Subgroups — #30641040

Order 16256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550309	C₂	1	2	yes	maximal, complemented
2	#22549713	C₂	1	2	yes	maximal, complemented
3	D₄₀₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861232	—	2	2	no	—
7	#16861152	—	2	2	no	—
8	D₂₀₃₂	C₂²	1	2	yes	—
9	C₁₆×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₁₂₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₂₇	C₂²	1	2	yes	—
12	C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
13	#14523249	—	4	2	no	—
14	#14523235	—	4	2	no	—
15	D₁₀₁₆	—	2	2	no	—
16	C₈×D₁₂₇	D₄	1	1	yes	characteristic
17	C₁₂₇⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₂₇	—	2	2	no	—
19	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
20	D₅₀₈	—	4	2	no	—
21	D₅₀₈	—	8	2	no	—
22	C₄×D₁₂₇	—	8	2	no	—
23	C₄×D₁₂₇	D₈	1	1	yes	characteristic
24	C₁₂₇⋊C₈	D₈	1	1	yes	characteristic
25	Q₈×C₁₂₇	—	4	2	no	—
26	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₂₇	—	8	2	no	—
28	C₂×D₁₂₇	D₁₆	1	1	yes	characteristic
29	C₄×C₁₂₇	—	8	2	no	—
30	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
31	Dic₁₂₇	D₁₆	1	1	yes	characteristic
32	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
33	D₁₂₇	—	2	1	no	—
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	D₁₂₇	1	1	yes	characteristic, complemented
39	SD₆₄	—	127	2	no	—
40	D₃₂	—	127	1	no	—
41	C₂×C₃₂	—	127	1	no	—
42	#135	—	254	2	no	—
43	Q₃₂	C₂×D₁₂₇	1	2	yes	—
44	SD₃₂	—	254	2	no	—
45	D₁₆	—	127	2	no	—
46	C₂×C₁₆	—	127	1	no	—
47	C₃₂	—	127	1	no	—
48	C₃₂	C₂×D₁₂₇	1	1	yes	characteristic
49	#41	—	508	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	508	2	no	—
52	D₈	—	254	2	no	—
53	C₂×C₈	—	127	1	no	—
54	C₁₆	—	127	1	no	—
55	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	1016	2	no	—
58	D₄	—	508	2	no	—
59	C₂×C₄	—	127	1	no	—
60	C₂×C₄	—	1016	2	no	—
61	C₈	—	127	1	no	—
62	C₈	D₄×D₁₂₇	1	1	yes	characteristic
63	C₂²	—	127	1	no	—
64	C₂²	—	1016	2	no	—
65	C₄	—	8	2	no	—
66	C₄	—	127	1	no	—
67	C₄	D₈×D₁₂₇	1	1	yes	characteristic
68	C₂	—	2032	2	no	—
69	C₂	—	254	1	no	—
70	C₂	D₁₆×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₂₇
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	Q₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₁₆
hypercenter	Q₆₄	D₁₂₇
hypocenter	C₁₂₇	#2262
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #22550309	C₂	semidirect
2. #22549713	C₂	semidirect
3. D₄₀₆₄	C₂	semidirect
4. C₃₂×D₁₂₇	C₂	semidirect
5. Q₆₄×C₁₂₇	C₂	semidirect
36. C₁₂₇	#2262	semidirect
38. Q₆₄	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550309	2
#22549713	2
D₄₀₆₄	2
C₃₂×D₁₂₇	2
Q₆₄×C₁₂₇	2
#2262	127