Subgroups — #28360948

Order 13952.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182004	C₂	1	1	yes	maximal, characteristic
2	C₈×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182004	C₄	1	2	yes	—
5	C₄×#11724784	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₀₉	C₄	1	2	yes	—
7	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#11724784	C₈	1	4	yes	—
9	#13182004	C₈	1	4	yes	complemented
10	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₀₉	C₈	1	4	yes	complemented
14	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
16	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
17	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
18	#11724784	—	2	4	no	—
19	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
20	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
21	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₀₉	M₃₂	1	4	yes	—
23	C₂×C₁₀₉	C₈²	1	1	yes	characteristic
24	C₂×C₁₀₉	#346	1	2	yes	—
25	#1311	—	109	1	no	maximal, Sylow
26	C₁₀₉	#1311	1	1	yes	characteristic, complemented, Sylow
27	C₄×C₁₆	—	109	1	no	—
28	C₄×C₁₆	D₁₀₉	1	1	yes	characteristic
29	C₈²	—	109	1	no	—
30	C₂×C₁₆	—	109	2	no	—
31	C₂×C₁₆	Dic₁₀₉	1	2	yes	—
32	C₄×C₈	—	109	2	no	—
33	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
34	C₂×C₈	—	109	4	no	—
35	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
37	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
38	C₁₆	—	109	4	no	—
39	C₁₆	#11724784	1	4	yes	complemented
40	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
41	C₂×C₄	C₈×D₁₀₉	1	2	yes	—
42	C₈	—	218	4	no	—
43	C₈	C₈×D₁₀₉	1	2	yes	—
44	C₈	C₂×#11724784	1	2	yes	—
45	C₂²	C₈×Dic₁₀₉	1	1	yes	characteristic
46	C₄	C₈×Dic₁₀₉	1	1	yes	characteristic
47	C₄	C₄×#11724784	1	1	yes	characteristic
48	C₄	#16812638	1	4	yes	—
49	C₂	C₈×#11724784	1	1	yes	characteristic
50	C₂	#20294801	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₈²
Fitting	C₄×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₄×C₈
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	C₄×C₁₆	D₁₀₉
hypocenter	C₁₀₉	#1311
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. #13182004	C₈	semidirect
13. C₁₆×C₁₀₉	C₈	semidirect
26. C₁₀₉	#1311	semidirect
39. C₁₆	#11724784	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182004	2
C₈×#11724784	2
C₄×C₁₆×C₁₀₉	2
#1311	109