Subgroups — #30641039

Order 16256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550308	C₂	1	2	yes	maximal, complemented
2	Dic₂₀₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549715	C₂	1	2	yes	maximal, complemented
4	D₄₀₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861231	—	2	2	no	—
7	Dic₁₀₁₆	C₂²	1	2	yes	—
8	#16861154	—	2	2	no	—
9	D₂₀₃₂	C₂²	1	2	yes	—
10	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
13	#14523248	—	4	2	no	—
14	Dic₅₀₈	—	2	2	no	—
15	#14523237	—	4	2	no	—
16	D₁₀₁₆	—	2	2	no	—
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
19	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
20	Dic₂₅₄	—	4	2	no	—
21	D₅₀₈	—	4	2	no	—
22	C₂²⋊D₁₂₇	—	8	2	no	—
23	C₄×D₁₂₇	—	8	2	no	—
24	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
25	C₈×C₁₂₇	D₈	1	1	yes	characteristic
26	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₂₇	—	8	2	no	—
29	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
30	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
31	Dic₁₂₇	—	8	2	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
34	D₁₂₇	—	16	2	no	—
35	#2262	—	127	1	no	maximal, Sylow
36	C₁₂₇	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	127	2	no	—
38	Q₆₄	—	127	1	no	—
39	SD₆₄	—	127	2	no	—
40	D₃₂	—	127	1	no	—
41	C₂×C₃₂	D₁₂₇	1	1	yes	characteristic, complemented
42	#135	—	254	2	no	—
43	Q₃₂	—	127	2	no	—
44	SD₃₂	—	254	2	no	—
45	D₁₆	—	127	2	no	—
46	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
47	C₃₂	C₂×D₁₂₇	1	1	yes	characteristic
48	C₃₂	C₂×D₁₂₇	1	1	yes	characteristic
49	#41	—	508	2	no	—
50	Q₁₆	—	254	2	no	—
51	SD₁₆	—	508	2	no	—
52	D₈	—	254	2	no	—
53	C₂×C₈	D₅₀₈	1	1	yes	characteristic
54	C₁₆	D₅₀₈	1	1	yes	characteristic
55	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
56	Q₈	—	508	2	no	—
57	D₄	—	1016	2	no	—
58	D₄	—	508	2	no	—
59	C₂×C₄	—	1016	2	no	—
60	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
61	C₈	D₁₀₁₆	1	1	yes	characteristic
62	C₈	C₂×D₅₀₈	1	1	yes	characteristic
63	C₂²	—	1016	2	no	—
64	C₂²	D₂₀₃₂	1	1	yes	characteristic
65	C₄	—	1016	2	no	—
66	C₄	D₂₀₃₂	1	1	yes	characteristic
67	C₄	C₂×D₁₀₁₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2032	2	no	—
70	C₂	C₂×D₂₀₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₀₃₂
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	C₂×C₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₁₆
hypercenter	C₂×C₃₂	D₁₂₇
hypocenter	C₁₂₇	#2262
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #22550308	C₂	semidirect
2. Dic₂₀₃₂	C₂	semidirect
3. #22549715	C₂	semidirect
4. D₄₀₆₄	C₂	semidirect
5. C₂×C₃₂×C₁₂₇	C₂	semidirect
36. C₁₂₇	#2262	semidirect
41. C₂×C₃₂	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550308	2
Dic₂₀₃₂	2
#22549715	2
D₄₀₆₄	2
C₂×C₃₂×C₁₂₇	2
#2262	127