	—	8	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	8	2	no	—
148	C₂×C₄×C₅₃	—	8	1	no	—
149	C₂×C₄×C₅₃	—	8	1	no	—
150	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
158	C₈×C₅₃	—	2	4	no	—
159	C₈×C₅₃	—	4	2	no	—
160	C₈×C₅₃	—	4	2	no	—
161	C₈×C₅₃	—	8	1	no	—
162	#10168	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	8	2	no	—
166	C₂²×C₅₃	—	8	1	no	—
167	C₂²×C₅₃	—	8	1	no	—
168	C₂²×C₅₃	—	8	2	no	—
169	C₂²×C₅₃	—	8	1	no	—
170	C₂²×C₅₃	—	8	2	no	—
171	C₂²×C₅₃	—	16	4	no	—
172	C₂²×C₅₃	#392	1	1	yes	characteristic
173	C₂²×C₅₃	#459	1	2	yes	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	8	1	no	—
177	C₄×C₅₃	—	8	2	no	—
178	C₄×C₅₃	#447	1	2	yes	—
179	C₄×C₅₃	#455	1	2	yes	—
180	C₄×C₅₃	#471	1	2	yes	—
181	C₄×C₅₃	#493	1	1	yes	characteristic
182	C₄×C₅₃	#496	1	1	yes	characteristic
183	#3205	C₂×C₅₃	1	1	yes	characteristic
184	#3152	C₂×C₅₃	1	1	yes	characteristic, complemented
185	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
186	#1965	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#1492	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
190	C₂×C₅₃	—	2	2	no	—
191	C₂×C₅₃	—	16	1	no	—
192	C₂×C₅₃	—	16	2	no	—
193	C₂×C₅₃	#2004	1	1	yes	characteristic
194	C₂×C₅₃	#2011	1	1	yes	characteristic
195	C₂×C₅₃	#2019	1	1	yes	characteristic
196	C₂×#137	—	2	2	no	—
197	#532	C₂²×C₅₃	1	1	yes	characteristic
198	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
199	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
200	#497	—	2	2	no	—
201	#494	—	2	1	no	—
202	#493	—	2	1	no	—
203	#492	—	2	1	no	—
204	#486	—	2	2	no	—
205	#486	—	2	1	no	—
206	#486	—	2	2	no	—
207	#459	—	2	2	no	—
208	#447	—	2	2	no	—
209	#431	C₂²×C₅₃	1	1	yes	characteristic
210	#423	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×#102	—	2	2	no	—
212	C₂×#100	—	2	2	no	—
213	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
215	#332	—	2	2	no	—
216	D₄⋊C₈	—	2	2	no	—
217	C₅₃	#10168	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×#41	—	2	1	no	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	—	2	1	no	—
222	C₂²×D₄	D₄×C₅₃	1	2	yes	—
223	#137	—	4	4	no	—
224	#137	—	4	4	no	—
225	C₂×Q₁₆	—	2	4	no	—
226	C₂×SD₁₆	—	2	4	no	—
227	C₂×D₈	—	2	4	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
230	C₂²×C₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
235	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	—	2	1	no	—
240	#127	D₄×C₅₃	1	1	yes	characteristic
241	#124	—	2	1	no	—
242	#121	—	2	1	no	—
243	#121	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
252	#102	—	2	8	no	—
253	#102	—	2	4	no	—
254	#100	—	4	4	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	#97	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	D₄×C₅₃	1	2	yes	—
261	C₂⁴	—	2	2	no	—
262	#41	—	4	2	no	—
263	#41	—	4	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	4	4	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	8	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
288	Q₁₆	—	4	8	no	—
289	SD₁₆	—	4	8	no	—
290	D₈	—	8	4	no	—
291	M₁₆	—	2	2	no	—
292	M₁₆	—	4	2	no	—
293	M₁₆	D₈×C₅₃	1	4	yes	complemented
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₂×C₈	—	4	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂³	—	4	1	no	—
312	C₂³	—	4	2	no	—
313	C₂³	—	4	4	no	—
314	C₂³	—	8	2	no	—
315	C₂³	—	8	2	no	—
316	C₂³	#121×C₅₃	1	1	yes	characteristic
317	Q₈	—	4	4	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	8	1	no	—
320	Q₈	—	8	1	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	4	no	—
325	D₄	—	4	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	1	no	—
329	D₄	—	8	1	no	—
330	D₄	—	8	2	no	—
331	D₄	—	8	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
340	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
341	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
342	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
343	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
344	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
346	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
347	C₈	—	2	4	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	16	4	no	—
360	C₂²	C₅₃×#392	1	1	yes	characteristic
361	C₂²	C₅₃×#459	1	2	yes	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	2	no	—
366	C₄	C₅₃×#447	1	2	yes	—
367	C₄	C₅₃×#455	1	2	yes	—
368	C₄	C₅₃×#471	1	2	yes	—
369	C₄	C₅₃×#493	1	1	yes	characteristic
370	C₄	C₅₃×#496	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	16	1	no	—
374	C₂	C₅₃×#2004	1	1	yes	characteristic
375	C₂	C₅₃×#2011	1	1	yes	characteristic
376	C₂	C₅₃×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#3205	C₂
LCS	G ▷ C₂×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
2. C₅₃×#3152	C₂	semidirect
3. C₂×C₅₃×#486	C₂	semidirect
4. C₅₃×#1965	C₂	semidirect
5. C₅₃×#1546	C₂	semidirect
6. C₅₃×#1492	C₂	semidirect
7. C₅₃×#1278	C₂	semidirect
104. M₁₆×C₅₃	D₈	semidirect
162. #10168	C₅₃	direct (N × H)
184. #3152	C₂×C₅₃	semidirect
185. C₂×#486	C₂×C₅₃	semidirect
186. #1965	C₂×C₅₃	semidirect
187. #1546	C₂×C₅₃	semidirect
188. #1492	C₂×C₅₃	semidirect
189. #1278	C₂×C₅₃	semidirect
217. C₅₃	#10168	direct (N × H)
293. M₁₆	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3205	2
C₅₃×#3152	2
C₂×C₅₃×#486	2
C₅₃×#1965	2
C₅₃×#1546	2
C₅₃×#1492	2
C₅₃×#1278	2
#10168	53

Subgroups — C53×#10168 #27217680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10168 #27217680