Subgroups — #22577771

Order 8768.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984080	C₂	1	1	yes	maximal, characteristic
2	#17984012	C₂	1	4	yes	maximal
3	#17983999	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#11764040	C₂²	1	2	yes	—
6	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
7	#14555489	—	2	2	no	—
8	C₂×Q₈×C₁₃₇	C₂²	1	2	yes	—
9	C₄⋊C₄×C₁₃₇	—	2	2	no	—
10	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₃₇	—	2	2	no	—
12	#11764040	—	2	2	no	—
13	Q₈×C₁₃₇	—	2	4	no	—
14	C₂×C₄×C₁₃₇	—	2	2	no	—
15	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
16	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
17	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
18	C₄×C₁₃₇	—	2	2	no	—
19	C₄×C₁₃₇	—	4	2	no	—
20	C₄×C₁₃₇	#41	1	2	yes	—
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	D₁₃₇	1	1	yes	characteristic
27	#125	—	137	1	no	—
28	#103	—	137	4	no	—
29	#97	—	137	1	no	—
30	C₂×Q₈	C₂×D₁₃₇	1	2	yes	—
31	C₂×C₈	—	137	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	274	2	no	—
34	C₄⋊C₄	—	137	2	no	—
35	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
36	Q₈	—	2	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	274	2	no	—
39	C₂×C₄	C₂²⋊D₁₃₇	1	2	yes	—
40	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
41	C₈	—	274	2	no	—
42	C₂²	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	548	2	no	—
46	C₄	#14555509	1	2	yes	—
47	C₂	#17984075	1	1	yes	characteristic
48	C₂	#17984119	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₄⋊Q₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	#491
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

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