Subgroups — C₅₃×#11329 #27218841

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#424	C₂	1	2	yes	maximal
3	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#339	C₂	1	2	yes	maximal
5	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#108×C₅₃	—	2	4	no	—
12	C₂×#108×C₅₃	—	2	2	no	—
13	C₅₃×#424	—	2	4	no	—
14	C₅₃×#424	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₃	—	2	4	no	—
16	C₅₃×#340	C₄	1	4	yes	complemented
17	C₅₃×#339	—	2	4	no	—
18	C₅₃×#336	—	2	2	no	—
19	C₂³×C₄×C₅₃	—	2	1	no	—
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	D₄	1	4	yes	—
41	#117×C₅₃	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
49	#108×C₅₃	—	2	8	no	—
50	C₄⋊C₈×C₅₃	—	2	16	no	—
51	C₂⁴×C₅₃	#41	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
86	C₂²⋊C₄×C₅₃	—	2	4	no	—
87	C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	4	1	no	—
92	C₄²×C₅₃	—	4	1	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
97	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₅₃	#123	1	2	yes	—
99	C₂³×C₅₃	#125	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	4	8	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#120	1	2	yes	—
113	C₂×C₄×C₅₃	#121	1	2	yes	—
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#125	1	2	yes	—
120	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
125	C₈×C₅₃	—	2	4	no	—
126	C₈×C₅₃	—	4	2	no	—
127	C₈×C₅₃	—	4	4	no	—
128	#11329	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₅₃	—	2	4	no	—
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	#385	1	2	yes	—
136	C₂²×C₅₃	#389	1	1	yes	characteristic
137	C₂²×C₅₃	#395	1	1	yes	characteristic
138	C₂²×C₅₃	#398	1	2	yes	—
139	C₂²×C₅₃	#399	1	1	yes	characteristic
140	C₂²×C₅₃	#443	1	2	yes	—
141	C₂²×C₅₃	#444	1	2	yes	—
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	—	4	8	no	—
144	C₄×C₅₃	#384	1	1	yes	characteristic
145	C₄×C₅₃	#385	1	1	yes	characteristic
146	C₄×C₅₃	#389	1	2	yes	—
147	C₄×C₅₃	#392	1	1	yes	characteristic
148	C₄×C₅₃	#395	1	2	yes	—
149	C₄×C₅₃	#398	1	1	yes	characteristic
150	C₂²×#108	C₂×C₅₃	1	1	yes	characteristic
151	C₂×#424	C₂×C₅₃	1	2	yes	—
152	#1735	C₂×C₅₃	1	1	yes	characteristic
153	C₂×#339	C₂×C₅₃	1	2	yes	—
154	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	2	no	—
158	C₂×C₅₃	#1444	1	1	yes	characteristic
159	C₂×C₅₃	#1866	1	4	yes	—
160	C₂×C₅₃	#1945	1	1	yes	characteristic
161	C₂×C₅₃	#1946	1	1	yes	characteristic
162	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
163	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×#117	C₂²×C₅₃	1	2	yes	—
166	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×#108	—	2	4	no	—
168	C₂×#108	—	2	2	no	—
169	#424	—	2	4	no	—
170	#424	—	2	4	no	—
171	C₂×C₄⋊C₈	—	2	4	no	—
172	#340	C₄×C₅₃	1	4	yes	complemented
173	#339	—	2	4	no	—
174	#336	—	2	2	no	—
175	C₅₃	#11329	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³×C₄	—	2	1	no	—
177	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
192	C₂²×C₈	D₄×C₅₃	1	2	yes	—
193	C₂²×C₈	D₄×C₅₃	1	2	yes	—
194	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	D₄×C₅₃	1	4	yes	—
198	#117	Q₈×C₅₃	1	4	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
206	#108	—	2	8	no	—
207	C₄⋊C₈	—	2	16	no	—
208	C₂⁴	#41×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
221	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	#41×C₅₃	1	2	yes	—
223	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
253	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
254	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
255	C₂³	#123×C₅₃	1	2	yes	—
256	C₂³	#125×C₅₃	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
266	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
267	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
268	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
269	C₂×C₄	#120×C₅₃	1	2	yes	—
270	C₂×C₄	#121×C₅₃	1	2	yes	—
271	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
273	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
274	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
275	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
276	C₂×C₄	#125×C₅₃	1	2	yes	—
277	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
279	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
280	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
281	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	4	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₅₃×#385	1	2	yes	—
292	C₂²	C₅₃×#389	1	1	yes	characteristic
293	C₂²	C₅₃×#395	1	1	yes	characteristic
294	C₂²	C₅₃×#398	1	2	yes	—
295	C₂²	C₅₃×#399	1	1	yes	characteristic
296	C₂²	C₅₃×#443	1	2	yes	—
297	C₂²	C₅₃×#444	1	2	yes	—
298	C₄	—	4	4	no	—
299	C₄	—	4	8	no	—
300	C₄	C₅₃×#384	1	1	yes	characteristic
301	C₄	C₅₃×#385	1	1	yes	characteristic
302	C₄	C₅₃×#389	1	2	yes	—
303	C₄	C₅₃×#392	1	1	yes	characteristic
304	C₄	C₅₃×#395	1	2	yes	—
305	C₄	C₅₃×#398	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	C₅₃×#1444	1	1	yes	characteristic
310	C₂	C₅₃×#1866	1	4	yes	—
311	C₂	C₅₃×#1945	1	1	yes	characteristic
312	C₂	C₅₃×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₅₃×#340	C₄	semidirect
128. #11329	C₅₃	direct (N × H)
172. #340	C₄×C₅₃	semidirect
175. C₅₃	#11329	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#108×C₅₃	2
C₂×C₅₃×#424	2
C₅₃×#1735	2
C₂×C₅₃×#339	2
C₂×C₅₃×#336	2
#11329	53

Subgroups — C53×#11329 #27218841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11329 #27218841