Subgroups — #22578443

Order 8768.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10587739	C₂	1	2	yes	maximal
2	#17984156	C₂	1	1	yes	maximal, characteristic
3	#17984151	C₂	1	2	yes	maximal
4	#17984142	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#14555518	C₂²	1	1	yes	characteristic
7	C₄⋊#10587739	—	2	2	no	—
8	C₄⋊#10587739	C₂²	1	2	yes	—
9	C₄×#10587739	—	2	2	no	—
10	C₄×#10587739	C₂²	1	1	yes	characteristic
11	Q₈×D₁₃₇	C₂²	1	2	yes	—
12	Dic₅₄₈	C₄	1	1	yes	characteristic, complemented
13	#14555500	C₄	1	2	yes	complemented
14	C₈×D₁₃₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#10587739	—	2	2	no	—
17	C₂×#10587739	D₄	1	2	yes	—
18	#11764047	—	2	1	no	—
19	Dic₂₇₄	—	2	2	no	—
20	Dic₂₇₄	C₂×C₄	1	2	yes	—
21	C₄×D₁₃₇	—	2	2	no	—
22	C₄×D₁₃₇	C₂³	1	1	yes	characteristic
23	#11764040	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
27	#10587739	—	2	2	no	—
28	#10587739	—	4	2	no	—
29	C₄×C₁₃₇	—	2	2	no	—
30	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₃₇	—	2	2	no	—
32	Dic₁₃₇	#41	1	1	yes	characteristic
33	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
34	D₁₃₇	#137	1	2	yes	—
35	C₁₃₇	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	137	1	no	maximal, Sylow
37	C₂×Q₁₆	—	137	1	no	—
38	C₄×Q₈	—	137	2	no	—
39	#106	—	137	1	no	—
40	#103	—	137	2	no	—
41	#97	—	137	1	no	—
42	C₂×Q₈	—	137	2	no	—
43	Q₁₆	—	137	2	no	—
44	Q₁₆	—	137	1	no	—
45	Q₁₆	#10587739	1	1	yes	characteristic, complemented
46	C₂×C₈	—	137	1	no	—
47	C₂×C₈	—	137	1	no	—
48	C₄⋊C₄	—	274	2	no	—
49	C₄⋊C₄	—	137	2	no	—
50	C₄²	—	274	2	no	—
51	C₄²	—	137	1	no	—
52	Q₈	—	274	2	no	—
53	Q₈	—	137	2	no	—
54	Q₈	C₂×#10587739	1	2	yes	—
55	C₂×C₄	—	274	2	no	—
56	C₂×C₄	—	274	2	no	—
57	C₂×C₄	—	137	2	no	—
58	C₂×C₄	—	137	1	no	—
59	C₈	—	274	1	no	—
60	C₈	—	137	1	no	—
61	C₈	C₂×#10587739	1	1	yes	characteristic
62	C₂²	—	137	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	274	2	no	—
65	C₄	—	274	2	no	—
66	C₄	—	548	2	no	—
67	C₄	—	137	1	no	—
68	C₄	C₂²×#10587739	1	1	yes	characteristic
69	C₂	—	137	2	no	—
70	C₂	D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#10587739
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₄	C₂²×#10587739
Fitting	Q₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₄×D₄
hypercenter	Q₁₆	#10587739
hypocenter	C₁₃₇	#441
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₅₄₈	C₄	semidirect
13. #14555500	C₄	semidirect
15. Q₁₆×C₁₃₇	C₄	semidirect
35. C₁₃₇	#441	semidirect
45. Q₁₆	#10587739	semidirect