Subgroups — #30641005

Order 16256.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550281	C₂	1	2	yes	maximal
2	#22549702	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861209	—	2	2	no	—
5	#16861143	—	2	2	no	—
6	#16861143	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
8	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₅₄	—	4	2	no	—
10	C₄⋊Dic₁₂₇	—	2	4	no	—
11	C₄⋊Dic₁₂₇	—	4	2	no	—
12	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
14	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₂₇	Q₈	1	4	yes	complemented
16	Dic₂₅₄	—	8	4	no	—
17	C₂×Dic₁₂₇	—	4	4	no	—
18	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₂₇	Q₁₆	1	2	yes	—
22	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
23	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
24	C₄×C₁₂₇	D₁₆	1	2	yes	—
25	C₄×C₁₂₇	Q₃₂	1	2	yes	—
26	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₂₇	—	8	4	no	—
29	C₂×C₁₂₇	#500	1	1	yes	characteristic
30	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
32	#2252	—	127	1	no	maximal, Sylow
33	C₁₂₇	#2252	1	1	yes	characteristic, complemented, Sylow
34	#500	—	127	2	no	—
35	#366	—	127	4	no	—
36	C₄×C₁₆	D₁₂₇	1	1	yes	characteristic
37	C₄⋊Q₈	—	254	2	no	—
38	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
39	#107	—	127	4	no	—
40	#107	—	254	2	no	—
41	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
42	C₂×Q₈	—	508	2	no	—
43	C₂×C₈	D₅₀₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
45	C₄⋊C₄	—	508	2	no	—
46	C₄⋊C₄	—	254	4	no	—
47	C₄²	D₅₀₈	1	1	yes	characteristic
48	C₁₆	Dic₂₅₄	1	4	yes	complemented
49	Q₈	—	1016	4	no	—
50	C₂×C₄	—	508	4	no	—
51	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
52	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
53	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
54	C₈	Dic₅₀₈	1	2	yes	—
55	C₈	C₂×Dic₂₅₄	1	2	yes	—
56	C₂²	C₂×D₁₀₁₆	1	1	yes	characteristic
57	C₄	—	1016	4	no	—
58	C₄	D₂₀₃₂	1	2	yes	—
59	C₄	Dic₁₀₁₆	1	2	yes	—
60	C₄	#16861209	1	1	yes	characteristic
61	C₄	C₂×Dic₅₀₈	1	1	yes	characteristic
62	C₂	#22550281	1	1	yes	characteristic
63	C₂	C₂×D₂₀₃₂	1	1	yes	characteristic
64	C₂	C₂×Dic₁₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₁₆
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₄×C₁₆×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#2252
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₁₂₇	Q₈	semidirect
33. C₁₂₇	#2252	semidirect
48. C₁₆	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
#22550281	2
#22549702	2
C₄×C₁₆×C₁₂₇	2
#2252	127