Subgroups — C₅₃×#21275 #27228787

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2293	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1970	C₂	1	2	yes	maximal
4	C₅₃×#1942	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1934	C₂	1	4	yes	maximal, complemented
6	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#102×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
11	C₂×#117×C₅₃	—	2	2	no	—
12	C₄×D₈×C₅₃	C₄	1	8	yes	complemented
13	C₂×#107×C₅₃	C₂²	1	2	yes	—
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#102×C₅₃	C₂²	1	4	yes	—
16	C₂×#102×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#390	C₂²	1	4	yes	—
22	C₅₃×#389	C₂²	1	4	yes	—
23	C₅₃×#388	—	2	4	no	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
26	C₄³×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#340	C₂²	1	2	yes	—
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	4	no	—
32	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₅₃	—	2	4	no	—
37	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
38	#117×C₅₃	—	4	8	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#107×C₅₃	—	2	4	no	—
52	#107×C₅₃	C₂×C₄	1	4	yes	—
53	#102×C₅₃	—	2	8	no	—
54	#102×C₅₃	C₂×C₄	1	8	yes	—
55	C₄×C₈×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	—	2	4	no	—
57	C₄×C₈×C₅₃	—	2	2	no	—
58	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
59	#95×C₅₃	—	2	4	no	—
60	#95×C₅₃	—	2	2	no	—
61	C₂⁴×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	2	4	no	—
63	C₂×D₄×C₅₃	—	2	4	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	8	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	4	4	no	—
70	C₂²×C₄×C₅₃	—	4	4	no	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	D₈×C₅₃	—	2	8	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	#41	1	4	yes	—
83	C₂×C₈×C₅₃	#41	1	4	yes	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	4	8	no	—
88	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂²⋊C₄×C₅₃	—	2	8	no	—
90	C₂²⋊C₄×C₅₃	—	4	8	no	—
91	C₂²⋊C₄×C₅₃	—	4	8	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	4	4	no	—
97	C₄²×C₅₃	D₈	1	4	yes	complemented
98	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	#41	1	2	yes	—
100	C₂³×C₅₃	—	2	4	no	—
101	C₂³×C₅₃	—	4	4	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	—	4	2	no	—
104	C₂³×C₅₃	—	4	4	no	—
105	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₅₃	—	2	4	no	—
107	D₄×C₅₃	—	2	4	no	—
108	D₄×C₅₃	—	4	4	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	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	16	no	—
118	C₂×C₄×C₅₃	—	4	4	no	—
119	C₂×C₄×C₅₃	—	4	4	no	—
120	C₂×C₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	4	yes	—
126	C₂×C₄×C₅₃	#135	1	2	yes	—
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	C₈×C₅₃	—	2	2	no	—
132	C₈×C₅₃	—	2	4	no	—
133	C₈×C₅₃	#117	1	4	yes	complemented
134	#21275	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₅₃	—	4	4	no	—
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	—	4	4	no	—
138	C₂²×C₅₃	—	8	4	no	—
139	C₂²×C₅₃	—	8	4	no	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₅₃	#532	1	2	yes	—
142	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#135	1	2	yes	—
145	C₄×C₅₃	—	2	4	no	—
146	C₄×C₅₃	—	2	4	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	4	4	no	—
149	C₄×C₅₃	—	8	4	no	—
150	C₄×C₅₃	C₄×D₈	1	4	yes	—
151	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
153	C₄×C₅₃	#529	1	2	yes	—
154	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
155	#2293	C₂×C₅₃	1	2	yes	complemented
156	#1970	C₂×C₅₃	1	2	yes	—
157	#1942	C₂×C₅₃	1	2	yes	complemented
158	#1934	C₂×C₅₃	1	4	yes	complemented
159	C₄×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
160	C₄×#102	C₂×C₅₃	1	2	yes	complemented
161	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
162	C₂×C₅₃	—	8	4	no	—
163	C₂×C₅₃	#2293	1	1	yes	characteristic
164	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
165	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
166	C₂×C₅₃	#3168	1	2	yes	—
167	C₂×C₅₃	#3169	1	2	yes	—
168	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
170	C₂×#117	—	2	2	no	—
171	C₄×D₈	C₄×C₅₃	1	8	yes	complemented
172	C₂×#107	C₂²×C₅₃	1	2	yes	—
173	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
174	C₂×#102	C₂²×C₅₃	1	4	yes	—
175	C₂×#102	C₂²×C₅₃	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
180	#390	C₂²×C₅₃	1	4	yes	—
181	#389	C₂²×C₅₃	1	4	yes	—
182	#388	—	2	4	no	—
183	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
184	C₄×C₂²⋊C₄	—	2	2	no	—
185	C₄³	C₂²×C₅₃	1	1	yes	characteristic
186	#340	C₂²×C₅₃	1	2	yes	—
187	#340	C₂²×C₅₃	1	4	yes	—
188	C₅₃	#21275	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
190	C₂³×C₄	—	2	2	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
193	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
195	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
198	#117	—	4	8	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₅₃	1	2	yes	—
208	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
210	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
211	#107	—	2	4	no	—
212	#107	C₂×C₄×C₅₃	1	4	yes	—
213	#102	—	2	8	no	—
214	#102	C₂×C₄×C₅₃	1	8	yes	—
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
219	#95	—	2	4	no	—
220	#95	—	2	2	no	—
221	C₂⁴	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
236	D₈	—	2	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
242	C₂×C₈	#41×C₅₃	1	4	yes	—
243	C₂×C₈	#41×C₅₃	1	4	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	8	no	—
248	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	4	8	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₄²	D₈×C₅₃	1	4	yes	complemented
258	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
259	C₄²	#41×C₅₃	1	2	yes	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	16	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
282	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
283	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
284	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
285	C₂×C₄	#135×C₅₃	1	4	yes	—
286	C₂×C₄	#135×C₅₃	1	2	yes	—
287	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
289	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
290	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	4	no	—
293	C₈	#117×C₅₃	1	4	yes	complemented
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
300	C₂²	C₅₃×#532	1	2	yes	—
301	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
302	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
303	C₂²	C₂×#135×C₅₃	1	2	yes	—
304	C₄	—	2	2	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×D₈×C₅₃	1	4	yes	—
310	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
311	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
312	C₄	C₅₃×#529	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₅₃×#2293	1	1	yes	characteristic
315	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
316	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
317	C₂	C₅₃×#3168	1	2	yes	—
318	C₂	C₅₃×#3169	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2293	C₂	semidirect
4. C₅₃×#1942	C₂	semidirect
5. C₅₃×#1934	C₂	semidirect
6. C₄×#107×C₅₃	C₂	semidirect
7. C₄×#102×C₅₃	C₂	semidirect
8. C₄²×C₈×C₅₃	C₂	semidirect
12. C₄×D₈×C₅₃	C₄	semidirect
97. C₄²×C₅₃	D₈	semidirect
133. C₈×C₅₃	#117	semidirect
134. #21275	C₅₃	direct (N × H)
155. #2293	C₂×C₅₃	semidirect
157. #1942	C₂×C₅₃	semidirect
158. #1934	C₂×C₅₃	semidirect
159. C₄×#107	C₂×C₅₃	semidirect
160. C₄×#102	C₂×C₅₃	semidirect
161. C₄²×C₈	C₂×C₅₃	semidirect
171. C₄×D₈	C₄×C₅₃	semidirect
188. C₅₃	#21275	direct (N × H)
257. C₄²	D₈×C₅₃	semidirect
293. C₈	#117×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₅₃	2
C₅₃×#2293	2
C₅₃×#1970	2
C₅₃×#1942	2
C₅₃×#1934	2
C₄×#107×C₅₃	2
C₄×#102×C₅₃	2
C₄²×C₈×C₅₃	2
#21275	53

Subgroups — C53×#21275 #27228787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21275 #27228787