Subgroups — #26104184

Order 12032.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241592	C₂	1	1	yes	maximal, characteristic
2	#20241590	C₂	1	1	yes	maximal, characteristic
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	#15688953	—	2	1	no	—
5	#15688949	—	2	1	no	—
6	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×C₆₄	C₄	1	2	yes	complemented
8	#13093960	—	4	1	no	—
9	#13093958	—	4	1	no	—
10	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
11	C₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₄₇	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
14	C₄⋊Dic₄₇	—	8	1	no	—
15	#10625329	—	8	1	no	—
16	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₄₇	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
19	C₂×Dic₄₇	—	16	1	no	—
20	C₈×C₄₇	#107	1	1	yes	characteristic
21	C₈×C₄₇	Q₃₂	1	1	yes	characteristic
22	C₄₇⋊C₈	—	16	1	no	—
23	#7548	—	47	1	no	maximal, Sylow
24	C₂²×C₄₇	SD₆₄	1	1	yes	characteristic
25	C₄×C₄₇	#366	1	1	yes	characteristic
26	C₄×C₄₇	SD₆₄	1	1	yes	characteristic
27	Dic₄₇	—	32	1	no	—
28	M₁₂₈	D₄₇	1	1	yes	characteristic
29	#1425	—	47	1	no	—
30	#1424	—	47	1	no	—
31	C₂×C₄₇	—	2	1	no	—
32	C₂×C₄₇	#1424	1	1	yes	characteristic
33	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
34	#368	—	94	1	no	—
35	#366	—	94	1	no	—
36	C₆₄	Dic₄₇	1	2	yes	complemented
37	C₄₇	#7548	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
39	#108	—	188	1	no	—
40	#107	—	188	1	no	—
41	C₃₂	Dic₉₄	1	1	yes	characteristic
42	C₃₂	C₂×Dic₄₇	1	1	yes	characteristic
43	M₁₆	—	376	1	no	—
44	C₂×C₈	D₃₇₆	1	1	yes	characteristic
45	C₄⋊C₄	—	376	1	no	—
46	C₁₆	Dic₁₈₈	1	1	yes	characteristic
47	C₁₆	C₄⋊Dic₄₇	1	1	yes	characteristic
48	C₂×C₄	—	752	1	no	—
49	C₂×C₄	D₇₅₂	1	1	yes	characteristic
50	C₈	—	752	1	no	—
51	C₈	#13093958	1	1	yes	characteristic
52	C₈	Dic₃₇₆	1	1	yes	characteristic
53	C₂²	#15688962	1	1	yes	characteristic
54	C₄	—	1504	1	no	—
55	C₄	#15688949	1	1	yes	characteristic
56	C₄	#15688962	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#20241590	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20241590
commutator	C₃₂×C₄₇	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄₇×M₁₂₈	C₂
socle	C₂×C₄₇	#1424
prime radical	C₂²×C₄₇	SD₆₄
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7548
LCS	G ▷ C₃₂×C₄₇ ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
7. C₄₇×C₆₄	C₄	semidirect
36. C₆₄	Dic₄₇	semidirect
37. C₄₇	#7548	semidirect

Maximal subgroups

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

subgroup	index
#20241592	2
#20241590	2
C₄₇×M₁₂₈	2
#7548	47