Subgroups — #30636866

Order 16256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	2	yes	maximal
2	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇⋊C₃₂	C₄	1	4	yes	complemented
4	C₂×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₂₇	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₁₂₇	C₈	1	1	yes	characteristic
10	C₁₆×C₁₂₇	C₈	1	2	yes	—
11	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₂₇	—	2	1	no	—
13	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
14	C₈×C₁₂₇	—	2	1	no	—
15	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
16	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₁₂₇	M₃₂	1	1	yes	characteristic
18	C₄×C₁₂₇	—	4	1	no	—
19	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
20	C₄×C₁₂₇	M₃₂	1	1	yes	characteristic
21	C₂×C₁₂₇	—	2	1	no	—
22	C₂×C₁₂₇	#346	1	1	yes	characteristic
23	#1398	—	127	1	no	maximal, Sylow
24	C₁₂₇	#1398	1	1	yes	characteristic, complemented, Sylow
25	M₆₄	—	127	2	no	—
26	#346	D₁₂₇	1	1	yes	characteristic
27	C₂×C₁₆	Dic₁₂₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
29	C₄×C₈	Dic₁₂₇	1	1	yes	characteristic
30	C₃₂	—	127	4	no	—
31	C₂×C₈	C₁₂₇⋊C₈	1	1	yes	characteristic
32	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
33	C₄²	C₁₂₇⋊C₈	1	1	yes	characteristic
34	C₁₆	C₁₂₇⋊C₈	1	2	yes	—
35	C₁₆	C₄×D₁₂₇	1	2	yes	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₄×Dic₁₂₇	1	1	yes	characteristic
40	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
41	C₂²	#16861149	1	1	yes	characteristic
42	C₄	—	4	1	no	—
43	C₄	C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
44	C₄	#16861149	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#22549652	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₂₇⋊C₃₂	C₄	semidirect
24. C₁₂₇	#1398	semidirect

Maximal subgroups

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

subgroup	index
#22549710	2
#346×C₁₂₇	2
#1398	127