Subgroups — C₅₃×#21172 #27228684

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#428	C₂	1	2	yes	maximal
2	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1949	C₂	1	8	yes	maximal, complemented
6	C₅₃×#1736	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×#117×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	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#428	—	2	8	no	—
13	C₂×#107×C₅₃	C₂²	1	4	yes	—
14	C₂×#106×C₅₃	C₂²	1	4	yes	—
15	C₅₃×#423	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#382	—	2	1	no	—
18	C₅₃×#382	—	2	1	no	—
19	C₅₃×#382	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
22	C₅₃×#376	—	2	1	no	—
23	C₂×#95×C₅₃	—	2	2	no	—
24	C₄³×C₅₃	—	2	1	no	—
25	C₅₃×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₅₃	—	2	2	no	—
27	C₂³×C₄×C₅₃	D₄	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	4	no	—
30	C₂×M₁₆×C₅₃	—	2	4	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
33	#117×C₅₃	—	2	2	no	—
34	#117×C₅₃	—	2	2	no	—
35	#117×C₅₃	D₄	1	4	yes	—
36	#117×C₅₃	Q₈	1	4	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	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×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	—	2	4	no	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
53	#107×C₅₃	—	2	8	no	—
54	#106×C₅₃	—	2	8	no	—
55	C₄⋊C₈×C₅₃	C₂×C₄	1	16	yes	complemented
56	#95×C₅₃	—	2	4	no	—
57	#95×C₅₃	—	2	4	no	—
58	#95×C₅₃	—	2	4	no	—
59	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	—	4	8	no	—
65	C₂²×C₄×C₅₃	—	4	4	no	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅₃	#41	1	2	yes	—
75	C₂²×C₄×C₅₃	#41	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₅₃	—	2	16	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	4	4	no	—
86	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₂²⋊C₄×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	4	8	no	—
93	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
94	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
97	C₂³×C₅₃	—	2	4	no	—
98	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₅₃	#121	1	2	yes	—
100	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	8	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	8	no	—
111	C₂×C₄×C₅₃	—	8	8	no	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
117	C₂×C₄×C₅₃	#121	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	#125	1	4	yes	—
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₈×C₅₃	—	2	8	no	—
128	#21172	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	4	no	—
132	C₂²×C₅₃	#389	1	4	yes	—
133	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₅₃	#575	1	4	yes	—
137	C₄×C₅₃	—	2	4	no	—
138	C₄×C₅₃	—	2	4	no	—
139	C₄×C₅₃	—	4	4	no	—
140	C₄×C₅₃	—	4	4	no	—
141	C₄×C₅₃	—	8	4	no	—
142	C₄×C₅₃	#389	1	4	yes	—
143	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂×#428	C₂×C₅₃	1	2	yes	—
148	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
149	C₂×#382	C₂×C₅₃	1	1	yes	characteristic
150	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
151	#1949	C₂×C₅₃	1	8	yes	complemented
152	#1736	C₂×C₅₃	1	2	yes	—
153	C₂×C₅₃	—	2	4	no	—
154	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
155	C₂×C₅₃	#2911	1	2	yes	—
156	C₂×C₅₃	#3040	1	2	yes	—
157	C₂×C₅₃	#3080	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
159	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
162	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
163	#428	—	2	8	no	—
164	C₂×#107	C₂²×C₅₃	1	4	yes	—
165	C₂×#106	C₂²×C₅₃	1	4	yes	—
166	#423	C₄×C₅₃	1	8	yes	complemented
167	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
168	#382	—	2	1	no	—
169	#382	—	2	1	no	—
170	#382	C₂²×C₅₃	1	4	yes	—
171	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
172	C₄×C₂²⋊C₄	—	2	2	no	—
173	#376	—	2	1	no	—
174	C₂×#95	—	2	2	no	—
175	C₄³	—	2	1	no	—
176	#340	C₂²×C₅₃	1	8	yes	—
177	C₅₃	#21172	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³×C₄	—	2	2	no	—
179	C₂³×C₄	D₄×C₅₃	1	2	yes	—
180	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
184	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	D₄×C₅₃	1	4	yes	—
188	#117	Q₈×C₅₃	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
201	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
203	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
204	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
205	#107	—	2	8	no	—
206	#106	—	2	8	no	—
207	C₄⋊C₈	C₂×C₄×C₅₃	1	16	yes	complemented
208	#95	—	2	4	no	—
209	#95	—	2	4	no	—
210	#95	—	2	4	no	—
211	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	8	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
219	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
226	C₂²×C₄	#41×C₅₃	1	2	yes	—
227	C₂²×C₄	#41×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
239	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	8	no	—
245	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
246	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
247	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
248	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
251	C₂³	#121×C₅₃	1	2	yes	—
252	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
253	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
268	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
269	C₂×C₄	#121×C₅₃	1	2	yes	—
270	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
271	C₂×C₄	#125×C₅₃	1	4	yes	—
272	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₅₃×#389	1	4	yes	—
284	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
285	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
286	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
287	C₂²	C₅₃×#575	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₅₃×#389	1	4	yes	—
294	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
295	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
296	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
297	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
300	C₂	C₅₃×#2911	1	2	yes	—
301	C₂	C₅₃×#3040	1	2	yes	—
302	C₂	C₅₃×#3080	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1949	C₂	semidirect
15. C₅₃×#423	C₄	semidirect
55. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
128. #21172	C₅₃	direct (N × H)
151. #1949	C₂×C₅₃	semidirect
166. #423	C₄×C₅₃	semidirect
177. C₅₃	#21172	direct (N × H)
207. C₄⋊C₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#428	2
C₂×C₅₃×#423	2
C₂×C₅₃×#382	2
C₄×#117×C₅₃	2
C₅₃×#1949	2
C₅₃×#1736	2
#21172	53

Subgroups — C53×#21172 #27228684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21172 #27228684