Subgroups — #15689866

Order 3008.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094011	C₂	1	3	yes	maximal, complemented
2	#13094010	C₂	1	6	yes	maximal, complemented
3	#13093993	C₂	1	2	yes	maximal, complemented
4	#13093992	C₂	1	3	yes	maximal, complemented
5	#120×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄₇	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₄₇	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₄×C₄₇	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₄₇	C₂²	1	6	yes	—
10	C₂×Dic₉₄	C₂²	1	3	yes	complemented
11	C₂²⋊C₄×C₄₇	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₄₇	C₂²	1	3	yes	complemented
13	C₂²⋊Dic₄₇	C₂²	1	6	yes	complemented
14	(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	complemented
15	C₄×Dic₄₇	C₂²	1	3	yes	complemented
16	C₂³×C₄₇	—	4	3	no	—
17	C₂³×C₄₇	C₂³	1	1	yes	characteristic
18	C₂³×C₄₇	C₂³	1	3	yes	—
19	C₂²×D₄₇	—	4	3	no	—
20	C₂²×D₄₇	C₂³	1	2	yes	—
21	D₄×C₄₇	—	4	3	no	—
22	C₂×C₄×C₄₇	C₂³	1	3	yes	—
23	C₂²⋊D₄₇	—	4	6	no	—
24	C₂×Dic₄₇	C₂³	1	6	yes	—
25	Dic₉₄	—	4	3	no	—
26	C₂²×C₄₇	—	2	6	no	—
27	C₂²×C₄₇	—	2	3	no	—
28	C₂²×C₄₇	—	2	3	no	—
29	C₂²×C₄₇	—	4	4	no	—
30	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₄₇	—	2	6	no	—
32	C₂×D₄₇	—	4	4	no	—
33	C₄×C₄₇	—	2	3	no	—
34	Dic₄₇	—	2	6	no	—
35	C₂×C₄₇	—	4	1	no	—
36	C₂×C₄₇	—	4	3	no	—
37	C₂×C₄₇	H(5,𝔽₂)	1	3	yes	—
38	D₄₇	—	4	2	no	—
39	H(3,𝔽₄)	—	47	1	no	maximal, Sylow
40	C₄₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
41	#124	—	47	3	no	—
42	#124	—	47	6	no	—
43	#120	—	47	2	no	—
44	#120	—	47	3	no	—
45	#120	D₄₇	1	1	yes	characteristic, complemented
46	C₂⁴	—	47	1	no	—
47	C₂⁴	C₂×D₄₇	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	47	3	no	—
49	C₂×D₄	—	47	6	no	—
50	C₂×D₄	C₂×D₄₇	1	3	yes	—
51	C₂²⋊C₄	—	47	3	no	—
52	C₂²⋊C₄	—	47	6	no	—
53	C₂²⋊C₄	—	47	6	no	—
54	C₂²⋊C₄	C₂×D₄₇	1	3	yes	complemented
55	C₄²	—	47	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	47	2	no	—
58	C₂³	—	188	3	no	—
59	C₂³	C₂²×D₄₇	1	1	yes	characteristic
60	C₂³	C₂²×D₄₇	1	3	yes	—
61	Q₈	—	188	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	188	6	no	—
64	C₂×C₄	—	47	6	no	—
65	C₂×C₄	C₂²×D₄₇	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	94	6	no	—
71	C₂²	—	188	4	no	—
72	C₂²	C₂³×D₄₇	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	94	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	188	2	no	—
78	C₂	#13094069	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#120×C₄₇	C₂
hypercenter	#120	D₄₇
hypocenter	C₄₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. #13094011	C₂	semidirect
2. #13094010	C₂	semidirect
3. #13093993	C₂	semidirect
4. #13093992	C₂	semidirect
5. #120×C₄₇	C₂	semidirect
6. C₂⁴×C₄₇	C₂²	semidirect
7. C₂³×D₄₇	C₂²	semidirect
10. C₂×Dic₉₄	C₂²	semidirect
11. C₂²⋊C₄×C₄₇	C₂²	semidirect
12. C₂²⋊Dic₄₇	C₂²	semidirect
13. C₂²⋊Dic₄₇	C₂²	semidirect
14. (C₂×D₄₇)⋊C₄	C₂²	semidirect
15. C₄×Dic₄₇	C₂²	semidirect
40. C₄₇	H(3,𝔽₄)	semidirect
45. #120	D₄₇	semidirect
47. C₂⁴	C₂×D₄₇	semidirect
54. C₂²⋊C₄	C₂×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#13094011	2
#13094010	2
#13093993	2
#13093992	2
#120×C₄₇	2
H(3,𝔽₄)	47