Subgroups — C₂×C₁₆×Dic₁₀₉ #28364618

Order 13952.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182004	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₆×Dic₁₀₉	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#13182004	C₄	1	4	yes	complemented
8	C₂×#13182004	C₂²	1	2	yes	—
9	C₈×Dic₁₀₉	C₄	1	2	yes	—
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₉	C₈	1	2	yes	—
15	C₂×#11724784	C₈	1	4	yes	—
16	C₂×#11724784	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₀₉	C₈	1	2	yes	—
18	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
19	#13182004	C₂×C₄	1	4	yes	complemented
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
26	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₁₆	1	8	yes	direct product factor
32	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
33	C₈×C₁₀₉	C₄²	1	2	yes	—
34	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	#11724784	C₂×C₈	1	4	yes	—
37	C₂²×C₁₀₉	C₄×C₈	1	2	yes	—
38	C₂²×C₁₀₉	C₂×C₁₆	1	4	yes	—
39	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₀₉	C₄×C₈	1	2	yes	—
41	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
42	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
43	Dic₁₀₉	C₂×C₁₆	1	8	yes	direct product factor
44	C₂×C₁₀₉	C₄×C₁₆	1	4	yes	complemented
45	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂²×C₁₆	1	2	yes	—
47	C₂×C₄×C₁₆	—	109	1	no	maximal, Sylow
48	C₁₀₉	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₁₆	—	109	1	no	—
50	C₂²×C₁₆	D₁₀₉	1	1	yes	characteristic
51	C₂×C₄×C₈	—	109	1	no	—
52	C₄×C₁₆	—	109	4	no	—
53	C₂²×C₈	—	109	1	no	—
54	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
55	C₂×C₄²	—	109	1	no	—
56	C₂×C₁₆	—	109	4	no	—
57	C₂×C₁₆	—	109	2	no	—
58	C₂×C₁₆	Dic₁₀₉	1	4	yes	direct product factor
59	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
60	C₄×C₈	—	109	2	no	—
61	C₄×C₈	—	109	2	no	—
62	C₂²×C₄	—	109	2	no	—
63	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
64	C₂×C₈	—	109	4	no	—
65	C₂×C₈	—	109	2	no	—
66	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
67	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
68	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
69	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
70	C₄²	—	109	2	no	—
71	C₄²	—	109	2	no	—
72	C₁₆	—	109	4	no	—
73	C₁₆	C₂×Dic₁₀₉	1	4	yes	direct product factor
74	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
75	C₂×C₄	—	109	8	no	—
76	C₂×C₄	—	109	4	no	—
77	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
78	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
79	C₂×C₄	C₈×D₁₀₉	1	2	yes	—
80	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
81	C₈	—	109	4	no	—
82	C₈	C₄×Dic₁₀₉	1	2	yes	—
83	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
84	C₈	C₂²×Dic₁₀₉	1	1	yes	characteristic
85	C₂²	C₈×Dic₁₀₉	1	2	yes	—
86	C₂²	C₁₆×D₁₀₉	1	4	yes	—
87	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
88	C₄	—	109	8	no	—
89	C₄	C₈×Dic₁₀₉	1	2	yes	—
90	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
91	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
92	C₂	C₁₆×Dic₁₀₉	1	4	yes	direct product factor
93	C₂	C₂×C₈×Dic₁₀₉	1	1	yes	characteristic
94	C₂	C₂×C₁₆×D₁₀₉	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	D₁₀₉
commutator, hypocenter	C₁₀₉	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₂²×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₁₆×Dic₁₀₉	C₂	direct (N × H)
7. C₂×#13182004	C₄	semidirect
12. C₂×C₁₆×C₁₀₉	C₄	semidirect
19. #13182004	C₂×C₄	semidirect
25. C₁₆×C₁₀₉	C₂×C₄	semidirect
31. C₂×Dic₁₀₉	C₁₆	direct (N × H)
43. Dic₁₀₉	C₂×C₁₆	direct (N × H)
44. C₂×C₁₀₉	C₄×C₁₆	semidirect
48. C₁₀₉	C₂×C₄×C₁₆	semidirect
58. C₂×C₁₆	Dic₁₀₉	direct (N × H)
73. C₁₆	C₂×Dic₁₀₉	direct (N × H)
92. C₂	C₁₆×Dic₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13182004	2
C₂×C₈×Dic₁₀₉	2
C₁₆×Dic₁₀₉	2
C₂²×C₁₆×C₁₀₉	2
C₂×C₄×C₁₆	109

Subgroups — C2×C16×Dic109 #28364618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×Dic₁₀₉ #28364618