Subgroups — #26103961

Order 12032.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241462	C₂	1	2	yes	maximal
2	C₄₇×#1397	C₂	1	1	yes	maximal, characteristic
3	#15688876	—	2	2	no	—
4	C₂×C₃₂×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₄₇	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
7	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
8	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
11	C₃₂×C₄₇	—	2	1	no	—
12	C₃₂×C₄₇	C₈	1	2	yes	complemented
13	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
15	C₄²×C₄₇	D₈	1	1	yes	characteristic
16	C₂×C₄₇⋊C₈	—	8	2	no	—
17	C₁₆×C₄₇	—	2	1	no	—
18	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
21	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
22	C₈×C₄₇	—	2	1	no	—
23	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₄₇	#108	1	1	yes	characteristic
25	C₄₇⋊C₈	—	16	2	no	—
26	#7463	—	47	1	no	maximal, Sylow
27	C₂²×C₄₇	#367	1	1	yes	characteristic
28	C₄×C₄₇	—	2	2	no	—
29	C₄×C₄₇	#335	1	1	yes	characteristic
30	C₄×C₄₇	#368	1	1	yes	characteristic
31	#1397	D₄₇	1	1	yes	characteristic
32	#1372	—	47	2	no	—
33	C₂×C₄₇	#1372	1	1	yes	characteristic
34	C₂×C₄₇	#1426	1	2	yes	—
35	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
36	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
37	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
38	#335	—	94	2	no	—
39	C₄₇	#7463	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
42	C₄⋊C₈	—	188	2	no	—
43	C₄×C₈	D₁₈₈	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	C₄₇⋊C₈	1	2	yes	complemented
46	C₂×C₈	—	376	2	no	—
47	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
48	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
49	C₄²	D₃₇₆	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×C₄₇⋊C₈	1	1	yes	characteristic
52	C₁₆	#10625329	1	1	yes	characteristic
53	C₂×C₄	#13093958	1	1	yes	characteristic
54	C₂×C₄	#13093969	1	2	yes	—
55	C₈	—	2	1	no	—
56	C₈	—	752	2	no	—
57	C₈	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
58	C₈	#13093960	1	1	yes	characteristic
59	C₂²	#15688951	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#15688876	1	1	yes	characteristic
62	C₄	#15688953	1	1	yes	characteristic
63	C₂	#20241462	1	1	yes	characteristic
64	C₂	#20241594	1	2	yes	—

	subgroup	G/H
center	C₂²	#15688951
commutator	C₁₆×C₄₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×#1397	C₂
socle, prime radical	C₂²×C₄₇	#367
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	#1397	D₄₇
hypocenter	C₄₇	#7463
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₄₇	C₈	semidirect
39. C₄₇	#7463	semidirect
45. C₃₂	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20241462	2
C₄₇×#1397	2
#7463	47