Subgroups — #22578449

Order 8768.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984176	C₂	1	2	yes	maximal, complemented
2	#17984158	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984153	C₂	1	2	yes	maximal, complemented
4	#17984142	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984111	C₂	1	1	yes	maximal, characteristic
6	#14555520	—	2	2	no	—
7	#14555520	C₂²	1	2	yes	—
8	#14555518	—	2	2	no	—
9	#14555518	C₂²	1	1	yes	characteristic
10	C₄×#10587739	C₂²	1	1	yes	characteristic
11	#14555511	C₂²	1	2	yes	—
12	#14555497	C₄	1	2	yes	complemented
13	D₁₀₉₆	C₄	1	1	yes	characteristic, complemented
14	C₈×D₁₃₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#10587739	—	2	1	no	—
17	#11764047	—	2	2	no	—
18	#11764047	D₄	1	2	yes	—
19	D₅₄₈	—	2	2	no	—
20	D₅₄₈	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	—	2	2	no	—
27	C₂×D₁₃₇	#41	1	1	yes	characteristic
28	#10587739	—	4	1	no	—
29	C₄×C₁₃₇	—	2	2	no	—
30	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
33	D₁₃₇	—	2	1	no	—
34	D₁₃₇	—	4	2	no	—
35	C₁₃₇	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	137	1	no	maximal, Sylow
37	#135	—	137	1	no	—
38	#131	—	137	2	no	—
39	#108	—	137	1	no	—
40	#104	—	137	2	no	—
41	#97	—	137	1	no	—
42	#41	—	137	2	no	—
43	Q₁₆	#10587739	1	1	yes	characteristic, complemented
44	SD₁₆	—	137	2	no	—
45	D₈	—	137	1	no	—
46	M₁₆	—	274	2	no	—
47	M₁₆	—	137	2	no	—
48	C₂×C₈	—	274	2	no	—
49	C₂×C₈	—	137	1	no	—
50	C₂×C₈	—	137	1	no	—
51	C₄²	—	137	1	no	—
52	Q₈	C₂×#10587739	1	2	yes	—
53	D₄	—	274	2	no	—
54	D₄	—	137	2	no	—
55	C₂×C₄	—	274	2	no	—
56	C₂×C₄	—	274	1	no	—
57	C₂×C₄	—	137	1	no	—
58	C₈	—	274	2	no	—
59	C₈	—	137	2	no	—
60	C₈	—	137	1	no	—
61	C₈	C₂×#10587739	1	1	yes	characteristic
62	C₂²	—	274	2	no	—
63	C₂²	—	137	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	548	1	no	—
66	C₄	—	137	1	no	—
67	C₄	C₂²×#10587739	1	1	yes	characteristic
68	C₂	—	274	1	no	—
69	C₂	—	548	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	#444
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
1. #17984176	C₂	semidirect
2. #17984158	C₂	semidirect
3. #17984153	C₂	semidirect
4. #17984142	C₂	semidirect
12. #14555497	C₄	semidirect
13. D₁₀₉₆	C₄	semidirect
15. Q₁₆×C₁₃₇	C₄	semidirect
35. C₁₃₇	#444	semidirect
43. Q₁₆	#10587739	semidirect