Subgroups — #31795815

Order 17536.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578264	C₂	1	1	yes	maximal, characteristic
2	#22578264	C₂	1	1	yes	maximal, characteristic
3	#22577178	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11764047	—	2	1	no	—
5	C₄⋊#11764047	—	2	1	no	—
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	—	4	1	no	—
10	C₂×#11764047	—	4	1	no	—
11	C₂×#11764040	Q₈	1	1	yes	characteristic
12	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
13	#14555485	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	—	2	1	no	—
16	#11764047	—	8	1	no	—
17	#11764047	—	8	1	no	—
18	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
19	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
20	#11764040	—	2	1	no	—
21	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
22	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₁₃₇	#106	1	1	yes	characteristic
25	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
26	C₄×C₁₃₇	#108	1	1	yes	characteristic
27	Dic₁₃₇	—	2	1	no	—
28	Dic₁₃₇	—	2	1	no	—
29	C₂×C₁₃₇	#334	1	1	yes	characteristic
30	C₂×C₁₃₇	#365	1	1	yes	characteristic
31	C₂×C₁₃₇	#365	1	1	yes	characteristic
32	C₁₃₇	#1368	1	1	yes	characteristic, complemented, Sylow
33	#1368	—	137	1	no	maximal, Sylow
34	#346	—	137	1	no	—
35	#334	—	137	1	no	—
36	#334	—	137	1	no	—
37	C₂×C₁₆	—	137	1	no	—
38	C₂×C₁₆	#10587739	1	1	yes	characteristic
39	C₄⋊C₈	—	274	1	no	—
40	C₄⋊C₈	—	274	1	no	—
41	C₄×C₈	—	137	1	no	—
42	C₂×C₈	—	548	1	no	—
43	C₂×C₈	—	548	1	no	—
44	C₂×C₈	—	137	1	no	—
45	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
46	C₄²	—	137	1	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	137	2	no	—
49	C₂×C₄	—	137	1	no	—
50	C₂×C₄	—	137	1	no	—
51	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
52	C₈	—	274	1	no	—
53	C₈	—	1096	1	no	—
54	C₈	—	1096	1	no	—
55	C₈	#14555518	1	1	yes	characteristic
56	C₈	#14555519	1	1	yes	characteristic
57	C₂²	#17984156	1	1	yes	characteristic
58	C₄	—	274	1	no	—
59	C₄	—	274	1	no	—
60	C₄	#17984155	1	1	yes	characteristic
61	C₄	#17984159	1	1	yes	characteristic
62	C₂	#22578265	1	1	yes	characteristic
63	C₂	#22578327	1	1	yes	characteristic
64	C₂	#22578328	1	1	yes	characteristic

	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
13. #14555485	C₈	semidirect
32. C₁₃₇	#1368	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