Subgroups — #20292669

Order 6848.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812068	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
3	#16812043	C₂	1	2	yes	maximal, complemented
4	#16812041	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₇	—	2	1	no	—
7	C₄⋊Dic₁₀₇	—	2	1	no	—
8	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
10	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₁₀₇	—	2	1	no	—
12	C₄×Dic₁₀₇	—	2	1	no	—
13	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₀₇	C₄	1	4	yes	complemented
15	C₂×C₈×C₁₀₇	C₂²	1	2	yes	—
16	C₂³×C₁₀₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	Q₈	1	2	yes	—
20	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₇	—	2	1	no	—
22	C₂×Dic₁₀₇	—	2	1	no	—
23	C₂×Dic₁₀₇	—	2	2	no	—
24	C₂×Dic₁₀₇	—	4	2	no	—
25	C₈×C₁₀₇	C₂×C₄	1	4	yes	complemented
26	C₂²×C₁₀₇	—	2	1	no	—
27	C₂²×C₁₀₇	—	2	1	no	—
28	C₂²×C₁₀₇	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
31	C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
33	Dic₁₀₇	—	4	1	no	—
34	Dic₁₀₇	—	4	2	no	—
35	Dic₁₀₇	—	4	1	no	—
36	C₂×C₁₀₇	—	2	2	no	—
37	C₂×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₁₀₇	#137	1	1	yes	characteristic
40	C₁₀₇	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	107	1	no	maximal, Sylow
42	C₂×M₁₆	D₁₀₇	1	1	yes	characteristic
43	#117	—	107	1	no	—
44	C₂×C₄⋊C₄	—	107	1	no	—
45	#107	—	107	2	no	—
46	#106	—	107	2	no	—
47	C₂²×C₄	—	214	1	no	—
48	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
49	M₁₆	Dic₁₀₇	1	4	yes	complemented
50	C₂×C₈	C₂×D₁₀₇	1	2	yes	—
51	C₄⋊C₄	—	107	1	no	—
52	C₄⋊C₄	—	107	2	no	—
53	C₄⋊C₄	—	107	1	no	—
54	C₄⋊C₄	—	214	1	no	—
55	C₂²⋊C₄	—	214	1	no	—
56	C₄²	—	214	1	no	—
57	C₂³	D₄₂₈	1	1	yes	characteristic
58	C₂×C₄	—	428	2	no	—
59	C₂×C₄	—	214	1	no	—
60	C₂×C₄	—	214	1	no	—
61	C₂×C₄	—	214	2	no	—
62	C₂×C₄	Dic₂₁₄	1	2	yes	—
63	C₂×C₄	D₄₂₈	1	1	yes	characteristic
64	C₂×C₄	C₂×Dic₁₀₇	1	2	yes	—
65	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
66	C₈	C₂×Dic₁₀₇	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₁₀₇	1	2	yes	—
70	C₂²	C₂×D₄₂₈	1	1	yes	characteristic
71	C₄	—	428	2	no	—
72	C₄	—	428	1	no	—
73	C₄	—	428	1	no	—
74	C₄	C₄⋊Dic₁₀₇	1	2	yes	—
75	C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
76	C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
79	C₂	#16812137	1	1	yes	characteristic
80	C₂	#16812142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₂₈
commutator	C₄×C₁₀₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₂×M₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	C₂×M₁₆	D₁₀₇
hypocenter	C₁₀₇	#428
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16812043	C₂	semidirect
4. #16812041	C₂	semidirect
14. M₁₆×C₁₀₇	C₄	semidirect
25. C₈×C₁₀₇	C₂×C₄	semidirect
40. C₁₀₇	#428	semidirect
49. M₁₆	Dic₁₀₇	semidirect
66. C₈	C₂×Dic₁₀₇	semidirect