Subgroups — C₅₃×#19332 #27226844

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3253	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×#1697	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1675	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1667	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1560	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×#493	C₂²	1	1	yes	characteristic
14	C₅₃×#486	C₂²	1	2	yes	—
15	C₅₃×#476	—	2	2	no	—
16	C₅₃×#476	C₂²	1	4	yes	—
17	C₅₃×#469	—	2	4	no	—
18	C₅₃×#466	—	2	1	no	—
19	C₅₃×#466	—	2	1	no	—
20	C₅₃×#459	—	2	2	no	—
21	C₅₃×#459	C₂²	1	4	yes	—
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×#423	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×#335	C₂²	1	2	yes	—
34	C₅₃×#333	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×D₈×C₅₃	—	2	1	no	—
39	C₂×D₈×C₅₃	—	2	1	no	—
40	C₂×D₈×C₅₃	—	2	4	no	—
41	#131×C₅₃	—	4	4	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	1	no	—
44	C₂×M₁₆×C₅₃	—	2	2	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	1	no	—
53	#124×C₅₃	—	2	2	no	—
54	#121×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	2	no	—
56	#121×C₅₃	—	2	1	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	#106×C₅₃	—	2	4	no	—
66	C₄⋊C₈×C₅₃	—	2	1	no	—
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
70	#102×C₅₃	—	2	2	no	—
71	#102×C₅₃	—	2	2	no	—
72	#102×C₅₃	—	2	4	no	—
73	#102×C₅₃	—	2	4	no	—
74	C₂²⋊C₈×C₅₃	—	2	2	no	—
75	C₂²⋊C₈×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	D₈×C₅₃	—	4	2	no	—
99	D₈×C₅₃	—	4	2	no	—
100	D₈×C₅₃	—	8	4	no	—
101	M₁₆×C₅₃	—	4	4	no	—
102	M₁₆×C₅₃	—	4	2	no	—
103	M₁₆×C₅₃	—	8	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	4	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	—	8	2	no	—
130	D₄×C₅₃	—	2	2	no	—
131	D₄×C₅₃	—	2	2	no	—
132	D₄×C₅₃	—	4	2	no	—
133	D₄×C₅₃	—	4	4	no	—
134	D₄×C₅₃	—	4	2	no	—
135	D₄×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	4	2	no	—
138	D₄×C₅₃	—	4	2	no	—
139	D₄×C₅₃	—	8	2	no	—
140	D₄×C₅₃	—	8	4	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	8	4	no	—
151	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	2	1	no	—
159	C₈×C₅₃	—	2	1	no	—
160	C₈×C₅₃	—	4	2	no	—
161	C₈×C₅₃	—	4	4	no	—
162	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
163	#19332	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	4	2	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	8	2	no	—
171	C₂²×C₅₃	—	8	2	no	—
172	C₂²×C₅₃	—	8	2	no	—
173	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	—	8	2	no	—
179	C₄×C₅₃	#472	1	2	yes	—
180	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₄×C₅₃	#575	1	1	yes	characteristic
182	C₄×C₅₃	#576	1	1	yes	characteristic
183	C₄×C₅₃	#578	1	1	yes	characteristic
184	#3253	C₂×C₅₃	1	2	yes	complemented
185	#3169	C₂×C₅₃	1	1	yes	characteristic, complemented
186	#2931	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#1697	C₂×C₅₃	1	4	yes	complemented
188	#1675	C₂×C₅₃	1	2	yes	complemented
189	#1667	C₂×C₅₃	1	2	yes	complemented
190	#1567	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#1560	C₂×C₅₃	1	2	yes	complemented
192	C₂×C₅₃	—	4	1	no	—
193	C₂×C₅₃	—	8	2	no	—
194	C₂×C₅₃	—	16	2	no	—
195	C₂×C₅₃	#3036	1	1	yes	characteristic
196	C₂×C₅₃	#3061	1	1	yes	characteristic
197	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
198	C₂×#131	—	2	1	no	—
199	#532	C₂²×C₅₃	1	2	yes	—
200	#532	C₂²×C₅₃	1	1	yes	characteristic
201	#500	C₂²×C₅₃	1	1	yes	characteristic
202	#493	C₂²×C₅₃	1	1	yes	characteristic
203	#486	C₂²×C₅₃	1	2	yes	—
204	#476	—	2	2	no	—
205	#476	C₂²×C₅₃	1	4	yes	—
206	#469	—	2	4	no	—
207	#466	—	2	1	no	—
208	#466	—	2	1	no	—
209	#459	—	2	2	no	—
210	#459	C₂²×C₅₃	1	4	yes	—
211	C₄×D₈	—	2	1	no	—
212	C₄×D₈	—	2	1	no	—
213	#436	—	2	1	no	—
214	#436	C₂²×C₅₃	1	2	yes	—
215	C₈×D₄	—	2	1	no	—
216	C₈×D₄	C₂²×C₅₃	1	2	yes	—
217	#423	C₂²×C₅₃	1	1	yes	characteristic
218	#423	C₂²×C₅₃	1	2	yes	—
219	#408	—	2	1	no	—
220	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
221	#335	C₂²×C₅₃	1	2	yes	—
222	#335	C₂²×C₅₃	1	2	yes	—
223	#333	C₂²×C₅₃	1	4	yes	—
224	#331	C₂²×C₅₃	1	4	yes	—
225	C₅₃	#19332	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×#41	—	2	1	no	—
227	C₂×#41	—	2	2	no	—
228	C₂×D₈	—	2	4	no	—
229	C₂×D₈	—	2	1	no	—
230	C₂×D₈	—	2	1	no	—
231	#131	—	4	4	no	—
232	C₂×M₁₆	—	2	1	no	—
233	C₂×M₁₆	—	2	2	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂²×C₈	—	2	2	no	—
236	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
237	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
238	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
239	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
240	#127	D₄×C₅₃	1	1	yes	characteristic
241	#127	C₂³×C₅₃	1	2	yes	—
242	#124	—	2	1	no	—
243	#124	—	2	2	no	—
244	#121	—	2	1	no	—
245	#121	—	2	2	no	—
246	#121	—	2	2	no	—
247	#121	D₄×C₅₃	1	2	yes	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	C₂³×C₅₃	1	2	yes	—
252	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
253	#107	—	2	1	no	—
254	#107	—	2	1	no	—
255	#106	—	2	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
259	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
260	#102	—	2	4	no	—
261	#102	—	2	2	no	—
262	#102	—	2	2	no	—
263	#102	—	2	4	no	—
264	C₂²⋊C₈	—	2	2	no	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
267	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
268	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
270	#41	—	4	2	no	—
271	#41	—	4	2	no	—
272	#41	—	8	4	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
288	D₈	—	4	2	no	—
289	D₈	—	4	2	no	—
290	D₈	—	8	4	no	—
291	M₁₆	—	4	4	no	—
292	M₁₆	—	4	2	no	—
293	M₁₆	—	8	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	4	4	no	—
299	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
300	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
301	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
302	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	4	2	no	—
310	C₂²⋊C₄	—	4	2	no	—
311	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₄²	#41×C₅₃	1	1	yes	characteristic
314	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
315	C₂³	—	2	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	#121×C₅₃	1	1	yes	characteristic
318	Q₈	—	4	2	no	—
319	Q₈	—	8	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	2	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	4	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	4	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
342	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
343	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	4	no	—
352	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	8	2	no	—
368	C₄	C₅₃×#472	1	2	yes	—
369	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
370	C₄	C₅₃×#575	1	1	yes	characteristic
371	C₄	C₅₃×#576	1	1	yes	characteristic
372	C₄	C₅₃×#578	1	1	yes	characteristic
373	C₂	—	4	1	no	—
374	C₂	—	8	2	no	—
375	C₂	—	16	2	no	—
376	C₂	C₅₃×#3036	1	1	yes	characteristic
377	C₂	C₅₃×#3061	1	1	yes	characteristic
378	C₂	C₂×C₅₃×#472	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×#3253	C₂	semidirect
2. C₅₃×#3169	C₂	semidirect
3. C₅₃×#2931	C₂	semidirect
4. C₅₃×#1697	C₂	semidirect
5. C₅₃×#1675	C₂	semidirect
6. C₅₃×#1667	C₂	semidirect
7. C₅₃×#1567	C₂	semidirect
8. C₅₃×#1560	C₂	semidirect
163. #19332	C₅₃	direct (N × H)
184. #3253	C₂×C₅₃	semidirect
185. #3169	C₂×C₅₃	semidirect
186. #2931	C₂×C₅₃	semidirect
187. #1697	C₂×C₅₃	semidirect
188. #1675	C₂×C₅₃	semidirect
189. #1667	C₂×C₅₃	semidirect
190. #1567	C₂×C₅₃	semidirect
191. #1560	C₂×C₅₃	semidirect
225. C₅₃	#19332	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×#3253	2
C₅₃×#3169	2
C₅₃×#2931	2
C₅₃×#1697	2
C₅₃×#1675	2
C₅₃×#1667	2
C₅₃×#1567	2
C₅₃×#1560	2
#19332	53

Subgroups — C53×#19332 #27226844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19332 #27226844