Subgroups — #31797212

Order 17536.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578335	C₂	1	3	yes	maximal, complemented
2	C₂×#17984138	C₂	1	3	yes	maximal, complemented
3	#22577820	C₂	1	1	yes	maximal, characteristic
4	#17984138	C₂²	1	1	yes	characteristic
5	#17984138	C₂²	1	3	yes	—
6	#17984028	C₄	1	3	yes	—
7	C₂×#14555485	C₂²	1	3	yes	—
8	#131×C₁₃₇	C₄	1	1	yes	characteristic
9	#14555485	C₂×C₄	1	3	yes	—
10	#14555485	C₂³	1	1	yes	characteristic
11	#41×C₁₃₇	C₈	1	1	yes	characteristic
12	M₁₆×C₁₃₇	C₈	1	3	yes	—
13	C₂×C₈×C₁₃₇	C₂×C₄	1	3	yes	—
14	Q₈×C₁₃₇	C₁₆	1	1	yes	characteristic
15	D₄×C₁₃₇	C₁₆	1	3	yes	—
16	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
17	C₈×C₁₃₇	C₂×C₈	1	3	yes	—
18	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₁₃₇	C₂×C₁₆	1	3	yes	—
20	C₄×C₁₃₇	C₂×C₁₆	1	3	yes	—
21	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₁₃₇	—	2	3	no	—
23	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
24	C₁₃₇	#2258	1	1	yes	characteristic, complemented, Sylow
25	#2258	—	137	1	no	maximal, Sylow
26	#504	—	137	1	no	—
27	M₆₄	—	137	3	no	—
28	C₂×C₃₂	—	137	3	no	—
29	#131	#10587739	1	1	yes	characteristic
30	M₃₂	—	137	3	no	—
31	C₂×C₁₆	—	137	3	no	—
32	C₃₂	—	137	1	no	—
33	C₃₂	—	137	3	no	—
34	#41	#11764047	1	1	yes	characteristic
35	M₁₆	#11764047	1	3	yes	—
36	C₂×C₈	C₂×#10587739	1	3	yes	—
37	C₁₆	—	137	3	no	—
38	C₁₆	—	137	1	no	—
39	Q₈	#14555513	1	1	yes	characteristic
40	D₄	#14555513	1	3	yes	—
41	C₂×C₄	C₂×#11764047	1	3	yes	—
42	C₈	C₂×#11764047	1	3	yes	—
43	C₈	C₂²×#10587739	1	1	yes	characteristic
44	C₂²	C₂×#14555513	1	3	yes	—
45	C₄	C₂×#14555513	1	3	yes	—
46	C₄	C₂²×#11764047	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×#14555513	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#10587739
commutator, socle	C₂×C₁₃₇	C₂²×C₁₆
Fitting	#131×C₁₃₇	C₄
prime radical	#41×C₁₃₇	C₈
cosocle	#14555485	C₂³
hypercenter	#131	#10587739
hypocenter	C₁₃₇	#2258
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #22578335	C₂	semidirect
2. C₂×#17984138	C₂	semidirect
24. C₁₃₇	#2258	semidirect

Maximal subgroups

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

subgroup	index
#22578335	2
C₂×#17984138	2
#22577820	2
#2258	137