Subgroups — Q₃₂×Dic₁₀₉ #28364891

Order 13952.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₀₉	C₂	1	2	yes	maximal
2	#20294852	C₂	1	1	yes	maximal, characteristic
3	#20294830	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₉	—	2	2	no	—
7	C₂×#13182004	C₂²	1	1	yes	characteristic
8	#16812633	C₂²	1	2	yes	—
9	#16812623	—	2	2	no	—
10	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
12	Q₃₂×C₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#11724784	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	—	2	2	no	—
16	C₄⋊Dic₁₀₉	—	4	2	no	—
17	C₄×Dic₁₀₉	—	4	2	no	—
18	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
19	#13182004	—	2	1	no	—
20	C₂×Q₈×C₁₀₉	—	2	2	no	—
21	Q₁₆×C₁₀₉	—	2	2	no	—
22	Q₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	Q₈×C₁₀₉	—	2	4	no	—
26	Q₈×C₁₀₉	—	4	2	no	—
27	C₂×C₄×C₁₀₉	—	4	2	no	—
28	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	4	2	no	—
30	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
31	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
32	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₀₉	#41	1	1	yes	characteristic
34	#11724784	—	2	1	no	—
35	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₀₉	—	4	4	no	—
37	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
38	C₄×C₁₀₉	#135	1	1	yes	characteristic
39	Dic₁₀₉	—	2	1	no	—
40	Dic₁₀₉	—	8	2	no	—
41	Dic₁₀₉	Q₃₂	1	2	yes	direct product factor
42	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
43	C₂×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₁₀₉	#508	1	1	yes	characteristic
45	C₄×Q₃₂	—	109	1	no	maximal, Sylow
46	C₁₀₉	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₃₂	D₁₀₉	1	1	yes	characteristic
48	C₄×Q₁₆	—	109	2	no	—
49	#366	—	109	1	no	—
50	#358	—	109	2	no	—
51	C₄×C₁₆	—	109	1	no	—
52	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
53	C₄×Q₈	—	218	2	no	—
54	Q₃₂	Dic₁₀₉	1	4	yes	direct product factor
55	C₂×C₁₆	—	109	1	no	—
56	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
57	#107	—	109	2	no	—
58	#103	—	218	2	no	—
59	C₄×C₈	—	109	1	no	—
60	C₂×Q₈	—	2	2	no	—
61	Q₁₆	—	2	2	no	—
62	Q₁₆	C₂×Dic₁₀₉	1	4	yes	—
63	C₂×C₈	—	109	1	no	—
64	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₄	—	436	2	no	—
66	C₄⋊C₄	—	218	2	no	—
67	C₄²	—	109	1	no	—
68	C₄²	—	436	2	no	—
69	C₁₆	—	218	1	no	—
70	C₁₆	C₂×Dic₁₀₉	1	2	yes	—
71	Q₈	—	2	4	no	—
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	109	1	no	—
75	C₂×C₄	—	109	1	no	—
76	C₂×C₄	—	436	2	no	—
77	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
78	C₈	—	218	1	no	—
79	C₈	C₂²×Dic₁₀₉	1	1	yes	characteristic
80	C₈	#13182028	1	1	yes	characteristic
81	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
82	C₄	—	4	4	no	—
83	C₄	—	872	2	no	—
84	C₄	—	109	2	no	—
85	C₄	—	218	1	no	—
86	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
87	C₄	#16812718	1	1	yes	characteristic
88	C₂	D₈×Dic₁₀₉	1	1	yes	characteristic
89	C₂	Q₃₂×D₁₀₉	1	1	yes	characteristic
90	C₂	#20295459	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₂×Q₃₂×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂×Q₃₂	D₁₀₉
hypocenter	C₁₀₉	C₄×Q₃₂
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
12. Q₃₂×C₁₀₉	C₄	semidirect
41. Dic₁₀₉	Q₃₂	direct (N × H)
46. C₁₀₉	C₄×Q₃₂	semidirect
54. Q₃₂	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
Q₁₆×Dic₁₀₉	2
#20294852	2
#20294830	2
C₁₆×Dic₁₀₉	2
C₂×Q₃₂×C₁₀₉	2
C₄×Q₃₂	109

Subgroups — Q32×Dic109 #28364891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₃₂×Dic₁₀₉ #28364891