Subgroups — C₅₃×#61880 #27269392

Order 13568.

2302 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	C₅₃×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3391	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3358	C₂	1	8	yes	maximal, complemented
5	C₅₃×#3205	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3100	C₂	1	6	yes	maximal, complemented
7	C₅₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
9	C₂²×#41×C₅₃	—	2	1	no	—
10	C₅₃×#550	—	2	4	no	—
11	C₅₃×#550	C₂²	1	8	yes	—
12	C₅₃×#550	C₂²	1	4	yes	complemented
13	D₄×Q₈×C₅₃	C₂²	1	4	yes	complemented
14	C₅₃×#548	—	2	4	no	—
15	C₅₃×#537	—	2	3	no	—
16	C₅₃×#532	C₂²	1	6	yes	—
17	C₂×#127×C₅₃	C₂²	1	3	yes	—
18	C₂×#121×C₅₃	—	2	3	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
21	C₅₃×#493	C₂²	1	12	yes	complemented
22	C₅₃×#486	C₂²	1	12	yes	complemented
23	C₅₃×#480	C₂²	1	4	yes	—
24	C₅₃×#474	C₂²	1	12	yes	complemented
25	C₅₃×#459	C₂²	1	24	yes	complemented
26	C₅₃×#459	C₂²	1	12	yes	complemented
27	C₅₃×#447	C₂²	1	12	yes	—
28	C₈×Q₈×C₅₃	C₂²	1	8	yes	complemented
29	C₄×D₈×C₅₃	C₂²	1	24	yes	complemented
30	C₅₃×#431	C₂²	1	6	yes	—
31	2₋¹⁺⁴×C₅₃	—	4	8	no	—
32	C₂×#41×C₅₃	—	2	4	no	—
33	C₂×#41×C₅₃	—	2	8	no	—
34	C₂×#41×C₅₃	—	2	6	no	—
35	C₂×#41×C₅₃	—	4	8	no	—
36	C₂²×Q₈×C₅₃	—	2	4	no	—
37	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₅₃	—	2	3	no	—
39	C₂²×D₄×C₅₃	—	2	3	no	—
40	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₅₃	—	2	3	no	—
42	C₂×D₈×C₅₃	C₂³	1	24	yes	—
43	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
44	#127×C₅₃	—	2	6	no	—
45	#127×C₅₃	C₂³	1	12	yes	—
46	#127×C₅₃	C₂³	1	6	yes	—
47	#124×C₅₃	—	2	6	no	—
48	C₂²⋊Q₈×C₅₃	—	2	12	no	—
49	#121×C₅₃	—	2	24	no	—
50	#121×C₅₃	—	2	12	no	—
51	#121×C₅₃	—	2	6	no	—
52	#121×C₅₃	C₂³	1	4	yes	—
53	C₄×Q₈×C₅₃	D₄	1	4	yes	—
54	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
55	C₄×D₄×C₅₃	—	2	6	no	—
56	C₄×D₄×C₅₃	—	2	6	no	—
57	C₄×D₄×C₅₃	C₂³	1	12	yes	—
58	C₄×D₄×C₅₃	C₂³	1	12	yes	—
59	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
62	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
63	#107×C₅₃	C₂³	1	8	yes	—
64	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
65	#102×C₅₃	C₂³	1	24	yes	—
66	#102×C₅₃	C₂³	1	8	yes	—
67	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
68	C₄×C₈×C₅₃	C₂³	1	12	yes	—
69	C₂⁴×C₅₃	—	2	3	no	—
70	#41×C₅₃	—	4	32	no	—
71	#41×C₅₃	—	4	16	no	—
72	#41×C₅₃	—	4	24	no	—
73	C₂×Q₈×C₅₃	—	2	6	no	—
74	C₂×Q₈×C₅₃	—	4	16	no	—
75	C₂×Q₈×C₅₃	D₈	1	8	yes	complemented
76	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
77	C₂×D₄×C₅₃	—	2	4	no	—
78	C₂×D₄×C₅₃	—	2	3	no	—
79	C₂×D₄×C₅₃	—	2	6	no	—
80	C₂×D₄×C₅₃	—	2	12	no	—
81	C₂×D₄×C₅₃	—	2	6	no	—
82	C₂×D₄×C₅₃	—	2	12	no	—
83	C₂×D₄×C₅₃	—	4	6	no	—
84	C₂×D₄×C₅₃	—	4	12	no	—
85	C₂×D₄×C₅₃	—	8	6	no	—
86	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
87	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	12	no	—
90	C₂²×C₄×C₅₃	—	2	12	no	—
91	C₂²×C₄×C₅₃	—	4	12	no	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	D₈×C₅₃	—	4	24	no	—
96	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	6	no	—
99	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
100	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
101	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
102	C₂²⋊C₄×C₅₃	—	2	12	no	—
103	C₂²⋊C₄×C₅₃	—	2	4	no	—
104	C₂²⋊C₄×C₅₃	—	2	12	no	—
105	C₄²×C₅₃	C₂×D₄	1	6	yes	—
106	C₄²×C₅₃	C₂⁴	1	6	yes	—
107	C₂³×C₅₃	—	2	4	no	—
108	C₂³×C₅₃	—	2	6	no	—
109	C₂³×C₅₃	—	4	6	no	—
110	C₂³×C₅₃	—	4	6	no	—
111	C₂³×C₅₃	—	4	6	no	—
112	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₅₃	—	4	12	no	—
114	Q₈×C₅₃	C₂×D₈	1	16	yes	—
115	D₄×C₅₃	—	2	12	no	—
116	D₄×C₅₃	—	2	8	no	—
117	D₄×C₅₃	—	4	6	no	—
118	D₄×C₅₃	—	4	8	no	—
119	D₄×C₅₃	—	4	24	no	—
120	D₄×C₅₃	—	4	12	no	—
121	D₄×C₅₃	—	8	6	no	—
122	D₄×C₅₃	—	8	12	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	4	3	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	24	no	—
129	C₂×C₄×C₅₃	—	4	24	no	—
130	C₂×C₄×C₅₃	C₂×D₈	1	12	yes	—
131	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
135	C₈×C₅₃	—	2	8	no	—
136	#61880	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	4	6	no	—
140	C₂²×C₅₃	—	4	4	no	—
141	C₂²×C₅₃	—	4	6	no	—
142	C₂²×C₅₃	—	4	4	no	—
143	C₂²×C₅₃	—	4	4	no	—
144	C₂²×C₅₃	—	4	6	no	—
145	C₂²×C₅₃	—	8	12	no	—
146	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
147	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
148	C₄×C₅₃	—	2	6	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	4	2	no	—
151	C₄×C₅₃	—	4	4	no	—
152	C₄×C₅₃	C₂²×D₈	1	12	yes	—
153	C₄×C₅₃	#585	1	2	yes	—
154	#3497	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₂×#550	C₂×C₅₃	1	1	yes	characteristic, complemented
156	#3391	C₂×C₅₃	1	8	yes	complemented
157	#3358	C₂×C₅₃	1	8	yes	complemented
158	#3205	C₂×C₅₃	1	6	yes	complemented
159	#3100	C₂×C₅₃	1	6	yes	complemented
160	#2982	C₂×C₅₃	1	1	yes	characteristic, complemented
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	—	8	4	no	—
163	C₂×C₅₃	—	8	6	no	—
164	C₂×C₅₃	#3487	1	1	yes	characteristic
165	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
166	C₂×C₅₃	#3585	1	1	yes	characteristic
167	C₂×2₋¹⁺⁴	—	2	1	no	—
168	C₂²×#41	—	2	1	no	—
169	#550	—	2	4	no	—
170	#550	C₂²×C₅₃	1	8	yes	—
171	#550	C₂²×C₅₃	1	4	yes	complemented
172	D₄×Q₈	C₂²×C₅₃	1	4	yes	complemented
173	#548	—	2	4	no	—
174	#537	—	2	3	no	—
175	#532	C₂²×C₅₃	1	6	yes	—
176	C₂×#127	C₂²×C₅₃	1	3	yes	—
177	C₂×#121	—	2	3	no	—
178	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
179	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
180	#493	C₂²×C₅₃	1	12	yes	complemented
181	#486	C₂²×C₅₃	1	12	yes	complemented
182	#480	C₂²×C₅₃	1	4	yes	—
183	#474	C₂²×C₅₃	1	12	yes	complemented
184	#459	C₂²×C₅₃	1	12	yes	complemented
185	#459	C₂²×C₅₃	1	24	yes	complemented
186	#447	C₂²×C₅₃	1	12	yes	—
187	C₈×Q₈	C₂²×C₅₃	1	8	yes	complemented
188	C₄×D₈	C₂²×C₅₃	1	24	yes	complemented
189	#431	C₂²×C₅₃	1	6	yes	—
190	C₅₃	#61880	1	1	yes	characteristic, direct product factor, Sylow
191	2₋¹⁺⁴	—	4	8	no	—
192	C₂×#41	—	2	4	no	—
193	C₂×#41	—	2	8	no	—
194	C₂×#41	—	2	6	no	—
195	C₂×#41	—	4	8	no	—
196	C₂²×Q₈	—	2	4	no	—
197	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
198	C₂²×D₄	—	2	3	no	—
199	C₂²×D₄	—	2	3	no	—
200	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
201	C₂³×C₄	—	2	3	no	—
202	C₂×D₈	C₂³×C₅₃	1	24	yes	—
203	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
204	#127	—	2	6	no	—
205	#127	C₂³×C₅₃	1	12	yes	—
206	#127	C₂³×C₅₃	1	6	yes	—
207	#124	—	2	6	no	—
208	C₂²⋊Q₈	—	2	12	no	—
209	#121	—	2	6	no	—
210	#121	—	2	24	no	—
211	#121	—	2	12	no	—
212	#121	C₂³×C₅₃	1	4	yes	—
213	C₄×Q₈	D₄×C₅₃	1	4	yes	—
214	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
215	C₄×D₄	—	2	6	no	—
216	C₄×D₄	—	2	6	no	—
217	C₄×D₄	C₂³×C₅₃	1	12	yes	—
218	C₄×D₄	C₂³×C₅₃	1	12	yes	—
219	C₂×C₄⋊C₄	D₄×C₅₃	1	3	yes	—
220	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	2	3	no	—
222	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
223	#107	C₂³×C₅₃	1	8	yes	—
224	C₄⋊C₈	C₂³×C₅₃	1	12	yes	—
225	#102	C₂³×C₅₃	1	8	yes	—
226	#102	C₂³×C₅₃	1	24	yes	—
227	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
228	C₄×C₈	C₂³×C₅₃	1	12	yes	—
229	C₂⁴	—	2	3	no	—
230	#41	—	4	32	no	—
231	#41	—	4	16	no	—
232	#41	—	4	24	no	—
233	C₂×Q₈	—	2	6	no	—
234	C₂×Q₈	—	4	16	no	—
235	C₂×Q₈	D₈×C₅₃	1	8	yes	complemented
236	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	3	no	—
239	C₂×D₄	—	2	6	no	—
240	C₂×D₄	—	2	12	no	—
241	C₂×D₄	—	2	6	no	—
242	C₂×D₄	—	2	12	no	—
243	C₂×D₄	—	4	6	no	—
244	C₂×D₄	—	4	12	no	—
245	C₂×D₄	—	8	6	no	—
246	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
247	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	12	no	—
250	C₂²×C₄	—	2	12	no	—
251	C₂²×C₄	—	4	12	no	—
252	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
253	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
254	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
255	D₈	—	4	24	no	—
256	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	6	no	—
259	C₄⋊C₄	C₂×D₄×C₅₃	1	12	yes	—
260	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
261	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
262	C₂²⋊C₄	—	2	12	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	12	no	—
265	C₄²	C₂×D₄×C₅₃	1	6	yes	—
266	C₄²	C₂⁴×C₅₃	1	6	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	6	no	—
269	C₂³	—	4	6	no	—
270	C₂³	—	4	6	no	—
271	C₂³	—	4	6	no	—
272	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
273	Q₈	—	4	12	no	—
274	Q₈	C₂×D₈×C₅₃	1	16	yes	—
275	D₄	—	2	8	no	—
276	D₄	—	2	12	no	—
277	D₄	—	4	6	no	—
278	D₄	—	4	8	no	—
279	D₄	—	4	24	no	—
280	D₄	—	4	12	no	—
281	D₄	—	8	6	no	—
282	D₄	—	8	12	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	3	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	24	no	—
289	C₂×C₄	—	4	24	no	—
290	C₂×C₄	C₂×D₈×C₅₃	1	12	yes	—
291	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
293	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
294	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	6	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	6	no	—
302	C₂²	—	4	6	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	8	12	no	—
305	C₂²	C₂²×D₈×C₅₃	1	2	yes	—
306	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	6	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	4	no	—
311	C₄	C₂²×D₈×C₅₃	1	12	yes	—
312	C₄	C₅₃×#585	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	8	6	no	—
315	C₂	—	8	4	no	—
316	C₂	C₅₃×#3487	1	1	yes	characteristic
317	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
318	C₂	C₅₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3497	C₂	semidirect
2. C₂×C₅₃×#550	C₂	semidirect
3. C₅₃×#3391	C₂	semidirect
4. C₅₃×#3358	C₂	semidirect
5. C₅₃×#3205	C₂	semidirect
6. C₅₃×#3100	C₂	semidirect
7. C₅₃×#2982	C₂	semidirect
12. C₅₃×#550	C₂²	semidirect
13. D₄×Q₈×C₅₃	C₂²	semidirect
21. C₅₃×#493	C₂²	semidirect
22. C₅₃×#486	C₂²	semidirect
24. C₅₃×#474	C₂²	semidirect
25. C₅₃×#459	C₂²	semidirect
26. C₅₃×#459	C₂²	semidirect
28. C₈×Q₈×C₅₃	C₂²	semidirect
29. C₄×D₈×C₅₃	C₂²	semidirect
75. C₂×Q₈×C₅₃	D₈	semidirect
136. #61880	C₅₃	direct (N × H)
154. #3497	C₂×C₅₃	semidirect
155. C₂×#550	C₂×C₅₃	semidirect
156. #3391	C₂×C₅₃	semidirect
157. #3358	C₂×C₅₃	semidirect
158. #3205	C₂×C₅₃	semidirect
159. #3100	C₂×C₅₃	semidirect
160. #2982	C₂×C₅₃	semidirect
171. #550	C₂²×C₅₃	semidirect
172. D₄×Q₈	C₂²×C₅₃	semidirect
180. #493	C₂²×C₅₃	semidirect
181. #486	C₂²×C₅₃	semidirect
183. #474	C₂²×C₅₃	semidirect
184. #459	C₂²×C₅₃	semidirect
185. #459	C₂²×C₅₃	semidirect
187. C₈×Q₈	C₂²×C₅₃	semidirect
188. C₄×D₈	C₂²×C₅₃	semidirect
190. C₅₃	#61880	direct (N × H)
235. C₂×Q₈	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3497	2
C₂×C₅₃×#550	2
C₅₃×#3391	2
C₅₃×#3358	2
C₅₃×#3205	2
C₅₃×#3100	2
C₅₃×#2982	2
#61880	53

Subgroups — C53×#61880 #27269392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61880 #27269392