Subgroups — #26104171

Order 12032.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241599	C₂	1	1	yes	maximal, characteristic
2	#20241592	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×Q₆₄	C₂	1	1	yes	maximal, characteristic
4	#15688953	—	2	1	no	—
5	C₄₇⋊C₆₄	—	2	1	no	—
6	C₂×Q₃₂×C₄₇	—	2	1	no	—
7	C₄₇×Q₆₄	—	2	1	no	—
8	C₄₇×Q₆₄	C₄	1	2	yes	—
9	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
10	#13093960	—	4	1	no	—
11	C₂×Q₁₆×C₄₇	—	4	1	no	—
12	Q₃₂×C₄₇	—	2	2	no	—
13	Q₃₂×C₄₇	—	4	1	no	—
14	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
15	C₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₄₇	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₄₇	—	8	1	no	—
18	Q₁₆×C₄₇	—	4	2	no	—
19	Q₁₆×C₄₇	—	8	1	no	—
20	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
21	#10625329	—	8	1	no	—
22	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
24	Q₈×C₄₇	—	8	2	no	—
25	Q₈×C₄₇	—	16	1	no	—
26	C₂×C₄×C₄₇	—	16	1	no	—
27	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
28	C₈×C₄₇	#102	1	1	yes	characteristic
29	C₈×C₄₇	SD₃₂	1	1	yes	characteristic
30	C₄₇⋊C₈	—	16	1	no	—
31	#7542	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	SD₆₄	1	1	yes	characteristic
33	C₄×C₄₇	—	16	2	no	—
34	C₄×C₄₇	#357	1	1	yes	characteristic
35	C₄×C₄₇	D₃₂	1	1	yes	characteristic
36	C₂×Q₆₄	D₄₇	1	1	yes	characteristic
37	M₁₂₈	—	47	1	no	—
38	#1425	—	47	1	no	—
39	C₂×C₄₇	—	2	1	no	—
40	C₂×C₄₇	#1415	1	1	yes	characteristic
41	C₂×Q₃₂	—	2	1	no	—
42	Q₆₄	—	2	1	no	—
43	Q₆₄	Dic₄₇	1	2	yes	—
44	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
45	#368	—	94	1	no	—
46	C₆₄	—	94	1	no	—
47	C₄₇	#7542	1	1	yes	characteristic, complemented, Sylow
48	C₂×Q₁₆	—	4	1	no	—
49	Q₃₂	—	2	2	no	—
50	Q₃₂	—	4	1	no	—
51	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
52	#108	—	188	1	no	—
53	C₃₂	C₂×Dic₄₇	1	1	yes	characteristic
54	C₃₂	C₂²⋊D₄₇	1	1	yes	characteristic
55	C₂×Q₈	—	8	1	no	—
56	Q₁₆	—	4	2	no	—
57	Q₁₆	—	8	1	no	—
58	M₁₆	—	376	1	no	—
59	C₂×C₈	#10625334	1	1	yes	characteristic
60	C₁₆	#10625335	1	1	yes	characteristic
61	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
62	Q₈	—	8	2	no	—
63	Q₈	—	16	1	no	—
64	C₂×C₄	—	16	1	no	—
65	C₂×C₄	#13093965	1	1	yes	characteristic
66	C₈	—	752	1	no	—
67	C₈	#13093945	1	1	yes	characteristic
68	C₈	#13093968	1	1	yes	characteristic
69	C₂²	#15688961	1	1	yes	characteristic
70	C₄	—	16	2	no	—
71	C₄	#15688924	1	1	yes	characteristic
72	C₄	#15688958	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#20241566	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20241566
commutator	C₃₂×C₄₇	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₂×C₄₇×Q₆₄	C₂
socle	C₂×C₄₇	#1415
prime radical	C₂²×C₄₇	SD₆₄
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	C₂×Q₆₄	D₄₇
hypocenter	C₄₇	#7542
LCS	G ▷ C₃₂×C₄₇ ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
47. C₄₇	#7542	semidirect

Maximal subgroups

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

subgroup	index
#20241599	2
#20241592	2
C₂×C₄₇×Q₆₄	2
#7542	47