Subgroups — #20295432

Order 6976.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812691	C₂	1	1	yes	maximal, characteristic
2	#16812632	C₂	1	2	yes	maximal
3	#16812630	C₂	1	2	yes	maximal
4	#16812610	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11724784	C₂²	1	2	yes	—
7	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#13182008	—	2	1	no	—
10	#13182008	—	2	1	no	—
11	C₂×Q₈×C₁₀₉	—	2	1	no	—
12	C₄⋊C₄×C₁₀₉	—	2	1	no	—
13	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	Q₈×C₁₀₉	—	4	2	no	—
16	C₂×C₄×C₁₀₉	—	2	2	no	—
17	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₀₉	—	2	1	no	—
21	C₂×Dic₁₀₉	—	2	1	no	—
22	#11724784	Q₈	1	4	yes	complemented
23	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₀₉	—	2	1	no	—
25	C₄×C₁₀₉	—	2	1	no	—
26	C₄×C₁₀₉	—	4	2	no	—
27	C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
28	Dic₁₀₉	—	4	1	no	—
29	Dic₁₀₉	—	4	1	no	—
30	C₂×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₁₀₉	#137	1	1	yes	characteristic
33	C₁₀₉	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	109	1	no	maximal, Sylow
35	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
36	#125	—	109	1	no	—
37	#107	—	109	2	no	—
38	#106	—	109	2	no	—
39	#97	—	109	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×C₈	—	109	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	109	1	no	—
44	C₄⋊C₄	—	109	1	no	—
45	C₄⋊C₄	—	218	1	no	—
46	C₄⋊C₄	—	218	1	no	—
47	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
48	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	218	1	no	—
52	C₂×C₄	—	218	1	no	—
53	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
56	C₈	—	109	4	no	—
57	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	436	1	no	—
62	C₄	—	436	1	no	—
63	C₄	Q₈×D₁₀₉	1	2	yes	—
64	C₂	#16812697	1	1	yes	characteristic
65	C₂	#16812725	1	1	yes	characteristic
66	C₂	#16812730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₄⋊Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#501
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. #11724784	Q₈	semidirect
33. C₁₀₉	#501	semidirect