Subgroups — #26104084

Order 12032.

74 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	#20241521	C₂	1	1	yes	maximal, characteristic
3	C₄₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	C₄	1	1	yes	characteristic
5	C₁₆×Dic₄₇	C₄	1	1	yes	characteristic
6	C₄₇⋊C₆₄	C₄	1	2	yes	complemented
7	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×C₆₄	C₄	1	2	yes	complemented
9	C₂×C₄₇⋊C₁₆	C₈	1	1	yes	characteristic
10	C₈×Dic₄₇	C₈	1	1	yes	characteristic
11	C₄₇⋊C₃₂	C₈	1	2	yes	—
12	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
16	C₄×Dic₄₇	C₁₆	1	1	yes	characteristic
17	C₂×C₄₇⋊C₈	C₁₆	1	1	yes	characteristic
18	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
19	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
20	C₄₇⋊C₁₆	C₁₆	1	2	yes	—
21	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₄₇	—	2	1	no	—
23	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
24	C₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
25	C₄₇⋊C₈	—	2	1	no	—
26	#7511	—	47	1	no	maximal, Sylow
27	C₂²×C₄₇	M₆₄	1	1	yes	characteristic
28	C₄×C₄₇	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₄₇	M₆₄	1	1	yes	characteristic
30	Dic₄₇	—	4	1	no	—
31	M₁₂₈	—	47	1	no	—
32	M₁₂₈	D₄₇	1	1	yes	characteristic
33	#1397	—	47	1	no	—
34	C₂×C₄₇	—	2	1	no	—
35	C₂×C₄₇	#1397	1	1	yes	characteristic
36	C₂×C₃₂	—	47	1	no	—
37	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
38	C₄×C₁₆	—	47	1	no	—
39	C₆₄	—	47	2	no	—
40	C₆₄	Dic₄₇	1	2	yes	complemented
41	C₄₇	#7511	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₁₆	—	47	1	no	—
43	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
44	C₄×C₈	—	47	1	no	—
45	C₃₂	—	47	2	no	—
46	C₃₂	C₄×D₄₇	1	1	yes	characteristic
47	C₃₂	C₂×Dic₄₇	1	1	yes	characteristic
48	C₂×C₈	—	47	1	no	—
49	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
50	C₄²	—	47	1	no	—
51	C₁₆	—	47	2	no	—
52	C₁₆	C₈×D₄₇	1	1	yes	characteristic
53	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
54	C₂×C₄	—	94	1	no	—
55	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
56	C₈	—	94	1	no	—
57	C₈	C₈×Dic₄₇	1	1	yes	characteristic
58	C₈	C₁₆×D₄₇	1	1	yes	characteristic
59	C₂²	#15688956	1	1	yes	characteristic
60	C₄	—	188	1	no	—
61	C₄	C₁₆×Dic₄₇	1	1	yes	characteristic
62	C₄	#15688956	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#20241521	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₄₇
commutator	C₄×C₄₇	C₄×C₁₆
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄₇×M₁₂₈	C₂
socle	C₂×C₄₇	#1397
prime radical	C₂²×C₄₇	M₆₄
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	M₁₂₈	D₄₇
hypocenter	C₄₇	#7511
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
6. C₄₇⋊C₆₄	C₄	semidirect
8. C₄₇×C₆₄	C₄	semidirect
40. C₆₄	Dic₄₇	semidirect
41. C₄₇	#7511	semidirect

Maximal subgroups

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

subgroup	index
#20241599	2
#20241521	2
C₄₇×M₁₂₈	2
#7511	47