Subgroups — C₂×#10624887 #13090373

Order 1458.

698 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624887	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×C₃×#737	C₃	1	1	yes	maximal, characteristic
3	C₂×#6900	C₃	1	3	yes	maximal, complemented
4	C₂×#6899	C₃	1	6	yes	maximal, complemented
5	C₂×#6896	C₃	1	9	yes	maximal, complemented
6	C₂×#6894	C₃	1	6	yes	maximal, complemented
7	C₂×#6894	C₃	1	9	yes	maximal, complemented
8	C₂×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal
9	C₂×#6887	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×#6887	C₃	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
12	C₃×#737	C₂×C₃	1	1	yes	characteristic
13	#6900	C₂×C₃	1	3	yes	complemented
14	#6899	C₂×C₃	1	6	yes	complemented
15	#6896	C₂×C₃	1	9	yes	complemented
16	#6894	C₂×C₃	1	6	yes	complemented
17	#6894	C₂×C₃	1	9	yes	complemented
18	C₉⋊H(3,𝔽₃)	C₂×C₃	1	3	yes	—
19	#6887	C₂×C₃	1	1	yes	characteristic, complemented
20	#6887	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₉⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×#737	—	3	9	no	—
23	C₂×C₃×3₋¹⁺²	C₃²	1	2	yes	—
24	C₂×C₃×3₋¹⁺²	C₃²	1	3	yes	—
25	C₂×C₃×3₋¹⁺²	C₃²	1	3	yes	—
26	C₂×C₃×3₋¹⁺²	C₃²	1	9	yes	—
27	C₂×C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₂×C₃²×C₉	C₃²	1	3	yes	—
29	C₂×C₃²×C₉	C₃²	1	3	yes	—
30	C₂×C₃²×C₉	C₃²	1	1	yes	characteristic
31	C₂×C₉⋊C₉	C₃²	1	3	yes	—
32	C₂×C₉⋊C₉	C₃²	1	3	yes	—
33	C₂×C₉⋊C₉	C₃²	1	2	yes	—
34	C₂×C₉⋊C₉	C₃²	1	1	yes	characteristic
35	C₂×C₉⋊C₉	C₃²	1	3	yes	—
36	C₂×C₉⋊C₉	C₃²	1	3	yes	—
37	C₂×C₉⋊C₉	C₃²	1	3	yes	—
38	C₂×C₉⋊C₉	C₃²	1	3	yes	—
39	C₂×C₉⋊C₉	C₃²	1	36	yes	—
40	C₂×C₉⋊C₉	C₃²	1	18	yes	—
41	C₂×C₃²⋊C₉	C₃²	1	3	yes	—
42	C₂×C₃²⋊C₉	C₃²	1	3	yes	—
43	C₂×C₃²⋊C₉	C₃²	1	18	yes	—
44	C₂×C₉²	C₃²	1	3	yes	—
45	C₂×C₉²	C₃²	1	3	yes	—
46	#737	—	3	9	no	—
47	C₃×3₋¹⁺²	C₂×C₃²	1	2	yes	—
48	C₃×3₋¹⁺²	C₂×C₃²	1	3	yes	—
49	C₃×3₋¹⁺²	C₂×C₃²	1	3	yes	—
50	C₃×3₋¹⁺²	C₂×C₃²	1	9	yes	—
51	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
52	C₃²×C₉	C₂×C₃²	1	3	yes	—
53	C₃²×C₉	C₂×C₃²	1	3	yes	—
54	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
55	C₉⋊C₉	C₂×C₃²	1	3	yes	—
56	C₉⋊C₉	C₂×C₃²	1	3	yes	—
57	C₉⋊C₉	C₂×C₃²	1	2	yes	—
58	C₉⋊C₉	C₂×C₃²	1	1	yes	characteristic
59	C₉⋊C₉	C₂×C₃²	1	3	yes	—
60	C₉⋊C₉	C₂×C₃²	1	3	yes	—
61	C₉⋊C₉	C₂×C₃²	1	3	yes	—
62	C₉⋊C₉	C₂×C₃²	1	3	yes	—
63	C₉⋊C₉	C₂×C₃²	1	36	yes	—
64	C₉⋊C₉	C₂×C₃²	1	18	yes	—
65	C₃²⋊C₉	C₂×C₃²	1	3	yes	—
66	C₃²⋊C₉	C₂×C₃²	1	3	yes	—
67	C₃²⋊C₉	C₂×C₃²	1	18	yes	—
68	C₉²	C₂×C₃²	1	3	yes	—
69	C₉²	C₂×C₃²	1	3	yes	—
70	C₂×C₃³	C₃³	1	3	yes	—
71	C₂×C₃³	C₃³	1	1	yes	characteristic
72	C₂×3₋¹⁺²	—	3	6	no	—
73	C₂×3₋¹⁺²	—	3	9	no	—
74	C₂×3₋¹⁺²	—	3	9	no	—
75	C₂×3₋¹⁺²	—	9	9	no	—
76	C₂×H(3,𝔽₃)	—	3	3	no	—
77	C₂×C₃×C₉	—	3	9	no	—
78	C₂×C₃×C₉	—	9	3	no	—
79	C₂×C₃×C₉	3₋¹⁺²	1	9	yes	complemented
80	C₂×C₃×C₉	C₃³	1	9	yes	—
81	C₂×C₃×C₉	C₃³	1	9	yes	—
82	C₂×C₃×C₉	C₃³	1	6	yes	—
83	C₂×C₃×C₉	C₃³	1	6	yes	—
84	C₂×C₃×C₉	C₃³	1	3	yes	—
85	C₂×C₃×C₉	C₃³	1	1	yes	characteristic
86	C₂×C₃×C₉	C₃³	1	1	yes	characteristic
87	C₂×C₃×C₉	C₃³	1	1	yes	characteristic
88	C₃³	C₂×C₃³	1	3	yes	—
89	C₃³	C₂×C₃³	1	1	yes	characteristic
90	3₋¹⁺²	—	3	6	no	—
91	3₋¹⁺²	—	3	9	no	—
92	3₋¹⁺²	—	3	9	no	—
93	3₋¹⁺²	—	9	9	no	—
94	H(3,𝔽₃)	—	3	3	no	—
95	C₃×C₉	—	3	9	no	—
96	C₃×C₉	—	9	3	no	—
97	C₃×C₉	C₂×3₋¹⁺²	1	9	yes	complemented
98	C₃×C₉	C₂×C₃³	1	9	yes	—
99	C₃×C₉	C₂×C₃³	1	9	yes	—
100	C₃×C₉	C₂×C₃³	1	6	yes	—
101	C₃×C₉	C₂×C₃³	1	6	yes	—
102	C₃×C₉	C₂×C₃³	1	3	yes	—
103	C₃×C₉	C₂×C₃³	1	1	yes	characteristic
104	C₃×C₉	C₂×C₃³	1	1	yes	characteristic
105	C₃×C₉	C₂×C₃³	1	1	yes	characteristic
106	C₂×C₃²	—	3	3	no	—
107	C₂×C₃²	—	3	3	no	—
108	C₂×C₃²	—	3	9	no	—
109	C₂×C₃²	C₃×3₋¹⁺²	1	3	yes	—
110	C₂×C₃²	C₃⁴	1	1	yes	characteristic
111	C₂×C₉	—	3	6	no	—
112	C₂×C₉	—	3	6	no	—
113	C₂×C₉	—	3	3	no	—
114	C₂×C₉	—	3	9	no	—
115	C₂×C₉	—	3	9	no	—
116	C₂×C₉	C₃×3₋¹⁺²	1	3	yes	—
117	C₂×C₉	C₃×3₋¹⁺²	1	3	yes	—
118	C₂×C₉	C₃×3₋¹⁺²	1	3	yes	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	3	no	—
121	C₃²	—	3	9	no	—
122	C₃²	C₂×C₃×3₋¹⁺²	1	3	yes	—
123	C₃²	C₂×C₃⁴	1	1	yes	characteristic
124	C₉	—	3	6	no	—
125	C₉	—	3	6	no	—
126	C₉	—	3	3	no	—
127	C₉	—	3	9	no	—
128	C₉	—	3	9	no	—
129	C₉	C₂×C₃×3₋¹⁺²	1	3	yes	—
130	C₉	C₂×C₃×3₋¹⁺²	1	3	yes	—
131	C₉	C₂×C₃×3₋¹⁺²	1	3	yes	—
132	C₂×C₃	—	3	3	no	—
133	C₂×C₃	—	9	3	no	—
134	C₂×C₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
135	C₂×C₃	3₋¹⁺⁴	1	3	yes	—
136	C₃	—	3	3	no	—
137	C₃	—	9	3	no	—
138	C₃	C₂×C₃²×3₋¹⁺²	1	1	yes	characteristic
139	C₃	C₂×3₋¹⁺⁴	1	3	yes	—
140	C₂	#10624887	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃²	C₃⁴
commutator, Frattini, cosocle	C₃²	C₂×C₃⁴
prime radical	C₂×C₃×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. #10624887	C₂	direct (N × H)
3. C₂×#6900	C₃	semidirect
4. C₂×#6899	C₃	semidirect
5. C₂×#6896	C₃	semidirect
6. C₂×#6894	C₃	semidirect
7. C₂×#6894	C₃	semidirect
9. C₂×#6887	C₃	semidirect
10. C₂×#6887	C₃	semidirect
11. C₂×C₃×C₉⋊C₉	C₃	semidirect
13. #6900	C₂×C₃	semidirect
14. #6899	C₂×C₃	semidirect
15. #6896	C₂×C₃	semidirect
16. #6894	C₂×C₃	semidirect
17. #6894	C₂×C₃	semidirect
19. #6887	C₂×C₃	semidirect
20. #6887	C₂×C₃	semidirect
21. C₃×C₉⋊C₉	C₂×C₃	semidirect
79. C₂×C₃×C₉	3₋¹⁺²	semidirect
97. C₃×C₉	C₂×3₋¹⁺²	semidirect
140. C₂	#10624887	direct (N × H)

Maximal subgroups

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

subgroup	index
#10624887	2
C₂×C₃×#737	3
C₂×#6900	3
C₂×#6899	3
C₂×#6896	3
C₂×#6894	3
C₂×C₉⋊H(3,𝔽₃)	3
C₂×#6887	3
C₂×C₃×C₉⋊C₉	3

Subgroups — C2×#10624887 #13090373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624887 #13090373