Subgroups — #16812761

Order 3488.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#89137	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₁₀₉	—	2	1	no	—
5	C₂×#89137	—	2	1	no	—
6	D₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
7	C₂²⋊D₁₀₉	—	2	1	no	—
8	D₄₃₆	C₄	1	1	yes	characteristic, complemented
9	C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	#11724786	—	2	1	no	—
11	C₂²×C₁₀₉	—	2	1	no	—
12	C₂×D₁₀₉	—	2	1	no	—
13	C₂×D₁₀₉	—	4	1	no	—
14	C₂×D₁₀₉	—	4	1	no	—
15	C₂×D₁₀₉	D₄	1	1	yes	characteristic
16	#89137	—	4	1	no	—
17	C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	Dic₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₁₀₉	—	4	1	no	—
20	C₂×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
21	D₁₀₉	—	4	1	no	—
22	D₁₀₉	D₈	1	1	yes	characteristic
23	D₁₀₉	SD₁₆	1	1	yes	characteristic
24	C₁₀₉	#102	1	1	yes	characteristic, complemented, Sylow
25	#102	—	109	1	no	maximal, Sylow
26	C₂×D₄	—	109	1	no	—
27	C₂×C₈	—	109	1	no	—
28	C₄⋊C₄	—	109	1	no	—
29	C₂³	—	218	1	no	—
30	D₄	—	109	1	no	—
31	D₄	—	218	1	no	—
32	D₄	#89137	1	1	yes	characteristic, complemented
33	C₂×C₄	—	109	1	no	—
34	C₂×C₄	—	218	1	no	—
35	C₈	—	218	1	no	—
36	C₂²	—	2	1	no	—
37	C₂²	—	109	1	no	—
38	C₂²	—	436	1	no	—
39	C₂²	—	436	1	no	—
40	C₂²	—	218	1	no	—
41	C₄	—	109	1	no	—
42	C₄	—	436	1	no	—
43	C₄	C₂×#89137	1	1	yes	characteristic
44	C₂	—	4	1	no	—
45	C₂	—	109	1	no	—
46	C₂	—	109	1	no	—
47	C₂	—	436	1	no	—
48	C₂	C₂²⋊#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₄
Frattini	C₄	C₂×#89137
Fitting	D₄×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂²⋊C₄
prime radical	D₄×D₁₀₉	C₂
cosocle	C₄×D₁₀₉	C₂²
hypercenter	D₄	#89137
hypocenter	C₁₀₉	#102
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊C₈	C₂	semidirect
2. C₄⋊#89137	C₂	semidirect
6. D₄×C₁₀₉	C₄	semidirect
8. D₄₃₆	C₄	semidirect
24. C₁₀₉	#102	semidirect
32. D₄	#89137	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₉⋊C₈	2
C₄⋊#89137	2
D₄×D₁₀₉	2
#102	109