Subgroups — C₅₃×#19629 #27227141

Order 13568.

1050 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×#3205	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3151	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1721	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1712	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
10	C₂²×D₈×C₅₃	—	2	2	no	—
11	C₅₃×#532	C₂²	1	2	yes	—
12	C₂×#127×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅₃×#496	—	2	4	no	—
15	C₅₃×#493	—	2	2	no	—
16	C₅₃×#493	C₂²	1	2	yes	—
17	C₅₃×#493	C₂²	1	2	yes	—
18	C₅₃×#493	C₂²	1	4	yes	complemented
19	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
20	C₅₃×#493	C₂²	1	4	yes	complemented
21	C₅₃×#492	C₂²	1	4	yes	complemented
22	C₅₃×#486	C₂²	1	4	yes	complemented
23	C₅₃×#486	C₂²	1	2	yes	complemented
24	C₅₃×#466	—	2	2	no	—
25	C₅₃×#459	—	2	4	no	—
26	C₅₃×#447	—	2	4	no	—
27	C₄×D₈×C₅₃	—	2	2	no	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	complemented
32	C₈²×C₅₃	C₂²	1	2	yes	complemented
33	C₂×#41×C₅₃	—	2	2	no	—
34	C₂²×D₄×C₅₃	—	2	2	no	—
35	C₂²×D₄×C₅₃	—	2	2	no	—
36	C₂²×D₄×C₅₃	—	2	2	no	—
37	AGL(1,ℤ/8)×C₅₃	—	4	16	no	—
38	C₂×SD₁₆×C₅₃	—	2	8	no	—
39	C₂×D₈×C₅₃	—	2	4	no	—
40	C₂×D₈×C₅₃	—	2	4	no	—
41	C₂×D₈×C₅₃	—	2	8	no	—
42	C₂×D₈×C₅₃	—	2	2	no	—
43	C₂×D₈×C₅₃	—	2	8	no	—
44	C₂×D₈×C₅₃	—	4	8	no	—
45	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
46	C₂²×C₈×C₅₃	D₄	1	2	yes	—
47	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
48	#127×C₅₃	—	2	2	no	—
49	#127×C₅₃	—	2	2	no	—
50	#127×C₅₃	—	2	2	no	—
51	#127×C₅₃	C₂³	1	4	yes	—
52	#127×C₅₃	C₂³	1	2	yes	—
53	#124×C₅₃	—	2	2	no	—
54	#121×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	2	no	—
56	C₄×D₄×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	#107×C₅₃	—	2	2	no	—
60	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
61	#102×C₅₃	—	2	4	no	—
62	#102×C₅₃	—	2	8	no	—
63	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
64	#97×C₅₃	D₄	1	2	yes	—
65	C₄×C₈×C₅₃	D₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	2	yes	—
68	C₄×C₈×C₅₃	C₂³	1	2	yes	—
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂⁴×C₅₃	—	4	2	no	—
71	#41×C₅₃	—	4	8	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	—	2	4	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	2	4	no	—
78	C₂×D₄×C₅₃	—	2	4	no	—
79	C₂×D₄×C₅₃	—	4	8	no	—
80	C₂×D₄×C₅₃	—	4	2	no	—
81	C₂×D₄×C₅₃	—	4	2	no	—
82	C₂×D₄×C₅₃	—	4	2	no	—
83	C₂×D₄×C₅₃	—	4	2	no	—
84	C₂×D₄×C₅₃	—	4	8	no	—
85	C₂×D₄×C₅₃	—	8	4	no	—
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	SD₁₆×C₅₃	—	4	16	no	—
91	D₈×C₅₃	—	4	8	no	—
92	D₈×C₅₃	—	4	16	no	—
93	D₈×C₅₃	—	4	4	no	—
94	D₈×C₅₃	—	4	8	no	—
95	D₈×C₅₃	—	8	8	no	—
96	M₁₆×C₅₃	D₈	1	8	yes	complemented
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	C₂²⋊C₄×C₅₃	—	4	2	no	—
105	C₂²⋊C₄×C₅₃	—	4	2	no	—
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂³×C₅₃	—	4	2	no	—
111	C₂³×C₅₃	—	4	4	no	—
112	C₂³×C₅₃	—	8	4	no	—
113	C₂³×C₅₃	—	8	4	no	—
114	C₂³×C₅₃	—	8	4	no	—
115	C₂³×C₅₃	#127	1	1	yes	characteristic
116	Q₈×C₅₃	—	4	4	no	—
117	D₄×C₅₃	—	4	8	no	—
118	D₄×C₅₃	—	4	4	no	—
119	D₄×C₅₃	—	4	8	no	—
120	D₄×C₅₃	—	4	4	no	—
121	D₄×C₅₃	—	4	8	no	—
122	D₄×C₅₃	—	4	4	no	—
123	D₄×C₅₃	—	8	4	no	—
124	D₄×C₅₃	—	8	4	no	—
125	D₄×C₅₃	—	8	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	—	4	2	no	—
130	C₂×C₄×C₅₃	—	8	4	no	—
131	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	—	2	4	no	—
140	C₈×C₅₃	C₂×D₈	1	8	yes	complemented
141	C₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	complemented
142	#19629	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	8	2	no	—
146	C₂²×C₅₃	—	8	4	no	—
147	C₂²×C₅₃	—	8	4	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	—	8	4	no	—
150	C₂²×C₅₃	—	8	2	no	—
151	C₂²×C₅₃	—	16	8	no	—
152	C₂²×C₅₃	#493	1	2	yes	—
153	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	#493	1	2	yes	—
158	C₄×C₅₃	C₂²×D₈	1	2	yes	—
159	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₄×C₅₃	#576	1	2	yes	—
161	#3205	C₂×C₅₃	1	2	yes	complemented
162	#3169	C₂×C₅₃	1	1	yes	characteristic, complemented
163	#3151	C₂×C₅₃	1	2	yes	complemented
164	C₂×#493	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1721	C₂×C₅₃	1	4	yes	complemented
166	#1712	C₂×C₅₃	1	2	yes	complemented
167	#1706	C₂×C₅₃	1	2	yes	complemented
168	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	—	16	2	no	—
171	C₂×C₅₃	—	16	4	no	—
172	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
173	C₂×C₅₃	#3151	1	1	yes	characteristic
174	C₂×C₅₃	#3155	1	1	yes	characteristic
175	C₂×AGL(1,ℤ/8)	—	2	4	no	—
176	C₂²×D₈	—	2	2	no	—
177	#532	C₂²×C₅₃	1	2	yes	—
178	C₂×#127	C₂²×C₅₃	1	2	yes	—
179	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
180	#496	—	2	4	no	—
181	#493	—	2	2	no	—
182	#493	C₂²×C₅₃	1	2	yes	—
183	#493	C₂²×C₅₃	1	2	yes	—
184	#493	C₂²×C₅₃	1	4	yes	complemented
185	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
186	#493	C₂²×C₅₃	1	4	yes	complemented
187	#492	C₂²×C₅₃	1	4	yes	complemented
188	#486	C₂²×C₅₃	1	2	yes	complemented
189	#486	C₂²×C₅₃	1	4	yes	complemented
190	#466	—	2	2	no	—
191	#459	—	2	4	no	—
192	#447	—	2	4	no	—
193	C₄×D₈	—	2	2	no	—
194	#431	C₂²×C₅₃	1	1	yes	characteristic
195	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
197	#322	C₂²×C₅₃	1	2	yes	complemented
198	C₈²	C₂²×C₅₃	1	2	yes	complemented
199	C₅₃	#19629	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#41	—	2	2	no	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	—	2	2	no	—
203	C₂²×D₄	—	2	2	no	—
204	AGL(1,ℤ/8)	—	4	16	no	—
205	C₂×SD₁₆	—	2	8	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	4	no	—
208	C₂×D₈	—	2	8	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	8	no	—
211	C₂×D₈	—	4	8	no	—
212	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
213	C₂²×C₈	D₄×C₅₃	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
215	#127	—	2	2	no	—
216	#127	—	2	2	no	—
217	#127	—	2	2	no	—
218	#127	C₂³×C₅₃	1	4	yes	—
219	#127	C₂³×C₅₃	1	2	yes	—
220	#124	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
228	#102	—	2	4	no	—
229	#102	—	2	8	no	—
230	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
231	#97	D₄×C₅₃	1	2	yes	—
232	C₄×C₈	D₄×C₅₃	1	2	yes	—
233	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₅₃	1	2	yes	—
235	C₄×C₈	C₂³×C₅₃	1	2	yes	—
236	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
237	C₂⁴	—	4	2	no	—
238	#41	—	4	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	4	8	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	8	no	—
252	C₂×D₄	—	8	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
257	SD₁₆	—	4	16	no	—
258	D₈	—	4	8	no	—
259	D₈	—	4	16	no	—
260	D₈	—	4	8	no	—
261	D₈	—	4	4	no	—
262	D₈	—	8	8	no	—
263	M₁₆	D₈×C₅₃	1	8	yes	complemented
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
268	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
276	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
277	C₂³	—	4	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	4	no	—
282	C₂³	#127×C₅₃	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	8	no	—
286	D₄	—	4	8	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	8	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
299	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
300	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
302	C₂×C₄	C₂²×D₄×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	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	C₂×D₈×C₅₃	1	8	yes	complemented
308	C₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	16	8	no	—
318	C₂²	C₅₃×#493	1	2	yes	—
319	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₅₃×#493	1	2	yes	—
324	C₄	C₂²×D₈×C₅₃	1	2	yes	—
325	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
326	C₄	C₅₃×#576	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	16	4	no	—
329	C₂	—	16	2	no	—
330	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
331	C₂	C₅₃×#3151	1	1	yes	characteristic
332	C₂	C₅₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
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×#3205	C₂	semidirect
2. C₅₃×#3169	C₂	semidirect
3. C₅₃×#3151	C₂	semidirect
4. C₂×C₅₃×#493	C₂	semidirect
5. C₅₃×#1721	C₂	semidirect
6. C₅₃×#1712	C₂	semidirect
7. C₅₃×#1706	C₂	semidirect
8. C₈×M₁₆×C₅₃	C₂	semidirect
13. C₅₃×#500	C₂²	semidirect
18. C₅₃×#493	C₂²	semidirect
19. C₅₃×#493	C₂²	semidirect
20. C₅₃×#493	C₂²	semidirect
21. C₅₃×#492	C₂²	semidirect
22. C₅₃×#486	C₂²	semidirect
23. C₅₃×#486	C₂²	semidirect
31. C₅₃×#322	C₂²	semidirect
32. C₈²×C₅₃	C₂²	semidirect
96. M₁₆×C₅₃	D₈	semidirect
140. C₈×C₅₃	C₂×D₈	semidirect
141. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
142. #19629	C₅₃	direct (N × H)
161. #3205	C₂×C₅₃	semidirect
162. #3169	C₂×C₅₃	semidirect
163. #3151	C₂×C₅₃	semidirect
164. C₂×#493	C₂×C₅₃	semidirect
165. #1721	C₂×C₅₃	semidirect
166. #1712	C₂×C₅₃	semidirect
167. #1706	C₂×C₅₃	semidirect
168. C₈×M₁₆	C₂×C₅₃	semidirect
179. #500	C₂²×C₅₃	semidirect
184. #493	C₂²×C₅₃	semidirect
185. #493	C₂²×C₅₃	semidirect
186. #493	C₂²×C₅₃	semidirect
187. #492	C₂²×C₅₃	semidirect
188. #486	C₂²×C₅₃	semidirect
189. #486	C₂²×C₅₃	semidirect
197. #322	C₂²×C₅₃	semidirect
198. C₈²	C₂²×C₅₃	semidirect
199. C₅₃	#19629	direct (N × H)
263. M₁₆	D₈×C₅₃	semidirect
307. C₈	C₂×D₈×C₅₃	semidirect
308. C₈	AGL(1,ℤ/8)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3205	2
C₅₃×#3169	2
C₅₃×#3151	2
C₂×C₅₃×#493	2
C₅₃×#1721	2
C₅₃×#1712	2
C₅₃×#1706	2
C₈×M₁₆×C₅₃	2
#19629	53

Subgroups — C53×#19629 #27227141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19629 #27227141