Subgroups — C₂²×#16810457 #27542198

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810457	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#13133123	C₂	1	2	yes	maximal, complemented
3	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810457	C₂²	1	16	yes	direct product factor
5	C₂×#13133123	C₂²	1	12	yes	complemented
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₄	1	6	yes	—
9	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	—
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	#13133123	C₂³	1	8	yes	complemented
12	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₈	1	6	yes	—
14	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	6	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
20	C₂×C₁₆×C₅₃	C₂³	1	3	yes	—
21	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
22	C₂⁴×C₅₃	C₁₆	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₁₆	1	6	yes	—
24	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	6	yes	—
29	C₂×C₈×C₅₃	C₂×C₈	1	3	yes	—
30	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
32	C₁₆×C₅₃	C₂²×C₄	1	3	yes	—
33	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
34	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₂³×C₅₃	—	2	4	no	—
36	C₂³×C₅₃	C₂×C₁₆	1	6	yes	—
37	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×C₁₆	1	12	yes	—
39	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
40	C₂×C₄×C₅₃	C₂×C₁₆	1	6	yes	—
41	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
42	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
43	C₈×C₅₃	C₂²×C₈	1	3	yes	—
44	C₈×C₅₃	C₂²×C₈	1	4	yes	—
45	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₂²×M₆₄	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	—	2	12	no	—
48	C₂²×C₅₃	M₆₄	1	4	yes	complemented
49	C₂²×C₅₃	C₂²×C₁₆	1	4	yes	—
50	C₂²×C₅₃	C₂²×C₁₆	1	3	yes	—
51	C₄×C₅₃	C₂²×C₁₆	1	3	yes	—
52	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
53	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
54	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
55	C₂×M₆₄	—	53	12	no	—
56	C₂²×C₃₂	—	53	2	no	—
57	C₂×C₅₃	—	2	4	no	—
58	C₂×C₅₃	C₂×M₆₄	1	6	yes	complemented
59	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
60	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
61	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
62	C₂²×C₁₆	Dic₅₃	1	6	yes	—
63	C₂²×C₁₆	C₂×D₅₃	1	6	yes	—
64	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
65	M₆₄	—	53	16	no	—
66	C₂×C₃₂	—	53	12	no	—
67	C₅₃	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₄	#88322	1	1	yes	characteristic
69	C₂²×C₈	#88322	1	1	yes	characteristic
70	C₂²×C₈	#88322	1	6	yes	—
71	C₂²×C₈	C₂×Dic₅₃	1	6	yes	—
72	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
73	C₂×C₁₆	C₂×Dic₅₃	1	6	yes	—
74	C₂×C₁₆	C₂×Dic₅₃	1	12	yes	—
75	C₂×C₁₆	C₂×Dic₅₃	1	3	yes	—
76	C₂×C₁₆	C₂²×D₅₃	1	3	yes	—
77	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
78	C₃₂	—	53	8	no	—
79	C₂⁴	#11719908	1	1	yes	characteristic
80	C₂²×C₄	#11719908	1	6	yes	—
81	C₂²×C₄	#11719908	1	1	yes	characteristic
82	C₂²×C₄	C₂×#88322	1	6	yes	—
83	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
84	C₂×C₈	C₂×#88322	1	12	yes	—
85	C₂×C₈	C₂×#88322	1	6	yes	—
86	C₂×C₈	C₂×#88322	1	3	yes	—
87	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
88	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
89	C₁₆	C₂²×Dic₅₃	1	4	yes	—
90	C₁₆	C₂²×Dic₅₃	1	3	yes	—
91	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
92	C₂³	—	2	4	no	—
93	C₂³	C₂×#11719908	1	6	yes	—
94	C₂³	C₂×#11719908	1	1	yes	characteristic
95	C₂×C₄	C₂×#11719908	1	12	yes	—
96	C₂×C₄	C₂×#11719908	1	3	yes	—
97	C₂×C₄	C₂×#11719908	1	6	yes	—
98	C₂×C₄	C₂²×#88322	1	4	yes	—
99	C₂×C₄	C₂²×#88322	1	3	yes	—
100	C₈	C₂²×#88322	1	4	yes	—
101	C₈	C₂²×#88322	1	3	yes	—
102	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
103	C₂²	—	2	12	no	—
104	C₂²	#16810457	1	4	yes	direct product factor
105	C₂²	C₂²×#11719908	1	4	yes	—
106	C₂²	C₂²×#11719908	1	3	yes	—
107	C₄	C₂²×#11719908	1	3	yes	—
108	C₄	C₂²×#11719908	1	4	yes	—
109	C₄	C₂³×#88322	1	1	yes	characteristic
110	C₂	—	2	4	no	—
111	C₂	C₂×#16810457	1	6	yes	direct product factor
112	C₂	C₂³×#11719908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₁₆
prime radical	C₂⁴×C₅₃	C₁₆
cosocle	C₁₆×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16810457	C₂	direct (N × H)
2. C₂²×#13133123	C₂	semidirect
4. #16810457	C₂²	direct (N × H)
5. C₂×#13133123	C₂²	semidirect
11. #13133123	C₂³	semidirect
48. C₂²×C₅₃	M₆₄	semidirect
58. C₂×C₅₃	C₂×M₆₄	semidirect
67. C₅₃	C₂²×M₆₄	semidirect
104. C₂²	#16810457	direct (N × H)
111. C₂	C₂×#16810457	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810457	2
C₂²×#13133123	2
C₂³×C₁₆×C₅₃	2
C₂²×M₆₄	53

Subgroups — C22×#16810457 #27542198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810457 #27542198