Subgroups — #31795814

Order 17536.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578264	C₂	1	2	yes	maximal
2	#22577178	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11764047	—	2	2	no	—
4	C₂×#14555485	C₄	1	1	yes	characteristic
5	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764047	—	4	2	no	—
8	C₂×#11764040	Q₈	1	1	yes	characteristic
9	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
10	#14555485	—	2	1	no	—
11	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₃₇	C₈	1	2	yes	complemented
13	#11764047	—	8	2	no	—
14	C₂×Dic₁₃₇	SD₁₆	1	2	yes	—
15	#11764040	—	2	1	no	—
16	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
19	C₂²×C₁₃₇	#106	1	1	yes	characteristic
20	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₃₇	#108	1	1	yes	characteristic
22	Dic₁₃₇	—	2	2	no	—
23	C₂×C₁₃₇	#334	1	1	yes	characteristic
24	C₂×C₁₃₇	#365	1	2	yes	—
25	C₁₃₇	#1368	1	1	yes	characteristic, complemented, Sylow
26	#1368	—	137	1	no	maximal, Sylow
27	#346	—	137	1	no	—
28	#334	—	137	2	no	—
29	C₂×C₁₆	—	137	1	no	—
30	C₂×C₁₆	#10587739	1	1	yes	characteristic
31	C₄⋊C₈	—	274	2	no	—
32	C₄×C₈	—	137	1	no	—
33	C₂×C₈	—	548	2	no	—
34	C₂×C₈	—	137	1	no	—
35	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
36	C₄²	—	137	1	no	—
37	C₁₆	—	274	1	no	—
38	C₁₆	#11764047	1	2	yes	complemented
39	C₂×C₄	—	137	2	no	—
40	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
41	C₈	—	274	1	no	—
42	C₈	—	1096	2	no	—
43	C₈	C₂×#11764047	1	1	yes	characteristic
44	C₈	#14555520	1	1	yes	characteristic
45	C₂²	#17984156	1	1	yes	characteristic
46	C₄	—	274	2	no	—
47	C₄	C₄⋊#11764047	1	1	yes	characteristic
48	C₄	#17984158	1	1	yes	characteristic
49	C₂	#22578264	1	1	yes	characteristic
50	C₂	#22578328	1	2	yes	—

	subgroup	G/H
center	C₂²	#17984156
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#106
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#1368
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₃₇	C₈	semidirect
25. C₁₃₇	#1368	semidirect
38. C₁₆	#11764047	semidirect

Maximal subgroups

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

subgroup	index
#22578264	2
#22577178	2
#1368	137