Subgroups — #28378182

Order 13952.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#20295952	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812768	—	2	1	no	—
5	#16812749	—	2	1	no	—
6	Q₁₆×D₁₀₉	—	2	1	no	—
7	Dic₈₇₂	C₄	1	1	yes	characteristic, complemented
8	#16812645	—	2	1	no	—
9	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#89137	—	4	1	no	—
12	Q₈×D₁₀₉	—	4	1	no	—
13	Dic₄₃₆	—	2	1	no	—
14	#13182019	—	4	1	no	—
15	C₈×D₁₀₉	D₄	1	1	yes	characteristic
16	#13182004	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₀₉	—	2	1	no	—
18	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×#89137	—	8	1	no	—
20	Q₈×C₁₀₉	—	4	1	no	—
21	C₄×D₁₀₉	—	8	1	no	—
22	C₄×D₁₀₉	D₈	1	1	yes	characteristic
23	Dic₂₁₈	—	4	1	no	—
24	Dic₂₁₈	—	8	1	no	—
25	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	#11724784	SD₁₆	1	1	yes	characteristic
27	C₂×D₁₀₉	D₁₆	1	1	yes	characteristic
28	#89137	—	16	1	no	—
29	C₄×C₁₀₉	—	8	1	no	—
30	C₄×C₁₀₉	#102	1	1	yes	characteristic
31	Dic₁₀₉	—	8	1	no	—
32	Dic₁₀₉	SD₃₂	1	1	yes	characteristic
33	C₂×C₁₀₉	#357	1	1	yes	characteristic
34	D₁₀₉	SD₆₄	1	1	yes	characteristic
35	D₁₀₉	Q₆₄	1	1	yes	characteristic
36	#1416	—	109	1	no	maximal, Sylow
37	C₁₀₉	#1416	1	1	yes	characteristic, complemented, Sylow
38	C₂×Q₃₂	—	109	1	no	—
39	C₂×C₃₂	—	109	1	no	—
40	#366	—	109	1	no	—
41	C₂×Q₁₆	—	218	1	no	—
42	Q₃₂	—	109	1	no	—
43	Q₃₂	—	218	1	no	—
44	Q₃₂	#89137	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	109	1	no	—
46	#107	—	218	1	no	—
47	C₃₂	—	218	1	no	—
48	C₂×Q₈	—	436	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	436	1	no	—
51	Q₁₆	—	218	1	no	—
52	C₂×C₈	—	109	1	no	—
53	C₄⋊C₄	—	436	1	no	—
54	C₁₆	—	109	1	no	—
55	C₁₆	C₂×#89137	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	872	1	no	—
58	Q₈	—	436	1	no	—
59	C₂×C₄	—	872	1	no	—
60	C₂×C₄	—	872	1	no	—
61	C₂×C₄	—	109	1	no	—
62	C₈	—	109	1	no	—
63	C₈	C₂²⋊#89137	1	1	yes	characteristic
64	C₂²	—	109	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	872	1	no	—
67	C₄	—	109	1	no	—
68	C₄	—	1744	1	no	—
69	C₄	#16812761	1	1	yes	characteristic
70	C₂	—	109	1	no	—
71	C₂	—	109	1	no	—
72	C₂	#20295938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295938
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₁₆	C₂×#89137
Fitting	Q₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	#357
prime radical	C₂×D₁₀₉	D₁₆
cosocle	C₁₆×D₁₀₉	C₂²
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	#1416
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
7. Dic₈₇₂	C₄	semidirect
10. Q₃₂×C₁₀₉	C₄	semidirect
37. C₁₀₉	#1416	semidirect
44. Q₃₂	#89137	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₉⋊C₃₂	2
#20295952	2
Q₃₂×D₁₀₉	2
#1416	109