Subgroups — #22577772

Order 8768.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984088	C₂	1	1	yes	maximal, characteristic
2	#17984080	C₂	1	1	yes	maximal, characteristic
3	#17984013	C₂	1	4	yes	maximal
4	#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₇₄	C₂²	1	2	yes	—
6	C₄⋊Dic₁₃₇	—	2	2	no	—
7	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
8	#14555489	—	2	2	no	—
9	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	Dic₂₇₄	—	2	4	no	—
12	C₂×Dic₁₃₇	—	2	2	no	—
13	C₂×Dic₁₃₇	—	2	2	no	—
14	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
15	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
16	C₈×C₁₃₇	—	2	2	no	—
17	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
18	C₄×C₁₃₇	—	2	2	no	—
19	C₄×C₁₃₇	#41	1	2	yes	—
20	Dic₁₃₇	—	4	2	no	—
21	Dic₁₃₇	—	4	2	no	—
22	C₂×C₁₃₇	#124	1	1	yes	characteristic
23	C₂×C₁₃₇	#137	1	2	yes	—
24	C₁₃₇	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	137	1	no	maximal, Sylow
26	C₄⋊Q₈	—	137	1	no	—
27	#125	—	137	1	no	—
28	#103	—	137	4	no	—
29	#97	D₁₃₇	1	1	yes	characteristic
30	C₂×Q₈	—	137	2	no	—
31	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
32	C₄⋊C₄	—	274	2	no	—
33	C₄⋊C₄	—	274	2	no	—
34	C₄⋊C₄	—	137	2	no	—
35	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
36	Q₈	—	274	4	no	—
37	C₂×C₄	—	274	2	no	—
38	C₂×C₄	—	274	2	no	—
39	C₂×C₄	D₅₄₈	1	2	yes	—
40	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	548	2	no	—
45	C₄	—	548	2	no	—
46	C₄	#14555510	1	2	yes	—
47	C₂	#17984076	1	1	yes	characteristic
48	C₂	#17984121	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#97×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#97	D₁₃₇
hypocenter	C₁₃₇	#491
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
24. C₁₃₇	#491	semidirect