Subgroups — #27305147

Order 13568.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277025	C₂	1	2	yes	maximal, complemented
2	#20272929	C₂	1	2	yes	maximal, complemented
3	D₃₃₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811056	—	2	2	no	—
7	#16810460	—	2	2	no	—
8	D₁₆₉₆	C₂²	1	2	yes	—
9	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
10	#16810342	C₂²	1	1	yes	characteristic
11	C₅₃×Q₆₄	C₂²	1	2	yes	—
12	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
13	#13133240	—	4	2	no	—
14	#13133160	—	4	2	no	—
15	D₈₄₈	—	2	2	no	—
16	C₁₆×D₅₃	D₄	1	1	yes	characteristic
17	#13133123	D₄	1	1	yes	characteristic
18	Q₃₂×C₅₃	—	2	2	no	—
19	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
20	#11719949	—	8	2	no	—
21	Q₁₆×C₅₃	—	4	2	no	—
22	#11719924	—	8	2	no	—
23	D₄₂₄	—	4	2	no	—
24	C₈×D₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
26	#11719908	D₈	1	1	yes	characteristic
27	Q₈×C₅₃	—	8	2	no	—
28	D₂₁₂	—	8	2	no	—
29	D₂₁₂	—	16	2	no	—
30	C₄×D₅₃	—	16	2	no	—
31	C₄×D₅₃	D₁₆	1	1	yes	characteristic
32	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
33	#88322	D₁₆	1	1	yes	characteristic
34	#13741	—	53	1	no	maximal, Sylow
35	C₂×D₅₃	—	16	2	no	—
36	C₂×D₅₃	D₃₂	1	1	yes	characteristic
37	C₄×C₅₃	—	16	2	no	—
38	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
39	Dic₅₃	D₃₂	1	1	yes	characteristic
40	#2262	—	53	2	no	—
41	Q₁₂₈	D₅₃	1	1	yes	characteristic, complemented
42	SD₁₂₈	—	53	2	no	—
43	D₆₄	—	53	1	no	—
44	C₂×C₆₄	—	53	1	no	—
45	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
46	D₅₃	—	2	1	no	—
47	D₅₃	—	32	2	no	—
48	#508	—	106	2	no	—
49	Q₆₄	C₂×D₅₃	1	2	yes	—
50	SD₆₄	—	106	2	no	—
51	D₃₂	—	53	2	no	—
52	C₂×C₃₂	—	53	1	no	—
53	C₆₄	—	53	1	no	—
54	C₆₄	C₂×D₅₃	1	1	yes	characteristic
55	C₅₃	#13741	1	1	yes	characteristic, complemented, Sylow
56	#135	—	212	2	no	—
57	Q₃₂	—	2	2	no	—
58	SD₃₂	—	212	2	no	—
59	D₁₆	—	106	2	no	—
60	C₂×C₁₆	—	53	1	no	—
61	C₃₂	—	53	1	no	—
62	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
63	#41	—	424	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	424	2	no	—
66	D₈	—	212	2	no	—
67	C₂×C₈	—	53	1	no	—
68	C₁₆	—	53	1	no	—
69	C₁₆	D₄×D₅₃	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	D₄	—	848	2	no	—
72	D₄	—	424	2	no	—
73	C₂×C₄	—	53	1	no	—
74	C₂×C₄	—	848	2	no	—
75	C₈	—	53	1	no	—
76	C₈	D₈×D₅₃	1	1	yes	characteristic
77	C₂²	—	53	1	no	—
78	C₂²	—	848	2	no	—
79	C₄	—	16	2	no	—
80	C₄	—	53	1	no	—
81	C₄	D₁₆×D₅₃	1	1	yes	characteristic
82	C₂	—	106	1	no	—
83	C₂	—	1696	2	no	—
84	C₂	D₃₂×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₂×D₅₃
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₅₃×Q₁₂₈	C₂
socle	C₂×C₅₃	C₂×D₃₂
hypercenter	Q₁₂₈	D₅₃
hypocenter	C₅₃	#13741
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₁₂₈

Complements

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

N	H ≅ G/N	type
1. #20277025	C₂	semidirect
2. #20272929	C₂	semidirect
3. D₃₃₉₂	C₂	semidirect
4. C₆₄×D₅₃	C₂	semidirect
5. C₅₃×Q₁₂₈	C₂	semidirect
41. Q₁₂₈	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
#20277025	2
#20272929	2
D₃₃₉₂	2
C₆₄×D₅₃	2
C₅₃×Q₁₂₈	2
#13741	53