Subgroups — #30636921

Order 16256.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₃₂	C₂	1	1	yes	maximal, characteristic
2	#22549706	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₁₆	—	2	1	no	—
5	D₂₀₃₂	—	2	1	no	—
6	D₂₀₃₂	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×D₅₀₈	—	4	1	no	—
11	D₁₀₁₆	—	2	2	no	—
12	D₁₀₁₆	—	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	C₂²×D₁₂₇	—	8	1	no	—
18	D₅₀₈	—	4	2	no	—
19	D₅₀₈	—	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	C₂×D₁₂₇	—	8	2	no	—
26	C₂×D₁₂₇	—	16	2	no	—
27	C₄×C₁₂₇	#102	1	1	yes	characteristic
28	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
29	C₂×C₁₂₇	—	2	1	no	—
30	C₂×C₁₂₇	#357	1	1	yes	characteristic
31	D₁₂₇	—	16	2	no	—
32	#1419	—	127	1	no	maximal, Sylow
33	C₁₂₇	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	127	1	no	—
35	M₆₄	D₁₂₇	1	1	yes	characteristic, complemented
36	#368	—	127	1	no	—
37	C₂×D₈	—	254	1	no	—
38	D₁₆	—	127	2	no	—
39	D₁₆	—	254	1	no	—
40	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
41	#108	—	254	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	508	1	no	—
44	D₈	—	508	1	no	—
45	D₈	—	254	2	no	—
46	M₁₆	—	508	1	no	—
47	C₂×C₈	D₅₀₈	1	1	yes	characteristic
48	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
50	C₂³	—	1016	1	no	—
51	D₄	—	1016	1	no	—
52	D₄	—	508	2	no	—
53	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
54	C₈	—	1016	1	no	—
55	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
56	C₈	#14523237	1	1	yes	characteristic
57	C₂²	—	2032	2	no	—
58	C₂²	—	1016	2	no	—
59	C₂²	#16861154	1	1	yes	characteristic
60	C₄	#16861131	1	1	yes	characteristic
61	C₄	D₂₀₃₂	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	2032	2	no	—
64	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	C₂×D₂₀₃₂	C₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#1419
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
2. #22549706	C₂	semidirect
3. M₆₄×C₁₂₇	C₂	semidirect
33. C₁₂₇	#1419	semidirect
35. M₆₄	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₃₂	2
#22549706	2
M₆₄×C₁₂₇	2
#1419	127