Subgroups — #27271987

Order 13568.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810342	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
6	#16810342	—	2	1	no	—
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₆₄	—	2	1	no	—
9	C₂×#11719908	C₈	1	1	yes	characteristic
10	C₈×Dic₅₃	C₈	1	1	yes	characteristic
11	#13133123	—	2	1	no	—
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	D₄	1	1	yes	characteristic
14	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
17	C₂×#88322	C₁₆	1	1	yes	characteristic
18	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
19	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
20	#11719908	—	2	1	no	—
21	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
23	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
24	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₅₃	M₃₂	1	1	yes	characteristic
26	#88322	—	2	1	no	—
27	C₄⋊C₆₄	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
29	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
30	C₄×C₅₃	M₆₄	1	1	yes	characteristic
31	Dic₅₃	—	2	1	no	—
32	Dic₅₃	C₆₄	1	2	yes	complemented
33	C₂×C₆₄	—	53	1	no	—
34	C₂×C₆₄	D₅₃	1	1	yes	characteristic
35	C₄×C₃₂	—	53	1	no	—
36	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
37	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
38	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
39	C₂×C₃₂	—	53	1	no	—
40	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
41	C₄×C₁₆	—	53	1	no	—
42	C₆₄	—	2	1	no	—
43	C₆₄	—	106	1	no	—
44	C₅₃	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₆	—	53	1	no	—
46	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
47	C₄×C₈	—	53	1	no	—
48	C₃₂	—	106	1	no	—
49	C₃₂	Dic₁₀₆	1	1	yes	characteristic
50	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
51	C₂×C₈	—	53	1	no	—
52	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
53	C₄²	—	53	1	no	—
54	C₁₆	—	106	1	no	—
55	C₁₆	#11719912	1	1	yes	characteristic
56	C₁₆	#11719919	1	1	yes	characteristic
57	C₂×C₄	—	53	1	no	—
58	C₂×C₄	—	53	1	no	—
59	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
60	C₈	—	106	1	no	—
61	C₈	#13133146	1	1	yes	characteristic
62	C₈	#13133156	1	1	yes	characteristic
63	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
64	C₄	—	53	2	no	—
65	C₄	—	106	1	no	—
66	C₄	#16810441	1	1	yes	characteristic
67	C₄	#16810456	1	1	yes	characteristic
68	C₂	#20272910	1	1	yes	characteristic
69	C₂	C₆₄×D₅₃	1	1	yes	characteristic
70	C₂	#20272925	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₆₄
Fitting	C₂×C₅₃×C₆₄	C₂
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	C₄⋊C₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
32. Dic₅₃	C₆₄	semidirect
44. C₅₃	C₄⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810342	2
C₃₂×Dic₅₃	2
C₂×C₅₃×C₆₄	2
C₄⋊C₆₄	53