Subgroups — #28361213

Order 13952.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812605	C₂	1	1	yes	maximal, characteristic
2	#20294852	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812633	—	2	1	no	—
5	#16812605	—	2	1	no	—
6	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
7	Q₃₂×C₁₀₉	—	2	1	no	—
8	Q₃₂×C₁₀₉	C₄	1	2	yes	complemented
9	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₀₉	—	4	1	no	—
11	C₂×Q₈×C₁₀₉	—	4	1	no	—
12	Q₁₆×C₁₀₉	—	2	2	no	—
13	Q₁₆×C₁₀₉	—	4	1	no	—
14	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
15	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
17	Q₈×C₁₀₉	—	4	2	no	—
18	Q₈×C₁₀₉	—	8	1	no	—
19	C₂×C₄×C₁₀₉	—	8	1	no	—
20	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	8	1	no	—
22	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₀₉	D₁₆	1	1	yes	characteristic
25	C₄×C₁₀₉	—	8	2	no	—
26	C₄×C₁₀₉	#102	1	1	yes	characteristic
27	C₄×C₁₀₉	SD₃₂	1	1	yes	characteristic
28	Dic₁₀₉	—	16	1	no	—
29	C₂×C₁₀₉	#357	1	1	yes	characteristic
30	C₂×C₁₀₉	SD₆₄	1	1	yes	characteristic
31	C₂×C₁₀₉	Q₆₄	1	1	yes	characteristic
32	#1416	—	109	1	no	maximal, Sylow
33	C₁₀₉	#1416	1	1	yes	characteristic, complemented, Sylow
34	C₂×Q₃₂	D₁₀₉	1	1	yes	characteristic
35	C₂×C₃₂	—	109	1	no	—
36	#366	—	109	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	Q₃₂	—	2	1	no	—
39	Q₃₂	Dic₁₀₉	1	2	yes	complemented
40	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
41	#107	—	218	1	no	—
42	C₃₂	—	218	1	no	—
43	C₂×Q₈	—	4	1	no	—
44	Q₁₆	—	2	2	no	—
45	Q₁₆	—	4	1	no	—
46	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
47	C₄⋊C₄	—	436	1	no	—
48	C₁₆	C₂×Dic₁₀₉	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₀₉	1	1	yes	characteristic
50	Q₈	—	4	2	no	—
51	Q₈	—	8	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	872	1	no	—
54	C₂×C₄	#13182014	1	1	yes	characteristic
55	C₈	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
56	C₈	#13182017	1	1	yes	characteristic
57	C₂²	#16812640	1	1	yes	characteristic
58	C₄	—	8	2	no	—
59	C₄	—	1744	1	no	—
60	C₄	#16812620	1	1	yes	characteristic
61	C₄	#16812643	1	1	yes	characteristic
62	C₂	#20294827	1	1	yes	characteristic
63	C₂	#20294863	1	1	yes	characteristic
64	C₂	#20294866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812640
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₉
Fitting	C₂×Q₃₂×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	D₁₆
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	C₂×Q₃₂	D₁₀₉
hypocenter	C₁₀₉	#1416
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
8. Q₃₂×C₁₀₉	C₄	semidirect
33. C₁₀₉	#1416	semidirect
39. Q₃₂	Dic₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16812605	2
#20294852	2
C₂×Q₃₂×C₁₀₉	2
#1416	109