Subgroups — C₂×D₃₄₂₄ #28338759

Order 13696.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₁₂	C₂	1	2	yes	maximal, complemented
2	D₃₄₂₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₅₆	—	2	2	no	—
5	D₁₇₁₂	—	2	2	no	—
6	D₁₇₁₂	C₂²	1	4	yes	complemented
7	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₀₇	C₂²	1	2	yes	complemented
9	C₂×D₄₂₈	—	4	2	no	—
10	D₈₅₆	—	2	4	no	—
11	D₈₅₆	—	4	2	no	—
12	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
13	C₁₆×C₁₀₇	D₄	1	1	yes	characteristic
14	C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
15	C₂²×D₁₀₇	—	8	2	no	—
16	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
17	D₄₂₈	—	4	4	no	—
18	D₄₂₈	—	8	2	no	—
19	C₈×C₁₀₇	D₈	1	1	yes	characteristic
20	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₁₀₇	D₁₆	1	1	yes	characteristic
22	C₂×D₁₀₇	—	8	4	no	—
23	C₂×D₁₀₇	—	16	4	no	—
24	C₄×C₁₀₇	D₁₆	1	1	yes	characteristic
25	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
26	C₂×C₁₀₇	D₃₂	1	2	yes	complemented
27	C₂×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
28	D₁₀₇	—	16	4	no	—
29	C₂×D₃₂	—	107	1	no	maximal, Sylow
30	C₁₀₇	C₂×D₃₂	1	1	yes	characteristic, complemented, Sylow
31	C₂×D₁₆	—	107	2	no	—
32	D₃₂	—	107	4	no	—
33	C₂×C₃₂	D₁₀₇	1	1	yes	characteristic, complemented
34	C₂×D₈	—	214	2	no	—
35	D₁₆	—	107	4	no	—
36	D₁₆	—	214	2	no	—
37	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
38	C₃₂	C₂×D₁₀₇	1	2	yes	complemented
39	C₂×D₄	—	428	2	no	—
40	D₈	—	428	2	no	—
41	D₈	—	214	4	no	—
42	C₂×C₈	D₄₂₈	1	1	yes	characteristic
43	C₁₆	D₄₂₈	1	1	yes	characteristic
44	C₁₆	C₂²×D₁₀₇	1	1	yes	characteristic
45	C₂³	—	856	2	no	—
46	D₄	—	856	2	no	—
47	D₄	—	428	4	no	—
48	C₂×C₄	D₈₅₆	1	1	yes	characteristic
49	C₈	D₈₅₆	1	1	yes	characteristic
50	C₈	C₂×D₄₂₈	1	1	yes	characteristic
51	C₂²	—	856	4	no	—
52	C₂²	—	1712	4	no	—
53	C₂²	D₁₇₁₂	1	1	yes	characteristic
54	C₄	D₁₇₁₂	1	1	yes	characteristic
55	C₄	C₂×D₈₅₆	1	1	yes	characteristic
56	C₂	—	1712	4	no	—
57	C₂	D₃₄₂₄	1	2	yes	direct product factor
58	C₂	C₂×D₁₇₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₇₁₂
commutator, cosocle	C₁₆×C₁₀₇	C₂³
Frattini	C₁₆	C₂²×D₁₀₇
Fitting	C₂×C₃₂×C₁₀₇	C₂
socle	C₂²×C₁₀₇	D₁₆
hypercenter	C₂×C₃₂	D₁₀₇
hypocenter	C₁₀₇	C₂×D₃₂
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₇₁₂	C₂	semidirect
2. D₃₄₂₄	C₂	direct (N × H)
3. C₂×C₃₂×C₁₀₇	C₂	semidirect
6. D₁₇₁₂	C₂²	semidirect
8. C₃₂×C₁₀₇	C₂²	semidirect
26. C₂×C₁₀₇	D₃₂	semidirect
30. C₁₀₇	C₂×D₃₂	semidirect
33. C₂×C₃₂	D₁₀₇	semidirect
38. C₃₂	C₂×D₁₀₇	semidirect
57. C₂	D₃₄₂₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₁₇₁₂	2
D₃₄₂₄	2
C₂×C₃₂×C₁₀₇	2
C₂×D₃₂	107

Subgroups — C2×D3424 #28338759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃₄₂₄ #28338759