Subgroups — #30636927

Order 16256.

58 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	C₁₆×Dic₁₂₇	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	—	2	1	no	—
7	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₂₇	—	2	1	no	—
9	C₂×C₁₂₇⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
11	C₁₂₇⋊C₁₆	—	2	1	no	—
12	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
14	C₁₆×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₂×Dic₁₂₇	—	2	1	no	—
16	C₁₂₇⋊C₈	C₁₆	1	2	yes	—
17	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
19	C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₂₇	M₃₂	1	1	yes	characteristic
21	C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
23	Dic₁₂₇	—	2	2	no	—
24	C₂×C₁₂₇	—	2	1	no	—
25	C₂×C₁₂₇	C₄⋊C₁₆	1	1	yes	characteristic
26	#1422	—	127	1	no	maximal, Sylow
27	C₁₂₇	#1422	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	127	1	no	—
29	M₆₄	D₁₂₇	1	1	yes	characteristic
30	C₄×C₁₆	—	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	—	2	1	no	—
35	C₃₂	—	254	1	no	—
36	C₂×C₈	—	127	1	no	—
37	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
38	C₄²	—	127	1	no	—
39	C₁₆	—	254	1	no	—
40	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
41	C₁₆	Dic₂₅₄	1	1	yes	characteristic
42	C₂×C₄	—	254	1	no	—
43	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
44	C₈	—	127	2	no	—
45	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
46	C₈	#14523231	1	1	yes	characteristic
47	C₂²	#16861148	1	1	yes	characteristic
48	C₄	—	254	2	no	—
49	C₄	Dic₁₂₇⋊C₈	1	1	yes	characteristic
50	C₄	C₁₆×D₁₂₇	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	Dic₁₂₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	M₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₄⋊C₁₆
prime radical	C₂²×C₁₂₇	M₃₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#1422
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
27. C₁₂₇	#1422	semidirect

Maximal subgroups

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

subgroup	index
#22549710	2
C₁₆×Dic₁₂₇	2
M₆₄×C₁₂₇	2
#1422	127