Subgroups — D₅₃×#1380 #27298967

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×D₅₃	C₂	1	2	yes	maximal
3	#20272807	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×D₅₃	—	2	2	no	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×D₅₃	—	2	4	no	—
10	C₂×C₁₆×D₅₃	C₄	1	4	yes	complemented
11	C₂×#13133151	C₂²	1	2	yes	—
12	C₂×#13133150	—	2	2	no	—
13	#107×D₅₃	—	2	4	no	—
14	#107×D₅₃	C₄	1	8	yes	complemented
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄×D₅₃	—	4	2	no	—
18	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
19	C₂²×#88322	D₄	1	1	yes	characteristic
20	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₅₃	D₄	1	4	yes	—
23	C₂×C₈×D₅₃	Q₈	1	2	yes	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×#11719919	—	4	2	no	—
26	C₄⋊C₄×D₅₃	—	2	8	no	—
27	C₄⋊C₄×D₅₃	—	4	4	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	C₂×C₄	1	2	yes	complemented
30	C₁₆×D₅₃	—	2	8	no	—
31	#13133151	—	2	2	no	—
32	#13133151	C₂×C₄	1	4	yes	complemented
33	#13133150	—	2	8	no	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
38	#107×C₅₃	—	2	2	no	—
39	#107×C₅₃	C₂×C₄	1	4	yes	complemented
40	C₂³×D₅₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	4	2	no	—
44	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
45	C₂×C₄×D₅₃	—	4	8	no	—
46	C₂×C₄×D₅₃	—	8	4	no	—
47	C₂×C₄×D₅₃	—	8	4	no	—
48	C₂×C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
51	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	#11719919	—	4	8	no	—
61	C₂×#88322	C₂²⋊C₄	1	4	yes	—
62	C₂×#88322	C₄⋊C₄	1	2	yes	—
63	C₈×D₅₃	C₄²	1	4	yes	—
64	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
65	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
66	C₁₆×C₅₃	—	2	2	no	—
67	#11719908	—	2	2	no	—
68	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
69	C₂²×D₅₃	#102	1	4	yes	—
70	C₂²×D₅₃	#107	1	2	yes	—
71	C₂²×D₅₃	D₁₆	1	2	yes	—
72	C₂²×D₅₃	SD₃₂	1	4	yes	—
73	C₂²×D₅₃	Q₃₂	1	2	yes	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	2	no	—
83	C₂×Dic₅₃	—	8	2	no	—
84	C₂×Dic₅₃	#102	1	4	yes	—
85	C₂×Dic₅₃	#107	1	2	yes	—
86	C₄×D₅₃	—	8	16	no	—
87	C₄×D₅₃	#95	1	4	yes	—
88	C₄×D₅₃	#103	1	8	yes	—
89	C₄×D₅₃	#106	1	4	yes	—
90	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
91	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	#88322	#95	1	4	yes	—
94	C₂×#1380	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	C₂×#102	1	2	yes	—
96	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
99	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
100	C₂×D₅₃	#340	1	4	yes	—
101	C₂×D₅₃	#357	1	8	yes	—
102	C₂×D₅₃	#358	1	8	yes	—
103	C₂×D₅₃	#366	1	4	yes	—
104	C₂×D₅₃	#367	1	4	yes	—
105	C₄×C₅₃	—	8	4	no	—
106	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#103	1	2	yes	—
108	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
109	Dic₅₃	—	8	4	no	—
110	Dic₅₃	#340	1	4	yes	—
111	C₂³×C₁₆	—	53	1	no	—
112	C₂²×#107	—	53	2	no	—
113	#1380	—	53	1	no	—
114	#1380	—	53	2	no	—
115	#1380	D₅₃	1	1	yes	characteristic, direct product factor
116	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#357	1	2	yes	—
118	C₂×C₅₃	C₂×#358	1	2	yes	—
119	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
121	D₅₃	#1380	1	8	yes	direct product factor
122	C₂³×C₈	—	53	1	no	—
123	C₂²×C₄⋊C₄	—	106	2	no	—
124	C₂²×C₁₆	—	53	1	no	—
125	C₂²×C₁₆	—	53	4	no	—
126	C₂²×C₁₆	—	106	4	no	—
127	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
128	C₂×#107	—	53	2	no	—
129	C₂×#107	—	53	8	no	—
130	C₂×#107	—	106	4	no	—
131	C₂×#107	—	106	2	no	—
132	C₂×#107	C₂×D₅₃	1	2	yes	—
133	C₅₃	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	53	1	no	—
135	C₂³×C₄	—	212	2	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	4	no	—
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	—	53	4	no	—
140	C₂²×C₈	—	53	2	no	—
141	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	106	8	no	—
144	C₂×C₄⋊C₄	—	106	2	no	—
145	C₂×C₄⋊C₄	—	212	4	no	—
146	C₂×C₄⋊C₄	—	212	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	53	2	no	—
149	C₂×C₁₆	—	106	8	no	—
150	C₂×C₁₆	—	106	2	no	—
151	C₂×C₁₆	C₄×D₅₃	1	2	yes	complemented
152	#107	—	2	2	no	—
153	#107	—	53	4	no	—
154	#107	—	106	8	no	—
155	#107	—	106	2	no	—
156	#107	C₄×D₅₃	1	4	yes	complemented
157	C₂⁴	—	53	1	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	4	no	—
161	C₂²×C₄	—	53	2	no	—
162	C₂²×C₄	—	53	4	no	—
163	C₂²×C₄	—	53	2	no	—
164	C₂²×C₄	—	424	4	no	—
165	C₂²×C₄	—	424	4	no	—
166	C₂²×C₄	—	212	8	no	—
167	C₂²×C₄	—	212	2	no	—
168	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	—	53	4	no	—
173	C₂×C₈	—	53	8	no	—
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
176	C₂×C₈	D₄×D₅₃	1	2	yes	—
177	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	106	4	no	—
181	C₄⋊C₄	—	212	8	no	—
182	C₄⋊C₄	—	212	2	no	—
183	C₁₆	—	2	2	no	—
184	C₁₆	—	106	2	no	—
185	C₂³	—	53	2	no	—
186	C₂³	—	53	4	no	—
187	C₂³	—	53	2	no	—
188	C₂³	—	53	4	no	—
189	C₂³	—	53	2	no	—
190	C₂³	D₈×D₅₃	1	1	yes	characteristic
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	8	no	—
197	C₂×C₄	—	53	4	no	—
198	C₂×C₄	—	53	4	no	—
199	C₂×C₄	—	424	16	no	—
200	C₂×C₄	—	424	2	no	—
201	C₂×C₄	—	424	2	no	—
202	C₂×C₄	—	212	4	no	—
203	C₂×C₄	C₂²⋊C₄×D₅₃	1	2	yes	—
204	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
205	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
206	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
207	C₈	—	53	4	no	—
208	C₈	C₄²×D₅₃	1	1	yes	characteristic
209	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
210	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
211	C₂²	—	53	4	no	—
212	C₂²	—	53	4	no	—
213	C₂²	—	53	8	no	—
214	C₂²	—	53	8	no	—
215	C₂²	—	53	4	no	—
216	C₂²	#102×D₅₃	1	2	yes	—
217	C₂²	#107×D₅₃	1	1	yes	characteristic
218	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
219	C₂²	SD₃₂×D₅₃	1	2	yes	—
220	C₂²	Q₃₂×D₅₃	1	1	yes	characteristic
221	C₄	—	8	4	no	—
222	C₄	—	53	4	no	—
223	C₄	—	424	4	no	—
224	C₄	#95×D₅₃	1	1	yes	characteristic
225	C₄	#103×D₅₃	1	2	yes	—
226	C₄	#106×D₅₃	1	1	yes	characteristic
227	C₂	—	53	8	no	—
228	C₂	#340×D₅₃	1	1	yes	characteristic
229	C₂	#357×D₅₃	1	2	yes	—
230	C₂	#358×D₅₃	1	2	yes	—
231	C₂	#366×D₅₃	1	1	yes	characteristic
232	C₂	#367×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1380	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1380	D₅₃
hypocenter	C₅₃	C₂×#1380
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
3. #20272807	C₂	semidirect
4. #20272806	C₂	semidirect
5. C₅₃×#1380	C₂	semidirect
10. C₂×C₁₆×D₅₃	C₄	semidirect
14. #107×D₅₃	C₄	semidirect
29. C₂×#11719908	C₂×C₄	semidirect
32. #13133151	C₂×C₄	semidirect
37. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
39. #107×C₅₃	C₂×C₄	semidirect
115. #1380	D₅₃	direct (N × H)
121. D₅₃	#1380	direct (N × H)
133. C₅₃	C₂×#1380	semidirect
151. C₂×C₁₆	C₄×D₅₃	semidirect
156. #107	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×#107×D₅₃	2
#20272807	2
#20272806	2
C₅₃×#1380	2
C₂×#1380	53

Subgroups — D53×#1380 #27298967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1380 #27298967