Subgroups — Q₁₆×#10587739 #22578441

Order 8768.

110 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	#17984157	C₂	1	1	yes	maximal, characteristic
3	#17984151	C₂	1	2	yes	maximal
4	C₈×#10587739	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	1	yes	characteristic
18	C₂×#10587739	D₄	1	1	yes	characteristic
19	#11764047	—	2	1	no	—
20	Dic₂₇₄	—	2	2	no	—
21	Dic₂₇₄	C₂×C₄	1	2	yes	—
22	C₄×D₁₃₇	—	2	2	no	—
23	C₄×D₁₃₇	C₂³	1	1	yes	characteristic
24	#11764040	C₂×C₄	1	1	yes	characteristic
25	Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
28	#10587739	—	2	1	no	—
29	#10587739	—	4	2	no	—
30	#10587739	Q₁₆	1	2	yes	direct product factor
31	C₄×C₁₃₇	—	2	2	no	—
32	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	Dic₁₃₇	—	2	2	no	—
34	Dic₁₃₇	#41	1	1	yes	characteristic
35	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
36	D₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
37	D₁₃₇	#135	1	1	yes	characteristic
38	C₁₃₇	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
39	C₄×Q₁₆	—	137	1	no	maximal, Sylow
40	C₂×Q₁₆	—	137	1	no	—
41	C₄×Q₈	—	137	2	no	—
42	#107	—	137	1	no	—
43	#103	—	137	2	no	—
44	C₄×C₈	—	137	1	no	—
45	C₂×Q₈	—	137	2	no	—
46	Q₁₆	—	137	2	no	—
47	Q₁₆	—	137	1	no	—
48	Q₁₆	#10587739	1	1	yes	characteristic, direct product factor
49	C₂×C₈	—	137	1	no	—
50	C₂×C₈	—	137	1	no	—
51	C₄⋊C₄	—	274	2	no	—
52	C₄⋊C₄	—	137	2	no	—
53	C₄²	—	274	2	no	—
54	C₄²	—	137	1	no	—
55	Q₈	—	274	2	no	—
56	Q₈	—	137	2	no	—
57	Q₈	C₂×#10587739	1	2	yes	—
58	C₂×C₄	—	274	2	no	—
59	C₂×C₄	—	274	2	no	—
60	C₂×C₄	—	137	1	no	—
61	C₂×C₄	—	137	1	no	—
62	C₂×C₄	—	137	1	no	—
63	C₈	—	274	1	no	—
64	C₈	—	137	1	no	—
65	C₈	C₂×#10587739	1	1	yes	characteristic
66	C₂²	—	137	1	no	—
67	C₄	—	2	2	no	—
68	C₄	—	274	1	no	—
69	C₄	—	274	2	no	—
70	C₄	—	548	2	no	—
71	C₄	—	137	2	no	—
72	C₄	—	137	1	no	—
73	C₄	C₂²×#10587739	1	1	yes	characteristic
74	C₂	—	137	1	no	—
75	C₂	—	137	1	no	—
76	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	C₄×Q₁₆
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
30. #10587739	Q₁₆	direct (N × H)
38. C₁₃₇	C₄×Q₁₆	semidirect
48. Q₁₆	#10587739	direct (N × H)

Subgroups — Q16×#10587739 #22578441

Conjugacy classes of subgroups

Complements

Subgroups — Q₁₆×#10587739 #22578441