Subgroups — #15750814

Order 3328.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718035	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
5	C₁₆×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂×#88034	C₄	1	1	yes	characteristic
7	C₁₃×C₆₄	—	2	1	no	—
8	#11718035	—	2	1	no	—
9	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
10	C₈×Dic₁₃	C₈	1	1	yes	characteristic
11	C₂×#6090	C₈	1	1	yes	characteristic
12	C₁₃×C₃₂	D₄	1	1	yes	characteristic
13	C₁₃×C₃₂	Q₈	1	1	yes	characteristic
14	#88034	—	2	1	no	—
15	C₄⋊C₆₄	—	13	1	no	maximal, Sylow
16	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
17	C₄×Dic₁₃	C₁₆	1	1	yes	characteristic
18	C₂×#1055	C₁₆	1	1	yes	characteristic
19	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
20	C₁₃×C₁₆	M₁₆	1	1	yes	characteristic
21	#6090	—	2	1	no	—
22	C₂×C₆₄	—	13	1	no	—
23	C₂×C₆₄	D₁₃	1	1	yes	characteristic
24	C₄×C₃₂	—	13	1	no	—
25	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
26	C₂×Dic₁₃	C₃₂	1	1	yes	characteristic
27	C₂×Dic₁₃	C₃₂	1	1	yes	characteristic
28	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
29	C₈×C₁₃	M₃₂	1	1	yes	characteristic
30	#1055	—	2	1	no	—
31	C₂×C₃₂	—	13	1	no	—
32	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
33	C₄×C₁₆	—	13	1	no	—
34	C₆₄	—	2	1	no	—
35	C₆₄	—	26	1	no	—
36	C₂²×C₁₃	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
38	C₄×C₁₃	M₆₄	1	1	yes	characteristic
39	Dic₁₃	—	2	1	no	—
40	Dic₁₃	C₆₄	1	2	yes	complemented
41	C₂×C₁₆	—	13	1	no	—
42	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
43	C₄×C₈	—	13	1	no	—
44	C₃₂	—	26	1	no	—
45	C₃₂	Dic₂₆	1	1	yes	characteristic
46	C₃₂	C₂²⋊D₁₃	1	1	yes	characteristic
47	C₂×C₁₃	C₄⋊C₃₂	1	1	yes	characteristic
48	C₂×C₁₃	C₂×C₆₄	1	1	yes	characteristic
49	C₂×C₁₃	M₁₂₈	1	1	yes	characteristic
50	C₂×C₈	—	13	1	no	—
51	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
52	C₄²	—	13	1	no	—
53	C₁₆	—	26	1	no	—
54	C₁₆	#6094	1	1	yes	characteristic
55	C₁₆	#6101	1	1	yes	characteristic
56	C₁₃	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄	—	13	1	no	—
58	C₂×C₄	—	13	1	no	—
59	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
60	C₈	—	26	1	no	—
61	C₈	#88038	1	1	yes	characteristic
62	C₈	#88054	1	1	yes	characteristic
63	C₂²	D₁₃×C₃₂	1	1	yes	characteristic
64	C₄	—	13	2	no	—
65	C₄	—	26	1	no	—
66	C₄	#11718039	1	1	yes	characteristic
67	C₄	#11718093	1	1	yes	characteristic
68	C₂	#13113031	1	1	yes	characteristic
69	C₂	D₁₃×C₆₄	1	1	yes	characteristic
70	C₂	#13113046	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₁₃
commutator	C₂×C₁₃	C₂×C₆₄
Fitting	C₂×C₁₃×C₆₄	C₂
socle, prime radical	C₂²×C₁₃	C₂×C₃₂
cosocle	C₂×C₁₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₁₃
hypocenter	C₁₃	C₄⋊C₆₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
40. Dic₁₃	C₆₄	semidirect
56. C₁₃	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×#11718035	2
C₃₂×Dic₁₃	2
C₂×C₁₃×C₆₄	2
C₄⋊C₆₄	13