Subgroups — #28361104

Order 13952.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294779	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11724784	—	2	2	no	—
4	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#11724784	—	4	2	no	—
8	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
10	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
11	C₁₆×C₁₀₉	—	2	1	no	—
12	C₁₆×C₁₀₉	C₈	1	2	yes	complemented
13	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
16	C₈×C₁₀₉	—	2	1	no	—
17	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
19	#11724784	—	8	2	no	—
20	C₂²×C₁₀₉	#107	1	1	yes	characteristic
21	C₄×C₁₀₉	—	2	1	no	—
22	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₀₉	#108	1	1	yes	characteristic
24	C₄×C₁₀₉	D₁₆	1	1	yes	characteristic
25	C₄×C₁₀₉	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₀₉	#335	1	1	yes	characteristic
27	C₂×C₁₀₉	#366	1	1	yes	characteristic
28	C₂×C₁₀₉	#368	1	1	yes	characteristic
29	#1371	—	109	1	no	maximal, Sylow
30	C₁₀₉	#1371	1	1	yes	characteristic, complemented, Sylow
31	C₄×C₁₆	D₁₀₉	1	1	yes	characteristic
32	#335	—	109	2	no	—
33	C₂×C₁₆	Dic₁₀₉	1	1	yes	characteristic
34	C₂×C₁₆	Dic₁₀₉	1	1	yes	characteristic
35	C₄⋊C₈	—	218	2	no	—
36	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
37	C₂×C₈	—	436	2	no	—
38	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
39	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
40	C₄²	D₄₃₆	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	#11724784	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
44	C₂×C₄	D₈₇₂	1	1	yes	characteristic
45	C₂×C₄	Dic₄₃₆	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	872	2	no	—
48	C₈	C₂×#11724784	1	1	yes	characteristic
49	C₈	#13182013	1	1	yes	characteristic
50	C₂²	#16812633	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊#11724784	1	1	yes	characteristic
53	C₄	#16812635	1	1	yes	characteristic
54	C₄	D₁₇₄₄	1	1	yes	characteristic
55	C₄	Dic₈₇₂	1	1	yes	characteristic
56	C₂	#20294779	1	1	yes	characteristic
57	C₂	#20294852	1	1	yes	characteristic
58	C₂	#20294856	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈₇₂
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₉
Fitting	C₄×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#107
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	C₄×C₁₆	D₁₀₉
hypocenter	C₁₀₉	#1371
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₀₉	C₈	semidirect
30. C₁₀₉	#1371	semidirect
42. C₁₆	#11724784	semidirect

Maximal subgroups

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

subgroup	index
#20294779	2
C₄×C₁₆×C₁₀₉	2
#1371	109