Subgroups — C₅₃×#60627 #27268139

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3142	C₂	1	3	yes	maximal, complemented
4	C₅₃×#3139	C₂	1	3	yes	maximal, complemented
5	C₅₃×#3137	C₂	1	12	yes	maximal, complemented
6	C₅₃×#3135	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₃×#491	C₂	1	3	yes	maximal, complemented
8	C₅₃×#2898	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
11	C₂×#131×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#557	—	2	2	no	—
13	C₅₃×#554	—	2	6	no	—
14	D₄×Q₈×C₅₃	—	2	2	no	—
15	C₅₃×#547	—	2	6	no	—
16	C₅₃×#544	C₂²	1	6	yes	—
17	C₅₃×#536	C₂²	1	6	yes	—
18	C₅₃×#533	C₂²	1	3	yes	—
19	C₅₃×#532	C₂²	1	3	yes	—
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
21	C₂×#125×C₅₃	C₂²	1	3	yes	—
22	C₅₃×#520	C₂²	1	2	yes	—
23	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#491	C₂²	1	3	yes	—
25	C₅₃×#491	C₂²	1	12	yes	—
26	C₅₃×#491	C₂²	1	6	yes	—
27	C₅₃×#491	C₂²	1	3	yes	—
28	C₅₃×#490	C₂²	1	1	yes	characteristic
29	C₅₃×#490	C₂²	1	3	yes	—
30	C₅₃×#489	C₂²	1	12	yes	—
31	C₅₃×#488	C₂²	1	12	yes	—
32	C₅₃×#487	C₂²	1	12	yes	—
33	C₅₃×#419	C₂²	1	12	yes	—
34	C₅₃×#417	C₂²	1	12	yes	—
35	C₅₃×#416	C₂²	1	12	yes	—
36	C₂×#103×C₅₃	C₂²	1	12	yes	—
37	C₅₃×#405	C₂²	1	6	yes	—
38	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
39	C₂×#97×C₅₃	C₂²	1	3	yes	—
40	2₋¹⁺⁴×C₅₃	—	2	16	no	—
41	C₂×#41×C₅₃	—	2	6	no	—
42	C₂×#41×C₅₃	C₂³	1	6	yes	—
43	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	—	2	2	no	—
45	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
46	#131×C₅₃	—	2	8	no	—
47	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
49	C₄⋊Q₈×C₅₃	—	2	6	no	—
50	C₄⋊Q₈×C₅₃	—	2	6	no	—
51	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
52	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
53	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
54	#127×C₅₃	C₂³	1	1	yes	characteristic
55	#125×C₅₃	—	2	6	no	—
56	#125×C₅₃	—	2	6	no	—
57	#125×C₅₃	C₂³	1	6	yes	—
58	#125×C₅₃	C₂³	1	3	yes	—
59	#125×C₅₃	C₂³	1	1	yes	characteristic
60	#124×C₅₃	C₂³	1	6	yes	—
61	#123×C₅₃	—	2	12	no	—
62	C₂²⋊Q₈×C₅₃	—	2	12	no	—
63	C₂²⋊Q₈×C₅₃	—	2	12	no	—
64	C₂²⋊Q₈×C₅₃	—	2	6	no	—
65	#121×C₅₃	—	2	6	no	—
66	C₄×Q₈×C₅₃	—	2	6	no	—
67	C₄×Q₈×C₅₃	—	2	2	no	—
68	C₄×Q₈×C₅₃	D₄	1	2	yes	—
69	C₄×D₄×C₅₃	—	2	6	no	—
70	C₄×D₄×C₅₃	—	2	6	no	—
71	C₄×D₄×C₅₃	D₄	1	6	yes	—
72	#117×C₅₃	C₂³	1	6	yes	—
73	#117×C₅₃	C₂³	1	3	yes	—
74	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
78	#103×C₅₃	C₂³	1	12	yes	—
79	#103×C₅₃	C₂³	1	24	yes	—
80	#103×C₅₃	C₂³	1	12	yes	—
81	#102×C₅₃	C₂³	1	12	yes	—
82	#102×C₅₃	C₂³	1	4	yes	—
83	#97×C₅₃	C₂³	1	6	yes	—
84	#97×C₅₃	C₂³	1	1	yes	characteristic
85	#97×C₅₃	C₂³	1	3	yes	—
86	C₄×C₈×C₅₃	C₂³	1	6	yes	—
87	#41×C₅₃	—	2	24	no	—
88	#41×C₅₃	—	4	24	no	—
89	#41×C₅₃	#41	1	8	yes	—
90	C₂×Q₈×C₅₃	—	2	6	no	—
91	C₂×Q₈×C₅₃	—	2	6	no	—
92	C₂×Q₈×C₅₃	—	2	24	no	—
93	C₂×Q₈×C₅₃	—	4	8	no	—
94	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
96	C₂×D₄×C₅₃	—	2	6	no	—
97	C₂×D₄×C₅₃	—	2	2	no	—
98	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
99	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
100	C₂²×C₄×C₅₃	—	2	6	no	—
101	C₂²×C₄×C₅₃	—	2	6	no	—
102	C₂²×C₄×C₅₃	—	2	6	no	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
104	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
105	M₁₆×C₅₃	—	2	12	no	—
106	C₂×C₈×C₅₃	—	2	12	no	—
107	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
108	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
109	C₄⋊C₄×C₅₃	—	2	6	no	—
110	C₄⋊C₄×C₅₃	—	2	6	no	—
111	C₄⋊C₄×C₅₃	—	2	12	no	—
112	C₄⋊C₄×C₅₃	—	2	12	no	—
113	C₄⋊C₄×C₅₃	—	2	6	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	6	no	—
116	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
117	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
118	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
119	C₂²⋊C₄×C₅₃	—	2	6	no	—
120	C₂²⋊C₄×C₅₃	—	2	12	no	—
121	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
122	C₄²×C₅₃	—	2	6	no	—
123	C₄²×C₅₃	C₂×D₄	1	6	yes	—
124	C₄²×C₅₃	C₂⁴	1	3	yes	—
125	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
128	Q₈×C₅₃	—	2	12	no	—
129	Q₈×C₅₃	—	4	12	no	—
130	Q₈×C₅₃	—	8	6	no	—
131	Q₈×C₅₃	#124	1	4	yes	—
132	D₄×C₅₃	—	2	4	no	—
133	D₄×C₅₃	—	4	4	no	—
134	D₄×C₅₃	—	4	12	no	—
135	D₄×C₅₃	#124	1	12	yes	—
136	C₂×C₄×C₅₃	—	2	6	no	—
137	C₂×C₄×C₅₃	—	2	6	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	6	no	—
140	C₂×C₄×C₅₃	—	4	12	no	—
141	C₂×C₄×C₅₃	—	4	12	no	—
142	C₂×C₄×C₅₃	—	8	6	no	—
143	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
145	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
146	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
147	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	2	6	no	—
150	#60627	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₅₃	—	2	6	no	—
152	C₂²×C₅₃	—	4	4	no	—
153	C₂²×C₅₃	—	4	2	no	—
154	C₂²×C₅₃	C₂×#124	1	6	yes	—
155	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	6	no	—
158	C₄×C₅₃	—	4	6	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	4	6	no	—
161	C₄×C₅₃	C₂×#124	1	6	yes	—
162	C₄×C₅₃	C₂²×#41	1	2	yes	—
163	#3504	C₂×C₅₃	1	1	yes	characteristic, complemented
164	#3499	C₂×C₅₃	1	1	yes	characteristic
165	#3142	C₂×C₅₃	1	3	yes	complemented
166	#3139	C₂×C₅₃	1	3	yes	complemented
167	#3137	C₂×C₅₃	1	12	yes	complemented
168	#3135	C₂×C₅₃	1	3	yes	complemented
169	C₂×#491	C₂×C₅₃	1	3	yes	complemented
170	#2898	C₂×C₅₃	1	4	yes	complemented
171	#2875	C₂×C₅₃	1	1	yes	characteristic, complemented
172	C₂×C₅₃	—	2	6	no	—
173	C₂×C₅₃	—	8	2	no	—
174	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
175	C₂×C₅₃	#3586	1	2	yes	—
176	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
177	C₂×#131	C₂²×C₅₃	1	2	yes	—
178	#557	—	2	2	no	—
179	#554	—	2	6	no	—
180	D₄×Q₈	—	2	2	no	—
181	#547	—	2	6	no	—
182	#544	C₂²×C₅₃	1	6	yes	—
183	#536	C₂²×C₅₃	1	6	yes	—
184	#533	C₂²×C₅₃	1	3	yes	—
185	#532	C₂²×C₅₃	1	3	yes	—
186	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
187	C₂×#125	C₂²×C₅₃	1	3	yes	—
188	#520	C₂²×C₅₃	1	2	yes	—
189	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
190	#491	C₂²×C₅₃	1	3	yes	—
191	#491	C₂²×C₅₃	1	12	yes	—
192	#491	C₂²×C₅₃	1	6	yes	—
193	#491	C₂²×C₅₃	1	3	yes	—
194	#490	C₂²×C₅₃	1	1	yes	characteristic
195	#490	C₂²×C₅₃	1	3	yes	—
196	#489	C₂²×C₅₃	1	12	yes	—
197	#488	C₂²×C₅₃	1	12	yes	—
198	#487	C₂²×C₅₃	1	12	yes	—
199	#419	C₂²×C₅₃	1	12	yes	—
200	#417	C₂²×C₅₃	1	12	yes	—
201	#416	C₂²×C₅₃	1	12	yes	—
202	C₂×#103	C₂²×C₅₃	1	12	yes	—
203	#405	C₂²×C₅₃	1	6	yes	—
204	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
205	C₂×#97	C₂²×C₅₃	1	3	yes	—
206	C₅₃	#60627	1	1	yes	characteristic, direct product factor, Sylow
207	2₋¹⁺⁴	—	2	16	no	—
208	C₂×#41	—	2	6	no	—
209	C₂×#41	C₂³×C₅₃	1	6	yes	—
210	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
211	C₂²×Q₈	—	2	2	no	—
212	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
213	#131	—	2	8	no	—
214	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
215	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
216	C₄⋊Q₈	—	2	6	no	—
217	C₄⋊Q₈	—	2	6	no	—
218	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
219	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
220	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
221	#127	C₂³×C₅₃	1	1	yes	characteristic
222	#125	—	2	6	no	—
223	#125	—	2	6	no	—
224	#125	C₂³×C₅₃	1	3	yes	—
225	#125	C₂³×C₅₃	1	6	yes	—
226	#125	C₂³×C₅₃	1	1	yes	characteristic
227	#124	C₂³×C₅₃	1	6	yes	—
228	#123	—	2	12	no	—
229	C₂²⋊Q₈	—	2	6	no	—
230	C₂²⋊Q₈	—	2	12	no	—
231	C₂²⋊Q₈	—	2	12	no	—
232	#121	—	2	6	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	6	no	—
235	C₄×Q₈	D₄×C₅₃	1	2	yes	—
236	C₄×D₄	—	2	6	no	—
237	C₄×D₄	—	2	6	no	—
238	C₄×D₄	D₄×C₅₃	1	6	yes	—
239	#117	C₂³×C₅₃	1	6	yes	—
240	#117	C₂³×C₅₃	1	3	yes	—
241	C₂×C₄⋊C₄	—	2	6	no	—
242	C₂×C₄⋊C₄	—	2	6	no	—
243	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
244	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
245	#103	C₂³×C₅₃	1	12	yes	—
246	#103	C₂³×C₅₃	1	24	yes	—
247	#103	C₂³×C₅₃	1	12	yes	—
248	#102	C₂³×C₅₃	1	4	yes	—
249	#102	C₂³×C₅₃	1	12	yes	—
250	#97	C₂³×C₅₃	1	1	yes	characteristic
251	#97	C₂³×C₅₃	1	3	yes	—
252	#97	C₂³×C₅₃	1	6	yes	—
253	C₄×C₈	C₂³×C₅₃	1	6	yes	—
254	#41	—	2	24	no	—
255	#41	—	4	24	no	—
256	#41	#41×C₅₃	1	8	yes	—
257	C₂×Q₈	—	2	6	no	—
258	C₂×Q₈	—	2	6	no	—
259	C₂×Q₈	—	2	24	no	—
260	C₂×Q₈	—	4	8	no	—
261	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
262	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
263	C₂×D₄	—	2	6	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
266	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
267	C₂²×C₄	—	2	6	no	—
268	C₂²×C₄	—	2	6	no	—
269	C₂²×C₄	—	2	6	no	—
270	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
271	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
272	M₁₆	—	2	12	no	—
273	C₂×C₈	—	2	12	no	—
274	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
275	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
276	C₄⋊C₄	—	2	6	no	—
277	C₄⋊C₄	—	2	6	no	—
278	C₄⋊C₄	—	2	12	no	—
279	C₄⋊C₄	—	2	12	no	—
280	C₄⋊C₄	—	2	6	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	6	no	—
283	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
284	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
285	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
286	C₂²⋊C₄	—	2	6	no	—
287	C₂²⋊C₄	—	2	12	no	—
288	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
289	C₄²	—	2	6	no	—
290	C₄²	C₂×D₄×C₅₃	1	6	yes	—
291	C₄²	C₂⁴×C₅₃	1	3	yes	—
292	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
293	C₂³	—	2	2	no	—
294	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
295	Q₈	—	2	12	no	—
296	Q₈	—	4	12	no	—
297	Q₈	—	8	6	no	—
298	Q₈	#124×C₅₃	1	4	yes	—
299	D₄	—	2	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	12	no	—
302	D₄	#124×C₅₃	1	12	yes	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	6	no	—
305	C₂×C₄	—	2	6	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	12	no	—
309	C₂×C₄	—	8	6	no	—
310	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
311	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
312	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
313	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
314	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	6	no	—
317	C₂²	—	2	6	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₂×#124×C₅₃	1	6	yes	—
321	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	6	no	—
324	C₄	—	4	6	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂×#124×C₅₃	1	6	yes	—
328	C₄	C₂²×#41×C₅₃	1	2	yes	—
329	C₂	—	2	6	no	—
330	C₂	—	8	2	no	—
331	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
332	C₂	C₅₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3504	C₂	semidirect
3. C₅₃×#3142	C₂	semidirect
4. C₅₃×#3139	C₂	semidirect
5. C₅₃×#3137	C₂	semidirect
6. C₅₃×#3135	C₂	semidirect
7. C₂×C₅₃×#491	C₂	semidirect
8. C₅₃×#2898	C₂	semidirect
9. C₅₃×#2875	C₂	semidirect
150. #60627	C₅₃	direct (N × H)
163. #3504	C₂×C₅₃	semidirect
165. #3142	C₂×C₅₃	semidirect
166. #3139	C₂×C₅₃	semidirect
167. #3137	C₂×C₅₃	semidirect
168. #3135	C₂×C₅₃	semidirect
169. C₂×#491	C₂×C₅₃	semidirect
170. #2898	C₂×C₅₃	semidirect
171. #2875	C₂×C₅₃	semidirect
206. C₅₃	#60627	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3504	2
C₅₃×#3499	2
C₅₃×#3142	2
C₅₃×#3139	2
C₅₃×#3137	2
C₅₃×#3135	2
C₂×C₅₃×#491	2
C₅₃×#2898	2
C₅₃×#2875	2
#60627	53

Subgroups — C53×#60627 #27268139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60627 #27268139