Subgroups — #31778987

Order 17536.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577237	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17983994	—	2	2	no	—
4	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
10	C₁₆×C₁₃₇	—	2	1	no	—
11	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
12	C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₃₇	—	2	1	no	—
14	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
15	C₈×C₁₃₇	C₁₆	1	2	yes	—
16	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
18	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
19	C₄×C₁₃₇	—	2	2	no	—
20	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₁₃₇	—	2	1	no	—
23	C₂×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
24	C₁₃₇	#1422	1	1	yes	characteristic, complemented, Sylow
25	#1422	—	137	1	no	maximal, Sylow
26	M₆₄	—	137	2	no	—
27	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic
28	C₂×C₁₆	Dic₁₃₇	1	1	yes	characteristic
29	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
30	C₄×C₈	Dic₁₃₇	1	1	yes	characteristic
31	C₃₂	—	274	2	no	—
32	C₂×C₈	#11764040	1	1	yes	characteristic
33	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
34	C₄²	#11764040	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₁₆	D₅₄₈	1	1	yes	characteristic
37	C₁₆	Dic₂₇₄	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂×#11764040	1	1	yes	characteristic
40	C₈	#14555485	1	2	yes	—
41	C₈	C₄⋊Dic₁₃₇	1	1	yes	characteristic
42	C₈	#14555494	1	1	yes	characteristic
43	C₂²	#17984028	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	C₄⋊#11764040	1	1	yes	characteristic
46	C₄	C₂×#14555485	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	C₄⋊#14555485	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₅₄₈
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	C₄×C₁₆×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₄⋊C₁₆
prime radical	C₂²×C₁₃₇	M₃₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	C₄×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#1422
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₃₇	#1422	semidirect

Maximal subgroups

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

subgroup	index
#22577237	2
C₄×C₁₆×C₁₃₇	2
#1422	137