Subgroups — #31795954

Order 17536.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578332	C₂	1	1	yes	maximal, characteristic
2	#22578330	C₂	1	1	yes	maximal, characteristic
3	#22577236	C₂	1	1	yes	maximal, characteristic
4	#17984159	—	2	1	no	—
5	#17984157	—	2	1	no	—
6	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
7	#17983994	C₄	1	1	yes	characteristic, complemented
8	C₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
9	#14555520	—	4	1	no	—
10	C₄⋊#10587739	—	4	1	no	—
11	C₈×D₁₃₇	D₄	1	1	yes	characteristic
12	#14555485	Q₈	1	1	yes	characteristic
13	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂×#10587739	—	8	1	no	—
15	#11764047	—	8	1	no	—
16	C₄×D₁₃₇	D₈	1	1	yes	characteristic
17	#11764040	Q₁₆	1	1	yes	characteristic
18	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₁₃₇	SD₃₂	1	1	yes	characteristic
20	#10587739	—	16	1	no	—
21	C₄×C₁₃₇	#107	1	1	yes	characteristic
22	Dic₁₃₇	SD₃₂	1	1	yes	characteristic
23	C₂×C₁₃₇	#367	1	1	yes	characteristic
24	D₁₃₇	—	2	1	no	—
25	C₁₃₇	#1426	1	1	yes	characteristic, complemented, Sylow
26	#1426	—	137	1	no	maximal, Sylow
27	M₆₄	—	137	1	no	—
28	#368	—	137	1	no	—
29	#367	—	137	1	no	—
30	C₂×C₁₆	—	137	1	no	—
31	#108	—	274	1	no	—
32	#107	—	274	1	no	—
33	C₃₂	—	137	1	no	—
34	C₃₂	#10587739	1	1	yes	characteristic, complemented
35	M₁₆	—	548	1	no	—
36	C₂×C₈	—	137	1	no	—
37	C₄⋊C₄	—	548	1	no	—
38	C₁₆	—	137	1	no	—
39	C₁₆	C₂×#10587739	1	1	yes	characteristic
40	C₂×C₄	—	1096	1	no	—
41	C₂×C₄	—	137	1	no	—
42	C₈	—	1096	1	no	—
43	C₈	—	137	1	no	—
44	C₈	C₄⋊#10587739	1	1	yes	characteristic
45	C₂²	—	137	1	no	—
46	C₄	—	137	1	no	—
47	C₄	—	2192	1	no	—
48	C₄	#17984157	1	1	yes	characteristic
49	C₂	—	274	1	no	—
50	C₂	#22578330	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578330
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₁₆	C₂×#10587739
Fitting	C₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	#367
prime radical	C₂×D₁₃₇	SD₃₂
cosocle	C₁₆×D₁₃₇	C₂²
hypercenter	C₃₂	#10587739
hypocenter	C₁₃₇	#1426
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
7. #17983994	C₄	semidirect
8. C₃₂×C₁₃₇	C₄	semidirect
25. C₁₃₇	#1426	semidirect
34. C₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578332	2
#22578330	2
#22577236	2
#1426	137