Subgroups — #27271971

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₉₆	C₂	1	1	yes	maximal, characteristic
2	#20272915	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₄₈	—	2	1	no	—
5	D₁₆₉₆	—	2	1	no	—
6	D₁₆₉₆	C₄	1	2	yes	complemented
7	#16810449	—	2	1	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×C₆₄	—	2	1	no	—
10	C₂×D₄₂₄	—	4	1	no	—
11	D₈₄₈	—	2	2	no	—
12	D₈₄₈	—	4	1	no	—
13	#13133151	—	4	1	no	—
14	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
15	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	D₄	1	1	yes	characteristic
17	C₂×D₂₁₂	—	8	1	no	—
18	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
19	C₄⋊Dic₅₃	—	8	1	no	—
20	D₄₂₄	—	4	2	no	—
21	D₄₂₄	—	8	1	no	—
22	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₂²×D₅₃	—	16	1	no	—
25	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
26	C₂×Dic₅₃	—	16	1	no	—
27	D₂₁₂	—	8	2	no	—
28	D₂₁₂	—	16	1	no	—
29	C₈×C₅₃	#102	1	1	yes	characteristic
30	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
31	#7537	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
33	C₂×D₅₃	—	16	2	no	—
34	C₂×D₅₃	—	32	1	no	—
35	C₂×D₅₃	—	32	1	no	—
36	C₄×C₅₃	#357	1	1	yes	characteristic
37	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
38	Dic₅₃	—	32	1	no	—
39	C₂×D₃₂	—	53	1	no	—
40	C₂×C₆₄	D₅₃	1	1	yes	characteristic, complemented
41	#1423	—	53	1	no	—
42	C₂×C₅₃	#1415	1	1	yes	characteristic
43	C₂×C₅₃	D₆₄	1	1	yes	characteristic
44	C₂×C₅₃	SD₁₂₈	1	1	yes	characteristic
45	D₅₃	—	32	2	no	—
46	C₂×D₁₆	—	106	1	no	—
47	D₃₂	—	53	2	no	—
48	D₃₂	—	106	1	no	—
49	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
50	#366	—	106	1	no	—
51	C₆₄	—	2	1	no	—
52	C₅₃	#7537	1	1	yes	characteristic, complemented, Sylow
53	C₂×D₈	—	212	1	no	—
54	D₁₆	—	106	2	no	—
55	D₁₆	—	212	1	no	—
56	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
57	#107	—	212	1	no	—
58	C₃₂	C₄×D₅₃	1	1	yes	characteristic
59	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
60	C₂×D₄	—	424	1	no	—
61	D₈	—	424	1	no	—
62	D₈	—	212	2	no	—
63	C₂×C₈	D₄₂₄	1	1	yes	characteristic
64	C₄⋊C₄	—	424	1	no	—
65	C₁₆	#11719913	1	1	yes	characteristic
66	C₁₆	#11719921	1	1	yes	characteristic
67	C₂³	—	848	1	no	—
68	D₄	—	848	1	no	—
69	D₄	—	424	2	no	—
70	C₂×C₄	—	848	1	no	—
71	C₂×C₄	D₈₄₈	1	1	yes	characteristic
72	C₈	#13133139	1	1	yes	characteristic
73	C₈	#13133162	1	1	yes	characteristic
74	C₂²	—	848	2	no	—
75	C₂²	—	1696	1	no	—
76	C₂²	—	1696	1	no	—
77	C₂²	D₁₆₉₆	1	1	yes	characteristic
78	C₄	—	1696	1	no	—
79	C₄	#16810425	1	1	yes	characteristic
80	C₄	#16810462	1	1	yes	characteristic
81	C₂	—	1696	2	no	—
82	C₂	#20272894	1	1	yes	characteristic
83	C₂	D₃₃₉₂	1	1	yes	characteristic
84	C₂	#20272931	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆₉₆
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂²×C₅₃	D₃₂
prime radical	C₂×D₁₆₉₆	C₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	#7537
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #20272915	C₂	semidirect
3. C₂×C₅₃×C₆₄	C₂	semidirect
6. D₁₆₉₆	C₄	semidirect
40. C₂×C₆₄	D₅₃	semidirect
52. C₅₃	#7537	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆₉₆	2
#20272915	2
C₂×C₅₃×C₆₄	2
#7537	53