Subgroups — C₅₃×#8662 #27216174

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1486	C₂	1	2	yes	maximal
5	C₅₃×#1301	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₅₃	—	2	1	no	—
7	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#436	—	2	1	no	—
9	C₅₃×#436	—	2	1	no	—
10	C₅₃×#436	C₄	1	2	yes	—
11	C₅₃×#436	C₄	1	2	yes	—
12	C₅₃×#431	C₂²	1	2	yes	—
13	C₂×#108×C₅₃	—	2	1	no	—
14	C₂×#108×C₅₃	—	2	1	no	—
15	C₅₃×#423	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	—	2	1	no	—
20	C₅₃×#334	—	2	1	no	—
21	C₅₃×D₄⋊C₈	—	2	4	no	—
22	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	—	2	1	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	—	4	2	no	—
28	C₂×M₁₆×C₅₃	—	4	2	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×D₄×C₅₃	—	2	1	no	—
33	C₄×D₄×C₅₃	—	2	1	no	—
34	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×D₄×C₅₃	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#108×C₅₃	—	4	2	no	—
40	#108×C₅₃	—	4	2	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₅₃	—	2	2	no	—
47	C₂²⋊C₈×C₅₃	—	2	2	no	—
48	C₂²⋊C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	—	2	2	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂⁴×C₅₃	—	2	1	no	—
55	C₂×D₄×C₅₃	—	2	1	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	M₁₆×C₅₃	—	4	4	no	—
69	M₁₆×C₅₃	—	4	4	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	D₈	1	2	yes	—
84	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
85	C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂²⋊C₄×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	4	1	no	—
97	C₂³×C₅₃	—	4	1	no	—
98	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	D₄×C₅₃	—	2	4	no	—
100	D₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	4	2	no	—
123	C₈×C₅₃	—	8	2	no	—
124	C₈×C₅₃	—	8	2	no	—
125	C₈×C₅₃	#102	1	2	yes	—
126	C₈×C₅₃	#102	1	2	yes	—
127	#8662	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	—	8	2	no	—
135	C₂²×C₅₃	#380	1	1	yes	characteristic
136	C₂²×C₅₃	#474	1	1	yes	characteristic
137	C₂²×C₅₃	#476	1	1	yes	characteristic
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
143	C₄×C₅₃	#417	1	1	yes	characteristic
144	C₄×C₅₃	#430	1	2	yes	—
145	C₄×C₅₃	#459	1	1	yes	characteristic
146	C₄×C₅₃	#461	1	1	yes	characteristic
147	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
148	#1848	C₂×C₅₃	1	1	yes	characteristic, complemented
149	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
150	#1486	C₂×C₅₃	1	2	yes	—
151	#1301	C₂×C₅₃	1	2	yes	complemented
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	8	2	no	—
155	C₂×C₅₃	#1802	1	1	yes	characteristic
156	C₂×C₅₃	#1811	1	1	yes	characteristic
157	C₂×C₅₃	#1814	1	1	yes	characteristic
158	C₂²×M₁₆	—	2	1	no	—
159	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
160	#436	—	2	1	no	—
161	#436	—	2	1	no	—
162	#436	C₄×C₅₃	1	2	yes	—
163	#436	C₄×C₅₃	1	2	yes	—
164	#431	C₂²×C₅₃	1	2	yes	—
165	C₂×#108	—	2	1	no	—
166	C₂×#108	—	2	1	no	—
167	#423	C₂²×C₅₃	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
171	#335	—	2	1	no	—
172	#334	—	2	1	no	—
173	D₄⋊C₈	—	2	4	no	—
174	C₅₃	#8662	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂×M₁₆	—	4	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
188	C₄×D₄	D₄×C₅₃	1	2	yes	—
189	C₂×C₄⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	2	1	no	—
191	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
192	#108	—	4	2	no	—
193	#108	—	4	2	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
206	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
207	C₂⁴	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
221	M₁₆	—	4	4	no	—
222	M₁₆	—	4	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₈	D₈×C₅₃	1	2	yes	—
237	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₄²	#41×C₅₃	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
252	D₄	—	2	4	no	—
253	D₄	—	4	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
266	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
267	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
271	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	2	no	—
276	C₈	—	8	2	no	—
277	C₈	—	8	2	no	—
278	C₈	#102×C₅₃	1	2	yes	—
279	C₈	#102×C₅₃	1	2	yes	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₅₃×#380	1	1	yes	characteristic
288	C₂²	C₅₃×#474	1	1	yes	characteristic
289	C₂²	C₅₃×#476	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	2	no	—
294	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
295	C₄	C₅₃×#417	1	1	yes	characteristic
296	C₄	C₅₃×#430	1	2	yes	—
297	C₄	C₅₃×#459	1	1	yes	characteristic
298	C₄	C₅₃×#461	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅₃×#1802	1	1	yes	characteristic
303	C₂	C₅₃×#1811	1	1	yes	characteristic
304	C₂	C₅₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₄×C₅₃	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1848	C₂	semidirect
3. C₅₃×#1564	C₂	semidirect
5. C₅₃×#1301	C₂	semidirect
127. #8662	C₅₃	direct (N × H)
148. #1848	C₂×C₅₃	semidirect
149. #1564	C₂×C₅₃	semidirect
151. #1301	C₂×C₅₃	semidirect
174. C₅₃	#8662	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₅₃×#1848	2
C₅₃×#1564	2
C₅₃×#1486	2
C₅₃×#1301	2
#8662	53

Subgroups — C53×#8662 #27216174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8662 #27216174