Subgroups — C₅₃×#12612 #27220124

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₅₃	—	2	2	no	—
10	C₂×M₃₂×C₅₃	—	2	2	no	—
11	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
20	C₅₃×C₂²⋊C₁₆	C₄	1	4	yes	complemented
21	C₅₃×#346	—	2	2	no	—
22	C₄×C₁₆×C₅₃	—	2	2	no	—
23	C₅₃×#336	C₄	1	1	yes	characteristic
24	C₅₃×#336	C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	M₃₂×C₅₃	—	2	8	no	—
40	C₂×C₁₆×C₅₃	—	2	1	no	—
41	C₂×C₁₆×C₅₃	—	2	1	no	—
42	C₂×C₁₆×C₅₃	—	2	1	no	—
43	C₂×C₁₆×C₅₃	—	2	1	no	—
44	C₂×C₁₆×C₅₃	—	2	1	no	—
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
51	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
52	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	#95×C₅₃	C₈	1	2	yes	—
58	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	C₄²	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₁₆×C₅₃	—	2	2	no	—
94	C₁₆×C₅₃	—	2	2	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
97	C₂³×C₅₃	—	2	1	no	—
98	C₂³×C₅₃	—	2	1	no	—
99	C₂³×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	C₄×C₈	1	2	yes	—
102	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
115	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₈×C₅₃	#117	1	1	yes	characteristic
127	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
129	#12612	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
137	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
139	C₂²×C₅₃	#504	1	2	yes	—
140	C₄×C₅₃	—	2	4	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	#405	1	1	yes	characteristic
145	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
146	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
147	C₂²×M₃₂	C₂×C₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
149	C₂×#346	C₂×C₅₃	1	1	yes	characteristic, complemented
150	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
151	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
152	#1374	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1374	C₂×C₅₃	1	1	yes	characteristic, complemented
154	C₂×C₅₃	—	4	2	no	—
155	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
157	C₂×C₅₃	#2108	1	1	yes	characteristic
158	C₂×C₅₃	#2168	1	2	yes	—
159	C₂×C₅₃	#2169	1	2	yes	—
160	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×M₃₂	—	2	2	no	—
162	C₂×M₃₂	—	2	2	no	—
163	C₂×M₃₂	C₄×C₅₃	1	4	yes	complemented
164	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
165	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
166	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
167	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	C₄×C₅₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
171	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
172	C₂²⋊C₁₆	C₄×C₅₃	1	4	yes	complemented
173	#346	—	2	2	no	—
174	C₄×C₁₆	—	2	2	no	—
175	#336	C₄×C₅₃	1	1	yes	characteristic
176	#336	C₄×C₅₃	1	1	yes	characteristic
177	C₅₃	#12612	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
184	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
185	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	C₈×C₅₃	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
192	M₃₂	—	2	8	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
200	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
201	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
202	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
203	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
204	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
205	C₂²⋊C₈	C₈×C₅₃	1	4	yes	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	#95	C₈×C₅₃	1	2	yes	—
211	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
217	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
220	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₄²×C₅₃	1	2	yes	—
233	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
234	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
239	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
240	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	C₂²⋊C₄×C₅₃	1	4	yes	complemented
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	C₄×C₈×C₅₃	1	2	yes	—
255	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
268	C₂×C₄	M₃₂×C₅₃	1	4	yes	complemented
269	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
273	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
274	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
279	C₈	#117×C₅₃	1	1	yes	characteristic
280	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
281	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
288	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
289	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
290	C₂²	C₂×M₃₂×C₅₃	1	2	yes	—
291	C₂²	C₅₃×#504	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
296	C₄	C₅₃×#405	1	1	yes	characteristic
297	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
298	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
299	C₂	—	4	2	no	—
300	C₂	C₈×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
301	C₂	C₄×M₃₂×C₅₃	1	1	yes	characteristic
302	C₂	C₅₃×#2108	1	1	yes	characteristic
303	C₂	C₅₃×#2168	1	2	yes	—
304	C₂	C₅₃×#2169	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#346	C₂	semidirect
4. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
6. C₅₃×#1374	C₂	semidirect
7. C₅₃×#1374	C₂	semidirect
11. C₂×M₃₂×C₅₃	C₄	semidirect
20. C₅₃×C₂²⋊C₁₆	C₄	semidirect
96. C₁₆×C₅₃	C₂²⋊C₄	semidirect
115. C₂×C₄×C₅₃	M₃₂	semidirect
129. #12612	C₅₃	direct (N × H)
149. C₂×#346	C₂×C₅₃	semidirect
150. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
152. #1374	C₂×C₅₃	semidirect
153. #1374	C₂×C₅₃	semidirect
163. C₂×M₃₂	C₄×C₅₃	semidirect
172. C₂²⋊C₁₆	C₄×C₅₃	semidirect
177. C₅₃	#12612	direct (N × H)
249. C₁₆	C₂²⋊C₄×C₅₃	semidirect
268. C₂×C₄	M₃₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₅₃	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₂×C₅₃×#346	2
C₂×C₄×C₁₆×C₅₃	2
C₈×C₂²⋊C₄×C₅₃	2
C₅₃×#1374	2
#12612	53

Subgroups — C53×#12612 #27220124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12612 #27220124