Subgroups — C₂×#543×C₁₀₉ #28360710

Order 13952.

1296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#543×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	12	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₁₀₉	C₂	1	3	yes	maximal
4	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₂²×Q₈×C₁₀₉	C₂²	1	4	yes	complemented
6	C₂³×C₄×C₁₀₉	C₂²	1	6	yes	—
7	C₂²⋊Q₈×C₁₀₉	C₂²	1	96	yes	complemented
8	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	12	yes	complemented
9	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	12	yes	—
10	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	24	yes	—
11	C₂⁴×C₁₀₉	—	4	4	no	—
12	C₂⁴×C₁₀₉	Q₈	1	8	yes	complemented
13	C₂⁴×C₁₀₉	C₂³	1	3	yes	—
14	C₂⁴×C₁₀₉	C₂³	1	4	yes	—
15	C₂×Q₈×C₁₀₉	—	4	8	no	—
16	C₂×Q₈×C₁₀₉	C₂³	1	16	yes	complemented
17	C₂²×C₄×C₁₀₉	—	4	12	no	—
18	C₂²×C₄×C₁₀₉	C₂³	1	12	yes	—
19	C₂²×C₄×C₁₀₉	C₂³	1	24	yes	—
20	C₄⋊C₄×C₁₀₉	C₂³	1	48	yes	complemented
21	C₂²⋊C₄×C₁₀₉	C₂³	1	48	yes	—
22	C₂³×C₁₀₉	—	2	12	no	—
23	C₂³×C₁₀₉	—	4	16	no	—
24	C₂³×C₁₀₉	—	4	8	no	—
25	C₂³×C₁₀₉	C₂×Q₈	1	12	yes	—
26	C₂³×C₁₀₉	C₂×Q₈	1	16	yes	complemented
27	C₂³×C₁₀₉	C₂⁴	1	6	yes	—
28	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
29	Q₈×C₁₀₉	—	4	16	no	—
30	C₂×C₄×C₁₀₉	—	2	24	no	—
31	C₂×C₄×C₁₀₉	—	4	24	no	—
32	C₂×C₄×C₁₀₉	C₂⁴	1	24	yes	—
33	C₂²×C₁₀₉	—	2	24	no	—
34	C₂²×C₁₀₉	—	2	12	no	—
35	C₂²×C₁₀₉	—	4	16	no	—
36	C₂²×C₁₀₉	C₂²×Q₈	1	12	yes	—
37	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
38	C₂²×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
39	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
40	C₄×C₁₀₉	—	2	24	no	—
41	C₂×C₁₀₉	—	2	12	no	—
42	C₂×C₁₀₉	#543	1	4	yes	direct product factor
43	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
44	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
45	C₂×#543	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₁₀₉	C₂×#543	1	1	yes	characteristic, direct product factor, Sylow
47	#543	C₂×C₁₀₉	1	16	yes	direct product factor
48	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	12	yes	complemented
49	C₂²×C₂²⋊C₄	C₂×C₁₀₉	1	3	yes	—
50	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
51	C₂²×Q₈	C₂²×C₁₀₉	1	4	yes	complemented
52	C₂³×C₄	C₂²×C₁₀₉	1	6	yes	—
53	C₂²⋊Q₈	C₂²×C₁₀₉	1	96	yes	complemented
54	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	12	yes	complemented
55	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	12	yes	—
56	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	24	yes	—
57	C₂⁴	—	4	4	no	—
58	C₂⁴	Q₈×C₁₀₉	1	8	yes	complemented
59	C₂⁴	C₂³×C₁₀₉	1	3	yes	—
60	C₂⁴	C₂³×C₁₀₉	1	4	yes	—
61	C₂×Q₈	—	4	8	no	—
62	C₂×Q₈	C₂³×C₁₀₉	1	16	yes	complemented
63	C₂²×C₄	—	4	12	no	—
64	C₂²×C₄	C₂³×C₁₀₉	1	12	yes	—
65	C₂²×C₄	C₂³×C₁₀₉	1	24	yes	—
66	C₄⋊C₄	C₂³×C₁₀₉	1	48	yes	complemented
67	C₂²⋊C₄	C₂³×C₁₀₉	1	48	yes	—
68	C₂³	—	2	12	no	—
69	C₂³	—	4	16	no	—
70	C₂³	—	4	8	no	—
71	C₂³	C₂×Q₈×C₁₀₉	1	12	yes	—
72	C₂³	C₂×Q₈×C₁₀₉	1	16	yes	complemented
73	C₂³	C₂⁴×C₁₀₉	1	6	yes	—
74	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
75	Q₈	—	4	16	no	—
76	C₂×C₄	—	2	24	no	—
77	C₂×C₄	—	4	24	no	—
78	C₂×C₄	C₂⁴×C₁₀₉	1	24	yes	—
79	C₂²	—	2	24	no	—
80	C₂²	—	2	12	no	—
81	C₂²	—	4	16	no	—
82	C₂²	C₂²×Q₈×C₁₀₉	1	12	yes	—
83	C₂²	C₂²×Q₈×C₁₀₉	1	2	yes	—
84	C₂²	H(5,𝔽₂)×C₁₀₉	1	4	yes	—
85	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
86	C₄	—	2	24	no	—
87	C₂	—	2	12	no	—
88	C₂	#543×C₁₀₉	1	4	yes	direct product factor
89	C₂	C₂³×Q₈×C₁₀₉	1	1	yes	characteristic
90	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
prime radical	C₂⁵×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #543×C₁₀₉	C₂	direct (N × H)
2. C₂×C₂²⋊Q₈×C₁₀₉	C₂	semidirect
5. C₂²×Q₈×C₁₀₉	C₂²	semidirect
7. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
8. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
12. C₂⁴×C₁₀₉	Q₈	semidirect
16. C₂×Q₈×C₁₀₉	C₂³	semidirect
20. C₄⋊C₄×C₁₀₉	C₂³	semidirect
26. C₂³×C₁₀₉	C₂×Q₈	semidirect
42. C₂×C₁₀₉	#543	direct (N × H)
45. C₂×#543	C₁₀₉	direct (N × H)
46. C₁₀₉	C₂×#543	direct (N × H)
47. #543	C₂×C₁₀₉	direct (N × H)
48. C₂×C₂²⋊Q₈	C₂×C₁₀₉	semidirect
51. C₂²×Q₈	C₂²×C₁₀₉	semidirect
53. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
54. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
58. C₂⁴	Q₈×C₁₀₉	semidirect
62. C₂×Q₈	C₂³×C₁₀₉	semidirect
66. C₄⋊C₄	C₂³×C₁₀₉	semidirect
72. C₂³	C₂×Q₈×C₁₀₉	semidirect
88. C₂	#543×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
#543×C₁₀₉	2
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
C₂×#543	109

Subgroups — C2×#543×C109 #28360710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#543×C₁₀₉ #28360710