Subgroups — #30636920

Order 16256.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549710	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549706	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861145	—	2	1	no	—
5	C₁₂₇⋊C₃₂	—	2	1	no	—
6	C₂×D₈×C₁₂₇	—	2	1	no	—
7	D₁₆×C₁₂₇	—	2	1	no	—
8	D₁₆×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	#14523232	—	4	1	no	—
11	C₂×D₄×C₁₂₇	—	4	1	no	—
12	D₈×C₁₂₇	—	2	2	no	—
13	D₈×C₁₂₇	—	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	C₁₂₇⋊C₈	—	8	1	no	—
18	C₂³×C₁₂₇	—	8	1	no	—
19	D₄×C₁₂₇	—	4	2	no	—
20	D₄×C₁₂₇	—	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	—	8	2	no	—
25	C₂²×C₁₂₇	—	16	2	no	—
26	C₂²×C₁₂₇	SD₃₂	1	1	yes	characteristic
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	—	16	2	no	—
31	C₂×C₁₂₇	#357	1	1	yes	characteristic
32	#1419	—	127	1	no	maximal, Sylow
33	C₁₂₇	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	D₁₂₇	1	1	yes	characteristic
35	M₆₄	—	127	1	no	—
36	#368	—	127	1	no	—
37	C₂×D₈	—	2	1	no	—
38	D₁₆	—	2	1	no	—
39	D₁₆	Dic₁₂₇	1	2	yes	—
40	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
41	#108	—	254	1	no	—
42	C₃₂	—	254	1	no	—
43	C₂×D₄	—	4	1	no	—
44	D₈	—	2	2	no	—
45	D₈	—	4	1	no	—
46	M₁₆	—	508	1	no	—
47	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
48	C₁₆	C₂×Dic₁₂₇	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
50	C₂³	—	8	1	no	—
51	D₄	—	4	2	no	—
52	D₄	—	8	1	no	—
53	C₂×C₄	#14523233	1	1	yes	characteristic
54	C₈	—	1016	1	no	—
55	C₈	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
56	C₈	#14523236	1	1	yes	characteristic
57	C₂²	—	8	2	no	—
58	C₂²	—	16	2	no	—
59	C₂²	#16861153	1	1	yes	characteristic
60	C₄	#16861130	1	1	yes	characteristic
61	C₄	#16861150	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	16	2	no	—
64	C₂	#22549677	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549677
commutator	C₁₆×C₁₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₂₇
Fitting, prime radical	C₂×D₁₆×C₁₂₇	C₂
socle	C₂×C₁₂₇	#357
cosocle	C₂×C₁₆×C₁₂₇	C₂²
hypercenter	C₂×D₁₆	D₁₂₇
hypocenter	C₁₂₇	#1419
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22549710	C₂	semidirect
2. #22549706	C₂	semidirect
33. C₁₂₇	#1419	semidirect

Maximal subgroups

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

subgroup	index
#22549710	2
#22549706	2
C₂×D₁₆×C₁₂₇	2
#1419	127