Subgroups — #376×Dic₅₃ #27305163

Order 13568.

2302 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810469	C₂	1	3	yes	maximal
2	C₂×C₄²×Dic₅₃	C₂	1	3	yes	maximal
3	#95×Dic₅₃	C₂	1	8	yes	maximal
4	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
6	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
8	#16810469	C₄	1	24	yes	complemented
9	C₂×#13133124	C₂²	1	12	yes	—
10	C₄²×Dic₅₃	C₄	1	24	yes	complemented
11	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
12	C₅₃×#376	C₄	1	8	yes	complemented
13	C₂×#95×C₅₃	C₂²	1	4	yes	—
14	C₂³×Dic₅₃	C₂³	1	6	yes	—
15	C₂³×Dic₅₃	C₂³	1	2	yes	—
16	C₂×C₄×Dic₅₃	—	2	48	no	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	24	yes	—
21	#13133124	C₂×C₄	1	48	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
27	#95×C₅₃	C₂×C₄	1	16	yes	—
28	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	—	2	24	no	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂²×Dic₅₃	—	2	24	no	—
37	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
38	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
39	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
40	C₂²×Dic₅₃	#41	1	16	yes	—
41	C₄²×C₅₃	C₄²	1	48	yes	complemented
42	C₄×Dic₅₃	—	2	96	no	—
43	C₄×Dic₅₃	C₄²	1	16	yes	—
44	C₄×Dic₅₃	C₄²	1	48	yes	complemented
45	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
46	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
47	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
48	C₂³×C₅₃	C₂×#41	1	8	yes	—
49	C₂×C₄×C₅₃	—	2	24	no	—
50	C₂×C₄×C₅₃	—	2	24	no	—
51	C₂×C₄×C₅₃	—	2	24	no	—
52	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
54	C₂×C₄×C₅₃	C₂×C₄²	1	24	yes	—
55	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
56	C₂×Dic₅₃	—	2	24	no	—
57	C₂×Dic₅₃	—	2	48	no	—
58	C₂×Dic₅₃	C₂×C₄²	1	24	yes	—
59	C₂×Dic₅₃	C₂×C₄²	1	8	yes	—
60	C₂×Dic₅₃	#117	1	48	yes	—
61	C₄×#376	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
63	C₂²×C₅₃	C₂²×C₄²	1	3	yes	—
64	C₂²×C₅₃	C₂²×C₄²	1	3	yes	—
65	C₂²×C₅₃	C₂×#117	1	12	yes	—
66	C₂²×C₅₃	C₄×#41	1	12	yes	—
67	C₂²×C₅₃	C₄×#41	1	4	yes	—
68	C₄×C₅₃	—	2	24	no	—
69	C₄×C₅₃	C₄³	1	8	yes	—
70	Dic₅₃	—	2	24	no	—
71	Dic₅₃	#376	1	16	yes	direct product factor
72	C₂×#376	—	53	3	no	—
73	C₂×#376	D₅₃	1	1	yes	characteristic
74	C₂×C₄³	—	53	3	no	—
75	C₄×#95	—	53	8	no	—
76	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
77	C₂×C₅₃	C₂×#376	1	2	yes	—
78	C₂×C₅₃	C₄×#117	1	6	yes	—
79	C₂×C₅₃	C₄×#117	1	6	yes	—
80	C₂²×C₄²	—	53	1	no	—
81	C₂²×C₄²	—	53	3	no	—
82	C₂²×C₄²	—	53	12	no	—
83	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
84	#376	—	53	24	no	—
85	#376	Dic₅₃	1	8	yes	direct product factor
86	C₂×#95	—	53	12	no	—
87	C₂×#95	C₂×D₅₃	1	4	yes	—
88	C₄³	—	53	24	no	—
89	C₅₃	C₄×#376	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	53	2	no	—
91	C₂³×C₄	—	53	6	no	—
92	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
93	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	C₂×C₄²	—	53	12	no	—
95	C₂×C₄²	—	53	12	no	—
96	C₂×C₄²	—	53	24	no	—
97	C₂×C₄²	—	53	48	no	—
98	C₂×C₄²	—	106	48	no	—
99	C₂×C₄²	C₄×D₅₃	1	12	yes	—
100	C₂×C₄²	C₄×D₅₃	1	12	yes	—
101	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
102	#95	—	53	48	no	—
103	#95	C₂×Dic₅₃	1	16	yes	—
104	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	24	no	—
106	C₂²×C₄	—	53	12	no	—
107	C₂²×C₄	—	53	16	no	—
108	C₂²×C₄	—	53	24	no	—
109	C₂²×C₄	—	53	12	no	—
110	C₂²×C₄	—	106	24	no	—
111	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
112	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
113	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
114	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
115	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
116	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
117	C₄²	—	53	16	no	—
118	C₄²	—	53	48	no	—
119	C₄²	—	106	96	no	—
120	C₄²	C₄×Dic₅₃	1	48	yes	complemented
121	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
122	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
123	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
124	C₂³	#41×D₅₃	1	8	yes	—
125	C₂×C₄	—	2	24	no	—
126	C₂×C₄	—	2	24	no	—
127	C₂×C₄	—	2	24	no	—
128	C₂×C₄	—	53	8	no	—
129	C₂×C₄	—	53	48	no	—
130	C₂×C₄	—	53	24	no	—
131	C₂×C₄	—	106	24	no	—
132	C₂×C₄	—	106	48	no	—
133	C₂×C₄	C₄²×D₅₃	1	4	yes	—
134	C₂×C₄	C₄²×D₅₃	1	12	yes	—
135	C₂×C₄	C₂×C₄×Dic₅₃	1	24	yes	—
136	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
137	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
138	C₂²	C₂×C₄²×D₅₃	1	3	yes	—
139	C₂²	C₂²×C₄×Dic₅₃	1	3	yes	—
140	C₂²	#117×D₅₃	1	12	yes	—
141	C₂²	#16811091	1	12	yes	—
142	C₂²	#41×Dic₅₃	1	4	yes	—
143	C₄	—	2	24	no	—
144	C₄	—	53	16	no	—
145	C₄	—	106	24	no	—
146	C₄	C₄²×Dic₅₃	1	8	yes	—
147	C₂	C₂×C₄²×Dic₅₃	1	1	yes	characteristic
148	C₂	#376×D₅₃	1	2	yes	—
149	C₂	#20277054	1	6	yes	—
150	C₂	#117×Dic₅₃	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄³
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#376	D₅₃
hypocenter	C₅₃	C₄×#376
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
8. #16810469	C₄	semidirect
10. C₄²×Dic₅₃	C₄	semidirect
12. C₅₃×#376	C₄	semidirect
41. C₄²×C₅₃	C₄²	semidirect
44. C₄×Dic₅₃	C₄²	semidirect
71. Dic₅₃	#376	direct (N × H)
85. #376	Dic₅₃	direct (N × H)
89. C₅₃	C₄×#376	semidirect
120. C₄²	C₄×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810469	2
C₂×C₄²×Dic₅₃	2
#95×Dic₅₃	2
C₂×C₅₃×#376	2
C₄×#376	53

Subgroups — #376×Dic53 #27305163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #376×Dic₅₃ #27305163