Subgroups — #34089710

Order 19744.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23754600	C₂	1	1	yes	maximal, characteristic
2	#23754583	C₂	1	2	yes	maximal, complemented
3	C₂×#19096921	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×D₆₁₇	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₂₃₄	C₄	1	1	yes	characteristic
7	D₂₄₆₈	C₄	1	1	yes	characteristic
8	C₂²⋊D₆₁₇	C₄	1	2	yes	—
9	C₄×D₆₁₇	C₂²	1	2	yes	—
10	#19096921	C₂²	1	1	yes	characteristic
11	#19096921	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
13	C₈×C₆₁₇	C₂²	1	2	yes	—
14	C₂²×C₆₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×D₆₁₇	C₂×C₄	1	2	yes	—
16	C₄×C₆₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₆₁₇	C₂³	1	1	yes	characteristic
18	Dic₆₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₆₁₇	—	2	1	no	—
20	C₂×C₆₁₇	C₂²×C₄	1	1	yes	characteristic
21	D₆₁₇	—	2	2	no	—
22	C₆₁₇	#131	1	1	yes	characteristic, complemented, Sylow
23	#131	—	617	1	no	maximal, Sylow
24	#41	—	617	1	no	—
25	M₁₆	—	617	2	no	—
26	M₁₆	D₆₁₇	1	1	yes	characteristic, complemented
27	C₂×C₈	—	617	1	no	—
28	C₂×C₈	—	617	2	no	—
29	Q₈	—	617	1	no	—
30	D₄	—	617	2	no	—
31	D₄	—	617	1	no	—
32	C₂×C₄	—	617	2	no	—
33	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
34	C₈	—	617	1	no	—
35	C₈	—	617	1	no	—
36	C₈	C₂×D₆₁₇	1	2	yes	—
37	C₂²	—	617	2	no	—
38	C₂²	C₄×D₆₁₇	1	1	yes	characteristic
39	C₄	—	617	2	no	—
40	C₄	C₄×D₆₁₇	1	1	yes	characteristic
41	C₄	C₂²×D₆₁₇	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	—	1234	2	no	—
44	C₂	C₂×C₄×D₆₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₆₁₇
commutator, socle	C₂×C₆₁₇	C₂²×C₄
Fitting	M₁₆×C₆₁₇	C₂
prime radical	#23754600	C₂
cosocle	C₄×C₆₁₇	C₂³
hypercenter	M₁₆	D₆₁₇
hypocenter	C₆₁₇	#131
LCS	G ▷ C₂×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #23754583	C₂	semidirect
3. C₂×#19096921	C₂	semidirect
4. C₈×D₆₁₇	C₂	semidirect
5. M₁₆×C₆₁₇	C₂	semidirect
22. C₆₁₇	#131	semidirect
26. M₁₆	D₆₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23754600	2
#23754583	2
C₂×#19096921	2
C₈×D₆₁₇	2
M₁₆×C₆₁₇	2
#131	617