Subgroups — C₅₃×#10706 #27218218

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#405	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#339	C₂	1	2	yes	maximal
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	C₅₃×#405	—	2	4	no	—
12	C₅₃×#405	C₄	1	8	yes	complemented
13	C₂×#97×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	—	2	4	no	—
15	C₂×C₄×C₈×C₅₃	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
17	C₅₃×#341	C₄	1	4	yes	complemented
18	C₅₃×#340	C₄	1	4	yes	complemented
19	C₅₃×#339	C₄	1	8	yes	complemented
20	C₂³×C₄×C₅₃	—	2	1	no	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₅₃	D₄	1	4	yes	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	—	2	2	no	—
42	#117×C₅₃	C₂×C₄	1	4	yes	—
43	#117×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	#97×C₅₃	—	2	8	no	—
52	C₄×C₈×C₅₃	—	2	8	no	—
53	C₄×C₈×C₅₃	—	2	8	no	—
54	C₂⁴×C₅₃	#41	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
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	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	C₄²	1	8	yes	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₄²	1	4	yes	—
81	C₂×C₈×C₅₃	C₄²	1	4	yes	—
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
85	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
87	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
90	C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂²⋊C₄×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	2	no	—
93	C₄²×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	C₄²	1	4	yes	—
97	C₂³×C₅₃	—	2	4	no	—
98	C₂³×C₅₃	#117	1	1	yes	characteristic
99	C₂³×C₅₃	#117	1	2	yes	—
100	C₂³×C₅₃	C₄×D₄	1	2	yes	—
101	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	16	no	—
104	C₂×C₄×C₅₃	—	2	8	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	—	4	8	no	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#117	1	2	yes	—
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
126	C₈×C₅₃	—	2	4	no	—
127	C₈×C₅₃	—	2	8	no	—
128	C₈×C₅₃	#95	1	4	yes	—
129	C₈×C₅₃	#95	1	4	yes	—
130	#10706	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	—	2	4	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	#376	1	1	yes	characteristic
136	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₅₃	#443	1	4	yes	—
141	C₄×C₅₃	—	2	8	no	—
142	C₄×C₅₃	—	4	8	no	—
143	C₄×C₅₃	C₄³	1	1	yes	characteristic
144	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
145	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
147	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
149	C₂×#405	C₂×C₅₃	1	2	yes	—
150	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
151	C₂×#341	C₂×C₅₃	1	1	yes	characteristic
152	#1735	C₂×C₅₃	1	1	yes	characteristic
153	C₂×#339	C₂×C₅₃	1	2	yes	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
158	C₂×C₅₃	#1764	1	4	yes	—
159	C₂×C₅₃	#1772	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
161	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×#117	C₂²×C₅₃	1	2	yes	—
164	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
165	#405	—	2	4	no	—
166	#405	C₄×C₅₃	1	8	yes	complemented
167	C₂×#97	—	2	2	no	—
168	C₂×C₄×C₈	—	2	4	no	—
169	C₂×C₄×C₈	—	2	2	no	—
170	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	complemented
171	#341	C₄×C₅₃	1	4	yes	complemented
172	#340	C₄×C₅₃	1	4	yes	complemented
173	#339	C₄×C₅₃	1	8	yes	complemented
174	C₅₃	#10706	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	—	2	1	no	—
176	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
181	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
182	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
187	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
188	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
189	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
190	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
191	C₂²×C₈	D₄×C₅₃	1	4	yes	—
192	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
194	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	C₂×C₄×C₅₃	1	4	yes	—
198	#117	C₂×C₄×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	1	no	—
204	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
205	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
206	#97	—	2	8	no	—
207	C₄×C₈	—	2	8	no	—
208	C₄×C₈	—	2	8	no	—
209	C₂⁴	#41×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
221	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	#41×C₅₃	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	C₄²×C₅₃	1	8	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₄²×C₅₃	1	4	yes	—
236	C₂×C₈	C₄²×C₅₃	1	4	yes	—
237	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
238	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
239	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
240	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
241	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
242	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	C₄²×C₅₃	1	8	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	C₄²×C₅₃	1	4	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	#117×C₅₃	1	2	yes	—
254	C₂³	#117×C₅₃	1	1	yes	characteristic
255	C₂³	C₄×D₄×C₅₃	1	2	yes	—
256	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
257	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
266	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
270	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
273	C₂×C₄	#117×C₅₃	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
277	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
278	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
279	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
280	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	8	no	—
283	C₈	#95×C₅₃	1	4	yes	—
284	C₈	#95×C₅₃	1	4	yes	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	C₅₃×#376	1	1	yes	characteristic
290	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
291	C₂²	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
292	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
293	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
294	C₂²	C₅₃×#443	1	4	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	4	8	no	—
297	C₄	C₄³×C₅₃	1	1	yes	characteristic
298	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
299	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
300	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
301	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
302	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
307	C₂	C₅₃×#1764	1	4	yes	—
308	C₂	C₅₃×#1772	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₅₃	D₄
commutator	C₄	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅₃×#405	C₄	semidirect
16. C₂×C₄×C₈×C₅₃	C₄	semidirect
17. C₅₃×#341	C₄	semidirect
18. C₅₃×#340	C₄	semidirect
19. C₅₃×#339	C₄	semidirect
130. #10706	C₅₃	direct (N × H)
166. #405	C₄×C₅₃	semidirect
170. C₂×C₄×C₈	C₄×C₅₃	semidirect
171. #341	C₄×C₅₃	semidirect
172. #340	C₄×C₅₃	semidirect
173. #339	C₄×C₅₃	semidirect
174. C₅₃	#10706	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×#405	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#341	2
C₅₃×#1735	2
C₂×C₅₃×#339	2
#10706	53

Subgroups — C53×#10706 #27218218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10706 #27218218