Subgroups — #26103949

Order 12032.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241524	C₂	1	1	yes	maximal, characteristic
2	#20241454	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1369	C₂	1	1	yes	maximal, characteristic
4	#15688957	C₄	1	1	yes	characteristic
5	#15688957	C₄	1	1	yes	characteristic
6	#15688874	—	2	1	no	—
7	C₄₇×#364	—	2	1	no	—
8	C₄₇×#346	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
10	C₄₇⋊C₃₂	—	2	1	no	—
11	C₄₇⋊C₃₂	C₈	1	2	yes	complemented
12	M₃₂×C₄₇	—	4	1	no	—
13	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
18	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
19	C₂×C₄₇⋊C₈	—	8	1	no	—
20	C₁₆×C₄₇	—	2	1	no	—
21	C₁₆×C₄₇	—	8	1	no	—
22	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₄₇	—	2	1	no	—
25	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
26	C₈×C₄₇	—	2	1	no	—
27	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₄₇	#108	1	1	yes	characteristic
29	C₄₇⋊C₈	—	16	1	no	—
30	#7457	—	47	1	no	maximal, Sylow
31	C₂²×C₄₇	#365	1	1	yes	characteristic
32	C₄×C₄₇	—	4	1	no	—
33	C₄×C₄₇	#334	1	1	yes	characteristic
34	C₄×C₄₇	#365	1	1	yes	characteristic
35	#1398	—	47	1	no	—
36	#1369	D₄₇	1	1	yes	characteristic
37	#1368	—	47	1	no	—
38	C₂×C₄₇	—	2	1	no	—
39	C₂×C₄₇	#1368	1	1	yes	characteristic
40	M₆₄	—	47	1	no	—
41	M₆₄	—	47	1	no	—
42	#364	—	2	1	no	—
43	#346	C₂×D₄₇	1	1	yes	characteristic
44	#334	—	94	1	no	—
45	C₄₇	#7457	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	4	1	no	—
47	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
48	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
49	C₄⋊C₈	—	188	1	no	—
50	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
51	C₃₂	—	47	2	no	—
52	C₃₂	—	94	1	no	—
53	C₂×C₈	—	376	1	no	—
54	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
55	C₂×C₈	#10625335	1	1	yes	characteristic
56	C₄²	#10625336	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	8	1	no	—
59	C₁₆	C₈×D₄₇	1	1	yes	characteristic
60	C₁₆	#10625324	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#13093955	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	752	1	no	—
65	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
66	C₈	#13093959	1	1	yes	characteristic
67	C₂²	#15688947	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#15688873	1	1	yes	characteristic
70	C₄	#15688945	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#20241453	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20241453
commutator	C₁₆×C₄₇	C₂×C₈
Frattini	#346	C₂×D₄₇
Fitting	C₄₇×#1369	C₂
socle	C₂×C₄₇	#1368
prime radical	C₂²×C₄₇	#365
cosocle	C₄₇×#346	C₂²
hypercenter	#1369	D₄₇
hypocenter	C₄₇	#7457
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1369

Complements

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

N	H ≅ G/N	type
11. C₄₇⋊C₃₂	C₈	semidirect
45. C₄₇	#7457	semidirect

Maximal subgroups

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

subgroup	index
#20241524	2
#20241454	2
C₄₇×#1369	2
#7457	47