Subgroups — #15689928

Order 3008.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094070	C₂	1	2	yes	maximal, complemented
2	#13094052	C₂	1	6	yes	maximal, complemented
3	#13094046	C₂	1	3	yes	maximal, complemented
4	C₂×D₃₇₆	C₂	1	3	yes	maximal, complemented
5	#131×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₇	C₂²	1	1	yes	characteristic
7	#10625354	—	2	2	no	—
8	D₄×D₄₇	—	2	6	no	—
9	#10625350	C₂²	1	6	yes	complemented
10	C₂×D₁₈₈	C₂²	1	6	yes	complemented
11	M₁₆×C₄₇	C₂²	1	3	yes	complemented
12	C₂×C₈×C₄₇	C₂²	1	3	yes	complemented
13	Dic₁₈₈	C₂²	1	1	yes	characteristic, complemented
14	D₃₇₆	C₂²	1	3	yes	complemented
15	D₃₇₆	C₂²	1	6	yes	complemented
16	#10625325	C₂²	1	6	yes	complemented
17	C₂²×D₄₇	—	2	6	no	—
18	Q₈×C₄₇	D₄	1	1	yes	characteristic
19	D₄×C₄₇	D₄	1	3	yes	—
20	C₂×C₄×C₄₇	C₂³	1	3	yes	—
21	C₂²⋊D₄₇	—	2	6	no	—
22	D₁₈₈	—	2	6	no	—
23	D₁₈₈	C₂³	1	6	yes	—
24	C₄×D₄₇	—	2	6	no	—
25	Dic₉₄	C₂³	1	2	yes	—
26	C₈×C₄₇	C₂³	1	3	yes	—
27	C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄₇	C₂×D₄	1	3	yes	—
29	C₂×D₄₇	—	2	6	no	—
30	C₂×D₄₇	—	8	6	no	—
31	C₄×C₄₇	C₂×D₄	1	3	yes	—
32	C₄×C₄₇	C₂⁴	1	1	yes	characteristic
33	Dic₄₇	—	2	2	no	—
34	C₂×C₄₇	—	2	3	no	—
35	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
36	D₄₇	—	4	6	no	—
37	#576	—	47	1	no	maximal, Sylow
38	C₄₇	#576	1	1	yes	characteristic, complemented, Sylow
39	H(5,𝔽₂)	—	47	2	no	—
40	AGL(1,ℤ/8)	—	47	6	no	—
41	#135	—	47	3	no	—
42	C₂×D₈	—	47	3	no	—
43	#131	D₄₇	1	1	yes	characteristic, complemented
44	#41	—	47	6	no	—
45	#41	—	94	2	no	—
46	#41	C₂×D₄₇	1	1	yes	characteristic
47	C₂×D₄	—	47	6	no	—
48	C₂×D₄	—	94	6	no	—
49	Q₁₆	—	47	1	no	—
50	SD₁₆	—	47	6	no	—
51	D₈	—	47	3	no	—
52	D₈	—	47	6	no	—
53	M₁₆	C₂×D₄₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₄₇	1	3	yes	complemented
55	C₂³	—	94	6	no	—
56	Q₈	—	47	2	no	—
57	Q₈	D₁₈₈	1	1	yes	characteristic
58	D₄	—	47	6	no	—
59	D₄	—	94	6	no	—
60	D₄	—	94	6	no	—
61	D₄	D₁₈₈	1	3	yes	—
62	C₂×C₄	—	94	6	no	—
63	C₂×C₄	C₂²×D₄₇	1	3	yes	—
64	C₈	C₂²×D₄₇	1	1	yes	characteristic
65	C₈	C₂²×D₄₇	1	3	yes	—
66	C₂²	—	94	6	no	—
67	C₂²	—	376	6	no	—
68	C₂²	C₂×D₁₈₈	1	3	yes	—
69	C₄	—	94	2	no	—
70	C₄	C₂×D₁₈₈	1	3	yes	—
71	C₄	C₂³×D₄₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	188	6	no	—
74	C₂	C₂²×D₁₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₈₈
commutator, cosocle	C₄×C₄₇	C₂⁴
Frattini	C₄	C₂³×D₄₇
Fitting	#131×C₄₇	C₂
socle	C₂×C₄₇	C₂²×D₄
hypercenter	#131	D₄₇
hypocenter	C₄₇	#576
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #13094070	C₂	semidirect
2. #13094052	C₂	semidirect
3. #13094046	C₂	semidirect
4. C₂×D₃₇₆	C₂	semidirect
5. #131×C₄₇	C₂	semidirect
9. #10625350	C₂²	semidirect
10. C₂×D₁₈₈	C₂²	semidirect
11. M₁₆×C₄₇	C₂²	semidirect
12. C₂×C₈×C₄₇	C₂²	semidirect
13. Dic₁₈₈	C₂²	semidirect
14. D₃₇₆	C₂²	semidirect
15. D₃₇₆	C₂²	semidirect
16. #10625325	C₂²	semidirect
38. C₄₇	#576	semidirect
43. #131	D₄₇	semidirect
53. M₁₆	C₂×D₄₇	semidirect
54. C₂×C₈	C₂×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#13094070	2
#13094052	2
#13094046	2
C₂×D₃₇₆	2
#131×C₄₇	2
#576	47