Subgroups — C₂×#518×C₁₀₉ #28360679

Order 13952.

1920 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator	C₂	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
prime radical	C₂³×D₄×C₁₀₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #518×C₁₀₉	C₂	direct (N × H)
3. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
4. C₂×#117×C₁₀₉	C₂	semidirect
5. C₂²×C₂²⋊C₄×C₁₀₉	C₂	semidirect
7. C₂²×D₄×C₁₀₉	C₄	semidirect
11. C₂³×C₄×C₁₀₉	C₂²	semidirect
12. C₄×D₄×C₁₀₉	C₂²	semidirect
13. #117×C₁₀₉	C₂²	semidirect
14. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
16. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
17. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
18. C₂×C₄²×C₁₀₉	C₂²	semidirect
20. C₂⁴×C₁₀₉	C₂×C₄	semidirect
24. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
25. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
28. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
32. C₂²×C₄×C₁₀₉	C₂³	semidirect
33. C₄⋊C₄×C₁₀₉	C₂³	semidirect
35. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
36. C₄²×C₁₀₉	C₂³	semidirect
43. C₂³×C₁₀₉	C₂²×C₄	semidirect
47. D₄×C₁₀₉	C₂²×C₄	semidirect
51. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
68. C₂×C₁₀₉	#518	direct (N × H)
71. C₂×#518	C₁₀₉	direct (N × H)
72. C₁₀₉	C₂×#518	direct (N × H)
74. #518	C₂×C₁₀₉	direct (N × H)
75. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
76. C₂×#117	C₂×C₁₀₉	semidirect
77. C₂²×C₂²⋊C₄	C₂×C₁₀₉	semidirect
79. C₂²×D₄	C₄×C₁₀₉	semidirect
83. C₂³×C₄	C₂²×C₁₀₉	semidirect
84. C₄×D₄	C₂²×C₁₀₉	semidirect
85. #117	C₂²×C₁₀₉	semidirect
86. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
88. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
89. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
90. C₂×C₄²	C₂²×C₁₀₉	semidirect
92. C₂⁴	C₂×C₄×C₁₀₉	semidirect
96. C₂×D₄	C₂×C₄×C₁₀₉	semidirect
97. C₂×D₄	C₂×C₄×C₁₀₉	semidirect
100. C₂²×C₄	C₂×C₄×C₁₀₉	semidirect
104. C₂²×C₄	C₂³×C₁₀₉	semidirect
105. C₄⋊C₄	C₂³×C₁₀₉	semidirect
107. C₂²⋊C₄	C₂³×C₁₀₉	semidirect
108. C₄²	C₂³×C₁₀₉	semidirect
115. C₂³	C₂²×C₄×C₁₀₉	semidirect
119. D₄	C₂²×C₄×C₁₀₉	semidirect
123. C₂×C₄	C₂²×C₄×C₁₀₉	semidirect
140. C₂	#518×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
C₂³×D₄×C₁₀₉	2
#518×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
C₂×#117×C₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
C₂×#518	109

Subgroups — C2×#518×C109 #28360679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#518×C₁₀₉ #28360679