Subgroups — #16861241

Order 4064.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14523249	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#14523238	C₂	1	1	yes	maximal, characteristic, complemented
4	#14523237	C₂	1	1	yes	maximal, characteristic, complemented
5	#14523235	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523231	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₂₅₄	—	2	1	no	—
9	Dic₂₅₄	C₂²	1	1	yes	characteristic
10	D₅₀₈	—	2	1	no	—
11	D₅₀₈	C₂²	1	1	yes	characteristic
12	C₄×D₁₂₇	—	2	1	no	—
13	C₄×D₁₂₇	—	2	1	no	—
14	C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×D₁₂₇	—	2	1	no	—
20	C₂×D₁₂₇	D₄	1	1	yes	characteristic
21	C₄×C₁₂₇	—	2	1	no	—
22	C₄×C₁₂₇	—	2	1	no	—
23	C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	Dic₁₂₇	—	2	1	no	—
25	Dic₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	D₁₂₇	—	2	1	no	—
28	D₁₂₇	—	4	1	no	—
29	C₁₂₇	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	127	1	no	maximal, Sylow
31	#41	—	127	1	no	—
32	C₂×Q₈	—	127	1	no	—
33	Q₁₆	—	127	1	no	—
34	Q₁₆	D₁₂₇	1	1	yes	characteristic, complemented
35	SD₁₆	—	127	1	no	—
36	SD₁₆	—	127	1	no	—
37	M₁₆	—	127	1	no	—
38	Q₈	—	127	1	no	—
39	Q₈	—	254	1	no	—
40	Q₈	C₂×D₁₂₇	1	1	yes	characteristic
41	Q₈	C₂×D₁₂₇	1	1	yes	characteristic
42	D₄	—	127	1	no	—
43	D₄	—	254	1	no	—
44	C₂×C₄	—	127	1	no	—
45	C₂×C₄	—	254	1	no	—
46	C₂×C₄	—	254	1	no	—
47	C₈	—	127	1	no	—
48	C₈	C₂×D₁₂₇	1	1	yes	characteristic
49	C₂²	—	127	1	no	—
50	C₂²	—	254	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	127	1	no	—
54	C₄	—	254	1	no	—
55	C₄	C₂²×D₁₂₇	1	1	yes	characteristic
56	C₂	—	508	1	no	—
57	C₂	—	254	1	no	—
58	C₂	D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₂₇
commutator, cosocle	C₄×C₁₂₇	C₂³
Frattini	C₄	C₂²×D₁₂₇
Fitting	Q₁₆×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₄
prime radical	#14523249	C₂
hypercenter	Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#137
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₁₂₇	C₂	semidirect
3. #14523238	C₂	semidirect
4. #14523237	C₂	semidirect
5. #14523235	C₂	semidirect
6. #14523231	C₂	semidirect
7. Q₁₆×C₁₂₇	C₂	semidirect
29. C₁₂₇	#137	semidirect
34. Q₁₆	D₁₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14523249	2
Q₈×D₁₂₇	2
#14523238	2
#14523237	2
#14523235	2
#14523231	2
Q₁₆×C₁₂₇	2
#137	127