Subgroups — #27305146

Order 13568.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277024	C₂	1	2	yes	maximal, complemented
2	Dic₁₆₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272931	C₂	1	2	yes	maximal, complemented
4	D₃₃₉₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811055	—	2	2	no	—
7	Dic₈₄₈	C₂²	1	2	yes	—
8	#16810462	—	2	2	no	—
9	D₁₆₉₆	C₂²	1	2	yes	—
10	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
12	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
13	#13133239	—	4	2	no	—
14	Dic₄₂₄	—	2	2	no	—
15	#13133162	—	4	2	no	—
16	D₈₄₈	—	2	2	no	—
17	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
20	#11719945	—	8	2	no	—
21	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
22	Dic₂₁₂	—	4	2	no	—
23	D₄₂₄	—	4	2	no	—
24	#11719913	—	8	2	no	—
25	C₁₆×C₅₃	D₈	1	1	yes	characteristic
26	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₅₃	—	16	2	no	—
29	D₂₁₂	—	8	2	no	—
30	C₄×D₅₃	—	16	2	no	—
31	Dic₁₀₆	—	8	2	no	—
32	C₈×C₅₃	D₁₆	1	1	yes	characteristic
33	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
34	#13741	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
36	C₂×D₅₃	—	16	2	no	—
37	C₄×C₅₃	D₃₂	1	1	yes	characteristic
38	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
39	Dic₅₃	—	16	2	no	—
40	#2262	—	53	2	no	—
41	Q₁₂₈	—	53	1	no	—
42	SD₁₂₈	—	53	2	no	—
43	D₆₄	—	53	1	no	—
44	C₂×C₆₄	D₅₃	1	1	yes	characteristic, complemented
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
47	D₅₃	—	32	2	no	—
48	#508	—	106	2	no	—
49	Q₆₄	—	53	2	no	—
50	SD₆₄	—	106	2	no	—
51	D₃₂	—	53	2	no	—
52	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
53	C₆₄	C₂×D₅₃	1	1	yes	characteristic
54	C₆₄	C₂×D₅₃	1	1	yes	characteristic
55	C₅₃	#13741	1	1	yes	characteristic, complemented, Sylow
56	#135	—	212	2	no	—
57	Q₃₂	—	106	2	no	—
58	SD₃₂	—	212	2	no	—
59	D₁₆	—	106	2	no	—
60	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
61	C₃₂	D₂₁₂	1	1	yes	characteristic
62	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
63	#41	—	424	2	no	—
64	Q₁₆	—	212	2	no	—
65	SD₁₆	—	424	2	no	—
66	D₈	—	212	2	no	—
67	C₂×C₈	D₄₂₄	1	1	yes	characteristic
68	C₁₆	D₄₂₄	1	1	yes	characteristic
69	C₁₆	C₂×D₂₁₂	1	1	yes	characteristic
70	Q₈	—	424	2	no	—
71	D₄	—	848	2	no	—
72	D₄	—	424	2	no	—
73	C₂×C₄	—	848	2	no	—
74	C₂×C₄	D₈₄₈	1	1	yes	characteristic
75	C₈	D₈₄₈	1	1	yes	characteristic
76	C₈	C₂×D₄₂₄	1	1	yes	characteristic
77	C₂²	—	848	2	no	—
78	C₂²	D₁₆₉₆	1	1	yes	characteristic
79	C₄	—	848	2	no	—
80	C₄	D₁₆₉₆	1	1	yes	characteristic
81	C₄	C₂×D₈₄₈	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	1696	2	no	—
84	C₂	C₂×D₁₆₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆₉₆
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂×C₅₃	C₂×D₃₂
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	#13741
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ 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
1. #20277024	C₂	semidirect
2. Dic₁₆₉₆	C₂	semidirect
3. #20272931	C₂	semidirect
4. D₃₃₉₂	C₂	semidirect
5. C₂×C₅₃×C₆₄	C₂	semidirect
44. C₂×C₆₄	D₅₃	semidirect
55. C₅₃	#13741	semidirect

Maximal subgroups

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

subgroup	index
#20277024	2
Dic₁₆₉₆	2
#20272931	2
D₃₃₉₂	2
C₂×C₅₃×C₆₄	2
#13741	53