Subgroups — C₅₃×#19688 #27227200

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3210	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3182	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1723	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1722	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1719	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1715	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#533	C₂²	1	2	yes	—
11	C₅₃×#532	C₂²	1	2	yes	—
12	C₅₃×#500	C₂²	1	1	yes	characteristic
13	C₅₃×#500	C₂²	1	1	yes	characteristic
14	C₅₃×#500	C₂²	1	2	yes	—
15	C₅₃×#500	C₂²	1	1	yes	characteristic
16	C₅₃×#499	—	2	2	no	—
17	C₅₃×#498	C₂²	1	2	yes	—
18	C₅₃×#494	C₂²	1	2	yes	—
19	C₅₃×#493	C₂²	1	1	yes	characteristic
20	C₅₃×#492	C₂²	1	2	yes	—
21	C₅₃×#492	C₂²	1	2	yes	—
22	C₅₃×#487	C₂²	1	2	yes	—
23	C₅₃×#487	C₂²	1	2	yes	—
24	C₅₃×#486	C₂²	1	2	yes	—
25	C₅₃×#486	C₂²	1	2	yes	—
26	C₅₃×#486	C₂²	1	2	yes	—
27	C₅₃×#468	—	2	4	no	—
28	C₅₃×#466	—	2	2	no	—
29	C₅₃×#464	—	2	4	no	—
30	C₅₃×#452	—	2	4	no	—
31	C₅₃×#440	—	2	4	no	—
32	C₄×D₈×C₅₃	—	2	2	no	—
33	C₅₃×#431	C₂²	1	1	yes	characteristic
34	C₅₃×#427	—	2	2	no	—
35	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#322	C₂²	1	2	yes	—
38	C₅₃×#322	C₂²	1	2	yes	—
39	C₂×#41×C₅₃	—	2	2	no	—
40	C₂×Q₁₆×C₅₃	—	2	4	no	—
41	C₂×SD₁₆×C₅₃	—	2	4	no	—
42	C₂×SD₁₆×C₅₃	—	2	4	no	—
43	C₂×D₈×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
49	#127×C₅₃	C₂³	1	2	yes	—
50	#125×C₅₃	—	2	2	no	—
51	#124×C₅₃	—	2	2	no	—
52	C₂²⋊Q₈×C₅₃	—	2	2	no	—
53	C₂²⋊Q₈×C₅₃	—	2	2	no	—
54	#121×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	—	2	2	no	—
60	#117×C₅₃	—	2	2	no	—
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	#107×C₅₃	—	2	2	no	—
63	#107×C₅₃	—	2	2	no	—
64	#107×C₅₃	—	2	2	no	—
65	#107×C₅₃	—	2	4	no	—
66	#106×C₅₃	—	2	4	no	—
67	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
68	#103×C₅₃	—	2	4	no	—
69	#103×C₅₃	—	2	4	no	—
70	#102×C₅₃	—	2	4	no	—
71	#102×C₅₃	—	2	4	no	—
72	#102×C₅₃	—	2	4	no	—
73	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
74	#97×C₅₃	D₄	1	2	yes	—
75	C₄×C₈×C₅₃	D₄	1	2	yes	—
76	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	#41×C₅₃	—	8	4	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	2	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₅₃	—	8	4	no	—
93	SD₁₆×C₅₃	—	4	8	no	—
94	SD₁₆×C₅₃	—	8	4	no	—
95	D₈×C₅₃	—	4	4	no	—
96	M₁₆×C₅₃	—	4	2	no	—
97	C₂×C₈×C₅₃	—	2	4	no	—
98	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	4	2	no	—
109	C₄⋊C₄×C₅₃	—	4	2	no	—
110	C₂²⋊C₄×C₅₃	—	4	2	no	—
111	C₂²⋊C₄×C₅₃	—	4	2	no	—
112	C₂²⋊C₄×C₅₃	—	4	2	no	—
113	C₄²×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂³×C₅₃	—	4	2	no	—
119	C₂³×C₅₃	#127	1	1	yes	characteristic
120	Q₈×C₅₃	—	4	4	no	—
121	Q₈×C₅₃	—	4	4	no	—
122	Q₈×C₅₃	—	4	4	no	—
123	D₄×C₅₃	—	4	4	no	—
124	D₄×C₅₃	—	4	4	no	—
125	D₄×C₅₃	—	4	4	no	—
126	D₄×C₅₃	—	8	4	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	8	4	no	—
134	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	—	2	4	no	—
142	C₈×C₅₃	—	2	4	no	—
143	C₈×C₅₃	#137	1	4	yes	—
144	#19688	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	—	8	2	no	—
150	C₂²×C₅₃	—	8	2	no	—
151	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	C₂×#137	1	2	yes	—
159	C₄×C₅₃	#575	1	2	yes	—
160	C₄×C₅₃	#576	1	2	yes	—
161	#3210	C₂×C₅₃	1	2	yes	complemented
162	#3182	C₂×C₅₃	1	2	yes	complemented
163	#3169	C₂×C₅₃	1	1	yes	characteristic
164	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1723	C₂×C₅₃	1	2	yes	complemented
166	#1722	C₂×C₅₃	1	2	yes	complemented
167	#1719	C₂×C₅₃	1	2	yes	complemented
168	#1715	C₂×C₅₃	1	2	yes	complemented
169	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
170	C₂×C₅₃	—	4	1	no	—
171	C₂×C₅₃	—	16	2	no	—
172	C₂×C₅₃	#3150	1	1	yes	characteristic
173	C₂×C₅₃	#3152	1	1	yes	characteristic
174	C₂×C₅₃	#3155	1	1	yes	characteristic
175	#533	C₂²×C₅₃	1	2	yes	—
176	#532	C₂²×C₅₃	1	2	yes	—
177	#500	C₂²×C₅₃	1	1	yes	characteristic
178	#500	C₂²×C₅₃	1	1	yes	characteristic
179	#500	C₂²×C₅₃	1	1	yes	characteristic
180	#500	C₂²×C₅₃	1	2	yes	—
181	#499	—	2	2	no	—
182	#498	C₂²×C₅₃	1	2	yes	—
183	#494	C₂²×C₅₃	1	2	yes	—
184	#493	C₂²×C₅₃	1	1	yes	characteristic
185	#492	C₂²×C₅₃	1	2	yes	—
186	#492	C₂²×C₅₃	1	2	yes	—
187	#487	C₂²×C₅₃	1	2	yes	—
188	#487	C₂²×C₅₃	1	2	yes	—
189	#486	C₂²×C₅₃	1	2	yes	—
190	#486	C₂²×C₅₃	1	2	yes	—
191	#486	C₂²×C₅₃	1	2	yes	—
192	#468	—	2	4	no	—
193	#466	—	2	2	no	—
194	#464	—	2	4	no	—
195	#452	—	2	4	no	—
196	#440	—	2	4	no	—
197	C₄×D₈	—	2	2	no	—
198	#431	C₂²×C₅₃	1	1	yes	characteristic
199	#427	—	2	2	no	—
200	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
202	#322	C₂²×C₅₃	1	2	yes	—
203	#322	C₂²×C₅₃	1	2	yes	—
204	C₅₃	#19688	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×#41	—	2	2	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
211	C₂²×C₈	D₄×C₅₃	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
215	#127	C₂³×C₅₃	1	2	yes	—
216	#125	—	2	2	no	—
217	#124	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	2	no	—
226	#117	—	2	2	no	—
227	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	—	2	2	no	—
231	#107	—	2	4	no	—
232	#106	—	2	4	no	—
233	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
234	#103	—	2	4	no	—
235	#103	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
240	#97	D₄×C₅₃	1	2	yes	—
241	C₄×C₈	D₄×C₅₃	1	2	yes	—
242	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₅₃	1	2	yes	—
245	C₄×C₈	C₂³×C₅₃	1	2	yes	—
246	#41	—	8	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	4	8	no	—
260	SD₁₆	—	8	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
266	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
283	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	#127×C₅₃	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
301	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
302	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	4	no	—
308	C₈	—	2	4	no	—
309	C₈	#137×C₅₃	1	4	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂×#137×C₅₃	1	2	yes	—
324	C₄	C₅₃×#575	1	2	yes	—
325	C₄	C₅₃×#576	1	2	yes	—
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₅₃×#3150	1	1	yes	characteristic
329	C₂	C₅₃×#3152	1	1	yes	characteristic
330	C₂	C₅₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3210	C₂	semidirect
2. C₅₃×#3182	C₂	semidirect
4. C₅₃×#3160	C₂	semidirect
5. C₅₃×#1723	C₂	semidirect
6. C₅₃×#1722	C₂	semidirect
7. C₅₃×#1719	C₂	semidirect
8. C₅₃×#1715	C₂	semidirect
9. C₅₃×#1451	C₂	semidirect
144. #19688	C₅₃	direct (N × H)
161. #3210	C₂×C₅₃	semidirect
162. #3182	C₂×C₅₃	semidirect
164. #3160	C₂×C₅₃	semidirect
165. #1723	C₂×C₅₃	semidirect
166. #1722	C₂×C₅₃	semidirect
167. #1719	C₂×C₅₃	semidirect
168. #1715	C₂×C₅₃	semidirect
169. #1451	C₂×C₅₃	semidirect
204. C₅₃	#19688	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×#3210	2
C₅₃×#3182	2
C₅₃×#3169	2
C₅₃×#3160	2
C₅₃×#1723	2
C₅₃×#1722	2
C₅₃×#1719	2
C₅₃×#1715	2
C₅₃×#1451	2
#19688	53

Subgroups — C53×#19688 #27227200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19688 #27227200