Subgroups — C₂×Dic₂₇₆₈ #36355868

Order 22144.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₈₄	C₂	1	2	yes	maximal
2	Dic₂₇₆₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃₂×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₉₂	—	2	2	no	—
5	Dic₁₃₈₄	—	2	2	no	—
6	Dic₁₃₈₄	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₇₃	C₂²	1	2	yes	—
9	C₂×Dic₃₄₆	—	4	2	no	—
10	Dic₆₉₂	—	2	4	no	—
11	Dic₆₉₂	—	4	2	no	—
12	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
13	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
14	C₁₆×C₁₇₃	C₂³	1	1	yes	characteristic
15	Dic₃₄₆	—	4	4	no	—
16	Dic₃₄₆	—	8	2	no	—
17	C₂×Dic₁₇₃	—	8	2	no	—
18	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
19	C₈×C₁₇₃	D₈	1	1	yes	characteristic
20	C₈×C₁₇₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₁₇₃	D₁₆	1	1	yes	characteristic
22	C₄×C₁₇₃	D₁₆	1	1	yes	characteristic
23	C₄×C₁₇₃	C₂×D₈	1	1	yes	characteristic
24	Dic₁₇₃	—	8	4	no	—
25	C₂×C₁₇₃	Q₆₄	1	2	yes	complemented
26	C₂×C₁₇₃	C₂×D₁₆	1	1	yes	characteristic
27	C₁₇₃	C₂×Q₆₄	1	1	yes	characteristic, complemented, Sylow
28	C₂×Q₆₄	—	173	1	no	maximal, Sylow
29	C₂×Q₃₂	—	173	2	no	—
30	Q₆₄	—	173	4	no	—
31	C₂×C₃₂	D₁₇₃	1	1	yes	characteristic
32	C₂×Q₁₆	—	346	2	no	—
33	Q₃₂	—	346	2	no	—
34	Q₃₂	—	173	4	no	—
35	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
36	C₃₂	C₂×D₁₇₃	1	2	yes	—
37	C₂×Q₈	—	692	2	no	—
38	Q₁₆	—	346	4	no	—
39	Q₁₆	—	692	2	no	—
40	C₂×C₈	D₆₉₂	1	1	yes	characteristic
41	C₁₆	D₆₉₂	1	1	yes	characteristic
42	C₁₆	C₂²×D₁₇₃	1	1	yes	characteristic
43	Q₈	—	1384	2	no	—
44	Q₈	—	692	4	no	—
45	C₂×C₄	—	1384	2	no	—
46	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
47	C₈	D₁₃₈₄	1	1	yes	characteristic
48	C₈	C₂×D₆₉₂	1	1	yes	characteristic
49	C₂²	D₂₇₆₈	1	1	yes	characteristic
50	C₄	—	1384	4	no	—
51	C₄	D₂₇₆₈	1	1	yes	characteristic
52	C₄	C₂×D₁₃₈₄	1	1	yes	characteristic
53	C₂	Dic₂₇₆₈	1	2	yes	direct product factor
54	C₂	C₂×D₂₇₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₇₆₈
commutator, cosocle	C₁₆×C₁₇₃	C₂³
Frattini	C₁₆	C₂²×D₁₇₃
Fitting	C₂×C₃₂×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	D₁₆
hypercenter	C₂×C₃₂	D₁₇₃
hypocenter	C₁₇₃	C₂×Q₆₄
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
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
2. Dic₂₇₆₈	C₂	direct (N × H)
25. C₂×C₁₇₃	Q₆₄	semidirect
27. C₁₇₃	C₂×Q₆₄	semidirect
53. C₂	Dic₂₇₆₈	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×Dic₁₃₈₄	2
Dic₂₇₆₈	2
C₂×C₃₂×C₁₇₃	2
C₂×Q₆₄	173

Subgroups — C2×Dic2768 #36355868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₂₇₆₈ #36355868