Subgroups — C₂²×Q₈×Dic₁₀₉ #28376208

Order 13952.

1664 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₀₉	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊Dic₁₀₉	C₂	1	3	yes	maximal
3	C₂²×C₄×Dic₁₀₉	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₉	C₂²	1	3	yes	—
6	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
7	Q₈×Dic₁₀₉	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	36	yes	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	36	yes	—
10	C₂²×Q₈×C₁₀₉	C₄	1	16	yes	complemented
11	C₂²×Q₈×C₁₀₉	C₂²	1	12	yes	—
12	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
13	C₂²×Dic₁₀₉	—	2	12	no	—
14	C₂²×Dic₁₀₉	Q₈	1	8	yes	direct product factor
15	C₂²×Dic₁₀₉	C₂³	1	18	yes	—
16	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
17	C₄⋊Dic₁₀₉	C₂³	1	48	yes	—
18	C₄×Dic₁₀₉	C₂³	1	48	yes	—
19	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₁₀₉	C₂×C₄	1	96	yes	complemented
21	C₂×Q₈×C₁₀₉	C₂³	1	16	yes	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	24	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	18	yes	—
24	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
25	C₂³×C₁₀₉	C₂×Q₈	1	4	yes	—
26	C₂³×C₁₀₉	#41	1	4	yes	—
27	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
28	Q₈×C₁₀₉	C₂²×C₄	1	64	yes	complemented
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	72	yes	—
30	C₂×C₄×C₁₀₉	C₂⁴	1	12	yes	—
31	C₂×Dic₁₀₉	—	2	36	no	—
32	C₂×Dic₁₀₉	C₂×Q₈	1	24	yes	direct product factor
33	C₂×Dic₁₀₉	C₂⁴	1	12	yes	—
34	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
35	C₂²×C₁₀₉	C₄×Q₈	1	16	yes	complemented
36	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
37	C₂²×C₁₀₉	C₂²×Q₈	1	6	yes	—
38	C₂²×C₁₀₉	C₂×#41	1	6	yes	—
39	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
40	C₄×C₁₀₉	C₂³×C₄	1	24	yes	—
41	Dic₁₀₉	—	2	12	no	—
42	Dic₁₀₉	C₂²×Q₈	1	8	yes	direct product factor
43	C₂×C₁₀₉	C₂×C₄×Q₈	1	12	yes	complemented
44	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
45	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
47	C₂²×C₄×Q₈	—	109	1	no	maximal, Sylow
48	C₁₀₉	C₂²×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
49	C₂³×Q₈	D₁₀₉	1	1	yes	characteristic
50	C₂×C₄×Q₈	—	109	24	no	—
51	C₂²×C₄⋊C₄	—	109	3	no	—
52	C₂²×C₄²	—	109	3	no	—
53	C₂²×Q₈	Dic₁₀₉	1	16	yes	direct product factor
54	C₂²×Q₈	C₂×D₁₀₉	1	12	yes	—
55	C₂³×C₄	—	109	1	no	—
56	C₂³×C₄	—	109	3	no	—
57	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
58	C₄×Q₈	—	109	64	no	—
59	C₂×C₄⋊C₄	—	109	36	no	—
60	C₂×C₄²	—	109	36	no	—
61	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
62	C₂×Q₈	C₂×Dic₁₀₉	1	96	yes	direct product factor
63	C₂×Q₈	C₂²×D₁₀₉	1	16	yes	—
64	C₂²×C₄	—	109	8	no	—
65	C₂²×C₄	—	109	6	no	—
66	C₂²×C₄	—	109	18	no	—
67	C₂²×C₄	—	218	12	no	—
68	C₂²×C₄	C₂×Dic₁₀₉	1	24	yes	—
69	C₂²×C₄	C₂²×D₁₀₉	1	18	yes	—
70	C₄⋊C₄	—	109	48	no	—
71	C₄²	—	109	48	no	—
72	C₂³	C₂²×Dic₁₀₉	1	4	yes	—
73	C₂³	Q₈×D₁₀₉	1	4	yes	—
74	C₂³	#13182030	1	4	yes	—
75	C₂³	C₂³×D₁₀₉	1	3	yes	—
76	Q₈	C₂²×Dic₁₀₉	1	64	yes	direct product factor
77	C₂×C₄	—	109	24	no	—
78	C₂×C₄	—	109	4	no	—
79	C₂×C₄	—	109	12	no	—
80	C₂×C₄	—	218	36	no	—
81	C₂×C₄	C₂²×Dic₁₀₉	1	72	yes	—
82	C₂×C₄	C₂³×D₁₀₉	1	12	yes	—
83	C₂²	Q₈×Dic₁₀₉	1	16	yes	direct product factor
84	C₂²	C₂³×Dic₁₀₉	1	6	yes	—
85	C₂²	C₂×Q₈×D₁₀₉	1	6	yes	—
86	C₂²	C₂×#13182030	1	6	yes	—
87	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
88	C₄	—	109	8	no	—
89	C₄	—	218	12	no	—
90	C₄	C₂³×Dic₁₀₉	1	24	yes	—
91	C₂	C₂×Q₈×Dic₁₀₉	1	12	yes	direct product factor
92	C₂	C₂⁴×Dic₁₀₉	1	1	yes	characteristic
93	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
94	C₂	C₂²×#13182030	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂³×Q₈×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	C₂³
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂³×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂²×C₄×Q₈
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₁₀₉	C₂	direct (N × H)
7. Q₈×Dic₁₀₉	C₂²	direct (N × H)
10. C₂²×Q₈×C₁₀₉	C₄	semidirect
14. C₂²×Dic₁₀₉	Q₈	direct (N × H)
20. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
28. Q₈×C₁₀₉	C₂²×C₄	semidirect
32. C₂×Dic₁₀₉	C₂×Q₈	direct (N × H)
35. C₂²×C₁₀₉	C₄×Q₈	semidirect
42. Dic₁₀₉	C₂²×Q₈	direct (N × H)
43. C₂×C₁₀₉	C₂×C₄×Q₈	semidirect
48. C₁₀₉	C₂²×C₄×Q₈	semidirect
53. C₂²×Q₈	Dic₁₀₉	direct (N × H)
62. C₂×Q₈	C₂×Dic₁₀₉	direct (N × H)
76. Q₈	C₂²×Dic₁₀₉	direct (N × H)
83. C₂²	Q₈×Dic₁₀₉	direct (N × H)
91. C₂	C₂×Q₈×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
C₂×Q₈×Dic₁₀₉	2
C₂²×C₄⋊Dic₁₀₉	2
C₂²×C₄×Dic₁₀₉	2
C₂³×Q₈×C₁₀₉	2
C₂²×C₄×Q₈	109

Subgroups — C22×Q8×Dic109 #28376208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×Dic₁₀₉ #28376208