Subgroups — #22578339

Order 8768.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10587739	C₂	1	2	yes	maximal
2	(C₂×#10587739)⋊C₄	C₂	1	4	yes	maximal
3	C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10587739	C₂²	1	4	yes	—
5	C₄×#10587739	C₄	1	8	yes	complemented
6	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
7	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
8	C₄×Dic₁₃₇	C₄	1	2	yes	complemented
9	C₄×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
10	C₄²×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#10587739	—	2	8	no	—
12	C₂×#10587739	C₂×C₄	1	8	yes	—
13	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
14	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
15	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₄×D₁₃₇	—	2	4	no	—
17	C₄×D₁₃₇	C₂×C₄	1	4	yes	—
18	C₂×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₁₃₇	C₂²×C₄	1	2	yes	—
22	C₂×D₁₃₇	#41	1	4	yes	—
23	#10587739	—	2	8	no	—
24	C₄×C₁₃₇	—	2	2	no	—
25	C₄×C₁₃₇	C₄²	1	2	yes	—
26	Dic₁₃₇	—	2	2	no	—
27	Dic₁₃₇	C₄²	1	2	yes	—
28	C₂×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₁₃₇	#117	1	2	yes	—
30	D₁₃₇	#117	1	4	yes	—
31	C₁₃₇	#376	1	1	yes	characteristic, complemented, Sylow
32	#376	—	137	1	no	maximal, Sylow
33	C₂×C₄²	—	137	2	no	—
34	C₂×C₄²	—	137	1	no	—
35	#95	—	137	4	no	—
36	C₂²×C₄	—	137	4	no	—
37	C₂²×C₄	—	137	2	no	—
38	C₂²×C₄	—	137	1	no	—
39	C₄²	—	137	8	no	—
40	C₄²	—	137	2	no	—
41	C₄²	—	137	1	no	—
42	C₄²	#10587739	1	1	yes	characteristic, complemented
43	C₂³	—	137	1	no	—
44	C₂×C₄	—	274	4	no	—
45	C₂×C₄	—	274	8	no	—
46	C₂×C₄	—	137	8	no	—
47	C₂×C₄	—	137	2	no	—
48	C₂×C₄	—	137	1	no	—
49	C₂×C₄	—	137	4	no	—
50	C₂×C₄	C₂×#10587739	1	2	yes	—
51	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
52	C₂²	—	137	2	no	—
53	C₂²	—	137	4	no	—
54	C₂²	C₂²×#10587739	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	274	8	no	—
57	C₄	—	274	2	no	—
58	C₄	—	137	2	no	—
59	C₄	C₄×#10587739	1	2	yes	—
60	C₂	—	137	4	no	—
61	C₂	C₂×C₄×#10587739	1	1	yes	characteristic
62	C₂	#17984167	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₄²
Frattini	C₂²	C₂²×#10587739
Fitting	C₄²×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₄
hypercenter	C₄²	#10587739
hypocenter	C₁₃₇	#376
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
5. C₄×#10587739	C₄	semidirect
8. C₄×Dic₁₃₇	C₄	semidirect
9. C₄×Dic₁₃₇	C₄	semidirect
10. C₄²×C₁₃₇	C₄	semidirect
31. C₁₃₇	#376	semidirect
42. C₄²	#10587739	semidirect