Subgroups — C₅₃×#11242 #27218754

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 350 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	1	yes	maximal, characteristic
3	C₂×C₅₃×#343	C₂	1	2	yes	maximal
4	C₅₃×#1736	C₂	1	2	yes	maximal
5	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂×#108×C₅₃	—	2	4	no	—
12	C₂×#108×C₅₃	—	2	2	no	—
13	C₅₃×#424	—	2	2	no	—
14	C₅₃×#424	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
16	C₅₃×#343	—	2	4	no	—
17	C₅₃×#340	—	2	2	no	—
18	C₅₃×#340	—	2	2	no	—
19	C₅₃×#340	—	2	2	no	—
20	C₂³×C₄×C₅₃	—	2	2	no	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	1	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	—	2	2	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	#117×C₅₃	—	2	1	no	—
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	C₂×C₄	1	2	yes	—
42	#117×C₅₃	Q₈	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₃	Q₈	1	4	yes	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	#108×C₅₃	—	2	8	no	—
54	C₄⋊C₈×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	—	8	2	no	—
62	C₂²×C₄×C₅₃	—	8	2	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	2	4	no	—
76	M₁₆×C₅₃	—	4	1	no	—
77	M₁₆×C₅₃	—	4	1	no	—
78	M₁₆×C₅₃	—	4	2	no	—
79	M₁₆×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
90	C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	—	2	4	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₅₃	—	2	2	no	—
99	C₂²⋊C₄×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	—	2	1	no	—
104	C₂³×C₅₃	—	2	1	no	—
105	C₂³×C₅₃	#117	1	1	yes	characteristic
106	C₂³×C₅₃	#121	1	1	yes	characteristic
107	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	C₂³×C₅₃	#126	1	2	yes	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	8	4	no	—
117	C₂×C₄×C₅₃	—	8	2	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	—	8	4	no	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#121	1	2	yes	—
126	C₂×C₄×C₅₃	#121	1	2	yes	—
127	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#123	1	2	yes	—
133	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#125	1	2	yes	—
135	C₂×C₄×C₅₃	#125	1	2	yes	—
136	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
138	C₈×C₅₃	—	2	4	no	—
139	C₈×C₅₃	—	4	2	no	—
140	C₈×C₅₃	—	4	4	no	—
141	#11242	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	#380	1	1	yes	characteristic
151	C₂²×C₅₃	#387	1	2	yes	—
152	C₂²×C₅₃	#395	1	1	yes	characteristic
153	C₂²×C₅₃	#396	1	2	yes	—
154	C₂²×C₅₃	#400	1	1	yes	characteristic
155	C₂²×C₅₃	#471	1	2	yes	—
156	C₂²×C₅₃	#472	1	1	yes	characteristic
157	C₂²×C₅₃	#473	1	1	yes	characteristic
158	C₄×C₅₃	—	4	2	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	8	4	no	—
161	C₄×C₅₃	#386	1	1	yes	characteristic
162	C₄×C₅₃	#389	1	2	yes	—
163	C₄×C₅₃	#391	1	1	yes	characteristic
164	C₄×C₅₃	#394	1	1	yes	characteristic
165	C₄×C₅₃	#398	1	2	yes	—
166	C₄×C₅₃	#398	1	1	yes	characteristic
167	C₂²×#108	C₂×C₅₃	1	1	yes	characteristic
168	C₂×#424	C₂×C₅₃	1	1	yes	characteristic
169	C₂×#343	C₂×C₅₃	1	2	yes	—
170	#1736	C₂×C₅₃	1	2	yes	—
171	C₂×#340	C₂×C₅₃	1	1	yes	characteristic
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	#1441	1	1	yes	characteristic
176	C₂×C₅₃	#1815	1	1	yes	characteristic
177	C₂×C₅₃	#1816	1	1	yes	characteristic
178	C₂×C₅₃	#2063	1	2	yes	—
179	C₂×C₅₃	#2064	1	2	yes	—
180	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
181	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
182	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
185	C₂×#108	—	2	4	no	—
186	C₂×#108	—	2	2	no	—
187	#424	—	2	2	no	—
188	#424	C₄×C₅₃	1	4	yes	complemented
189	C₂×C₄⋊C₈	—	2	2	no	—
190	#343	—	2	4	no	—
191	#340	—	2	2	no	—
192	#340	—	2	2	no	—
193	#340	—	2	2	no	—
194	C₅₃	#11242	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³×C₄	—	2	2	no	—
196	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
206	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
212	C₂²×C₈	D₄×C₅₃	1	2	yes	—
213	C₂²×C₈	D₄×C₅₃	1	2	yes	—
214	#117	—	2	1	no	—
215	#117	—	2	1	no	—
216	#117	C₂×C₄×C₅₃	1	2	yes	—
217	#117	Q₈×C₅₃	1	2	yes	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄⋊C₄	Q₈×C₅₃	1	4	yes	—
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	2	1	no	—
228	#108	—	2	8	no	—
229	C₄⋊C₈	—	2	8	no	—
230	C₂⁴	#41×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	8	2	no	—
237	C₂²×C₄	—	8	2	no	—
238	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
243	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
245	C₂²×C₄	#41×C₅₃	1	2	yes	—
246	C₂²×C₄	#41×C₅₃	1	2	yes	—
247	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
272	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	#117×C₅₃	1	1	yes	characteristic
281	C₂³	#121×C₅₃	1	1	yes	characteristic
282	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
283	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
284	C₂³	#126×C₅₃	1	2	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
298	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
299	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
300	C₂×C₄	#121×C₅₃	1	2	yes	—
301	C₂×C₄	#121×C₅₃	1	2	yes	—
302	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
304	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
305	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
306	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
307	C₂×C₄	#123×C₅₃	1	2	yes	—
308	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
309	C₂×C₄	#125×C₅₃	1	2	yes	—
310	C₂×C₄	#125×C₅₃	1	2	yes	—
311	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
312	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
313	C₈	—	2	4	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	C₅₃×#380	1	1	yes	characteristic
325	C₂²	C₅₃×#387	1	2	yes	—
326	C₂²	C₅₃×#395	1	1	yes	characteristic
327	C₂²	C₅₃×#396	1	2	yes	—
328	C₂²	C₅₃×#400	1	1	yes	characteristic
329	C₂²	C₅₃×#471	1	2	yes	—
330	C₂²	C₅₃×#472	1	1	yes	characteristic
331	C₂²	C₅₃×#473	1	1	yes	characteristic
332	C₄	—	4	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	4	no	—
335	C₄	C₅₃×#386	1	1	yes	characteristic
336	C₄	C₅₃×#389	1	2	yes	—
337	C₄	C₅₃×#391	1	1	yes	characteristic
338	C₄	C₅₃×#394	1	1	yes	characteristic
339	C₄	C₅₃×#398	1	2	yes	—
340	C₄	C₅₃×#398	1	1	yes	characteristic
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	C₅₃×#1441	1	1	yes	characteristic
345	C₂	C₅₃×#1815	1	1	yes	characteristic
346	C₂	C₅₃×#1816	1	1	yes	characteristic
347	C₂	C₅₃×#2063	1	2	yes	—
348	C₂	C₅₃×#2064	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₅₃×#424	C₄	semidirect
141. #11242	C₅₃	direct (N × H)
188. #424	C₄×C₅₃	semidirect
194. C₅₃	#11242	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
C₂×C₅₃×#343	2
C₅₃×#1736	2
C₂×C₅₃×#340	2
#11242	53

Subgroups — C53×#11242 #27218754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11242 #27218754