Subgroups — #26104179

Order 12032.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇⋊C₆₄	C₂	1	1	yes	maximal, characteristic
2	#20241588	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1423	C₂	1	1	yes	maximal, characteristic
4	#15688949	—	2	1	no	—
5	C₄₇⋊C₆₄	C₄	1	2	yes	complemented
6	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#366	—	2	1	no	—
8	#13093958	—	4	1	no	—
9	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
10	#107×C₄₇	—	4	1	no	—
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⋊C₄×C₄₇	—	8	1	no	—
15	C₄⋊Dic₄₇	—	8	1	no	—
16	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₄₇	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₄₇	—	16	1	no	—
19	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
20	C₂×Dic₄₇	—	16	1	no	—
21	C₈×C₄₇	#107	1	1	yes	characteristic
22	C₈×C₄₇	Q₃₂	1	1	yes	characteristic
23	#7546	—	47	1	no	maximal, Sylow
24	C₂²×C₄₇	D₃₂	1	1	yes	characteristic
25	C₄×C₄₇	—	32	1	no	—
26	C₄×C₄₇	#366	1	1	yes	characteristic
27	C₄×C₄₇	Q₆₄	1	1	yes	characteristic
28	Dic₄₇	—	32	1	no	—
29	C₂×C₆₄	—	47	1	no	—
30	#1423	—	47	1	no	—
31	#1423	D₄₇	1	1	yes	characteristic
32	C₂×C₄₇	#1423	1	1	yes	characteristic
33	C₂×C₄₇	SD₁₂₈	1	1	yes	characteristic
34	C₂×C₄₇	SD₁₂₈	1	1	yes	characteristic
35	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
36	#366	—	2	1	no	—
37	#366	—	94	1	no	—
38	C₆₄	—	47	2	no	—
39	C₄₇	#7546	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
41	#107	—	4	1	no	—
42	#107	—	188	1	no	—
43	C₃₂	Dic₉₄	1	1	yes	characteristic
44	C₃₂	C₄×D₄₇	1	1	yes	characteristic
45	C₂×C₈	#10625334	1	1	yes	characteristic
46	C₄⋊C₄	—	8	1	no	—
47	C₄⋊C₄	—	376	1	no	—
48	C₁₆	Dic₄₇⋊C₄	1	1	yes	characteristic
49	C₁₆	#10625337	1	1	yes	characteristic
50	C₂×C₄	—	16	1	no	—
51	C₂×C₄	—	752	1	no	—
52	C₂×C₄	#13093965	1	1	yes	characteristic
53	C₈	#13093957	1	1	yes	characteristic
54	C₈	#13093970	1	1	yes	characteristic
55	C₂²	#15688958	1	1	yes	characteristic
56	C₄	—	32	1	no	—
57	C₄	—	1504	1	no	—
58	C₄	#15688948	1	1	yes	characteristic
59	C₄	#15688963	1	1	yes	characteristic
60	C₂	#20241587	1	1	yes	characteristic
61	C₂	#20241602	1	1	yes	characteristic
62	C₂	#20241603	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688958
commutator	C₃₂×C₄₇	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄₇×#1423	C₂
socle, prime radical	C₂²×C₄₇	D₃₂
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	#1423	D₄₇
hypocenter	C₄₇	#7546
LCS	G ▷ C₃₂×C₄₇ ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
5. C₄₇⋊C₆₄	C₄	semidirect
39. C₄₇	#7546	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇⋊C₆₄	2
#20241588	2
C₄₇×#1423	2
#7546	47