Subgroups — #22577509

Order 8768.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅₄₈	C₂	1	1	yes	maximal, characteristic
2	#17984053	C₂	1	2	yes	maximal
3	#17984011	C₂	1	2	yes	maximal
4	#17983998	C₂	1	1	yes	maximal, characteristic
5	#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₇₄	C₂²	1	2	yes	—
7	Dic₅₄₈	C₄	1	4	yes	complemented
8	C₂×#11764040	C₂²	1	1	yes	characteristic
9	#14555489	—	2	2	no	—
10	C₄×Dic₁₃₇	—	2	2	no	—
11	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
14	Dic₂₇₄	—	2	2	no	—
15	Dic₂₇₄	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₃₇	—	2	2	no	—
17	C₂×Dic₁₃₇	D₄	1	2	yes	—
18	#11764040	—	2	1	no	—
19	C₂×C₄×C₁₃₇	—	2	2	no	—
20	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₈×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₃₇	—	4	2	no	—
24	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₃₇	#41	1	1	yes	characteristic
26	Dic₁₃₇	—	2	4	no	—
27	Dic₁₃₇	—	2	2	no	—
28	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
29	C₂×C₁₃₇	#137	1	2	yes	—
30	C₁₃₇	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	137	1	no	maximal, Sylow
32	C₂×Q₁₆	—	137	1	no	—
33	C₄×Q₈	—	137	2	no	—
34	#106	D₁₃₇	1	1	yes	characteristic
35	#103	—	137	2	no	—
36	#97	—	137	1	no	—
37	C₂×Q₈	—	137	2	no	—
38	Q₁₆	—	137	4	no	—
39	C₂×C₈	—	137	1	no	—
40	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
41	C₄⋊C₄	—	274	2	no	—
42	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
43	C₄²	—	274	2	no	—
44	C₄²	—	137	1	no	—
45	Q₈	—	274	2	no	—
46	Q₈	—	137	4	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	274	2	no	—
49	C₂×C₄	—	137	2	no	—
50	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
51	C₈	—	274	1	no	—
52	C₈	C₄×D₁₃₇	1	2	yes	—
53	C₂²	D₄×D₁₃₇	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	274	4	no	—
56	C₄	—	274	2	no	—
57	C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
58	C₄	#14555511	1	1	yes	characteristic
59	C₂	#17984050	1	1	yes	characteristic
60	C₂	#17984117	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#106×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#106	D₁₃₇
hypocenter	C₁₃₇	#441
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
7. Dic₅₄₈	C₄	semidirect
30. C₁₃₇	#441	semidirect