Subgroups — #30638340

Order 16256.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550281	C₂	1	2	yes	maximal
2	#22550227	C₂	1	4	yes	maximal
3	C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861209	C₂²	1	4	yes	—
5	#16861143	—	2	4	no	—
6	#16861134	—	2	8	no	—
7	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×Dic₂₅₄	—	2	4	no	—
10	C₄⋊Dic₁₂₇	—	2	4	no	—
11	C₄⋊Dic₁₂₇	—	2	4	no	—
12	C₂×C₈×C₁₂₇	D₄	1	4	yes	—
13	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
14	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
15	Dic₂₅₄	—	4	8	no	—
16	C₂×Dic₁₂₇	—	4	4	no	—
17	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
19	C₈×C₁₂₇	—	2	4	no	—
20	C₈×C₁₂₇	Q₁₆	1	4	yes	—
21	C₂²×C₁₂₇	#127	1	1	yes	characteristic
22	C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
23	C₄×C₁₂₇	#135	1	4	yes	—
24	Dic₁₂₇	—	8	4	no	—
25	C₂×C₁₂₇	#494	1	1	yes	characteristic
26	C₂×C₁₂₇	#495	1	2	yes	—
27	#1710	—	127	1	no	maximal, Sylow
28	C₁₂₇	#1710	1	1	yes	characteristic, complemented, Sylow
29	#500	—	127	2	no	—
30	#487	—	127	4	no	—
31	C₈²	D₁₂₇	1	1	yes	characteristic
32	C₄⋊Q₈	—	127	4	no	—
33	#107	—	254	4	no	—
34	#103	—	254	8	no	—
35	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
36	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
37	C₂×Q₈	—	254	4	no	—
38	C₂×C₈	D₅₀₈	1	4	yes	—
39	C₂×C₈	D₅₀₈	1	2	yes	—
40	C₄⋊C₄	—	254	4	no	—
41	C₄⋊C₄	—	254	4	no	—
42	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
43	Q₈	—	508	8	no	—
44	C₂×C₄	—	508	4	no	—
45	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	Dic₅₀₈	1	4	yes	—
49	C₂²	#16861206	1	1	yes	characteristic
50	C₄	—	1016	4	no	—
51	C₄	C₂×Dic₅₀₈	1	2	yes	—
52	C₄	#16861231	1	4	yes	—
53	C₂	#22550256	1	1	yes	characteristic
54	C₂	#22550260	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861206
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	C₈²×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#127
hypercenter	C₈²	D₁₂₇
hypocenter	C₁₂₇	#1710
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
28. C₁₂₇	#1710	semidirect

Maximal subgroups

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

subgroup	index
#22550281	2
#22550227	2
C₈²×C₁₂₇	2
#1710	127