Subgroups — #26137317

Order 12032.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241599	C₂	1	3	yes	maximal, complemented
2	C₂×C₄₇⋊C₆₄	C₂	1	3	yes	maximal, complemented
3	C₄₇×#2258	C₂	1	1	yes	maximal, characteristic
4	C₄₇⋊C₆₄	C₂²	1	1	yes	characteristic
5	C₄₇⋊C₆₄	C₂²	1	3	yes	—
6	C₄₇×#504	C₄	1	1	yes	characteristic
7	C₄₇×M₆₄	C₄	1	3	yes	—
8	C₂×C₃₂×C₄₇	C₂²	1	3	yes	—
9	#131×C₄₇	C₈	1	1	yes	characteristic
10	M₃₂×C₄₇	C₈	1	3	yes	—
11	C₂×C₁₆×C₄₇	C₂×C₄	1	3	yes	—
12	C₃₂×C₄₇	C₂×C₄	1	3	yes	—
13	C₃₂×C₄₇	C₂³	1	1	yes	characteristic
14	#41×C₄₇	C₁₆	1	1	yes	characteristic
15	M₁₆×C₄₇	C₁₆	1	3	yes	—
16	C₂×C₈×C₄₇	C₂×C₈	1	3	yes	—
17	C₁₆×C₄₇	C₂×C₈	1	3	yes	—
18	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
19	Q₈×C₄₇	C₃₂	1	1	yes	characteristic
20	D₄×C₄₇	C₃₂	1	3	yes	—
21	C₂×C₄×C₄₇	C₂×C₁₆	1	3	yes	—
22	C₈×C₄₇	C₂×C₁₆	1	3	yes	—
23	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
24	#13737	—	47	1	no	maximal, Sylow
25	C₂²×C₄₇	C₂×C₃₂	1	3	yes	—
26	C₄×C₄₇	C₂×C₃₂	1	3	yes	—
27	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
28	#2258	D₄₇	1	1	yes	characteristic
29	M₁₂₈	—	47	3	no	—
30	C₂×C₆₄	—	47	3	no	—
31	C₂×C₄₇	—	2	3	no	—
32	C₂×C₄₇	C₂²×C₃₂	1	1	yes	characteristic
33	#504	Dic₄₇	1	1	yes	characteristic
34	M₆₄	Dic₄₇	1	3	yes	—
35	C₂×C₃₂	C₂×D₄₇	1	3	yes	—
36	C₆₄	—	47	1	no	—
37	C₆₄	—	47	3	no	—
38	C₄₇	#13737	1	1	yes	characteristic, complemented, Sylow
39	#131	C₄₇⋊C₈	1	1	yes	characteristic
40	M₃₂	C₄₇⋊C₈	1	3	yes	—
41	C₂×C₁₆	C₂×Dic₄₇	1	3	yes	—
42	C₃₂	C₂×Dic₄₇	1	3	yes	—
43	C₃₂	C₂²×D₄₇	1	1	yes	characteristic
44	#41	C₄₇⋊C₁₆	1	1	yes	characteristic
45	M₁₆	C₄₇⋊C₁₆	1	3	yes	—
46	C₂×C₈	C₂×C₄₇⋊C₈	1	3	yes	—
47	C₁₆	C₂×C₄₇⋊C₈	1	3	yes	—
48	C₁₆	C₂²×Dic₄₇	1	1	yes	characteristic
49	Q₈	C₄₇⋊C₃₂	1	1	yes	characteristic
50	D₄	C₄₇⋊C₃₂	1	3	yes	—
51	C₂×C₄	C₂×C₄₇⋊C₁₆	1	3	yes	—
52	C₈	C₂×C₄₇⋊C₁₆	1	3	yes	—
53	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
54	C₂²	C₂×C₄₇⋊C₃₂	1	3	yes	—
55	C₄	C₂×C₄₇⋊C₃₂	1	3	yes	—
56	C₄	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₄₇⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₄₇
commutator, socle	C₂×C₄₇	C₂²×C₃₂
Fitting	C₄₇×#2258	C₂
prime radical	#41×C₄₇	C₁₆
cosocle	C₃₂×C₄₇	C₂³
hypercenter	#2258	D₄₇
hypocenter	C₄₇	#13737
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #20241599	C₂	semidirect
2. C₂×C₄₇⋊C₆₄	C₂	semidirect
38. C₄₇	#13737	semidirect

Maximal subgroups

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

subgroup	index
#20241599	2
C₂×C₄₇⋊C₆₄	2
C₄₇×#2258	2
#13737	47