Subgroups — #20295068

Order 6976.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812658	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812633	C₂	1	2	yes	maximal, complemented
4	#16812631	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₉	—	2	1	no	—
7	C₄⋊Dic₁₀₉	—	2	1	no	—
8	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₁₀₉	—	2	1	no	—
12	C₄×Dic₁₀₉	—	2	1	no	—
13	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₀₉	C₄	1	4	yes	complemented
15	C₂×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	Q₈	1	2	yes	—
20	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	1	no	—
22	C₂×Dic₁₀₉	—	2	1	no	—
23	C₂×Dic₁₀₉	—	2	2	no	—
24	C₂×Dic₁₀₉	—	4	2	no	—
25	C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
26	C₂²×C₁₀₉	—	2	1	no	—
27	C₂²×C₁₀₉	—	2	1	no	—
28	C₂²×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	Dic₁₀₉	—	4	1	no	—
34	Dic₁₀₉	—	4	2	no	—
35	Dic₁₀₉	—	4	1	no	—
36	C₂×C₁₀₉	—	2	2	no	—
37	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₁₀₉	#137	1	1	yes	characteristic
40	C₁₀₉	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	109	1	no	maximal, Sylow
42	C₂×M₁₆	D₁₀₉	1	1	yes	characteristic
43	#117	—	109	1	no	—
44	C₂×C₄⋊C₄	—	109	1	no	—
45	#107	—	109	2	no	—
46	#106	—	109	2	no	—
47	C₂²×C₄	—	218	1	no	—
48	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
49	M₁₆	Dic₁₀₉	1	4	yes	complemented
50	C₂×C₈	C₂×D₁₀₉	1	2	yes	—
51	C₄⋊C₄	—	109	1	no	—
52	C₄⋊C₄	—	109	2	no	—
53	C₄⋊C₄	—	109	1	no	—
54	C₄⋊C₄	—	218	1	no	—
55	C₂²⋊C₄	—	218	1	no	—
56	C₄²	—	218	1	no	—
57	C₂³	D₄₃₆	1	1	yes	characteristic
58	C₂×C₄	—	436	2	no	—
59	C₂×C₄	—	218	1	no	—
60	C₂×C₄	—	218	1	no	—
61	C₂×C₄	—	218	2	no	—
62	C₂×C₄	Dic₂₁₈	1	2	yes	—
63	C₂×C₄	D₄₃₆	1	1	yes	characteristic
64	C₂×C₄	C₂×Dic₁₀₉	1	2	yes	—
65	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
66	C₈	C₂×Dic₁₀₉	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₁₀₉	1	2	yes	—
70	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
71	C₄	—	436	2	no	—
72	C₄	—	436	1	no	—
73	C₄	—	436	1	no	—
74	C₄	C₄⋊Dic₁₀₉	1	2	yes	—
75	C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
76	C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
79	C₂	#16812727	1	1	yes	characteristic
80	C₂	#16812732	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₃₆
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#428
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16812633	C₂	semidirect
4. #16812631	C₂	semidirect
14. M₁₆×C₁₀₉	C₄	semidirect
25. C₈×C₁₀₉	C₂×C₄	semidirect
40. C₁₀₉	#428	semidirect
49. M₁₆	Dic₁₀₉	semidirect
66. C₈	C₂×Dic₁₀₉	semidirect