Subgroups — #22577819

Order 8768.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984094	C₂	1	1	yes	maximal, characteristic
2	#17984028	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984027	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14555510	C₄	1	1	yes	characteristic
7	#14555494	C₄	1	1	yes	characteristic
8	#14555493	C₄	1	2	yes	—
9	C₈×D₁₃₇	C₂²	1	2	yes	—
10	#14555485	C₂²	1	2	yes	—
11	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	Dic₂₇₄	C₈	1	1	yes	characteristic
15	D₅₄₈	C₈	1	1	yes	characteristic
16	C₂²⋊D₁₃₇	C₈	1	2	yes	—
17	C₄×D₁₃₇	C₂×C₄	1	2	yes	—
18	#11764040	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₈×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₂×D₁₃₇	C₂×C₈	1	2	yes	—
24	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
25	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
26	Dic₁₃₇	C₂×C₈	1	2	yes	—
27	C₂×C₁₃₇	—	2	1	no	—
28	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
29	D₁₃₇	—	2	2	no	—
30	C₁₃₇	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	137	1	no	maximal, Sylow
32	#131	—	137	1	no	—
33	M₃₂	—	137	1	no	—
34	M₃₂	—	137	2	no	—
35	C₂×C₁₆	—	137	2	no	—
36	C₂×C₁₆	D₁₃₇	1	1	yes	characteristic, complemented
37	#41	—	137	1	no	—
38	M₁₆	—	137	1	no	—
39	M₁₆	—	137	2	no	—
40	C₂×C₈	—	137	2	no	—
41	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
42	C₁₆	—	137	2	no	—
43	C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
44	C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
45	Q₈	—	137	1	no	—
46	D₄	—	137	2	no	—
47	D₄	—	137	1	no	—
48	C₂×C₄	—	137	2	no	—
49	C₂×C₄	C₄×D₁₃₇	1	1	yes	characteristic
50	C₈	—	137	2	no	—
51	C₈	C₄×D₁₃₇	1	1	yes	characteristic
52	C₈	C₂²×D₁₃₇	1	1	yes	characteristic
53	C₂²	—	137	2	no	—
54	C₂²	C₈×D₁₃₇	1	1	yes	characteristic
55	C₄	—	137	2	no	—
56	C₄	C₈×D₁₃₇	1	1	yes	characteristic
57	C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	274	2	no	—
60	C₂	C₂×C₈×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×D₁₃₇
commutator, socle	C₂×C₁₃₇	C₂²×C₈
Frattini	C₈	C₂²×D₁₃₇
Fitting	C₂×C₁₆×C₁₃₇	C₂
hypercenter	C₂×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#504
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #17984028	C₂	semidirect
3. #17984027	C₂	semidirect
4. C₁₆×D₁₃₇	C₂	semidirect
5. C₂×C₁₆×C₁₃₇	C₂	semidirect
30. C₁₃₇	#504	semidirect
36. C₂×C₁₆	D₁₃₇	semidirect