Subgroups — C₅₃×#19329 #27226841

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3214	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1701	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1673	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1667	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1557	C₂	1	2	yes	maximal, complemented
9	C₂×#131×C₅₃	—	2	1	no	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₅₃×#500	C₂²	1	1	yes	characteristic
13	C₅₃×#496	—	2	2	no	—
14	C₅₃×#493	C₂²	1	1	yes	characteristic
15	C₅₃×#489	—	2	2	no	—
16	C₅₃×#486	C₂²	1	2	yes	—
17	C₅₃×#466	—	2	1	no	—
18	C₅₃×#466	—	2	1	no	—
19	C₅₃×#461	C₂²	1	4	yes	—
20	C₅₃×#459	C₂²	1	4	yes	—
21	C₅₃×#449	—	2	4	no	—
22	C₄×D₈×C₅₃	—	2	1	no	—
23	C₄×D₈×C₅₃	—	2	1	no	—
24	C₅₃×#436	—	2	1	no	—
25	C₅₃×#436	C₂²	1	2	yes	—
26	C₈×D₄×C₅₃	—	2	1	no	—
27	C₈×D₄×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#432	C₂²	1	2	yes	—
29	C₅₃×#423	C₂²	1	1	yes	characteristic
30	C₅₃×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	2	yes	—
34	C₅₃×#332	C₂²	1	4	yes	—
35	C₅₃×#331	C₂²	1	4	yes	—
36	C₂×#41×C₅₃	—	2	1	no	—
37	C₂×#41×C₅₃	—	2	2	no	—
38	C₂×SD₁₆×C₅₃	—	2	4	no	—
39	C₂×D₈×C₅₃	—	2	1	no	—
40	C₂×D₈×C₅₃	—	2	1	no	—
41	C₂×D₈×C₅₃	—	2	4	no	—
42	#131×C₅₃	—	4	4	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
50	#127×C₅₃	D₄	1	1	yes	characteristic
51	#127×C₅₃	C₂³	1	2	yes	—
52	#124×C₅₃	—	2	2	no	—
53	#124×C₅₃	—	2	1	no	—
54	#121×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	1	no	—
56	#121×C₅₃	—	2	2	no	—
57	#121×C₅₃	D₄	1	2	yes	—
58	C₄×D₄×C₅₃	—	2	1	no	—
59	C₄×D₄×C₅₃	—	2	2	no	—
60	C₄×D₄×C₅₃	—	2	2	no	—
61	C₄×D₄×C₅₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#107×C₅₃	—	2	1	no	—
64	#107×C₅₃	—	2	1	no	—
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
67	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
68	#103×C₅₃	—	2	4	no	—
69	#102×C₅₃	—	2	2	no	—
70	#102×C₅₃	—	2	2	no	—
71	#102×C₅₃	—	2	4	no	—
72	C₂²⋊C₈×C₅₃	—	2	2	no	—
73	C₂²⋊C₈×C₅₃	—	2	2	no	—
74	C₂²⋊C₈×C₅₃	—	2	2	no	—
75	#97×C₅₃	—	2	2	no	—
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	#41×C₅₃	—	4	2	no	—
81	#41×C₅₃	—	4	2	no	—
82	#41×C₅₃	—	8	4	no	—
83	C₂×Q₈×C₅₃	—	2	1	no	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	2	no	—
86	C₂×D₄×C₅₃	—	2	1	no	—
87	C₂×D₄×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	4	2	no	—
92	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	4	2	no	—
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
98	SD₁₆×C₅₃	—	8	4	no	—
99	D₈×C₅₃	—	4	2	no	—
100	D₈×C₅₃	—	4	2	no	—
101	D₈×C₅₃	—	8	4	no	—
102	M₁₆×C₅₃	—	4	4	no	—
103	M₁₆×C₅₃	—	4	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	4	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	4	1	no	—
108	C₂×C₈×C₅₃	—	4	4	no	—
109	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄×C₅₃	—	2	2	no	—
118	C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₂²⋊C₄×C₅₃	—	4	2	no	—
120	C₂²⋊C₄×C₅₃	—	4	2	no	—
121	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅₃	#41	1	1	yes	characteristic
124	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	4	2	no	—
127	C₂³×C₅₃	#121	1	1	yes	characteristic
128	Q₈×C₅₃	—	4	2	no	—
129	Q₈×C₅₃	—	4	4	no	—
130	D₄×C₅₃	—	2	2	no	—
131	D₄×C₅₃	—	2	2	no	—
132	D₄×C₅₃	—	4	2	no	—
133	D₄×C₅₃	—	4	2	no	—
134	D₄×C₅₃	—	4	4	no	—
135	D₄×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	4	4	no	—
138	D₄×C₅₃	—	8	2	no	—
139	D₄×C₅₃	—	8	4	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	8	4	no	—
149	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	4	2	no	—
159	C₈×C₅₃	—	4	4	no	—
160	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
161	#19329	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	4	2	no	—
166	C₂²×C₅₃	—	4	2	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	8	2	no	—
169	C₂²×C₅₃	—	8	2	no	—
170	C₂²×C₅₃	—	8	2	no	—
171	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	4	2	no	—
177	C₄×C₅₃	—	8	2	no	—
178	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₄×C₅₃	#575	1	1	yes	characteristic
180	C₄×C₅₃	#576	1	2	yes	—
181	#3214	C₂×C₅₃	1	2	yes	complemented
182	#3169	C₂×C₅₃	1	1	yes	characteristic, complemented
183	#2931	C₂×C₅₃	1	1	yes	characteristic, complemented
184	#1701	C₂×C₅₃	1	4	yes	complemented
185	#1673	C₂×C₅₃	1	2	yes	complemented
186	#1667	C₂×C₅₃	1	2	yes	complemented
187	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#1557	C₂×C₅₃	1	2	yes	complemented
189	C₂×C₅₃	—	4	1	no	—
190	C₂×C₅₃	—	8	2	no	—
191	C₂×C₅₃	—	16	2	no	—
192	C₂×C₅₃	#3036	1	1	yes	characteristic
193	C₂×C₅₃	#3058	1	1	yes	characteristic
194	C₂×C₅₃	#3067	1	1	yes	characteristic
195	C₂×#131	—	2	1	no	—
196	#532	C₂²×C₅₃	1	2	yes	—
197	#532	C₂²×C₅₃	1	1	yes	characteristic
198	#500	C₂²×C₅₃	1	1	yes	characteristic
199	#496	—	2	2	no	—
200	#493	C₂²×C₅₃	1	1	yes	characteristic
201	#489	—	2	2	no	—
202	#486	C₂²×C₅₃	1	2	yes	—
203	#466	—	2	1	no	—
204	#466	—	2	1	no	—
205	#461	C₂²×C₅₃	1	4	yes	—
206	#459	C₂²×C₅₃	1	4	yes	—
207	#449	—	2	4	no	—
208	C₄×D₈	—	2	1	no	—
209	C₄×D₈	—	2	1	no	—
210	#436	—	2	1	no	—
211	#436	C₂²×C₅₃	1	2	yes	—
212	C₈×D₄	—	2	1	no	—
213	C₈×D₄	C₂²×C₅₃	1	2	yes	—
214	#432	C₂²×C₅₃	1	2	yes	—
215	#423	C₂²×C₅₃	1	1	yes	characteristic
216	#408	—	2	1	no	—
217	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
218	#335	C₂²×C₅₃	1	2	yes	—
219	#334	C₂²×C₅₃	1	2	yes	—
220	#332	C₂²×C₅₃	1	4	yes	—
221	#331	C₂²×C₅₃	1	4	yes	—
222	C₅₃	#19329	1	1	yes	characteristic, direct product factor, Sylow
223	C₂×#41	—	2	1	no	—
224	C₂×#41	—	2	2	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×D₈	—	2	4	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×D₈	—	2	1	no	—
229	#131	—	4	4	no	—
230	C₂×M₁₆	—	2	1	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
234	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
235	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
236	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
237	#127	D₄×C₅₃	1	1	yes	characteristic
238	#127	C₂³×C₅₃	1	2	yes	—
239	#124	—	2	2	no	—
240	#124	—	2	1	no	—
241	#121	—	2	1	no	—
242	#121	—	2	2	no	—
243	#121	—	2	2	no	—
244	#121	D₄×C₅₃	1	2	yes	—
245	C₄×D₄	—	2	1	no	—
246	C₄×D₄	—	2	2	no	—
247	C₄×D₄	—	2	2	no	—
248	C₄×D₄	C₂³×C₅₃	1	2	yes	—
249	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
250	#107	—	2	1	no	—
251	#107	—	2	1	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
254	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
255	#103	—	2	4	no	—
256	#102	—	2	2	no	—
257	#102	—	2	2	no	—
258	#102	—	2	4	no	—
259	C₂²⋊C₈	—	2	2	no	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₂²⋊C₈	—	2	2	no	—
262	#97	—	2	2	no	—
263	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
264	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
265	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
266	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	#41	—	8	4	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	4	2	no	—
282	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
285	SD₁₆	—	8	4	no	—
286	D₈	—	4	2	no	—
287	D₈	—	4	2	no	—
288	D₈	—	8	4	no	—
289	M₁₆	—	4	4	no	—
290	M₁₆	—	4	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	4	4	no	—
296	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
297	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
298	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
299	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₄²	#41×C₅₃	1	1	yes	characteristic
311	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	—	4	2	no	—
314	C₂³	#121×C₅₃	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	4	4	no	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	4	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	4	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	4	no	—
336	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
337	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
338	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	4	no	—
347	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	8	2	no	—
364	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
365	C₄	C₅₃×#575	1	1	yes	characteristic
366	C₄	C₅₃×#576	1	2	yes	—
367	C₂	—	4	1	no	—
368	C₂	—	8	2	no	—
369	C₂	—	16	2	no	—
370	C₂	C₅₃×#3036	1	1	yes	characteristic
371	C₂	C₅₃×#3058	1	1	yes	characteristic
372	C₂	C₅₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
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×#3214	C₂	semidirect
2. C₅₃×#3169	C₂	semidirect
3. C₅₃×#2931	C₂	semidirect
4. C₅₃×#1701	C₂	semidirect
5. C₅₃×#1673	C₂	semidirect
6. C₅₃×#1667	C₂	semidirect
7. C₅₃×#1568	C₂	semidirect
8. C₅₃×#1557	C₂	semidirect
161. #19329	C₅₃	direct (N × H)
181. #3214	C₂×C₅₃	semidirect
182. #3169	C₂×C₅₃	semidirect
183. #2931	C₂×C₅₃	semidirect
184. #1701	C₂×C₅₃	semidirect
185. #1673	C₂×C₅₃	semidirect
186. #1667	C₂×C₅₃	semidirect
187. #1568	C₂×C₅₃	semidirect
188. #1557	C₂×C₅₃	semidirect
222. C₅₃	#19329	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×#3214	2
C₅₃×#3169	2
C₅₃×#2931	2
C₅₃×#1701	2
C₅₃×#1673	2
C₅₃×#1667	2
C₅₃×#1568	2
C₅₃×#1557	2
#19329	53

Subgroups — C53×#19329 #27226841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19329 #27226841