Subgroups — #31795895

Order 17536.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578334	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₃₂×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984161	C₄	1	1	yes	characteristic
5	C₈×#10587739	C₄	1	1	yes	characteristic
6	#17984138	C₄	1	2	yes	complemented
7	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
8	#17983994	C₄	1	1	yes	characteristic, complemented
9	C₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
10	#14555518	C₈	1	1	yes	characteristic
11	C₄×#10587739	C₈	1	1	yes	characteristic
12	#14555513	C₈	1	2	yes	—
13	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
14	#14555485	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#10587739	C₁₆	1	2	yes	—
17	#11764047	C₁₆	1	2	yes	—
18	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
19	#11764040	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
21	C₂×D₁₃₇	C₂×C₁₆	1	1	yes	characteristic
22	#10587739	—	2	2	no	—
23	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
24	Dic₁₃₇	C₂×C₁₆	1	1	yes	characteristic
25	C₂×C₁₃₇	C₄×C₁₆	1	1	yes	characteristic
26	D₁₃₇	M₆₄	1	2	yes	—
27	C₁₃₇	#1397	1	1	yes	characteristic, complemented, Sylow
28	#1397	—	137	1	no	maximal, Sylow
29	C₂×C₃₂	—	137	1	no	—
30	C₂×C₃₂	—	137	1	no	—
31	C₄×C₁₆	—	137	1	no	—
32	C₂×C₁₆	—	137	1	no	—
33	C₂×C₁₆	—	137	1	no	—
34	C₄×C₈	—	137	1	no	—
35	C₃₂	—	137	2	no	—
36	C₃₂	—	137	1	no	—
37	C₃₂	#10587739	1	1	yes	characteristic, complemented
38	C₂×C₈	—	137	1	no	—
39	C₂×C₈	—	137	1	no	—
40	C₄²	—	137	1	no	—
41	C₁₆	—	137	2	no	—
42	C₁₆	—	137	1	no	—
43	C₁₆	C₂×#10587739	1	1	yes	characteristic
44	C₂×C₄	—	137	2	no	—
45	C₂×C₄	—	137	1	no	—
46	C₈	—	137	2	no	—
47	C₈	—	137	1	no	—
48	C₈	C₄×#10587739	1	1	yes	characteristic
49	C₂²	—	137	1	no	—
50	C₄	—	274	2	no	—
51	C₄	—	137	1	no	—
52	C₄	C₈×#10587739	1	1	yes	characteristic
53	C₂	—	137	2	no	—
54	C₂	C₁₆×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂×#10587739
commutator, socle	C₂×C₁₃₇	C₄×C₁₆
Fitting	C₃₂×C₁₃₇	C₄
prime radical	C₂×D₁₃₇	C₂×C₁₆
cosocle	C₁₆×D₁₃₇	C₂²
hypercenter	C₃₂	#10587739
hypocenter	C₁₃₇	#1397
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
6. #17984138	C₄	semidirect
8. #17983994	C₄	semidirect
9. C₃₂×C₁₃₇	C₄	semidirect
27. C₁₃₇	#1397	semidirect
37. C₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578334	2
C₁₆×#10587739	2
C₃₂×D₁₃₇	2
#1397	137