Subgroups — #28378140

Order 13952.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812749	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812749	C₄	1	4	yes	complemented
4	C₂×#13182004	C₂²	1	1	yes	characteristic
5	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#11724784	C₈	1	1	yes	characteristic
8	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
9	#13182004	C₂×C₄	1	2	yes	—
10	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
11	C₁₆×C₁₀₉	C₈	1	2	yes	—
12	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
13	C₂×Dic₁₀₉	C₁₆	1	2	yes	—
14	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
15	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
16	#11724784	C₁₆	1	2	yes	—
17	C₂²×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
20	Dic₁₀₉	—	2	2	no	—
21	C₂×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
22	C₂×C₁₀₉	M₆₄	1	2	yes	—
23	#1397	—	109	1	no	maximal, Sylow
24	C₁₀₉	#1397	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃₂	—	109	2	no	—
26	C₄×C₁₆	—	109	1	no	—
27	C₂×C₁₆	—	109	1	no	—
28	C₂×C₁₆	#89137	1	1	yes	characteristic
29	C₄×C₈	—	109	1	no	—
30	C₃₂	—	109	4	no	—
31	C₂×C₈	—	109	1	no	—
32	C₂×C₈	C₂×#89137	1	1	yes	characteristic
33	C₄²	—	109	1	no	—
34	C₁₆	—	109	2	no	—
35	C₁₆	#11724786	1	2	yes	—
36	C₂×C₄	—	109	2	no	—
37	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
38	C₈	—	109	2	no	—
39	C₈	C₄×#89137	1	1	yes	characteristic
40	C₈	C₂×#11724786	1	1	yes	characteristic
41	C₂²	D₁₀₉⋊C₁₆	1	1	yes	characteristic
42	C₄	—	218	2	no	—
43	C₄	C₄×#11724786	1	1	yes	characteristic
44	C₄	D₁₀₉⋊C₁₆	1	1	yes	characteristic
45	C₂	#20295912	1	1	yes	characteristic
46	C₂	#20295959	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₄×C₁₆
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₂×C₁₆
cosocle	C₂×#13182004	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1397
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
3. #16812749	C₄	semidirect
24. C₁₀₉	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812749	2
C₁₆×Dic₁₀₉	2
#1397	109