Subgroups — #13094050

Order 1504.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625350	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625335	C₂	1	2	yes	maximal, complemented
4	#10625334	C₂	1	2	yes	maximal, complemented
5	#10625329	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄₇	—	2	1	no	—
7	D₄×C₄₇	—	2	1	no	—
8	D₄×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²⋊D₄₇	—	2	1	no	—
11	D₁₈₈	C₂²	1	1	yes	characteristic, complemented
12	C₄×D₄₇	—	2	1	no	—
13	Dic₉₄	C₂²	1	1	yes	characteristic, complemented
14	C₄₇⋊C₈	C₂²	1	2	yes	complemented
15	C₂²×C₄₇	—	2	2	no	—
16	C₂²×C₄₇	—	4	2	no	—
17	C₂²×C₄₇	D₄	1	1	yes	characteristic
18	C₂×D₄₇	—	2	1	no	—
19	C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₄×C₄₇	C₂³	1	1	yes	characteristic
21	Dic₄₇	—	2	1	no	—
22	C₂×C₄₇	—	2	1	no	—
23	C₂×C₄₇	—	4	2	no	—
24	C₂×C₄₇	C₂×D₄	1	1	yes	characteristic
25	D₄₇	—	4	1	no	—
26	C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	47	1	no	maximal, Sylow
28	#41	—	47	1	no	—
29	C₂×D₄	D₄₇	1	1	yes	characteristic, complemented
30	SD₁₆	—	47	2	no	—
31	D₈	—	47	2	no	—
32	M₁₆	—	47	1	no	—
33	C₂³	—	2	1	no	—
34	Q₈	—	47	1	no	—
35	D₄	—	2	1	no	—
36	D₄	—	47	1	no	—
37	D₄	—	94	1	no	—
38	D₄	C₂×D₄₇	1	2	yes	—
39	C₂×C₄	—	94	1	no	—
40	C₂×C₄	C₂×D₄₇	1	1	yes	characteristic
41	C₈	—	47	2	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	4	2	no	—
44	C₂²	—	94	1	no	—
45	C₂²	C₂²⋊D₄₇	1	1	yes	characteristic
46	C₄	—	94	1	no	—
47	C₄	C₂²⋊D₄₇	1	1	yes	characteristic
48	C₄	C₂²×D₄₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	2	no	—
51	C₂	—	188	1	no	—
52	C₂	C₂×C₂²⋊D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₄₇
commutator, cosocle	C₄×C₄₇	C₂³
Frattini	C₄	C₂²×D₄₇
Fitting	C₂×D₄×C₄₇	C₂
socle	C₂×C₄₇	C₂×D₄
hypercenter	C₂×D₄	D₄₇
hypocenter	C₄₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₄₇	C₂	semidirect
2. #10625350	C₂	semidirect
3. #10625335	C₂	semidirect
4. #10625334	C₂	semidirect
5. #10625329	C₂	semidirect
11. D₁₈₈	C₂²	semidirect
13. Dic₉₄	C₂²	semidirect
14. C₄₇⋊C₈	C₂²	semidirect
26. C₄₇	AGL(1,ℤ/8)	semidirect
29. C₂×D₄	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₄₇	2
#10625350	2
#10625335	2
#10625334	2
#10625329	2
AGL(1,ℤ/8)	47