Subgroups — #26103962

Order 12032.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241522	C₂	1	1	yes	maximal, characteristic
2	#20241464	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₄₇⋊C₃₂	C₄	1	1	yes	characteristic
6	#15688944	—	2	1	no	—
7	C₄₇×#364	—	2	1	no	—
8	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	#13093964	—	4	1	no	—
10	C₄₇⋊C₃₂	—	2	1	no	—
11	C₄₇⋊C₃₂	C₈	1	2	yes	—
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	D₈	1	1	yes	characteristic
18	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
19	C₁₆×C₄₇	—	2	1	no	—
20	C₁₆×C₄₇	—	8	1	no	—
21	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
23	C₄₇⋊C₁₆	—	8	1	no	—
24	C₂×C₄×C₄₇	—	2	1	no	—
25	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
26	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
27	C₈×C₄₇	#108	1	1	yes	characteristic
28	C₈×C₄₇	SD₃₂	1	1	yes	characteristic
29	C₈×C₄₇	SD₃₂	1	1	yes	characteristic
30	#7464	—	47	1	no	maximal, Sylow
31	C₂²×C₄₇	#368	1	1	yes	characteristic
32	C₄×C₄₇	—	4	1	no	—
33	C₄×C₄₇	#335	1	1	yes	characteristic
34	C₄×C₄₇	#367	1	1	yes	characteristic
35	#1397	—	47	1	no	—
36	#1373	—	47	1	no	—
37	#1373	D₄₇	1	1	yes	characteristic
38	C₂×C₄₇	—	2	1	no	—
39	C₂×C₄₇	#1372	1	1	yes	characteristic
40	C₂×C₃₂	—	47	1	no	—
41	C₂×C₃₂	—	47	1	no	—
42	#364	—	2	1	no	—
43	#364	—	94	1	no	—
44	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
45	C₄₇	#7464	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	4	1	no	—
47	M₃₂	—	188	1	no	—
48	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
49	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
50	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
51	C₃₂	—	47	2	no	—
52	C₃₂	—	94	1	no	—
53	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
54	C₂×C₈	#10625334	1	1	yes	characteristic
55	C₄²	#10625337	1	1	yes	characteristic
56	C₁₆	—	2	1	no	—
57	C₁₆	—	8	1	no	—
58	C₁₆	—	376	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	#13093957	1	1	yes	characteristic
63	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
64	C₈	#13093959	1	1	yes	characteristic
65	C₈	#13093967	1	1	yes	characteristic
66	C₈	#13093968	1	1	yes	characteristic
67	C₂²	#15688952	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#15688875	1	1	yes	characteristic
70	C₄	#15688950	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#20241461	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15688950
commutator	C₁₆×C₄₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×#1373	C₂
socle	C₂×C₄₇	#1372
prime radical	C₂²×C₄₇	#368
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	#1373	D₄₇
hypocenter	C₄₇	#7464
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
45. C₄₇	#7464	semidirect

Maximal subgroups

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

subgroup	index
#20241522	2
#20241464	2
C₄₇×#1373	2
#7464	47