Subgroups — #30636924

Order 16256.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	#22549706	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₀₈	—	2	1	no	—
5	Dic₁₀₁₆	—	2	1	no	—
6	Dic₁₀₁₆	C₄	1	2	yes	—
7	#16861145	—	2	1	no	—
8	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₂₇	—	2	1	no	—
10	C₂×Dic₂₅₄	—	4	1	no	—
11	Dic₅₀₈	—	2	2	no	—
12	Dic₅₀₈	—	4	1	no	—
13	#14523232	—	4	1	no	—
14	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
17	Dic₂₅₄	—	4	2	no	—
18	Dic₂₅₄	—	8	1	no	—
19	C₂×Dic₁₂₇	—	8	1	no	—
20	C₁₂₇⋊C₈	—	8	1	no	—
21	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
22	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₂₇	SD₃₂	1	1	yes	characteristic
25	C₄×C₁₂₇	#102	1	1	yes	characteristic
26	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
27	Dic₁₂₇	—	8	2	no	—
28	C₂×C₁₂₇	—	2	1	no	—
29	C₂×C₁₂₇	#357	1	1	yes	characteristic
30	#1420	—	127	1	no	maximal, Sylow
31	C₁₂₇	#1420	1	1	yes	characteristic, complemented, Sylow
32	C₂×Q₃₂	—	127	1	no	—
33	M₆₄	D₁₂₇	1	1	yes	characteristic
34	#368	—	127	1	no	—
35	C₂×Q₁₆	—	254	1	no	—
36	Q₃₂	—	127	2	no	—
37	Q₃₂	—	254	1	no	—
38	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
39	#108	—	254	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	508	1	no	—
42	Q₁₆	—	508	1	no	—
43	Q₁₆	—	254	2	no	—
44	M₁₆	—	508	1	no	—
45	C₂×C₈	D₅₀₈	1	1	yes	characteristic
46	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
48	Q₈	—	1016	1	no	—
49	Q₈	—	508	2	no	—
50	C₂×C₄	—	1016	1	no	—
51	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
52	C₈	—	1016	1	no	—
53	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
54	C₈	#14523237	1	1	yes	characteristic
55	C₂²	#16861154	1	1	yes	characteristic
56	C₄	—	1016	2	no	—
57	C₄	#16861131	1	1	yes	characteristic
58	C₄	D₂₀₃₂	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#22549678	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549678
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting	M₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	#357
prime radical	C₂²×C₁₂₇	SD₃₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#1420
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
31. C₁₂₇	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₁₆	2
#22549706	2
M₆₄×C₁₂₇	2
#1420	127