Subgroups — C₅₃×#8644 #27216156

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1802	C₂	1	2	yes	maximal
4	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1296	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₅₃	—	2	1	no	—
7	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₅₃	—	2	2	no	—
9	C₈×D₄×C₅₃	C₄	1	4	yes	complemented
10	C₂×#107×C₅₃	—	2	1	no	—
11	C₂×#107×C₅₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#102×C₅₃	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₅₃×#340	—	2	2	no	—
19	C₅₃×#335	—	2	2	no	—
20	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₈×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	4	2	no	—
26	C₂²×C₈×C₅₃	—	4	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×D₄×C₅₃	—	2	1	no	—
30	C₄×D₄×C₅₃	—	2	1	no	—
31	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₅₃	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	4	2	no	—
40	#107×C₅₃	—	4	2	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
45	#102×C₅₃	—	4	4	no	—
46	C₂²⋊C₈×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₂⁴×C₅₃	—	2	1	no	—
50	C₂×D₄×C₅₃	—	2	1	no	—
51	C₂×D₄×C₅₃	—	2	1	no	—
52	C₂×D₄×C₅₃	—	2	1	no	—
53	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₅₃	D₈	1	2	yes	—
55	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	8	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₃	D₈	1	2	yes	—
79	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊C₄×C₅₃	—	4	2	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	—	8	2	no	—
86	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂²⋊C₄×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	—	2	2	no	—
96	C₂³×C₅₃	—	4	1	no	—
97	C₂³×C₅₃	—	4	1	no	—
98	C₂³×C₅₃	—	4	1	no	—
99	C₂³×C₅₃	—	4	1	no	—
100	C₂³×C₅₃	#121	1	1	yes	characteristic
101	D₄×C₅₃	—	2	2	no	—
102	D₄×C₅₃	—	2	2	no	—
103	D₄×C₅₃	—	2	2	no	—
104	D₄×C₅₃	#107	1	4	yes	complemented
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	8	2	no	—
115	C₂×C₄×C₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	—	8	2	no	—
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
129	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
131	C₈×C₅₃	—	2	2	no	—
132	C₈×C₅₃	—	4	2	no	—
133	C₈×C₅₃	—	8	2	no	—
134	C₈×C₅₃	#102	1	4	yes	complemented
135	#8644	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	—	4	2	no	—
138	C₂²×C₅₃	—	4	2	no	—
139	C₂²×C₅₃	—	4	2	no	—
140	C₂²×C₅₃	—	4	2	no	—
141	C₂²×C₅₃	—	4	2	no	—
142	C₂²×C₅₃	—	4	2	no	—
143	C₂²×C₅₃	#380	1	1	yes	characteristic
144	C₂²×C₅₃	#459	1	1	yes	characteristic
145	C₂²×C₅₃	#461	1	1	yes	characteristic
146	C₂²×C₅₃	#466	1	1	yes	characteristic
147	C₂²×C₅₃	#467	1	1	yes	characteristic
148	C₂²×C₅₃	#472	1	1	yes	characteristic
149	C₂²×C₅₃	#473	1	1	yes	characteristic
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	4	2	no	—
153	C₄×C₅₃	—	16	2	no	—
154	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
155	C₄×C₅₃	#417	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
157	C₄×C₅₃	#427	1	1	yes	characteristic
158	C₄×C₅₃	#466	1	1	yes	characteristic
159	C₄×C₅₃	#467	1	1	yes	characteristic
160	C₄×C₅₃	#474	1	1	yes	characteristic
161	C₄×C₅₃	#476	1	1	yes	characteristic
162	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
163	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
164	#1802	C₂×C₅₃	1	2	yes	—
165	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#1296	C₂×C₅₃	1	2	yes	complemented
167	C₂×C₅₃	—	4	4	no	—
168	C₂×C₅₃	#1667	1	1	yes	characteristic
169	C₂×C₅₃	#1669	1	1	yes	characteristic
170	C₂×C₅₃	#1673	1	1	yes	characteristic
171	C₂×C₅₃	#1675	1	1	yes	characteristic
172	C₂×C₅₃	#1802	1	1	yes	characteristic
173	C₂×C₅₃	#1808	1	1	yes	characteristic
174	C₂×C₅₃	#1816	1	1	yes	characteristic
175	C₂³×C₈	—	2	1	no	—
176	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
177	C₈×D₄	—	2	2	no	—
178	C₈×D₄	C₄×C₅₃	1	4	yes	complemented
179	C₂×#107	—	2	1	no	—
180	C₂×#107	—	2	1	no	—
181	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
182	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×#102	—	2	2	no	—
184	C₂×C₂²⋊C₈	—	2	1	no	—
185	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
186	#384	C₂²×C₅₃	1	2	yes	—
187	#340	—	2	2	no	—
188	#335	—	2	2	no	—
189	C₅₃	#8644	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
191	C₂³×C₄	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	4	2	no	—
196	C₂²×C₈	—	4	2	no	—
197	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
202	C₄×D₄	Q₈×C₅₃	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
209	#107	—	4	2	no	—
210	#107	—	4	2	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
215	#102	—	4	4	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
218	C₄×C₈	D₄×C₅₃	1	2	yes	—
219	C₂⁴	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
224	C₂×D₄	D₈×C₅₃	1	2	yes	—
225	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
247	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
248	C₂×C₈	D₈×C₅₃	1	2	yes	—
249	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	8	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
261	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
262	C₄²	#41×C₅₃	1	1	yes	characteristic
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	#121×C₅₃	1	1	yes	characteristic
271	D₄	—	2	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	#107×C₅₃	1	4	yes	complemented
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
290	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
297	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
298	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
299	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
300	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	8	2	no	—
304	C₈	#102×C₅₃	1	4	yes	complemented
305	C₂²	—	2	4	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	C₅₃×#380	1	1	yes	characteristic
313	C₂²	C₅₃×#459	1	1	yes	characteristic
314	C₂²	C₅₃×#461	1	1	yes	characteristic
315	C₂²	C₅₃×#466	1	1	yes	characteristic
316	C₂²	C₅₃×#467	1	1	yes	characteristic
317	C₂²	C₅₃×#472	1	1	yes	characteristic
318	C₂²	C₅₃×#473	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	16	2	no	—
323	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
324	C₄	C₅₃×#417	1	1	yes	characteristic
325	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
326	C₄	C₅₃×#427	1	1	yes	characteristic
327	C₄	C₅₃×#466	1	1	yes	characteristic
328	C₄	C₅₃×#467	1	1	yes	characteristic
329	C₄	C₅₃×#474	1	1	yes	characteristic
330	C₄	C₅₃×#476	1	1	yes	characteristic
331	C₂	—	4	4	no	—
332	C₂	C₅₃×#1667	1	1	yes	characteristic
333	C₂	C₅₃×#1669	1	1	yes	characteristic
334	C₂	C₅₃×#1673	1	1	yes	characteristic
335	C₂	C₅₃×#1675	1	1	yes	characteristic
336	C₂	C₅₃×#1802	1	1	yes	characteristic
337	C₂	C₅₃×#1808	1	1	yes	characteristic
338	C₂	C₅₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ 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×#1845	C₂	semidirect
4. C₂×C₅₃×#335	C₂	semidirect
5. C₅₃×#1296	C₂	semidirect
9. C₈×D₄×C₅₃	C₄	semidirect
104. D₄×C₅₃	#107	semidirect
134. C₈×C₅₃	#102	semidirect
135. #8644	C₅₃	direct (N × H)
163. #1845	C₂×C₅₃	semidirect
165. C₂×#335	C₂×C₅₃	semidirect
166. #1296	C₂×C₅₃	semidirect
178. C₈×D₄	C₄×C₅₃	semidirect
189. C₅₃	#8644	direct (N × H)
274. D₄	#107×C₅₃	semidirect
304. C₈	#102×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₅₃	2
C₅₃×#1845	2
C₅₃×#1802	2
C₂×C₅₃×#335	2
C₅₃×#1296	2
#8644	53

Subgroups — C53×#8644 #27216156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8644 #27216156