Subgroups — C₅₃×#10443 #27217955

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#460	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1801	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₅₃	—	2	1	no	—
9	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#460	—	2	2	no	—
12	C₅₃×#460	—	2	2	no	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₅₃×#423	C₂²	1	1	yes	characteristic
16	C₅₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#102×C₅₃	—	2	1	no	—
19	C₂×#101×C₅₃	—	2	1	no	—
20	C₅₃×#343	—	2	1	no	—
21	C₅₃×#334	—	2	2	no	—
22	C₅₃×#331	—	2	2	no	—
23	C₅₃×#331	—	2	2	no	—
24	C₅₃×Q₈⋊C₈	—	2	2	no	—
25	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₅₃	—	2	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	2	1	no	—
29	C₂×SD₁₆×C₅₃	—	2	4	no	—
30	C₂×SD₁₆×C₅₃	—	4	2	no	—
31	C₂×SD₁₆×C₅₃	—	4	2	no	—
32	C₂×M₁₆×C₅₃	—	2	1	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	#127×C₅₃	—	2	1	no	—
36	#127×C₅₃	—	2	1	no	—
37	#127×C₅₃	—	2	1	no	—
38	#127×C₅₃	D₄	1	2	yes	—
39	C₄×Q₈×C₅₃	—	2	1	no	—
40	C₄×Q₈×C₅₃	—	2	1	no	—
41	C₄×Q₈×C₅₃	—	2	1	no	—
42	C₄×Q₈×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	1	no	—
48	C₄⋊C₈×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	D₄	1	2	yes	—
54	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
55	#102×C₅₃	—	2	4	no	—
56	#101×C₅₃	—	4	2	no	—
57	C₂²⋊C₈×C₅₃	—	2	1	no	—
58	C₂²⋊C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₂⁴×C₅₃	—	4	1	no	—
62	C₂×Q₈×C₅₃	—	2	1	no	—
63	C₂×Q₈×C₅₃	—	2	1	no	—
64	C₂×Q₈×C₅₃	—	2	1	no	—
65	C₂×Q₈×C₅₃	—	2	1	no	—
66	C₂×Q₈×C₅₃	—	2	1	no	—
67	C₂×Q₈×C₅₃	SD₁₆	1	2	yes	—
68	C₂×D₄×C₅₃	—	2	2	no	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	—	4	1	no	—
72	C₂×D₄×C₅₃	—	4	1	no	—
73	C₂×D₄×C₅₃	—	4	1	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂×D₄×C₅₃	—	4	1	no	—
76	C₂×D₄×C₅₃	—	4	1	no	—
77	C₂×D₄×C₅₃	—	4	1	no	—
78	C₂×D₄×C₅₃	—	8	2	no	—
79	C₂×D₄×C₅₃	—	8	2	no	—
80	C₂²×C₄×C₅₃	—	2	1	no	—
81	C₂²×C₄×C₅₃	—	2	1	no	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
85	SD₁₆×C₅₃	—	4	8	no	—
86	M₁₆×C₅₃	—	4	1	no	—
87	M₁₆×C₅₃	—	4	1	no	—
88	M₁₆×C₅₃	—	4	1	no	—
89	M₁₆×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	4	1	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	—	4	1	no	—
95	C₂×C₈×C₅₃	—	4	1	no	—
96	C₂×C₈×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	1	no	—
102	C₄²×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	#41	1	1	yes	characteristic
107	C₂³×C₅₃	—	4	2	no	—
108	C₂³×C₅₃	—	8	2	no	—
109	C₂³×C₅₃	—	8	2	no	—
110	C₂³×C₅₃	—	8	1	no	—
111	C₂³×C₅₃	—	8	1	no	—
112	C₂³×C₅₃	#123	1	1	yes	characteristic
113	Q₈×C₅₃	—	2	2	no	—
114	Q₈×C₅₃	—	2	2	no	—
115	Q₈×C₅₃	—	4	1	no	—
116	Q₈×C₅₃	—	4	1	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	4	2	no	—
119	D₄×C₅₃	—	8	2	no	—
120	D₄×C₅₃	—	8	2	no	—
121	D₄×C₅₃	—	8	2	no	—
122	D₄×C₅₃	—	8	1	no	—
123	D₄×C₅₃	—	8	1	no	—
124	D₄×C₅₃	—	8	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	8	1	no	—
132	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
140	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₈×C₅₃	—	4	2	no	—
143	C₈×C₅₃	—	4	2	no	—
144	C₈×C₅₃	—	8	1	no	—
145	C₈×C₅₃	—	8	1	no	—
146	C₈×C₅₃	—	8	1	no	—
147	#10443	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	8	2	no	—
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	—	8	2	no	—
153	C₂²×C₅₃	—	16	2	no	—
154	C₂²×C₅₃	—	16	2	no	—
155	C₂²×C₅₃	#394	1	1	yes	characteristic
156	C₂²×C₅₃	#481	1	1	yes	characteristic
157	C₂²×C₅₃	#482	1	1	yes	characteristic
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	4	2	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	#449	1	1	yes	characteristic
164	C₄×C₅₃	#450	1	1	yes	characteristic
165	C₄×C₅₃	#465	1	1	yes	characteristic
166	C₄×C₅₃	#469	1	1	yes	characteristic
167	C₄×C₅₃	#471	1	2	yes	—
168	C₂×#460	C₂×C₅₃	1	1	yes	characteristic
169	#1801	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1570	C₂×C₅₃	1	1	yes	characteristic, complemented
171	#1545	C₂×C₅₃	1	1	yes	characteristic
172	#1545	C₂×C₅₃	1	1	yes	characteristic
173	#1479	C₂×C₅₃	1	1	yes	characteristic, complemented
174	#1301	C₂×C₅₃	1	1	yes	characteristic, complemented
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	16	2	no	—
178	C₂×C₅₃	#2034	1	1	yes	characteristic
179	C₂×C₅₃	#2046	1	1	yes	characteristic
180	C₂×C₅₃	#2048	1	1	yes	characteristic
181	C₂²×SD₁₆	—	2	1	no	—
182	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
184	#460	—	2	2	no	—
185	#460	—	2	2	no	—
186	#431	C₂²×C₅₃	1	1	yes	characteristic
187	#431	C₂²×C₅₃	1	1	yes	characteristic
188	#423	C₂²×C₅₃	1	1	yes	characteristic
189	#423	C₂²×C₅₃	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
191	C₂×#102	—	2	1	no	—
192	C₂×#101	—	2	1	no	—
193	#343	—	2	1	no	—
194	#334	—	2	2	no	—
195	#331	—	2	2	no	—
196	#331	—	2	2	no	—
197	Q₈⋊C₈	—	2	2	no	—
198	C₅₃	#10443	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	4	2	no	—
205	C₂×SD₁₆	—	4	2	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	#127	—	2	1	no	—
210	#127	—	2	1	no	—
211	#127	—	2	1	no	—
212	#127	D₄×C₅₃	1	2	yes	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
228	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
229	#102	—	2	4	no	—
230	#101	—	4	2	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₂⁴	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	SD₁₆×C₅₃	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	8	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
259	SD₁₆	—	4	8	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₄²	#41×C₅₃	1	1	yes	characteristic
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	#123×C₅₃	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
307	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
308	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
309	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
310	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
311	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
313	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
314	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
315	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₈	—	8	1	no	—
319	C₈	—	8	1	no	—
320	C₈	—	8	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	C₅₃×#394	1	1	yes	characteristic
329	C₂²	C₅₃×#481	1	1	yes	characteristic
330	C₂²	C₅₃×#482	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	1	no	—
336	C₄	C₅₃×#449	1	1	yes	characteristic
337	C₄	C₅₃×#450	1	1	yes	characteristic
338	C₄	C₅₃×#465	1	1	yes	characteristic
339	C₄	C₅₃×#469	1	1	yes	characteristic
340	C₄	C₅₃×#471	1	2	yes	—
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₅₃×#2034	1	1	yes	characteristic
345	C₂	C₅₃×#2046	1	1	yes	characteristic
346	C₂	C₅₃×#2048	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
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×#1801	C₂	semidirect
3. C₅₃×#1570	C₂	semidirect
6. C₅₃×#1479	C₂	semidirect
7. C₅₃×#1301	C₂	semidirect
147. #10443	C₅₃	direct (N × H)
169. #1801	C₂×C₅₃	semidirect
170. #1570	C₂×C₅₃	semidirect
173. #1479	C₂×C₅₃	semidirect
174. #1301	C₂×C₅₃	semidirect
198. C₅₃	#10443	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×#460	2
C₅₃×#1801	2
C₅₃×#1570	2
C₅₃×#1545	2
C₅₃×#1479	2
C₅₃×#1301	2
#10443	53

Subgroups — C53×#10443 #27217955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10443 #27217955