Subgroups — C₅₃×#21356 #27228868

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1939	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1851	C₂	1	4	yes	maximal
6	C₅₃×#1767	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×#117×C₅₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#432	—	2	4	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	8	yes	—
13	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₄×M₁₆×C₅₃	—	2	4	no	—
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#376	C₄	1	2	yes	—
24	C₄³×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#336	C₂²	1	8	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	8	no	—
30	C₂×M₁₆×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
32	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
33	#117×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	4	yes	—
43	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	16	no	—
48	C₄⋊C₈×C₅₃	—	2	4	no	—
49	C₂²⋊C₈×C₅₃	—	2	4	no	—
50	#97×C₅₃	—	2	4	no	—
51	#97×C₅₃	—	2	4	no	—
52	C₄×C₈×C₅₃	—	2	4	no	—
53	#95×C₅₃	C₂×C₄	1	4	yes	—
54	#95×C₅₃	C₂×C₄	1	2	yes	—
55	#95×C₅₃	C₂×C₄	1	2	yes	—
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	16	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	8	no	—
78	C₂×C₈×C₅₃	#41	1	8	yes	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	M₁₆	1	8	yes	complemented
81	C₂²⋊C₄×C₅₃	—	2	8	no	—
82	C₄²×C₅₃	—	2	8	no	—
83	C₄²×C₅₃	—	2	8	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₄²×C₅₃	—	4	2	no	—
86	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
87	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
88	C₄²×C₅₃	C₂×D₄	1	2	yes	—
89	C₄²×C₅₃	C₂×D₄	1	2	yes	—
90	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	1	no	—
95	C₂³×C₅₃	—	2	1	no	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	8	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
119	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
123	C₈×C₅₃	—	2	8	no	—
124	C₈×C₅₃	—	4	4	no	—
125	#21356	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₄×#41	1	2	yes	—
133	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
134	C₂²×C₅₃	#568	1	1	yes	characteristic
135	C₂²×C₅₃	#568	1	1	yes	characteristic
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	8	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	4	4	no	—
142	C₄×C₅₃	#423	1	4	yes	—
143	C₄×C₅₃	#525	1	2	yes	—
144	C₄×C₅₃	#533	1	2	yes	—
145	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
146	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
147	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
148	#1939	C₂×C₅₃	1	4	yes	complemented
149	#1851	C₂×C₅₃	1	4	yes	—
150	#1767	C₂×C₅₃	1	4	yes	complemented
151	C₂×C₅₃	—	4	2	no	—
152	C₂×C₅₃	#2296	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
154	C₂×C₅₃	#2905	1	1	yes	characteristic
155	C₂×C₅₃	#2970	1	2	yes	—
156	C₂×C₅₃	#2971	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
158	C₂×#117	C₄×C₅₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
160	#432	—	2	4	no	—
161	C₂×C₄⋊C₈	C₂²×C₅₃	1	8	yes	—
162	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
164	C₄×M₁₆	—	2	4	no	—
165	C₂×#97	C₂²×C₅₃	1	2	yes	—
166	C₂×#97	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
168	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
169	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
170	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
171	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
172	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
173	#376	C₄×C₅₃	1	2	yes	—
174	C₄³	C₂²×C₅₃	1	2	yes	—
175	#336	C₂²×C₅₃	1	8	yes	—
176	C₅₃	#21356	1	1	yes	characteristic, direct product factor, Sylow
177	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
183	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
184	#117	—	2	8	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
188	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₅₃	1	4	yes	—
194	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
195	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
196	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
198	C₄⋊C₈	—	2	16	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	—	2	4	no	—
201	#97	—	2	4	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	#95	C₂×C₄×C₅₃	1	4	yes	—
205	#95	C₂×C₄×C₅₃	1	2	yes	—
206	#95	C₂×C₄×C₅₃	1	2	yes	—
207	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
213	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
222	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
224	M₁₆	—	2	16	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	#41×C₅₃	1	8	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	M₁₆×C₅₃	1	8	yes	complemented
232	C₂²⋊C₄	—	2	8	no	—
233	C₄²	—	2	8	no	—
234	C₄²	—	2	8	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
238	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
239	C₄²	C₂×D₄×C₅₃	1	2	yes	—
240	C₄²	C₂×D₄×C₅₃	1	2	yes	—
241	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
242	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
248	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
269	C₂×C₄	C₂×M₁₆×C₅₃	1	8	yes	—
270	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
271	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
274	C₈	—	2	8	no	—
275	C₈	—	4	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
282	C₂²	C₄×#41×C₅₃	1	2	yes	—
283	C₂²	C₂²×M₁₆×C₅₃	1	2	yes	—
284	C₂²	C₅₃×#568	1	1	yes	characteristic
285	C₂²	C₅₃×#568	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	C₅₃×#423	1	4	yes	—
293	C₄	C₅₃×#525	1	2	yes	—
294	C₄	C₅₃×#533	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₅₃×#2296	1	1	yes	characteristic
297	C₂	C₂×C₅₃×#423	1	1	yes	characteristic
298	C₂	C₅₃×#2905	1	1	yes	characteristic
299	C₂	C₅₃×#2970	1	2	yes	—
300	C₂	C₅₃×#2971	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1939	C₂	semidirect
6. C₅₃×#1767	C₂	semidirect
80. C₄⋊C₄×C₅₃	M₁₆	semidirect
125. #21356	C₅₃	direct (N × H)
148. #1939	C₂×C₅₃	semidirect
150. #1767	C₂×C₅₃	semidirect
176. C₅₃	#21356	direct (N × H)
231. C₄⋊C₄	M₁₆×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×#432	2
C₂×C₄×M₁₆×C₅₃	2
C₄×#117×C₅₃	2
C₅₃×#1939	2
C₅₃×#1851	2
C₅₃×#1767	2
#21356	53

Subgroups — C53×#21356 #27228868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21356 #27228868