Subgroups — #28360950

Order 13952.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182004	C₂	1	2	yes	maximal
2	C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182004	C₄	1	4	yes	—
4	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
5	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	#13182004	—	2	4	no	—
8	C₂×C₈×C₁₀₉	C₈	1	2	yes	—
9	C₂×C₈×C₁₀₉	C₈	1	2	yes	—
10	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
11	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
13	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
15	C₈×C₁₀₉	—	2	2	no	—
16	C₈×C₁₀₉	C₁₆	1	4	yes	complemented
17	C₈×C₁₀₉	M₁₆	1	4	yes	—
18	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₀₉	#97	1	2	yes	—
20	C₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
21	C₄×C₁₀₉	M₃₂	1	2	yes	—
22	C₂×C₁₀₉	#322	1	1	yes	characteristic
23	C₂×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₁₀₉	#346	1	1	yes	characteristic
25	#1312	—	109	1	no	maximal, Sylow
26	C₁₀₉	#1312	1	1	yes	characteristic, complemented, Sylow
27	C₄×C₁₆	—	109	2	no	—
28	C₈²	D₁₀₉	1	1	yes	characteristic
29	C₂×C₁₆	—	109	4	no	—
30	C₄×C₈	Dic₁₀₉	1	1	yes	characteristic
31	C₄×C₈	Dic₁₀₉	1	1	yes	characteristic
32	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
33	C₂×C₈	#11724784	1	2	yes	—
34	C₂×C₈	#11724784	1	2	yes	—
35	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
36	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
37	C₁₆	—	218	4	no	—
38	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
39	C₂×C₄	C₂×#11724784	1	1	yes	characteristic
40	C₂×C₄	C₂×#11724784	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₈	#13182004	1	4	yes	complemented
43	C₈	#13182012	1	4	yes	—
44	C₂²	C₄×#11724784	1	1	yes	characteristic
45	C₄	#16812609	1	2	yes	—
46	C₄	C₂×#13182004	1	2	yes	—
47	C₄	#16812639	1	2	yes	—
48	C₂	#20294747	1	1	yes	characteristic
49	C₂	C₄×#13182004	1	1	yes	characteristic
50	C₂	#20294802	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₄×C₁₆
Fitting	C₈²×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₄×C₈
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	C₈²	D₁₀₉
hypocenter	C₁₀₉	#1312
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
16. C₈×C₁₀₉	C₁₆	semidirect
26. C₁₀₉	#1312	semidirect
42. C₈	#13182004	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
C₈²×C₁₀₉	2
#1312	109