Subgroups — #15689522

Order 3008.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094016	C₂	1	1	yes	maximal, characteristic
2	#13093957	C₂	1	2	yes	maximal
3	#13093955	C₂	1	2	yes	maximal
4	C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#125×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×C₄₇	—	2	2	no	—
7	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
8	C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	Dic₄₇⋊C₄	—	2	2	no	—
11	C₂×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇	—	2	2	no	—
14	C₂×C₄×C₄₇	D₄	1	2	yes	—
15	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×Dic₄₇	—	2	2	no	—
17	C₄₇⋊C₈	Q₈	1	2	yes	—
18	C₄₇⋊C₈	Q₈	1	2	yes	—
19	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₄₇	—	2	2	no	—
21	C₄×C₄₇	—	4	2	no	—
22	C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
24	Dic₄₇	—	4	2	no	—
25	C₂×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₄₇	#135	1	2	yes	—
27	#499	—	47	1	no	maximal, Sylow
28	C₄₇	#499	1	1	yes	characteristic, complemented, Sylow
29	#125	—	47	1	no	—
30	#125	D₄₇	1	1	yes	characteristic
31	#107	—	47	2	no	—
32	#106	—	47	2	no	—
33	C₄×C₈	—	47	1	no	—
34	C₂×C₈	—	47	1	no	—
35	C₂×C₈	—	47	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	47	2	no	—
38	C₄⋊C₄	—	94	2	no	—
39	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
40	C₄²	C₂×D₄₇	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	94	2	no	—
43	C₂×C₄	C₂²⋊D₄₇	1	2	yes	—
44	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
45	C₈	—	47	2	no	—
46	C₈	—	47	2	no	—
47	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	188	2	no	—
51	C₄	Q₈×D₄₇	1	1	yes	characteristic
52	C₄	Q₈×D₄₇	1	1	yes	characteristic
53	C₂	#13094022	1	1	yes	characteristic
54	C₂	#13094044	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂³
Frattini	C₂×C₄	C₂²×D₄₇
Fitting	#125×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₄
hypercenter	#125	D₄₇
hypocenter	C₄₇	#499
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
28. C₄₇	#499	semidirect

Maximal subgroups

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

subgroup	index
#13094016	2
#13093957	2
#13093955	2
C₄×C₄₇⋊C₈	2
#125×C₄₇	2
#499	47