Subgroups — #16812134

Order 3424.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133805	C₂	1	2	yes	maximal, complemented
3	#13133803	C₂	1	2	yes	maximal, complemented
4	#13133802	C₂	1	1	yes	maximal, characteristic, complemented
5	#41×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₇	—	2	1	no	—
7	Q₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₁₀₇	—	2	1	no	—
9	D₄×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₄×C₁₀₇	—	2	1	no	—
11	C₂×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
12	D₄₂₈	—	2	1	no	—
13	D₄₂₈	C₂²	1	2	yes	—
14	C₁₀₇⋊C₈	C₂²	1	2	yes	complemented
15	C₂²×C₁₀₇	—	2	1	no	—
16	C₂²×C₁₀₇	D₄	1	1	yes	characteristic
17	C₂×D₁₀₇	—	2	2	no	—
18	C₂×D₁₀₇	—	4	2	no	—
19	C₄×C₁₀₇	—	2	1	no	—
20	C₄×C₁₀₇	D₄	1	1	yes	characteristic
21	C₄×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₂×C₁₀₇	—	2	1	no	—
23	C₂×C₁₀₇	—	4	1	no	—
24	C₂×C₁₀₇	C₂×D₄	1	1	yes	characteristic
25	D₁₀₇	—	4	2	no	—
26	C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	107	1	no	maximal, Sylow
28	#41	D₁₀₇	1	1	yes	characteristic, complemented
29	C₂×D₄	—	107	1	no	—
30	SD₁₆	—	107	2	no	—
31	D₈	—	107	2	no	—
32	M₁₆	—	107	1	no	—
33	C₂³	—	214	1	no	—
34	Q₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
35	D₄	—	2	1	no	—
36	D₄	—	107	2	no	—
37	D₄	—	214	1	no	—
38	D₄	C₂×D₁₀₇	1	1	yes	characteristic, complemented
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	C₂×D₁₀₇	1	1	yes	characteristic
41	C₈	—	107	2	no	—
42	C₂²	—	2	1	no	—
43	C₂²	—	428	2	no	—
44	C₂²	—	214	2	no	—
45	C₂²	C₂²⋊D₁₀₇	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
48	C₄	C₂²×D₁₀₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	1	no	—
51	C₂	—	428	2	no	—
52	C₂	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₁₀₇
commutator, cosocle	C₄×C₁₀₇	C₂³
Frattini	C₄	C₂²×D₁₀₇
Fitting	#41×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×D₄
hypercenter	#41	D₁₀₇
hypocenter	C₁₀₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₂₈	C₂	semidirect
2. #13133805	C₂	semidirect
3. #13133803	C₂	semidirect
4. #13133802	C₂	semidirect
5. #41×C₁₀₇	C₂	semidirect
7. Q₈×C₁₀₇	C₂²	semidirect
9. D₄×C₁₀₇	C₂²	semidirect
14. C₁₀₇⋊C₈	C₂²	semidirect
26. C₁₀₇	AGL(1,ℤ/8)	semidirect
28. #41	D₁₀₇	semidirect
34. Q₈	C₂×D₁₀₇	semidirect
38. D₄	C₂×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₂₈	2
#13133805	2
#13133803	2
#13133802	2
#41×C₁₀₇	2
AGL(1,ℤ/8)	107