Subgroups — #31797849

Order 17536.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578492	C₂	1	2	yes	maximal
2	#22578290	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11764049	—	2	2	no	—
4	#17984161	C₄	1	1	yes	characteristic
5	C₈×#10587739	C₂²	1	1	yes	characteristic
6	C₁₆×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764049	—	4	2	no	—
8	#14555518	Q₈	1	1	yes	characteristic
9	C₄×#10587739	D₄	1	1	yes	characteristic
10	#14555513	—	2	1	no	—
11	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
12	#14555485	C₈	1	1	yes	characteristic, complemented
13	C₁₆×C₁₃₇	C₈	1	1	yes	characteristic, complemented
14	#11764049	—	8	2	no	—
15	C₂×#10587739	SD₁₆	1	2	yes	—
16	#11764047	—	2	1	no	—
17	C₄×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	#11764040	M₁₆	1	1	yes	characteristic
19	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
20	C₂×D₁₃₇	#106	1	1	yes	characteristic
21	#10587739	—	2	2	no	—
22	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
23	Dic₁₃₇	#108	1	1	yes	characteristic
24	C₂×C₁₃₇	#334	1	1	yes	characteristic
25	D₁₃₇	#365	1	2	yes	—
26	C₁₃₇	#1368	1	1	yes	characteristic, complemented, Sylow
27	#1368	—	137	1	no	maximal, Sylow
28	#346	—	137	1	no	—
29	#334	—	137	2	no	—
30	C₂×C₁₆	—	137	1	no	—
31	C₂×C₁₆	—	137	1	no	—
32	C₄⋊C₈	—	274	2	no	—
33	C₄×C₈	—	137	1	no	—
34	C₂×C₈	—	548	2	no	—
35	C₂×C₈	—	137	1	no	—
36	C₂×C₈	—	137	1	no	—
37	C₄²	—	137	1	no	—
38	C₁₆	—	274	1	no	—
39	C₁₆	—	137	1	no	—
40	C₁₆	#11764049	1	1	yes	characteristic, complemented
41	C₂×C₄	—	137	2	no	—
42	C₂×C₄	—	137	1	no	—
43	C₈	—	274	1	no	—
44	C₈	—	1096	2	no	—
45	C₈	—	137	1	no	—
46	C₈	C₂×#11764049	1	1	yes	characteristic
47	C₂²	—	137	1	no	—
48	C₄	—	274	2	no	—
49	C₄	—	137	1	no	—
50	C₄	C₄⋊#11764049	1	1	yes	characteristic
51	C₂	—	137	2	no	—
52	C₂	#22578492	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578492
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₈	C₂×#11764049
Fitting	C₁₆×C₁₃₇	C₈
socle	C₂×C₁₃₇	#334
prime radical	C₂×D₁₃₇	#106
cosocle	C₈×#10587739	C₂²
hypercenter	C₁₆	#11764049
hypocenter	C₁₃₇	#1368
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
12. #14555485	C₈	semidirect
13. C₁₆×C₁₃₇	C₈	semidirect
26. C₁₃₇	#1368	semidirect
40. C₁₆	#11764049	semidirect

Maximal subgroups

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

subgroup	index
#22578492	2
#22578290	2
#1368	137