Subgroups — D₅₃×#1455 #27349796

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₁₆×D₅₃	C₂	1	3	yes	maximal
2	#20273019	C₂	1	3	yes	maximal, complemented
3	#20273018	C₂	1	3	yes	maximal, complemented
4	#20273017	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×D₅₃	C₂	1	3	yes	maximal, complemented
6	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	C₄	1	6	yes	—
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	C₄×#11719917	C₂²	1	3	yes	—
11	#16810575	C₄	1	6	yes	—
12	M₁₆×Dic₅₃	C₄	1	6	yes	—
13	C₄×#11719912	C₂²	1	3	yes	—
14	C₄×#11719912	C₂²	1	3	yes	—
15	#97×D₅₃	C₄	1	6	yes	—
16	C₄×C₈×D₅₃	C₂²	1	3	yes	—
17	C₄×C₈×D₅₃	C₂²	1	3	yes	—
18	#16810345	C₂²	1	6	yes	complemented
19	#16810344	C₂²	1	3	yes	complemented
20	C₈×#88322	C₂²	1	3	yes	complemented
21	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
22	C₅₃×#322	C₂²	1	3	yes	complemented
23	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
25	C₂×#11719917	C₂×C₄	1	6	yes	—
26	C₂×#11719912	C₂×C₄	1	6	yes	—
27	C₂×#11719912	C₂×C₄	1	6	yes	—
28	M₁₆×D₅₃	—	2	24	no	—
29	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
32	C₄²×D₅₃	C₂×C₄	1	3	yes	—
33	C₄²×D₅₃	C₂×C₄	1	3	yes	—
34	C₄²×D₅₃	C₂³	1	1	yes	characteristic
35	C₄²×D₅₃	C₂³	1	1	yes	characteristic
36	#13133128	C₂×C₄	1	6	yes	—
37	#13133127	C₂×C₄	1	6	yes	—
38	#13133127	C₂×C₄	1	6	yes	—
39	C₄×#88322	C₂³	1	3	yes	—
40	C₄×#88322	C₂³	1	3	yes	—
41	C₈×Dic₅₃	C₂×C₄	1	6	yes	—
42	C₈×Dic₅₃	C₂×C₄	1	6	yes	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	6	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#97×C₅₃	C₂×C₄	1	6	yes	—
46	C₄×C₈×C₅₃	C₂³	1	3	yes	—
47	C₄×C₈×C₅₃	C₂³	1	3	yes	—
48	C₂³×D₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₅₃	C₄²	1	3	yes	—
51	C₂×C₄×D₅₃	—	2	12	no	—
52	C₂×C₄×D₅₃	C₄²	1	3	yes	—
53	C₂×C₄×D₅₃	C₄²	1	3	yes	—
54	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
55	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
56	M₁₆×C₅₃	—	2	12	no	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
60	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₄×Dic₅₃	C₄²	1	6	yes	—
62	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
63	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
64	#11719917	—	2	12	no	—
65	C₂×#88322	C₂²×C₄	1	6	yes	—
66	C₂×#88322	C₂²×C₄	1	6	yes	—
67	#11719912	—	2	12	no	—
68	#11719912	—	2	12	no	—
69	C₈×D₅₃	—	2	12	no	—
70	C₈×D₅₃	M₁₆	1	24	yes	complemented
71	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂²×D₅₃	—	2	6	no	—
73	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	6	no	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
78	C₂×Dic₅₃	—	2	6	no	—
79	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
80	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
81	C₄×D₅₃	—	2	6	no	—
82	C₄×D₅₃	—	2	6	no	—
83	C₄×D₅₃	—	2	6	no	—
84	C₄×D₅₃	C₂×M₁₆	1	12	yes	—
85	C₈×C₅₃	—	2	6	no	—
86	C₈×C₅₃	C₂×M₁₆	1	12	yes	complemented
87	#88322	—	2	6	no	—
88	#88322	C₂×M₁₆	1	12	yes	complemented
89	C₂×#1455	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	3	no	—
91	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
92	C₂×D₅₃	—	2	3	no	—
93	C₂×D₅₃	—	4	4	no	—
94	C₂×D₅₃	C₄×M₁₆	1	6	yes	—
95	C₄×C₅₃	—	2	3	no	—
96	C₄×C₅₃	C₂²×M₁₆	1	6	yes	—
97	Dic₅₃	—	2	3	no	—
98	Dic₅₃	C₄×M₁₆	1	6	yes	—
99	C₂×C₄×M₁₆	—	53	3	no	—
100	#1455	—	53	3	no	—
101	#1455	—	53	3	no	—
102	#1455	—	53	1	no	—
103	#1455	D₅₃	1	1	yes	characteristic, direct product factor
104	C₂×#322	—	53	3	no	—
105	C₂×C₈²	—	53	1	no	—
106	C₂×C₅₃	—	4	1	no	—
107	C₂×C₅₃	C₂×C₄×M₁₆	1	3	yes	—
108	D₅₃	—	4	1	no	—
109	D₅₃	#1455	1	4	yes	direct product factor
110	C₂²×M₁₆	—	53	6	no	—
111	C₂²×C₄²	—	53	1	no	—
112	C₄×M₁₆	—	53	6	no	—
113	C₄×M₁₆	—	53	3	no	—
114	C₄×M₁₆	—	53	6	no	—
115	C₄×M₁₆	—	53	3	no	—
116	C₄×M₁₆	—	53	3	no	—
117	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
118	C₂×#97	—	53	6	no	—
119	C₂×C₄×C₈	—	53	3	no	—
120	C₂×C₄×C₈	—	53	3	no	—
121	#322	—	53	3	no	—
122	#322	—	53	6	no	—
123	#322	C₂×D₅₃	1	3	yes	complemented
124	C₈²	—	53	3	no	—
125	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
126	C₅₃	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	—	53	3	no	—
128	C₂×M₁₆	—	53	6	no	—
129	C₂×M₁₆	—	53	6	no	—
130	C₂×M₁₆	—	53	6	no	—
131	C₂×M₁₆	—	106	24	no	—
132	C₂×M₁₆	C₄×D₅₃	1	6	yes	—
133	C₂²×C₈	—	53	6	no	—
134	C₂²×C₈	—	53	6	no	—
135	C₂×C₄²	—	53	3	no	—
136	C₂×C₄²	—	53	3	no	—
137	C₂×C₄²	—	53	3	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	#97	—	53	6	no	—
142	#97	—	53	6	no	—
143	#97	—	53	6	no	—
144	#97	C₄×D₅₃	1	6	yes	—
145	C₄×C₈	—	53	6	no	—
146	C₄×C₈	—	53	6	no	—
147	C₄×C₈	—	53	3	no	—
148	C₄×C₈	—	53	3	no	—
149	C₄×C₈	C₂²×D₅₃	1	3	yes	—
150	C₄×C₈	C₂²×D₅₃	1	3	yes	—
151	C₂⁴	—	53	1	no	—
152	C₂²×C₄	—	53	3	no	—
153	C₂²×C₄	—	53	3	no	—
154	C₂²×C₄	—	53	3	no	—
155	C₂²×C₄	—	53	3	no	—
156	C₂²×C₄	—	53	3	no	—
157	C₂²×C₄	—	106	12	no	—
158	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
159	M₁₆	—	2	12	no	—
160	M₁₆	—	106	12	no	—
161	M₁₆	—	106	12	no	—
162	M₁₆	—	106	12	no	—
163	C₂×C₈	—	53	6	no	—
164	C₂×C₈	—	53	24	no	—
165	C₂×C₈	—	53	6	no	—
166	C₂×C₈	—	106	12	no	—
167	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
168	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
169	C₄²	—	53	6	no	—
170	C₄²	—	53	3	no	—
171	C₄²	—	53	3	no	—
172	C₄²	C₂×C₄×D₅₃	1	3	yes	—
173	C₄²	C₂³×D₅₃	1	1	yes	characteristic
174	C₂³	—	53	1	no	—
175	C₂³	—	53	1	no	—
176	C₂³	—	106	6	no	—
177	C₂³	C₄²×D₅₃	1	1	yes	characteristic
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	53	3	no	—
180	C₂×C₄	—	53	12	no	—
181	C₂×C₄	—	53	3	no	—
182	C₂×C₄	—	106	6	no	—
183	C₂×C₄	—	106	6	no	—
184	C₂×C₄	—	106	6	no	—
185	C₂×C₄	—	106	6	no	—
186	C₂×C₄	C₄²×D₅₃	1	3	yes	—
187	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
188	C₈	—	2	6	no	—
189	C₈	—	53	12	no	—
190	C₈	—	106	6	no	—
191	C₈	M₁₆×D₅₃	1	12	yes	complemented
192	C₂²	—	2	3	no	—
193	C₂²	—	53	6	no	—
194	C₂²	—	106	3	no	—
195	C₂²	—	212	4	no	—
196	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	53	6	no	—
199	C₄	—	106	3	no	—
200	C₄	C₂×M₁₆×D₅₃	1	6	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	53	4	no	—
203	C₂	—	212	1	no	—
204	C₂	C₄×M₁₆×D₅₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1455	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1455	D₅₃
hypocenter	C₅₃	C₂×#1455
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
2. #20273019	C₂	semidirect
3. #20273018	C₂	semidirect
4. #20273017	C₂	semidirect
5. #322×D₅₃	C₂	semidirect
6. C₈²×D₅₃	C₂	semidirect
7. C₅₃×#1455	C₂	semidirect
18. #16810345	C₂²	semidirect
19. #16810344	C₂²	semidirect
20. C₈×#88322	C₂²	semidirect
22. C₅₃×#322	C₂²	semidirect
23. C₈²×C₅₃	C₂²	semidirect
70. C₈×D₅₃	M₁₆	semidirect
86. C₈×C₅₃	C₂×M₁₆	semidirect
88. #88322	C₂×M₁₆	semidirect
103. #1455	D₅₃	direct (N × H)
109. D₅₃	#1455	direct (N × H)
123. #322	C₂×D₅₃	semidirect
125. C₈²	C₂×D₅₃	semidirect
126. C₅₃	C₂×#1455	semidirect
191. C₈	M₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₁₆×D₅₃	2
#20273019	2
#20273018	2
#20273017	2
#322×D₅₃	2
C₈²×D₅₃	2
C₅₃×#1455	2
C₂×#1455	53

Subgroups — D53×#1455 #27349796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1455 #27349796