Subgroups — #30636865

Order 16256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	1	yes	maximal, characteristic
2	#22549651	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×Dic₁₂₇	C₄	1	1	yes	characteristic
6	C₁₂₇⋊C₃₂	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₂₇	C₄	1	2	yes	complemented
9	C₂×C₁₂₇⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
11	C₁₂₇⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₁₂₇	—	2	1	no	—
16	C₁₂₇⋊C₈	—	2	1	no	—
17	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
19	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
20	C₂²×C₁₂₇	M₃₂	1	1	yes	characteristic
21	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
23	Dic₁₂₇	—	4	1	no	—
24	C₂×C₁₂₇	—	2	1	no	—
25	C₂×C₁₂₇	#346	1	1	yes	characteristic
26	#1398	—	127	1	no	maximal, Sylow
27	C₁₂₇	#1398	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	127	1	no	—
29	M₆₄	D₁₂₇	1	1	yes	characteristic
30	#346	—	127	1	no	—
31	C₂×C₁₆	—	127	1	no	—
32	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
33	C₄×C₈	—	127	1	no	—
34	C₃₂	—	127	2	no	—
35	C₃₂	Dic₁₂₇	1	2	yes	complemented
36	C₂×C₈	—	127	1	no	—
37	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
38	C₄²	—	127	1	no	—
39	C₁₆	—	127	2	no	—
40	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
41	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
42	C₂×C₄	—	254	1	no	—
43	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
44	C₈	—	254	1	no	—
45	C₈	C₄×Dic₁₂₇	1	1	yes	characteristic
46	C₈	C₈×D₁₂₇	1	1	yes	characteristic
47	C₂²	#16861148	1	1	yes	characteristic
48	C₄	—	508	1	no	—
49	C₄	C₈×Dic₁₂₇	1	1	yes	characteristic
50	C₄	#16861148	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#22549651	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₂₇
commutator	C₄×C₁₂₇	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	M₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	#346
prime radical	C₂²×C₁₂₇	M₃₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#1398
LCS	G ▷ C₄×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
6. C₁₂₇⋊C₃₂	C₄	semidirect
8. C₃₂×C₁₂₇	C₄	semidirect
27. C₁₂₇	#1398	semidirect
35. C₃₂	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549710	2
#22549651	2
M₆₄×C₁₂₇	2
#1398	127