Subgroups — #15689860

Order 3008.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094018	C₂	1	2	yes	maximal, complemented
2	#13094005	C₂	1	3	yes	maximal, complemented
3	#13093996	C₂	1	6	yes	maximal, complemented
4	#13093992	C₂	1	3	yes	maximal, complemented
5	#127×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₇	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₄₇	C₂²	1	3	yes	—
8	C₂×D₄×C₄₇	C₂²	1	3	yes	complemented
9	C₂×C₂²⋊D₄₇	C₂²	1	6	yes	—
10	C₂²×Dic₄₇	C₂²	1	3	yes	complemented
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊Dic₄₇	C₂²	1	3	yes	—
13	C₂²⋊Dic₄₇	C₂²	1	3	yes	complemented
14	(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
15	Dic₄₇⋊C₄	C₂²	1	6	yes	complemented
16	C₂³×C₄₇	C₂³	1	1	yes	characteristic
17	C₂³×C₄₇	C₂³	1	3	yes	—
18	C₂²×D₄₇	—	4	3	no	—
19	C₂²×D₄₇	C₂³	1	2	yes	—
20	D₄×C₄₇	—	4	3	no	—
21	D₄×C₄₇	—	4	3	no	—
22	C₂×C₄×C₄₇	C₂³	1	3	yes	—
23	C₂²⋊D₄₇	—	4	6	no	—
24	C₂×Dic₄₇	—	4	3	no	—
25	C₂×Dic₄₇	C₂³	1	6	yes	—
26	C₂²×C₄₇	—	2	3	no	—
27	C₂²×C₄₇	—	2	6	no	—
28	C₂²×C₄₇	—	2	3	no	—
29	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₄₇	—	2	6	no	—
31	C₂×D₄₇	—	4	4	no	—
32	C₄×C₄₇	—	2	3	no	—
33	Dic₄₇	—	2	6	no	—
34	C₂×C₄₇	—	4	1	no	—
35	C₂×C₄₇	—	4	3	no	—
36	C₂×C₄₇	H(5,𝔽₂)	1	3	yes	—
37	D₄₇	—	4	2	no	—
38	#560	—	47	1	no	maximal, Sylow
39	C₄₇	#560	1	1	yes	characteristic, complemented, Sylow
40	#127	D₄₇	1	1	yes	characteristic, complemented
41	#126	—	47	2	no	—
42	#123	—	47	3	no	—
43	#121	—	47	6	no	—
44	#120	—	47	3	no	—
45	C₂⁴	—	47	1	no	—
46	C₂×D₄	—	47	6	no	—
47	C₂×D₄	C₂×D₄₇	1	3	yes	—
48	C₂×D₄	C₂×D₄₇	1	3	yes	complemented
49	C₂²×C₄	—	47	3	no	—
50	C₄⋊C₄	—	47	6	no	—
51	C₂²⋊C₄	—	47	3	no	—
52	C₂²⋊C₄	—	47	3	no	—
53	C₂²⋊C₄	—	47	6	no	—
54	C₄²	C₂×D₄₇	1	1	yes	characteristic, complemented
55	C₂³	—	47	2	no	—
56	C₂³	—	188	3	no	—
57	C₂³	C₂²×D₄₇	1	1	yes	characteristic
58	C₂³	C₂²×D₄₇	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	188	6	no	—
62	C₂×C₄	—	47	6	no	—
63	C₂×C₄	—	188	3	no	—
64	C₂×C₄	C₂²×D₄₇	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	94	6	no	—
69	C₂²	—	188	4	no	—
70	C₂²	C₂³×D₄₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	94	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	188	2	no	—
76	C₂	#13094069	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#127×C₄₇	C₂
hypercenter	#127	D₄₇
hypocenter	C₄₇	#560
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #13094018	C₂	semidirect
2. #13094005	C₂	semidirect
3. #13093996	C₂	semidirect
4. #13093992	C₂	semidirect
5. #127×C₄₇	C₂	semidirect
8. C₂×D₄×C₄₇	C₂²	semidirect
10. C₂²×Dic₄₇	C₂²	semidirect
11. C₄²×C₄₇	C₂²	semidirect
13. C₂²⋊Dic₄₇	C₂²	semidirect
15. Dic₄₇⋊C₄	C₂²	semidirect
39. C₄₇	#560	semidirect
40. #127	D₄₇	semidirect
48. C₂×D₄	C₂×D₄₇	semidirect
54. C₄²	C₂×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#13094018	2
#13094005	2
#13093996	2
#13093992	2
#127×C₄₇	2
#560	47