Subgroups — #30636918

Order 16256.

74 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	#22549704	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861231	—	2	1	no	—
5	Dic₁₀₁₆	C₄	1	1	yes	characteristic, complemented
6	#16861154	—	2	1	no	—
7	D₂₀₃₂	C₄	1	1	yes	characteristic, complemented
8	#16861143	—	2	1	no	—
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₂₇	—	2	1	no	—
11	#14523248	—	4	1	no	—
12	Dic₅₀₈	—	2	1	no	—
13	#14523237	—	4	1	no	—
14	D₁₀₁₆	—	2	1	no	—
15	C₄⋊Dic₁₂₇	—	4	1	no	—
16	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
19	Dic₂₅₄	—	4	1	no	—
20	D₅₀₈	—	4	1	no	—
21	C₂²⋊D₁₂₇	—	8	1	no	—
22	C₂×Dic₁₂₇	—	8	1	no	—
23	C₄×D₁₂₇	—	8	1	no	—
24	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
25	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₂₇	—	8	1	no	—
29	C₄×C₁₂₇	#102	1	1	yes	characteristic
30	C₄×C₁₂₇	SD₃₂	1	1	yes	characteristic
31	Dic₁₂₇	—	8	1	no	—
32	Dic₁₂₇	—	16	1	no	—
33	C₂×C₁₂₇	—	2	1	no	—
34	C₂×C₁₂₇	#357	1	1	yes	characteristic
35	D₁₂₇	—	16	1	no	—
36	#1418	—	127	1	no	maximal, Sylow
37	C₁₂₇	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	—	127	1	no	—
39	M₆₄	D₁₂₇	1	1	yes	characteristic, complemented
40	#367	—	127	1	no	—
41	#135	—	254	1	no	—
42	Q₃₂	—	127	1	no	—
43	SD₃₂	—	254	1	no	—
44	D₁₆	—	127	1	no	—
45	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
46	#107	—	254	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	508	1	no	—
49	Q₁₆	—	254	1	no	—
50	SD₁₆	—	508	1	no	—
51	D₈	—	254	1	no	—
52	C₂×C₈	D₅₀₈	1	1	yes	characteristic
53	C₄⋊C₄	—	508	1	no	—
54	C₁₆	C₄×D₁₂₇	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	Q₈	—	508	1	no	—
57	D₄	—	1016	1	no	—
58	D₄	—	508	1	no	—
59	C₂×C₄	—	1016	1	no	—
60	C₂×C₄	—	1016	1	no	—
61	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
62	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
63	C₈	#14523237	1	1	yes	characteristic
64	C₂²	—	1016	1	no	—
65	C₂²	D₂₀₃₂	1	1	yes	characteristic
66	C₄	—	2032	1	no	—
67	C₄	—	1016	1	no	—
68	C₄	#16861131	1	1	yes	characteristic
69	C₄	#16861154	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2032	1	no	—
72	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	#22550308	C₂
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	M₆₄	D₁₂₇
hypocenter	C₁₂₇	#1418
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. #22549704	C₂	semidirect
3. M₆₄×C₁₂₇	C₂	semidirect
5. Dic₁₀₁₆	C₄	semidirect
7. D₂₀₃₂	C₄	semidirect
37. C₁₂₇	#1418	semidirect
39. 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
#22549704	2
M₆₄×C₁₂₇	2
#1418	127