Subgroups — #31778925

Order 17536.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577237	C₂	1	2	yes	maximal
2	#346×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17983994	C₄	1	4	yes	complemented
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	C₈	1	2	yes	—
11	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₃₇	—	2	1	no	—
13	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
14	C₈×C₁₃₇	—	2	1	no	—
15	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
16	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
18	C₄×C₁₃₇	—	4	1	no	—
19	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
20	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
21	C₂×C₁₃₇	—	2	1	no	—
22	C₂×C₁₃₇	#346	1	1	yes	characteristic
23	C₁₃₇	#1398	1	1	yes	characteristic, complemented, Sylow
24	#1398	—	137	1	no	maximal, Sylow
25	M₆₄	—	137	2	no	—
26	#346	D₁₃₇	1	1	yes	characteristic
27	C₂×C₁₆	Dic₁₃₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
29	C₄×C₈	Dic₁₃₇	1	1	yes	characteristic
30	C₃₂	—	137	4	no	—
31	C₂×C₈	#11764040	1	1	yes	characteristic
32	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
33	C₄²	#11764040	1	1	yes	characteristic
34	C₁₆	#11764040	1	2	yes	—
35	C₁₆	C₄×D₁₃₇	1	2	yes	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×#11764040	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₄×Dic₁₃₇	1	1	yes	characteristic
40	C₈	C₂×#11764040	1	1	yes	characteristic
41	C₂²	#17984028	1	1	yes	characteristic
42	C₄	—	4	1	no	—
43	C₄	C₄×#11764040	1	1	yes	characteristic
44	C₄	#17984028	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#22577179	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₃₇
commutator	C₄×C₁₃₇	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	#346×C₁₃₇	C₂
socle	C₂×C₁₃₇	#346
prime radical	C₂²×C₁₃₇	M₃₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	#346	D₁₃₇
hypocenter	C₁₃₇	#1398
LCS	G ▷ C₄×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
3. #17983994	C₄	semidirect
23. C₁₃₇	#1398	semidirect

Maximal subgroups

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

subgroup	index
#22577237	2
#346×C₁₃₇	2
#1398	137