Subgroups — #20294797

Order 6976.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182013	C₂	1	1	yes	maximal, characteristic
2	#16812658	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#13182013	—	2	1	no	—
5	#13182013	C₄	1	2	yes	complemented
6	C₂×#11724784	—	2	1	no	—
7	C₄⋊Dic₁₀₉	—	2	1	no	—
8	C₂²⋊Dic₁₀₉	C₄	1	2	yes	—
9	C₄×Dic₁₀₉	—	2	1	no	—
10	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₁₀₉	—	2	1	no	—
12	M₁₆×C₁₀₉	—	2	1	no	—
13	C₂×C₈×C₁₀₉	C₄	1	2	yes	complemented
14	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	2	no	—
22	C₈×C₁₀₉	—	2	2	no	—
23	#11724784	—	2	2	no	—
24	C₂²×C₁₀₉	—	4	1	no	—
25	C₂²×C₁₀₉	C₄²	1	1	yes	characteristic
26	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	Dic₁₀₉	—	4	2	no	—
33	C₂×C₁₀₉	—	2	1	no	—
34	C₂×C₁₀₉	—	2	1	no	—
35	C₂×C₁₀₉	—	2	1	no	—
36	C₂×C₁₀₉	#95	1	1	yes	characteristic
37	C₁₀₉	#344	1	1	yes	characteristic, complemented, Sylow
38	#344	—	109	1	no	maximal, Sylow
39	C₂×M₁₆	—	109	1	no	—
40	C₂×M₁₆	D₁₀₉	1	1	yes	characteristic
41	#117	—	109	1	no	—
42	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
43	M₁₆	—	2	1	no	—
44	M₁₆	—	2	1	no	—
45	M₁₆	—	109	2	no	—
46	M₁₆	—	218	1	no	—
47	C₂×C₈	—	218	1	no	—
48	C₂×C₈	Dic₁₀₉	1	2	yes	complemented
49	C₄⋊C₄	—	218	1	no	—
50	C₂²⋊C₄	—	109	2	no	—
51	C₄²	—	218	1	no	—
52	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
53	C₂×C₄	—	218	2	no	—
54	C₂×C₄	Dic₂₁₈	1	1	yes	characteristic
55	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
56	C₂×C₄	D₄₃₆	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	218	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₄×Dic₁₀₉	1	1	yes	characteristic
64	C₂²	#13182006	1	1	yes	characteristic
65	C₂²	#13182008	1	1	yes	characteristic
66	C₄	—	436	2	no	—
67	C₄	#13182006	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
69	C₄	#13182008	1	1	yes	characteristic
70	C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	#16812606	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₁₀₉
commutator	C₂²×C₁₀₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₀₉
Fitting	C₂×M₁₆×C₁₀₉	C₂
socle	C₂×C₁₀₉	#95
hypercenter	C₂×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#344
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. #13182013	C₄	semidirect
13. C₂×C₈×C₁₀₉	C₄	semidirect
37. C₁₀₉	#344	semidirect
48. C₂×C₈	Dic₁₀₉	semidirect