Subgroups — #31795705

Order 17536.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578292	C₂	1	1	yes	maximal, characteristic
2	#22578244	C₂	1	1	yes	maximal, characteristic
3	#22577178	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	C₄	1	2	yes	—
5	C₄×#11764047	C₄	1	2	yes	—
6	C₂×#14555485	C₄	1	1	yes	characteristic
7	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
9	C₂×#11764047	—	2	2	no	—
10	#14555513	—	2	2	no	—
11	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
13	#14555485	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	C₈	1	2	yes	complemented
16	#11764047	—	4	2	no	—
17	C₂×Dic₁₃₇	M₁₆	1	2	yes	—
18	#11764040	M₁₆	1	2	yes	—
19	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
20	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
21	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
22	C₂²×C₁₃₇	#97	1	1	yes	characteristic
23	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
24	C₄×C₁₃₇	#97	1	1	yes	characteristic
25	Dic₁₃₇	—	2	2	no	—
26	C₂×C₁₃₇	#322	1	1	yes	characteristic
27	C₂×C₁₃₇	#347	1	2	yes	—
28	C₁₃₇	#1313	1	1	yes	characteristic, complemented, Sylow
29	#1313	—	137	1	no	maximal, Sylow
30	#346	—	137	1	no	—
31	#346	—	137	1	no	—
32	#322	—	137	1	no	—
33	C₂×C₁₆	—	137	2	no	—
34	C₂×C₁₆	—	137	1	no	—
35	C₂×C₁₆	#10587739	1	1	yes	characteristic
36	C₄×C₈	—	137	2	no	—
37	C₄×C₈	—	137	1	no	—
38	C₂×C₈	—	274	2	no	—
39	C₂×C₈	—	137	1	no	—
40	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
41	C₄²	—	137	1	no	—
42	C₁₆	—	274	2	no	—
43	C₁₆	—	137	2	no	—
44	C₁₆	#11764047	1	2	yes	complemented
45	C₂×C₄	—	137	2	no	—
46	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
47	C₈	—	548	2	no	—
48	C₈	—	137	2	no	—
49	C₈	C₂×#11764047	1	1	yes	characteristic
50	C₈	#14555518	1	1	yes	characteristic
51	C₂²	#17984142	1	1	yes	characteristic
52	C₄	—	274	2	no	—
53	C₄	C₄×#11764047	1	1	yes	characteristic
54	C₄	#17984142	1	1	yes	characteristic
55	C₂	#22578244	1	1	yes	characteristic
56	C₂	#22578293	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×#10587739
commutator	C₄×C₁₃₇	C₄×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#97
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#1313
LCS	G ▷ C₄×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
13. #14555485	C₈	semidirect
15. C₁₆×C₁₃₇	C₈	semidirect
28. C₁₃₇	#1313	semidirect
44. C₁₆	#11764047	semidirect

Maximal subgroups

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

subgroup	index
#22578292	2
#22578244	2
#22577178	2
#1313	137