Subgroups — C₅₃×#21733 #27229245

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2961	C₂	1	2	yes	maximal
3	C₅₃×#1995	C₂	1	2	yes	maximal
4	C₅₃×#1946	C₂	1	2	yes	maximal
5	C₅₃×#1945	C₂	1	2	yes	maximal
6	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1773	C₂	1	2	yes	maximal
8	C₅₃×#1772	C₂	1	2	yes	maximal
9	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#533	C₂²	1	2	yes	—
11	C₅₃×#500	C₄	1	2	yes	complemented
12	C₅₃×#499	C₄	1	2	yes	complemented
13	C₅₃×#499	C₄	1	2	yes	complemented
14	C₅₃×#498	C₄	1	2	yes	complemented
15	C₅₃×#446	—	2	8	no	—
16	C₂×#108×C₅₃	C₂²	1	2	yes	—
17	C₂×#108×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#427	C₂²	1	2	yes	—
19	C₅₃×#427	C₂²	1	2	yes	—
20	C₅₃×#424	C₂²	1	4	yes	—
21	C₅₃×#423	C₂²	1	2	yes	—
22	C₅₃×#405	C₂²	1	4	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#376	C₂²	1	1	yes	characteristic
30	C₅₃×#339	C₂²	1	4	yes	—
31	C₅₃×#339	C₂²	1	4	yes	—
32	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
38	#125×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²⋊Q₈×C₅₃	—	2	4	no	—
40	C₄×Q₈×C₅₃	—	2	4	no	—
41	#117×C₅₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#108×C₅₃	—	2	4	no	—
46	#108×C₅₃	—	2	4	no	—
47	#107×C₅₃	C₂×C₄	1	4	yes	—
48	#107×C₅₃	C₂×C₄	1	4	yes	—
49	#106×C₅₃	C₂×C₄	1	4	yes	—
50	#106×C₅₃	C₂×C₄	1	4	yes	—
51	C₄⋊C₈×C₅₃	—	2	8	no	—
52	C₄⋊C₈×C₅₃	—	2	4	no	—
53	#97×C₅₃	—	2	4	no	—
54	#97×C₅₃	—	2	2	no	—
55	#97×C₅₃	D₄	1	4	yes	—
56	#97×C₅₃	Q₈	1	4	yes	complemented
57	C₄×C₈×C₅₃	—	2	4	no	—
58	C₄×C₈×C₅₃	—	2	2	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
64	#95×C₅₃	—	2	2	no	—
65	C₂×Q₈×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₅₃	#41	1	2	yes	—
86	C₂×C₈×C₅₃	#41	1	2	yes	—
87	C₄⋊C₄×C₅₃	—	2	8	no	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	2	yes	—
97	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
98	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
99	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
100	Q₈×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	—	4	4	no	—
117	C₈×C₅₃	C₄×Q₈	1	2	yes	—
118	C₈×C₅₃	C₄×Q₈	1	2	yes	—
119	C₈×C₅₃	C₄×Q₈	1	2	yes	—
120	C₈×C₅₃	C₄×Q₈	1	2	yes	—
121	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
122	#21733	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₅₃	#533	1	1	yes	characteristic
126	C₂²×C₅₃	#533	1	1	yes	characteristic
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	—	4	8	no	—
132	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₄×C₅₃	#532	1	1	yes	characteristic
136	#3160	C₂×C₅₃	1	1	yes	characteristic
137	#2961	C₂×C₅₃	1	2	yes	—
138	#1995	C₂×C₅₃	1	2	yes	—
139	#1946	C₂×C₅₃	1	2	yes	—
140	#1945	C₂×C₅₃	1	2	yes	—
141	#1848	C₂×C₅₃	1	1	yes	characteristic
142	#1773	C₂×C₅₃	1	2	yes	—
143	#1772	C₂×C₅₃	1	2	yes	—
144	#1726	C₂×C₅₃	1	1	yes	characteristic
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₅₃	#2957	1	2	yes	—
149	#533	C₂²×C₅₃	1	2	yes	—
150	#500	C₄×C₅₃	1	2	yes	complemented
151	#499	C₄×C₅₃	1	2	yes	complemented
152	#499	C₄×C₅₃	1	2	yes	complemented
153	#498	C₄×C₅₃	1	2	yes	complemented
154	#446	—	2	8	no	—
155	C₂×#108	C₂²×C₅₃	1	2	yes	—
156	C₂×#108	C₂²×C₅₃	1	2	yes	—
157	#427	C₂²×C₅₃	1	2	yes	—
158	#427	C₂²×C₅₃	1	2	yes	—
159	#424	C₂²×C₅₃	1	4	yes	—
160	#423	C₂²×C₅₃	1	2	yes	—
161	#405	C₂²×C₅₃	1	4	yes	—
162	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
164	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
168	#376	C₂²×C₅₃	1	1	yes	characteristic
169	#339	C₂²×C₅₃	1	4	yes	—
170	#339	C₂²×C₅₃	1	4	yes	—
171	C₅₃	#21733	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
173	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
177	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
178	#125	C₂×C₄×C₅₃	1	4	yes	—
179	C₂²⋊Q₈	—	2	4	no	—
180	C₄×Q₈	—	2	4	no	—
181	#117	C₂³×C₅₃	1	4	yes	—
182	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
185	#108	—	2	4	no	—
186	#108	—	2	4	no	—
187	#107	C₂×C₄×C₅₃	1	4	yes	—
188	#107	C₂×C₄×C₅₃	1	4	yes	—
189	#106	C₂×C₄×C₅₃	1	4	yes	—
190	#106	C₂×C₄×C₅₃	1	4	yes	—
191	C₄⋊C₈	—	2	8	no	—
192	C₄⋊C₈	—	2	4	no	—
193	#97	—	2	4	no	—
194	#97	—	2	2	no	—
195	#97	D₄×C₅₃	1	4	yes	—
196	#97	Q₈×C₅₃	1	4	yes	complemented
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
203	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
204	#95	—	2	2	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	#41×C₅₃	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
221	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
222	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
223	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
224	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
225	C₂×C₈	#41×C₅₃	1	2	yes	—
226	C₂×C₈	#41×C₅₃	1	2	yes	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₅₃	1	2	yes	—
237	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
238	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
239	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
248	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
249	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	C₄×Q₈×C₅₃	1	2	yes	—
258	C₈	C₄×Q₈×C₅₃	1	2	yes	—
259	C₈	C₄×Q₈×C₅₃	1	2	yes	—
260	C₈	C₄×Q₈×C₅₃	1	2	yes	—
261	C₈	C₄⋊Q₈×C₅₃	1	4	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
264	C₂²	C₅₃×#533	1	1	yes	characteristic
265	C₂²	C₅₃×#533	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	8	no	—
271	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
272	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
273	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
274	C₄	C₅₃×#532	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
278	C₂	C₅₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#500	C₄	semidirect
12. C₅₃×#499	C₄	semidirect
13. C₅₃×#499	C₄	semidirect
14. C₅₃×#498	C₄	semidirect
56. #97×C₅₃	Q₈	semidirect
63. C₄×C₈×C₅₃	Q₈	semidirect
122. #21733	C₅₃	direct (N × H)
150. #500	C₄×C₅₃	semidirect
151. #499	C₄×C₅₃	semidirect
152. #499	C₄×C₅₃	semidirect
153. #498	C₄×C₅₃	semidirect
171. C₅₃	#21733	direct (N × H)
196. #97	Q₈×C₅₃	semidirect
203. C₄×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3160	2
C₅₃×#2961	2
C₅₃×#1995	2
C₅₃×#1946	2
C₅₃×#1945	2
C₅₃×#1848	2
C₅₃×#1773	2
C₅₃×#1772	2
C₅₃×#1726	2
#21733	53

Subgroups — C53×#21733 #27229245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21733 #27229245