Subgroups — #22578316

Order 8768.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984161	C₂	1	1	yes	maximal, characteristic
2	#17984157	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#10587739	—	2	1	no	—
5	#14555513	—	2	1	no	—
6	Q₈×D₁₃₇	—	2	1	no	—
7	Dic₅₄₈	C₄	1	1	yes	characteristic, complemented
8	#14555500	—	2	1	no	—
9	C₈×D₁₃₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#10587739	—	4	1	no	—
12	Dic₂₇₄	—	2	1	no	—
13	Dic₂₇₄	—	4	1	no	—
14	C₄×D₁₃₇	—	4	1	no	—
15	C₄×D₁₃₇	D₄	1	1	yes	characteristic
16	#11764040	D₄	1	1	yes	characteristic
17	Q₈×C₁₃₇	—	2	1	no	—
18	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₁₃₇	D₈	1	1	yes	characteristic
20	#10587739	—	8	1	no	—
21	C₄×C₁₃₇	—	4	1	no	—
22	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₁₃₇	—	4	1	no	—
24	Dic₁₃₇	SD₁₆	1	1	yes	characteristic
25	C₂×C₁₃₇	#102	1	1	yes	characteristic
26	D₁₃₇	SD₃₂	1	1	yes	characteristic
27	D₁₃₇	Q₃₂	1	1	yes	characteristic
28	C₁₃₇	#358	1	1	yes	characteristic, complemented, Sylow
29	#358	—	137	1	no	maximal, Sylow
30	C₂×Q₁₆	—	137	1	no	—
31	C₂×C₁₆	—	137	1	no	—
32	#107	—	137	1	no	—
33	C₂×Q₈	—	274	1	no	—
34	Q₁₆	—	274	1	no	—
35	Q₁₆	—	137	1	no	—
36	Q₁₆	#10587739	1	1	yes	characteristic, complemented
37	C₂×C₈	—	137	1	no	—
38	C₄⋊C₄	—	274	1	no	—
39	C₁₆	—	274	1	no	—
40	Q₈	—	2	1	no	—
41	Q₈	—	274	1	no	—
42	Q₈	—	548	1	no	—
43	C₂×C₄	—	548	1	no	—
44	C₂×C₄	—	548	1	no	—
45	C₂×C₄	—	137	1	no	—
46	C₈	—	137	1	no	—
47	C₈	C₂×#10587739	1	1	yes	characteristic
48	C₂²	—	137	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	548	1	no	—
51	C₄	—	1096	1	no	—
52	C₄	—	137	1	no	—
53	C₄	C₂²⋊#10587739	1	1	yes	characteristic
54	C₂	—	137	1	no	—
55	C₂	—	137	1	no	—
56	C₂	#17984150	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17984150
commutator	C₈×C₁₃₇	C₂×C₄
Frattini	C₈	C₂×#10587739
Fitting	Q₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	#102
hypercenter	Q₁₆	#10587739
hypocenter	C₁₃₇	#358
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
7. Dic₅₄₈	C₄	semidirect
10. Q₁₆×C₁₃₇	C₄	semidirect
28. C₁₃₇	#358	semidirect
36. Q₁₆	#10587739	semidirect