Subgroups — C₄×H(5,𝔽₂)×C₁₀₉ #28360683

Order 13952.

1660 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#520×C₁₀₉	C₂	1	6	yes	maximal, complemented
3	#518×C₁₀₉	C₂	1	9	yes	maximal, complemented
4	C₄×#41×C₁₀₉	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₄×C₁₀₉	C₂	1	9	yes	maximal, complemented
6	H(5,𝔽₂)×C₁₀₉	C₄	1	16	yes	direct product factor
7	C₂×#41×C₁₀₉	C₂²	1	6	yes	—
8	C₂²×D₄×C₁₀₉	C₂²	1	9	yes	—
9	C₂³×C₄×C₁₀₉	C₂²	1	6	yes	complemented
10	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
11	C₄×Q₈×C₁₀₉	C₂²	1	6	yes	complemented
12	C₄×D₄×C₁₀₉	C₂²	1	18	yes	complemented
13	C₄×D₄×C₁₀₉	C₂²	1	36	yes	complemented
14	C₄×D₄×C₁₀₉	C₂²	1	18	yes	complemented
15	#117×C₁₀₉	C₂²	1	18	yes	complemented
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	9	yes	complemented
17	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	18	yes	complemented
18	C₂×C₄²×C₁₀₉	C₂²	1	9	yes	complemented
19	C₂⁴×C₁₀₉	C₂³	1	6	yes	—
20	#41×C₁₀₉	C₂×C₄	1	48	yes	complemented
21	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
22	C₂×D₄×C₁₀₉	C₂×C₄	1	72	yes	complemented
23	C₂×D₄×C₁₀₉	C₂³	1	18	yes	—
24	C₂²×C₄×C₁₀₉	—	4	12	no	—
25	C₂²×C₄×C₁₀₉	C₂³	1	9	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	9	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	18	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	9	yes	—
29	C₄⋊C₄×C₁₀₉	C₂³	1	18	yes	—
30	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
31	C₂²⋊C₄×C₁₀₉	C₂³	1	36	yes	—
32	C₄²×C₁₀₉	C₂³	1	18	yes	—
33	C₄²×C₁₀₉	C₂³	1	6	yes	—
34	C₂³×C₁₀₉	—	4	6	no	—
35	C₂³×C₁₀₉	—	4	6	no	—
36	C₂³×C₁₀₉	C₂²×C₄	1	24	yes	complemented
37	C₂³×C₁₀₉	C₂⁴	1	9	yes	—
38	Q₈×C₁₀₉	C₂²×C₄	1	8	yes	complemented
39	D₄×C₁₀₉	C₂²×C₄	1	72	yes	complemented
40	C₂×C₄×C₁₀₉	—	2	18	no	—
41	C₂×C₄×C₁₀₉	—	4	18	no	—
42	C₂×C₄×C₁₀₉	—	4	9	no	—
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	36	yes	complemented
44	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
45	C₂×C₄×C₁₀₉	C₂⁴	1	9	yes	—
46	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
47	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂²×C₁₀₉	—	2	9	no	—
49	C₂²×C₁₀₉	—	2	9	no	—
50	C₂²×C₁₀₉	—	4	24	no	—
51	C₂²×C₁₀₉	C₂³×C₄	1	18	yes	—
52	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
53	C₄×C₁₀₉	—	2	9	no	—
54	C₄×C₁₀₉	—	2	6	no	—
55	C₄×C₁₀₉	C₂³×C₄	1	12	yes	—
56	C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	direct product factor
57	C₂×C₁₀₉	—	2	18	no	—
58	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
59	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
60	C₂×C₁₀₉	#585	1	1	yes	characteristic
61	C₄×H(5,𝔽₂)	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₁₀₉	C₄×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×H(5,𝔽₂)	C₂×C₁₀₉	1	1	yes	characteristic
64	#520	C₂×C₁₀₉	1	6	yes	complemented
65	#518	C₂×C₁₀₉	1	9	yes	complemented
66	C₄×#41	C₂×C₁₀₉	1	6	yes	complemented
67	C₂×C₄×D₄	C₂×C₁₀₉	1	9	yes	complemented
68	H(5,𝔽₂)	C₄×C₁₀₉	1	16	yes	direct product factor
69	C₂×#41	C₂²×C₁₀₉	1	6	yes	—
70	C₂²×D₄	C₂²×C₁₀₉	1	9	yes	—
71	C₂³×C₄	C₂²×C₁₀₉	1	6	yes	complemented
72	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	complemented
73	C₄×Q₈	C₂²×C₁₀₉	1	6	yes	complemented
74	C₄×D₄	C₂²×C₁₀₉	1	18	yes	complemented
75	C₄×D₄	C₂²×C₁₀₉	1	36	yes	complemented
76	C₄×D₄	C₂²×C₁₀₉	1	18	yes	complemented
77	#117	C₂²×C₁₀₉	1	18	yes	complemented
78	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	9	yes	complemented
79	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	18	yes	complemented
80	C₂×C₄²	C₂²×C₁₀₉	1	9	yes	complemented
81	C₂⁴	C₂³×C₁₀₉	1	6	yes	—
82	#41	C₂×C₄×C₁₀₉	1	48	yes	complemented
83	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
84	C₂×D₄	C₂×C₄×C₁₀₉	1	72	yes	complemented
85	C₂×D₄	C₂³×C₁₀₉	1	18	yes	—
86	C₂²×C₄	—	4	12	no	—
87	C₂²×C₄	C₂³×C₁₀₉	1	9	yes	—
88	C₂²×C₄	C₂³×C₁₀₉	1	9	yes	—
89	C₂²×C₄	C₂³×C₁₀₉	1	18	yes	—
90	C₂²×C₄	C₂³×C₁₀₉	1	9	yes	—
91	C₄⋊C₄	C₂³×C₁₀₉	1	18	yes	—
92	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
93	C₂²⋊C₄	C₂³×C₁₀₉	1	36	yes	—
94	C₄²	C₂³×C₁₀₉	1	6	yes	—
95	C₄²	C₂³×C₁₀₉	1	18	yes	—
96	C₂³	—	4	6	no	—
97	C₂³	—	4	6	no	—
98	C₂³	C₂²×C₄×C₁₀₉	1	24	yes	complemented
99	C₂³	C₂⁴×C₁₀₉	1	9	yes	—
100	Q₈	C₂²×C₄×C₁₀₉	1	8	yes	complemented
101	D₄	C₂²×C₄×C₁₀₉	1	72	yes	complemented
102	C₂×C₄	—	2	18	no	—
103	C₂×C₄	—	4	18	no	—
104	C₂×C₄	—	4	9	no	—
105	C₂×C₄	C₂²×C₄×C₁₀₉	1	36	yes	complemented
106	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
107	C₂×C₄	C₂⁴×C₁₀₉	1	9	yes	—
108	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
109	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
110	C₂²	—	2	9	no	—
111	C₂²	—	2	9	no	—
112	C₂²	—	4	24	no	—
113	C₂²	C₂³×C₄×C₁₀₉	1	18	yes	—
114	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
115	C₄	—	2	9	no	—
116	C₄	—	2	6	no	—
117	C₄	C₂³×C₄×C₁₀₉	1	12	yes	—
118	C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	direct product factor
119	C₂	—	2	18	no	—
120	C₂	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
121	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
122	C₂	#585×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator	C₂	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
socle	C₂²×C₁₀₉	C₂⁵
prime radical	C₂×H(5,𝔽₂)×C₁₀₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #520×C₁₀₉	C₂	semidirect
3. #518×C₁₀₉	C₂	semidirect
4. C₄×#41×C₁₀₉	C₂	semidirect
5. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
6. H(5,𝔽₂)×C₁₀₉	C₄	direct (N × H)
9. C₂³×C₄×C₁₀₉	C₂²	semidirect
10. C₄×Q₈×C₁₀₉	C₂²	semidirect
11. C₄×Q₈×C₁₀₉	C₂²	semidirect
12. C₄×D₄×C₁₀₉	C₂²	semidirect
13. C₄×D₄×C₁₀₉	C₂²	semidirect
14. C₄×D₄×C₁₀₉	C₂²	semidirect
15. #117×C₁₀₉	C₂²	semidirect
16. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
17. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
18. C₂×C₄²×C₁₀₉	C₂²	semidirect
20. #41×C₁₀₉	C₂×C₄	semidirect
22. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
36. C₂³×C₁₀₉	C₂²×C₄	semidirect
38. Q₈×C₁₀₉	C₂²×C₄	semidirect
39. D₄×C₁₀₉	C₂²×C₄	semidirect
43. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
56. C₄×C₁₀₉	H(5,𝔽₂)	direct (N × H)
61. C₄×H(5,𝔽₂)	C₁₀₉	direct (N × H)
62. C₁₀₉	C₄×H(5,𝔽₂)	direct (N × H)
64. #520	C₂×C₁₀₉	semidirect
65. #518	C₂×C₁₀₉	semidirect
66. C₄×#41	C₂×C₁₀₉	semidirect
67. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
68. H(5,𝔽₂)	C₄×C₁₀₉	direct (N × H)
71. C₂³×C₄	C₂²×C₁₀₉	semidirect
72. C₄×Q₈	C₂²×C₁₀₉	semidirect
73. C₄×Q₈	C₂²×C₁₀₉	semidirect
74. C₄×D₄	C₂²×C₁₀₉	semidirect
75. C₄×D₄	C₂²×C₁₀₉	semidirect
76. C₄×D₄	C₂²×C₁₀₉	semidirect
77. #117	C₂²×C₁₀₉	semidirect
78. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
79. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
80. C₂×C₄²	C₂²×C₁₀₉	semidirect
82. #41	C₂×C₄×C₁₀₉	semidirect
84. C₂×D₄	C₂×C₄×C₁₀₉	semidirect
98. C₂³	C₂²×C₄×C₁₀₉	semidirect
100. Q₈	C₂²×C₄×C₁₀₉	semidirect
101. D₄	C₂²×C₄×C₁₀₉	semidirect
105. C₂×C₄	C₂²×C₄×C₁₀₉	semidirect
118. C₄	H(5,𝔽₂)×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
C₂×H(5,𝔽₂)×C₁₀₉	2
#520×C₁₀₉	2
#518×C₁₀₉	2
C₄×#41×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
C₄×H(5,𝔽₂)	109

Subgroups — C4×H(5,𝔽2)×C109 #28360683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×H(5,𝔽₂)×C₁₀₉ #28360683