Subgroups — #504×C₁₃₇ #22577039

Order 8768.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	M₃₂×C₁₃₇	C₂	1	3	yes	maximal, complemented
3	C₂×C₁₆×C₁₃₇	C₂	1	3	yes	maximal, complemented
4	#41×C₁₃₇	C₄	1	1	yes	characteristic
5	M₁₆×C₁₃₇	C₄	1	3	yes	—
6	C₂×C₈×C₁₃₇	C₂²	1	3	yes	—
7	C₁₆×C₁₃₇	C₂²	1	3	yes	—
8	C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	Q₈×C₁₃₇	C₈	1	1	yes	characteristic
10	D₄×C₁₃₇	C₈	1	3	yes	—
11	C₂×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
12	C₈×C₁₃₇	C₂×C₄	1	3	yes	—
13	C₈×C₁₃₇	C₂³	1	1	yes	characteristic
14	C₂²×C₁₃₇	C₂×C₈	1	3	yes	—
15	C₄×C₁₃₇	C₂×C₈	1	3	yes	—
16	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₁₃₇	—	2	3	no	—
18	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
19	C₁₃₇	#504	1	1	yes	characteristic, direct product factor, Sylow
20	#504	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	#131	C₂×C₁₃₇	1	1	yes	characteristic
22	M₃₂	C₂×C₁₃₇	1	3	yes	complemented
23	C₂×C₁₆	C₂×C₁₃₇	1	3	yes	complemented
24	#41	C₄×C₁₃₇	1	1	yes	characteristic
25	M₁₆	C₄×C₁₃₇	1	3	yes	—
26	C₂×C₈	C₂²×C₁₃₇	1	3	yes	—
27	C₁₆	C₂²×C₁₃₇	1	3	yes	—
28	C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
29	Q₈	C₈×C₁₃₇	1	1	yes	characteristic
30	D₄	C₈×C₁₃₇	1	3	yes	—
31	C₂×C₄	C₂×C₄×C₁₃₇	1	3	yes	—
32	C₈	C₂×C₄×C₁₃₇	1	3	yes	—
33	C₈	C₂³×C₁₃₇	1	1	yes	characteristic
34	C₂²	C₂×C₈×C₁₃₇	1	3	yes	—
35	C₄	C₂×C₈×C₁₃₇	1	3	yes	—
36	C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
37	C₂	—	2	3	no	—
38	C₂	C₂²×C₈×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₁₃₇	C₂²
commutator	C₂	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₈	C₂³×C₁₃₇
socle	C₂×C₁₃₇	C₂²×C₈
prime radical	#41×C₁₃₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₃₂×C₁₃₇	C₂	semidirect
3. C₂×C₁₆×C₁₃₇	C₂	semidirect
19. C₁₃₇	#504	direct (N × H)
20. #504	C₁₃₇	direct (N × H)
22. M₃₂	C₂×C₁₃₇	semidirect
23. C₂×C₁₆	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#131×C₁₃₇	2
M₃₂×C₁₃₇	2
C₂×C₁₆×C₁₃₇	2
#504	137

Subgroups — #504×C137 #22577039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #504×C₁₃₇ #22577039