Subgroups — C₂×C₁₁×#518 #13066263

Order 1408.

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×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#518	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×D₄×C₁₁	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₁×#117	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₁×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₂⁵×C₁₁	C₂²	1	2	yes	—
7	C₂²×D₄×C₁₁	C₄	1	16	yes	complemented
8	C₂²×D₄×C₁₁	C₂²	1	4	yes	—
9	C₂²×D₄×C₁₁	C₂²	1	8	yes	—
10	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₁	C₂²	1	8	yes	complemented
12	C₄×D₄×C₁₁	C₂²	1	64	yes	complemented
13	C₁₁×#117	C₂²	1	16	yes	complemented
14	C₂×C₁₁×C₄⋊C₄	C₂²	1	4	yes	complemented
15	C₂×C₁₁×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₂×C₁₁×C₂²⋊C₄	C₂²	1	32	yes	complemented
17	C₂×C₁₁×C₂²⋊C₄	C₂²	1	8	yes	complemented
18	C₂×C₄²×C₁₁	C₂²	1	4	yes	complemented
19	C₂⁴×C₁₁	—	4	8	no	—
20	C₂⁴×C₁₁	C₂×C₄	1	16	yes	complemented
21	C₂⁴×C₁₁	C₂³	1	1	yes	characteristic
22	C₂⁴×C₁₁	C₂³	1	4	yes	—
23	C₂⁴×C₁₁	C₂³	1	8	yes	—
24	C₂×D₄×C₁₁	C₂×C₄	1	32	yes	complemented
25	C₂×D₄×C₁₁	C₂×C₄	1	64	yes	complemented
26	C₂×D₄×C₁₁	C₂³	1	16	yes	—
27	C₂²×C₄×C₁₁	—	4	16	no	—
28	C₂²×C₄×C₁₁	C₂×C₄	1	8	yes	complemented
29	C₂²×C₄×C₁₁	C₂³	1	8	yes	—
30	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₁	C₂³	1	32	yes	complemented
33	C₁₁×C₄⋊C₄	C₂³	1	16	yes	complemented
34	C₁₁×C₂²⋊C₄	C₂³	1	16	yes	—
35	C₁₁×C₂²⋊C₄	C₂³	1	32	yes	complemented
36	C₄²×C₁₁	C₂³	1	16	yes	complemented
37	C₂×#518	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂³×C₁₁	—	2	16	no	—
39	C₂³×C₁₁	—	2	4	no	—
40	C₂³×C₁₁	—	4	32	no	—
41	C₂³×C₁₁	—	4	16	no	—
42	C₂³×C₁₁	C₂²×C₄	1	4	yes	—
43	C₂³×C₁₁	C₂²×C₄	1	16	yes	—
44	C₂³×C₁₁	C₂²×C₄	1	32	yes	complemented
45	C₂³×C₁₁	C₂⁴	1	2	yes	—
46	C₂³×C₁₁	C₂⁴	1	8	yes	—
47	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
48	D₄×C₁₁	C₂²×C₄	1	64	yes	complemented
49	C₂×C₄×C₁₁	—	2	16	no	—
50	C₂×C₄×C₁₁	—	4	32	no	—
51	C₂×C₄×C₁₁	C₂²×C₄	1	8	yes	—
52	C₂×C₄×C₁₁	C₂²×C₄	1	16	yes	complemented
53	C₂×C₄×C₁₁	C₂⁴	1	16	yes	—
54	C₂×C₄×C₁₁	C₂⁴	1	4	yes	—
55	C₂³×D₄	C₂×C₁₁	1	1	yes	characteristic
56	#518	C₂×C₁₁	1	16	yes	direct product factor
57	C₂×C₄×D₄	C₂×C₁₁	1	8	yes	complemented
58	C₂×#117	C₂×C₁₁	1	2	yes	complemented
59	C₂²×C₂²⋊C₄	C₂×C₁₁	1	4	yes	complemented
60	C₂²×C₁₁	—	2	8	no	—
61	C₂²×C₁₁	—	2	32	no	—
62	C₂²×C₁₁	—	2	16	no	—
63	C₂²×C₁₁	—	2	4	no	—
64	C₂²×C₁₁	—	4	32	no	—
65	C₂²×C₁₁	C₂³×C₄	1	4	yes	—
66	C₂²×C₁₁	C₂³×C₄	1	16	yes	—
67	C₂²×C₁₁	C₂³×C₄	1	2	yes	—
68	C₂²×C₁₁	H(5,𝔽₂)	1	4	yes	—
69	C₂²×C₁₁	C₂⁵	1	1	yes	characteristic
70	C₄×C₁₁	—	2	16	no	—
71	C₄×C₁₁	C₂³×C₄	1	8	yes	—
72	C₂⁵	C₂²×C₁₁	1	2	yes	—
73	C₂²×D₄	C₄×C₁₁	1	16	yes	complemented
74	C₂²×D₄	C₂²×C₁₁	1	4	yes	—
75	C₂²×D₄	C₂²×C₁₁	1	8	yes	—
76	C₂³×C₄	C₂²×C₁₁	1	1	yes	characteristic
77	C₂³×C₄	C₂²×C₁₁	1	8	yes	complemented
78	C₄×D₄	C₂²×C₁₁	1	64	yes	complemented
79	#117	C₂²×C₁₁	1	16	yes	complemented
80	C₂×C₄⋊C₄	C₂²×C₁₁	1	4	yes	complemented
81	C₂×C₂²⋊C₄	C₂²×C₁₁	1	4	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₁₁	1	32	yes	complemented
83	C₂×C₂²⋊C₄	C₂²×C₁₁	1	8	yes	complemented
84	C₂×C₄²	C₂²×C₁₁	1	4	yes	complemented
85	C₂×C₁₁	—	2	4	no	—
86	C₂×C₁₁	—	2	16	no	—
87	C₂×C₁₁	#518	1	4	yes	direct product factor
88	C₂×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
89	C₂×C₁₁	C₂×H(5,𝔽₂)	1	2	yes	—
90	C₂⁴	—	4	8	no	—
91	C₂⁴	C₂×C₄×C₁₁	1	16	yes	complemented
92	C₂⁴	C₂³×C₁₁	1	1	yes	characteristic
93	C₂⁴	C₂³×C₁₁	1	4	yes	—
94	C₂⁴	C₂³×C₁₁	1	8	yes	—
95	C₂×D₄	C₂×C₄×C₁₁	1	32	yes	complemented
96	C₂×D₄	C₂×C₄×C₁₁	1	64	yes	complemented
97	C₂×D₄	C₂³×C₁₁	1	16	yes	—
98	C₂²×C₄	—	4	16	no	—
99	C₂²×C₄	C₂×C₄×C₁₁	1	8	yes	complemented
100	C₂²×C₄	C₂³×C₁₁	1	8	yes	—
101	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
102	C₂²×C₄	C₂³×C₁₁	1	4	yes	—
103	C₂²×C₄	C₂³×C₁₁	1	32	yes	complemented
104	C₄⋊C₄	C₂³×C₁₁	1	16	yes	complemented
105	C₂²⋊C₄	C₂³×C₁₁	1	16	yes	—
106	C₂²⋊C₄	C₂³×C₁₁	1	32	yes	complemented
107	C₄²	C₂³×C₁₁	1	16	yes	complemented
108	C₁₁	C₂×#518	1	1	yes	characteristic, direct product factor, Sylow
109	C₂³	—	2	16	no	—
110	C₂³	—	2	4	no	—
111	C₂³	—	4	32	no	—
112	C₂³	—	4	16	no	—
113	C₂³	C₂²×C₄×C₁₁	1	4	yes	—
114	C₂³	C₂²×C₄×C₁₁	1	16	yes	—
115	C₂³	C₂²×C₄×C₁₁	1	32	yes	complemented
116	C₂³	C₂⁴×C₁₁	1	2	yes	—
117	C₂³	C₂⁴×C₁₁	1	8	yes	—
118	C₂³	C₂⁴×C₁₁	1	1	yes	characteristic
119	D₄	C₂²×C₄×C₁₁	1	64	yes	complemented
120	C₂×C₄	—	2	16	no	—
121	C₂×C₄	—	4	32	no	—
122	C₂×C₄	C₂²×C₄×C₁₁	1	8	yes	—
123	C₂×C₄	C₂²×C₄×C₁₁	1	16	yes	complemented
124	C₂×C₄	C₂⁴×C₁₁	1	16	yes	—
125	C₂×C₄	C₂⁴×C₁₁	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×C₁₁	1	4	yes	—
132	C₂²	C₂³×C₄×C₁₁	1	16	yes	—
133	C₂²	C₂³×C₄×C₁₁	1	2	yes	—
134	C₂²	C₁₁×H(5,𝔽₂)	1	4	yes	—
135	C₂²	C₂⁵×C₁₁	1	1	yes	characteristic
136	C₄	—	2	16	no	—
137	C₄	C₂³×C₄×C₁₁	1	8	yes	—
138	C₂	—	2	4	no	—
139	C₂	—	2	16	no	—
140	C₂	C₁₁×#518	1	4	yes	direct product factor
141	C₂	C₂⁴×C₄×C₁₁	1	1	yes	characteristic
142	C₂	C₂×C₁₁×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂⁴
commutator	C₂	C₂⁴×C₄×C₁₁
Frattini, cosocle	C₂²	C₂⁵×C₁₁
prime radical	C₂³×D₄×C₁₁	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁×#518	C₂	direct (N × H)
3. C₂×C₄×D₄×C₁₁	C₂	semidirect
4. C₂×C₁₁×#117	C₂	semidirect
5. C₂²×C₁₁×C₂²⋊C₄	C₂	semidirect
7. C₂²×D₄×C₁₁	C₄	semidirect
11. C₂³×C₄×C₁₁	C₂²	semidirect
12. C₄×D₄×C₁₁	C₂²	semidirect
13. C₁₁×#117	C₂²	semidirect
14. C₂×C₁₁×C₄⋊C₄	C₂²	semidirect
16. C₂×C₁₁×C₂²⋊C₄	C₂²	semidirect
17. C₂×C₁₁×C₂²⋊C₄	C₂²	semidirect
18. C₂×C₄²×C₁₁	C₂²	semidirect
20. C₂⁴×C₁₁	C₂×C₄	semidirect
24. C₂×D₄×C₁₁	C₂×C₄	semidirect
25. C₂×D₄×C₁₁	C₂×C₄	semidirect
28. C₂²×C₄×C₁₁	C₂×C₄	semidirect
32. C₂²×C₄×C₁₁	C₂³	semidirect
33. C₁₁×C₄⋊C₄	C₂³	semidirect
35. C₁₁×C₂²⋊C₄	C₂³	semidirect
36. C₄²×C₁₁	C₂³	semidirect
37. C₂×#518	C₁₁	direct (N × H)
44. C₂³×C₁₁	C₂²×C₄	semidirect
48. D₄×C₁₁	C₂²×C₄	semidirect
52. C₂×C₄×C₁₁	C₂²×C₄	semidirect
56. #518	C₂×C₁₁	direct (N × H)
57. C₂×C₄×D₄	C₂×C₁₁	semidirect
58. C₂×#117	C₂×C₁₁	semidirect
59. C₂²×C₂²⋊C₄	C₂×C₁₁	semidirect
73. C₂²×D₄	C₄×C₁₁	semidirect
77. C₂³×C₄	C₂²×C₁₁	semidirect
78. C₄×D₄	C₂²×C₁₁	semidirect
79. #117	C₂²×C₁₁	semidirect
80. C₂×C₄⋊C₄	C₂²×C₁₁	semidirect
82. C₂×C₂²⋊C₄	C₂²×C₁₁	semidirect
83. C₂×C₂²⋊C₄	C₂²×C₁₁	semidirect
84. C₂×C₄²	C₂²×C₁₁	semidirect
87. C₂×C₁₁	#518	direct (N × H)
91. C₂⁴	C₂×C₄×C₁₁	semidirect
95. C₂×D₄	C₂×C₄×C₁₁	semidirect
96. C₂×D₄	C₂×C₄×C₁₁	semidirect
99. C₂²×C₄	C₂×C₄×C₁₁	semidirect
103. C₂²×C₄	C₂³×C₁₁	semidirect
104. C₄⋊C₄	C₂³×C₁₁	semidirect
106. C₂²⋊C₄	C₂³×C₁₁	semidirect
107. C₄²	C₂³×C₁₁	semidirect
108. C₁₁	C₂×#518	direct (N × H)
115. C₂³	C₂²×C₄×C₁₁	semidirect
119. D₄	C₂²×C₄×C₁₁	semidirect
123. C₂×C₄	C₂²×C₄×C₁₁	semidirect
140. C₂	C₁₁×#518	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×C₁₁	2
C₁₁×#518	2
C₂×C₄×D₄×C₁₁	2
C₂×C₁₁×#117	2
C₂²×C₁₁×C₂²⋊C₄	2
C₂×#518	11

Subgroups — C2×C11×#518 #13066263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#518 #13066263