Subgroups — Q₈×Dic₂₁₄ #20293358

Order 6848.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812109	C₂	1	3	yes	maximal
2	#16812108	C₂	1	6	yes	maximal
3	Q₈×Dic₁₀₇	C₂	1	2	yes	maximal
4	C₄×Dic₂₁₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₄	C₂²	1	6	yes	—
7	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₀₇	C₂²	1	3	yes	—
9	C₄⋊Dic₁₀₇	C₂²	1	6	yes	—
10	Dic₁₀₇⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₁₀₇	C₂²	1	6	yes	—
12	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₇	C₂²	1	3	yes	—
14	C₄²×C₁₀₇	C₂²	1	3	yes	—
15	Q₈×C₁₀₇	Q₈	1	4	yes	complemented
16	C₂×C₄×C₁₀₇	C₂³	1	3	yes	—
17	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₁₀₇	C₂³	1	3	yes	—
19	C₂×Dic₁₀₇	C₂³	1	6	yes	—
20	C₂×Dic₁₀₇	C₂³	1	2	yes	—
21	Dic₂₁₄	—	4	6	no	—
22	Dic₂₁₄	Q₈	1	4	yes	direct product factor
23	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
24	C₄×C₁₀₇	—	2	3	no	—
25	C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
26	C₄×C₁₀₇	C₂×Q₈	1	6	yes	—
27	Dic₁₀₇	—	2	6	no	—
28	Dic₁₀₇	C₂×Q₈	1	4	yes	—
29	C₂×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
31	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
32	C₁₀₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	107	1	no	maximal, Sylow
34	C₄⋊Q₈	—	107	6	no	—
35	C₄⋊Q₈	—	107	3	no	—
36	C₄×Q₈	—	107	2	no	—
37	C₄×Q₈	—	107	3	no	—
38	C₄×Q₈	D₁₀₇	1	1	yes	characteristic
39	C₂×Q₈	—	107	1	no	—
40	C₂×Q₈	—	107	6	no	—
41	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
42	C₄⋊C₄	—	107	6	no	—
43	C₄⋊C₄	—	107	3	no	—
44	C₄⋊C₄	—	107	6	no	—
45	C₄⋊C₄	C₂×D₁₀₇	1	3	yes	—
46	C₄²	—	107	6	no	—
47	C₄²	C₂×D₁₀₇	1	3	yes	—
48	Q₈	—	107	4	no	—
49	Q₈	—	428	6	no	—
50	Q₈	Dic₂₁₄	1	4	yes	direct product factor
51	C₂×C₄	—	107	2	no	—
52	C₂×C₄	—	107	6	no	—
53	C₂×C₄	C₂²×D₁₀₇	1	3	yes	—
54	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₁₀₇	1	3	yes	—
56	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
57	C₄	—	2	3	no	—
58	C₄	—	107	4	no	—
59	C₄	—	214	6	no	—
60	C₄	Q₈×D₁₀₇	1	2	yes	—
61	C₄	C₂×Dic₂₁₄	1	6	yes	—
62	C₂	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
63	C₂	C₂²×Dic₂₁₄	1	1	yes	characteristic
64	C₂	#16812155	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₄×Q₈×C₁₀₇	C₂
hypercenter	C₄×Q₈	D₁₀₇
hypocenter	C₁₀₇	Q₈²
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
15. Q₈×C₁₀₇	Q₈	semidirect
22. Dic₂₁₄	Q₈	direct (N × H)
32. C₁₀₇	Q₈²	semidirect
50. Q₈	Dic₂₁₄	direct (N × H)

Subgroups — Q8×Dic214 #20293358

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×Dic₂₁₄ #20293358