Subgroups — #28378181

Order 13952.

82 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, complemented
2	#20295952	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812768	—	2	1	no	—
5	#16812749	—	2	1	no	—
6	D₈×D₁₀₉	—	2	1	no	—
7	D₁₇₄₄	C₄	1	1	yes	characteristic, complemented
8	#16812640	—	2	1	no	—
9	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
10	D₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#89137	—	4	1	no	—
12	D₄×D₁₀₉	—	4	1	no	—
13	D₈₇₂	—	2	1	no	—
14	#13182014	—	4	1	no	—
15	C₈×D₁₀₉	D₄	1	1	yes	characteristic
16	#13182004	D₄	1	1	yes	characteristic
17	D₈×C₁₀₉	—	2	1	no	—
18	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₁₀₉	—	8	1	no	—
20	C₂×#89137	—	8	1	no	—
21	D₄×C₁₀₉	—	4	1	no	—
22	C₂²⋊D₁₀₉	—	8	1	no	—
23	D₄₃₆	—	4	1	no	—
24	C₄×D₁₀₉	D₈	1	1	yes	characteristic
25	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	#11724784	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₀₉	—	8	1	no	—
28	C₂×D₁₀₉	—	8	1	no	—
29	C₂×D₁₀₉	—	16	1	no	—
30	C₂×D₁₀₉	—	16	1	no	—
31	C₂×D₁₀₉	D₁₆	1	1	yes	characteristic
32	#89137	—	16	1	no	—
33	C₄×C₁₀₉	#102	1	1	yes	characteristic
34	Dic₁₀₉	SD₃₂	1	1	yes	characteristic
35	C₂×C₁₀₉	—	16	1	no	—
36	C₂×C₁₀₉	#357	1	1	yes	characteristic
37	D₁₀₉	—	16	1	no	—
38	D₁₀₉	D₃₂	1	1	yes	characteristic
39	D₁₀₉	SD₆₄	1	1	yes	characteristic
40	#1415	—	109	1	no	maximal, Sylow
41	C₁₀₉	#1415	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₁₆	—	109	1	no	—
43	C₂×C₃₂	—	109	1	no	—
44	#366	—	109	1	no	—
45	C₂×D₈	—	218	1	no	—
46	D₁₆	—	109	1	no	—
47	D₁₆	—	218	1	no	—
48	D₁₆	#89137	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	109	1	no	—
50	#107	—	218	1	no	—
51	C₃₂	—	218	1	no	—
52	C₂×D₄	—	436	1	no	—
53	D₈	—	2	1	no	—
54	D₈	—	436	1	no	—
55	D₈	—	218	1	no	—
56	C₂×C₈	—	109	1	no	—
57	C₄⋊C₄	—	436	1	no	—
58	C₁₆	—	109	1	no	—
59	C₁₆	C₂×#89137	1	1	yes	characteristic
60	C₂³	—	872	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	872	1	no	—
63	D₄	—	436	1	no	—
64	C₂×C₄	—	872	1	no	—
65	C₂×C₄	—	109	1	no	—
66	C₈	—	109	1	no	—
67	C₈	C₂²⋊#89137	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	872	1	no	—
70	C₂²	—	109	1	no	—
71	C₂²	—	1744	1	no	—
72	C₂²	—	1744	1	no	—
73	C₄	—	109	1	no	—
74	C₄	—	1744	1	no	—
75	C₄	#16812761	1	1	yes	characteristic
76	C₂	—	16	1	no	—
77	C₂	—	109	1	no	—
78	C₂	—	109	1	no	—
79	C₂	—	1744	1	no	—
80	C₂	#20295938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295938
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₁₆	C₂×#89137
Fitting	D₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#357
prime radical	D₁₆×D₁₀₉	C₂
cosocle	C₁₆×D₁₀₉	C₂²
hypercenter	D₁₆	#89137
hypocenter	C₁₀₉	#1415
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊C₃₂	C₂	semidirect
2. #20295952	C₂	semidirect
7. D₁₇₄₄	C₄	semidirect
10. D₁₆×C₁₀₉	C₄	semidirect
41. C₁₀₉	#1415	semidirect
48. D₁₆	#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
D₁₆×D₁₀₉	2
#1415	109