Subgroups — #11763714

Order 1088.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587560	C₂	1	1	yes	maximal, characteristic
2	#10587539	C₂	1	2	yes	maximal, complemented
3	C₂×D₆₈	—	2	2	no	—
4	C₂×D₆₈	C₄	1	1	yes	characteristic, complemented
5	C₂×D₆₈	C₂²	1	1	yes	characteristic
6	C₂²⋊#595	—	2	2	no	—
7	C₄²×C₁₇	C₄	1	1	yes	characteristic, complemented
8	C₂²×D₁₇	—	2	1	no	—
9	C₂²×D₁₇	D₄	1	2	yes	—
10	C₂×#595	—	4	2	no	—
11	C₂×C₄×C₁₇	—	2	1	no	—
12	C₂×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
13	D₆₈	—	4	1	no	—
14	D₆₈	—	4	1	no	—
15	D₆₈	—	4	4	no	—
16	C₂²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
17	C₂×D₁₇	—	2	2	no	—
18	C₂×D₁₇	—	4	2	no	—
19	C₂×D₁₇	—	4	2	no	—
20	C₂×D₁₇	—	4	1	no	—
21	#595	—	8	2	no	—
22	C₄×C₁₇	—	2	2	no	—
23	C₄×C₁₇	—	2	1	no	—
24	#353	—	17	1	no	maximal, Sylow
25	C₂×C₁₇	—	2	1	no	—
26	C₂×C₁₇	#99	1	1	yes	characteristic
27	D₁₇	—	4	2	no	—
28	D₁₇	—	8	1	no	—
29	#127	—	17	1	no	—
30	#99	—	17	2	no	—
31	C₁₇	#353	1	1	yes	characteristic, complemented, Sylow
32	C₂×D₄	—	17	1	no	—
33	C₂×D₄	—	17	1	no	—
34	C₂×D₄	—	34	2	no	—
35	C₂²⋊C₄	—	34	2	no	—
36	C₄²	#595	1	1	yes	characteristic, complemented
37	C₂³	—	17	2	no	—
38	C₂³	—	34	1	no	—
39	D₄	—	68	1	no	—
40	D₄	—	68	1	no	—
41	D₄	—	68	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	68	2	no	—
44	C₂×C₄	C₂×#595	1	1	yes	characteristic
45	C₂²	—	34	2	no	—
46	C₂²	—	68	2	no	—
47	C₂²	—	68	2	no	—
48	C₂²	—	68	1	no	—
49	C₂²	C₂²⋊#595	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	136	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	68	2	no	—
55	C₂	—	136	1	no	—
56	C₂	#10587539	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10587539
commutator	C₂×C₄×C₁₇	C₂×C₄
Frattini	C₂×C₄	C₂×#595
Fitting	C₄²×C₁₇	C₄
socle	C₂×C₁₇	#99
prime radical	#10587560	C₂
cosocle	C₂×D₆₈	C₂²
hypercenter	C₄²	#595
hypocenter	C₁₇	#353
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #10587539	C₂	semidirect
4. C₂×D₆₈	C₄	semidirect
7. C₄²×C₁₇	C₄	semidirect
31. C₁₇	#353	semidirect
36. C₄²	#595	semidirect

Maximal subgroups

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

subgroup	index
#10587560	2
#10587539	2
#353	17