Subgroups — #30641050

Order 16256.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550308	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₀₃₂	C₂	1	2	yes	maximal, complemented
4	#22549715	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861231	—	2	1	no	—
7	C₂×Dic₅₀₈	—	2	1	no	—
8	Dic₁₀₁₆	—	2	1	no	—
9	Dic₁₀₁₆	C₂²	1	2	yes	—
10	Dic₁₀₁₆	C₂²	1	1	yes	characteristic
11	#16861154	—	2	1	no	—
12	D₂₀₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₂₇	C₂²	1	2	yes	—
15	#14523248	—	4	1	no	—
16	C₂×Dic₂₅₄	—	4	1	no	—
17	Dic₅₀₈	—	2	2	no	—
18	Dic₅₀₈	—	2	1	no	—
19	Dic₅₀₈	—	4	1	no	—
20	#14523237	—	4	1	no	—
21	D₁₀₁₆	—	2	1	no	—
22	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
24	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
25	Dic₂₅₄	—	4	1	no	—
26	Dic₂₅₄	—	4	2	no	—
27	Dic₂₅₄	—	8	1	no	—
28	D₅₀₈	—	4	1	no	—
29	C₂²⋊D₁₂₇	—	8	1	no	—
30	C₂×Dic₁₂₇	—	8	1	no	—
31	C₄×D₁₂₇	—	8	1	no	—
32	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
33	C₈×C₁₂₇	D₈	1	1	yes	characteristic
34	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
36	C₂×D₁₂₇	—	8	1	no	—
37	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
38	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
39	Dic₁₂₇	—	8	1	no	—
40	Dic₁₂₇	—	8	2	no	—
41	C₂×C₁₂₇	—	2	1	no	—
42	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
43	D₁₂₇	—	16	1	no	—
44	#2264	—	127	1	no	maximal, Sylow
45	C₁₂₇	#2264	1	1	yes	characteristic, complemented, Sylow
46	#508	—	127	1	no	—
47	C₂×Q₃₂	—	127	1	no	—
48	Q₆₄	—	127	2	no	—
49	SD₆₄	—	127	2	no	—
50	M₆₄	D₁₂₇	1	1	yes	characteristic, complemented
51	#135	—	254	1	no	—
52	C₂×Q₁₆	—	254	1	no	—
53	Q₃₂	—	127	2	no	—
54	Q₃₂	—	127	1	no	—
55	Q₃₂	—	254	1	no	—
56	SD₃₂	—	254	1	no	—
57	D₁₆	—	127	1	no	—
58	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
59	C₃₂	C₂×D₁₂₇	1	2	yes	—
60	#41	—	508	1	no	—
61	C₂×Q₈	—	508	1	no	—
62	Q₁₆	—	508	1	no	—
63	Q₁₆	—	254	1	no	—
64	Q₁₆	—	254	2	no	—
65	SD₁₆	—	508	1	no	—
66	D₈	—	254	1	no	—
67	C₂×C₈	D₅₀₈	1	1	yes	characteristic
68	C₁₆	D₅₀₈	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
70	Q₈	—	1016	1	no	—
71	Q₈	—	508	1	no	—
72	Q₈	—	508	2	no	—
73	D₄	—	1016	1	no	—
74	D₄	—	508	1	no	—
75	C₂×C₄	—	1016	1	no	—
76	C₂×C₄	—	1016	1	no	—
77	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
78	C₈	D₁₀₁₆	1	1	yes	characteristic
79	C₈	C₂×D₅₀₈	1	1	yes	characteristic
80	C₂²	—	1016	1	no	—
81	C₂²	D₂₀₃₂	1	1	yes	characteristic
82	C₄	—	1016	1	no	—
83	C₄	—	1016	2	no	—
84	C₄	D₂₀₃₂	1	1	yes	characteristic
85	C₄	C₂×D₁₀₁₆	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2032	1	no	—
88	C₂	C₂×D₂₀₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₀₃₂
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	M₆₄×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₁₆
prime radical	#22550308	C₂
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#2264
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. C₂×Dic₁₀₁₆	C₂	semidirect
3. Dic₂₀₃₂	C₂	semidirect
4. #22549715	C₂	semidirect
5. M₆₄×C₁₂₇	C₂	semidirect
45. C₁₂₇	#2264	semidirect
50. M₆₄	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
C₂×Dic₁₀₁₆	2
Dic₂₀₃₂	2
#22549715	2
M₆₄×C₁₂₇	2
#2264	127