Subgroups — D₅₃×#1454 #27349714

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×D₅₃	C₂	1	3	yes	maximal
2	#20273016	C₂	1	3	yes	maximal, complemented
3	#20273015	C₂	1	3	yes	maximal, complemented
4	#20273014	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×#1454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	#16810676	C₂²	1	3	yes	—
10	#16810675	C₄	1	6	yes	—
11	#16810674	C₂²	1	3	yes	—
12	#16810674	C₂²	1	3	yes	—
13	#16810673	C₄	1	6	yes	—
14	C₄⋊C₈×D₅₃	C₄	1	6	yes	—
15	C₂²⋊C₈×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	3	yes	complemented
19	#16810345	C₂²	1	3	yes	complemented
20	#16810344	C₂²	1	3	yes	complemented
21	C₈×#88322	C₂²	1	3	yes	complemented
22	C₅₃×#431	C₂²	1	3	yes	—
23	C₅₃×#322	C₂²	1	3	yes	complemented
24	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
27	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
29	C₄²×D₅₃	C₂×C₄	1	3	yes	—
30	C₄²×D₅₃	C₂×C₄	1	3	yes	—
31	C₄²×D₅₃	C₂³	1	1	yes	characteristic
32	C₄²×D₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊#88322	C₂×C₄	1	6	yes	—
34	#13133146	C₂×C₄	1	6	yes	—
35	#13133146	C₂×C₄	1	6	yes	—
36	C₂²⋊#88322	C₂×C₄	1	6	yes	—
37	#13133129	C₂×C₄	1	6	yes	—
38	#13133129	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×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
45	C₂²⋊C₈×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	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
58	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
59	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₄×Dic₅₃	C₄²	1	6	yes	—
61	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
62	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
63	C₂×#88322	C₂²×C₄	1	6	yes	—
64	C₂×#88322	C₂²×C₄	1	6	yes	—
65	C₈×D₅₃	—	2	12	no	—
66	C₈×D₅₃	—	2	12	no	—
67	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂²×D₅₃	—	2	6	no	—
69	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	6	no	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
73	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
74	C₂×Dic₅₃	—	2	6	no	—
75	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
76	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
77	C₄×D₅₃	—	2	6	no	—
78	C₄×D₅₃	—	2	6	no	—
79	C₄×D₅₃	—	2	6	no	—
80	C₄×D₅₃	#131	1	12	yes	—
81	C₈×C₅₃	—	2	6	no	—
82	C₈×C₅₃	—	2	6	no	—
83	#88322	—	2	6	no	—
84	#88322	—	2	6	no	—
85	C₂×#1454	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	3	no	—
87	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
88	C₂×D₅₃	—	2	3	no	—
89	C₂×D₅₃	—	4	4	no	—
90	C₂×D₅₃	#405	1	6	yes	—
91	C₄×C₅₃	—	2	3	no	—
92	C₄×C₅₃	C₂×#131	1	6	yes	—
93	Dic₅₃	—	2	3	no	—
94	Dic₅₃	#405	1	6	yes	—
95	C₂×#431	—	53	3	no	—
96	#1454	—	53	3	no	—
97	#1454	—	53	3	no	—
98	#1454	—	53	1	no	—
99	#1454	D₅₃	1	1	yes	characteristic, direct product factor
100	C₂×#322	—	53	3	no	—
101	C₂×C₈²	—	53	1	no	—
102	C₂×C₅₃	—	4	1	no	—
103	C₂×C₅₃	C₂×#405	1	3	yes	—
104	D₅₃	—	4	1	no	—
105	D₅₃	#1454	1	4	yes	direct product factor
106	C₂²×C₄²	—	53	1	no	—
107	#431	—	53	3	no	—
108	#431	—	53	6	no	—
109	#431	—	53	6	no	—
110	#431	—	53	3	no	—
111	#431	—	53	3	no	—
112	#431	C₂×D₅₃	1	3	yes	—
113	C₂×C₄⋊C₈	—	53	6	no	—
114	C₂×C₂²⋊C₈	—	53	6	no	—
115	C₂×C₄×C₈	—	53	3	no	—
116	C₂×C₄×C₈	—	53	3	no	—
117	#322	—	53	3	no	—
118	#322	—	53	3	no	—
119	#322	—	53	3	no	—
120	#322	C₂×D₅₃	1	3	yes	complemented
121	C₈²	—	53	3	no	—
122	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
123	C₅₃	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₄	—	53	3	no	—
125	C₂²×C₈	—	53	6	no	—
126	C₂²×C₈	—	53	6	no	—
127	C₂×C₄²	—	53	3	no	—
128	C₂×C₄²	—	53	3	no	—
129	C₂×C₄²	—	53	3	no	—
130	C₂×C₄²	—	53	1	no	—
131	C₂×C₄²	—	53	1	no	—
132	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
133	C₄⋊C₈	—	53	6	no	—
134	C₄⋊C₈	—	53	6	no	—
135	C₄⋊C₈	—	53	6	no	—
136	C₄⋊C₈	C₄×D₅₃	1	6	yes	—
137	C₂²⋊C₈	—	53	6	no	—
138	C₂²⋊C₈	—	53	6	no	—
139	C₂²⋊C₈	—	53	6	no	—
140	C₂²⋊C₈	C₄×D₅₃	1	6	yes	—
141	C₄×C₈	—	53	6	no	—
142	C₄×C₈	—	53	6	no	—
143	C₄×C₈	—	53	3	no	—
144	C₄×C₈	—	53	3	no	—
145	C₄×C₈	C₂²×D₅₃	1	3	yes	—
146	C₄×C₈	C₂²×D₅₃	1	3	yes	—
147	C₂⁴	—	53	1	no	—
148	C₂²×C₄	—	53	3	no	—
149	C₂²×C₄	—	53	3	no	—
150	C₂²×C₄	—	53	3	no	—
151	C₂²×C₄	—	53	3	no	—
152	C₂²×C₄	—	53	3	no	—
153	C₂²×C₄	—	106	12	no	—
154	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
155	C₂×C₈	—	53	6	no	—
156	C₂×C₈	—	53	6	no	—
157	C₂×C₈	—	106	12	no	—
158	C₂×C₈	—	106	12	no	—
159	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
160	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
161	C₄²	—	53	6	no	—
162	C₄²	—	53	3	no	—
163	C₄²	—	53	3	no	—
164	C₄²	C₂×C₄×D₅₃	1	3	yes	—
165	C₄²	C₂³×D₅₃	1	1	yes	characteristic
166	C₂³	—	53	1	no	—
167	C₂³	—	53	1	no	—
168	C₂³	—	106	6	no	—
169	C₂³	C₄²×D₅₃	1	1	yes	characteristic
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	53	3	no	—
172	C₂×C₄	—	53	12	no	—
173	C₂×C₄	—	53	3	no	—
174	C₂×C₄	—	106	6	no	—
175	C₂×C₄	—	106	6	no	—
176	C₂×C₄	—	106	6	no	—
177	C₂×C₄	—	106	6	no	—
178	C₂×C₄	C₄²×D₅₃	1	3	yes	—
179	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
180	C₈	—	2	6	no	—
181	C₈	—	2	6	no	—
182	C₈	—	106	6	no	—
183	C₈	—	106	6	no	—
184	C₂²	—	2	3	no	—
185	C₂²	—	53	6	no	—
186	C₂²	—	106	3	no	—
187	C₂²	—	212	4	no	—
188	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
189	C₄	—	2	3	no	—
190	C₄	—	53	6	no	—
191	C₄	—	106	3	no	—
192	C₄	#131×D₅₃	1	6	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	53	4	no	—
195	C₂	—	212	1	no	—
196	C₂	#405×D₅₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1454	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1454	D₅₃
hypocenter	C₅₃	C₂×#1454
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1454

Complements

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

N	H ≅ G/N	type
2. #20273016	C₂	semidirect
3. #20273015	C₂	semidirect
4. #20273014	C₂	semidirect
5. #322×D₅₃	C₂	semidirect
6. C₈²×D₅₃	C₂	semidirect
7. C₅₃×#1454	C₂	semidirect
18. #16810345	C₂²	semidirect
19. #16810345	C₂²	semidirect
20. #16810344	C₂²	semidirect
21. C₈×#88322	C₂²	semidirect
23. C₅₃×#322	C₂²	semidirect
24. C₈²×C₅₃	C₂²	semidirect
99. #1454	D₅₃	direct (N × H)
105. D₅₃	#1454	direct (N × H)
120. #322	C₂×D₅₃	semidirect
122. C₈²	C₂×D₅₃	semidirect
123. C₅₃	C₂×#1454	semidirect

Maximal subgroups

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

subgroup	index
#431×D₅₃	2
#20273016	2
#20273015	2
#20273014	2
#322×D₅₃	2
C₈²×D₅₃	2
C₅₃×#1454	2
C₂×#1454	53

Subgroups — D53×#1454 #27349714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1454 #27349714