Subgroups — C₂×C₁₁×#1398 #14590743

Order 2816.

222 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₁₁×#346	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1398	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₁×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
7	C₁₁×M₆₄	C₄	1	4	yes	complemented
8	C₁₁×M₆₄	C₂²	1	4	yes	—
9	C₂×C₁₁×C₃₂	C₄	1	4	yes	complemented
10	C₁₁×#346	C₄	1	2	yes	—
11	C₁₁×#346	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₁	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₁	C₈	1	1	yes	characteristic
15	C₂×C₁₁×C₁₆	C₈	1	2	yes	—
16	C₂×C₁₁×C₁₆	C₈	1	2	yes	—
17	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₁₁	C₈	1	2	yes	—
23	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
24	C₁₁×C₃₂	C₂×C₄	1	8	yes	complemented
25	C₂×#1398	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₄×C₁₁	—	2	1	no	—
27	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₁	—	2	1	no	—
29	C₂×C₈×C₁₁	—	2	1	no	—
30	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₁₁	—	2	1	no	—
38	C₄²×C₁₁	C₂×C₈	1	2	yes	—
39	C₁₁×C₁₆	C₄²	1	2	yes	—
40	C₁₁×C₁₆	C₂×C₈	1	4	yes	—
41	C₁₁×C₁₆	C₂²×C₄	1	2	yes	—
42	C₂×M₆₄	C₂×C₁₁	1	2	yes	—
43	C₂×#346	C₂×C₁₁	1	1	yes	characteristic
44	#1398	C₂×C₁₁	1	4	yes	direct product factor
45	C₂³×C₁₁	M₃₂	1	1	yes	characteristic
46	C₂×C₄×C₁₁	—	2	2	no	—
47	C₂×C₄×C₁₁	—	4	2	no	—
48	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₁	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₁₁	M₃₂	1	1	yes	characteristic
52	C₂×C₄×C₁₁	M₃₂	1	1	yes	characteristic
53	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₁₁	—	2	2	no	—
55	C₈×C₁₁	C₄×C₈	1	1	yes	characteristic
56	C₈×C₁₁	C₄×C₈	1	1	yes	characteristic
57	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
58	C₈×C₁₁	C₂²×C₈	1	1	yes	characteristic
59	C₂²×C₁₆	C₄×C₁₁	1	1	yes	characteristic
60	C₂²×C₁₆	C₂²×C₁₁	1	1	yes	characteristic
61	C₂×C₄×C₈	C₄×C₁₁	1	1	yes	characteristic
62	M₆₄	C₄×C₁₁	1	4	yes	complemented
63	M₆₄	C₂²×C₁₁	1	4	yes	—
64	C₂×C₃₂	C₄×C₁₁	1	4	yes	complemented
65	#346	C₄×C₁₁	1	2	yes	—
66	#346	C₂²×C₁₁	1	2	yes	—
67	C₂²×C₁₁	—	2	2	no	—
68	C₂²×C₁₁	#346	1	1	yes	characteristic
69	C₂²×C₁₁	#346	1	1	yes	characteristic
70	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
71	C₄×C₁₁	—	4	2	no	—
72	C₄×C₁₁	#346	1	1	yes	characteristic
73	C₄×C₁₁	#346	1	1	yes	characteristic
74	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₁₁	C₂×M₃₂	1	1	yes	characteristic
76	C₂²×C₈	C₈×C₁₁	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
78	C₂×C₄²	C₈×C₁₁	1	1	yes	characteristic
79	C₂×C₁₆	C₈×C₁₁	1	2	yes	—
80	C₂×C₁₆	C₈×C₁₁	1	2	yes	—
81	C₂×C₁₆	C₂×C₄×C₁₁	1	1	yes	characteristic
82	C₂×C₁₆	C₂×C₄×C₁₁	1	2	yes	—
83	C₂×C₁₆	C₂×C₄×C₁₁	1	2	yes	—
84	C₂×C₁₆	C₂×C₄×C₁₁	1	2	yes	—
85	C₂×C₁₆	C₂³×C₁₁	1	1	yes	characteristic
86	C₄×C₈	C₈×C₁₁	1	2	yes	—
87	C₄×C₈	C₂×C₄×C₁₁	1	2	yes	—
88	C₃₂	C₂×C₄×C₁₁	1	8	yes	complemented
89	C₂×C₁₁	—	2	1	no	—
90	C₂×C₁₁	—	2	1	no	—
91	C₂×C₁₁	#1398	1	2	yes	direct product factor
92	C₂×C₁₁	C₂×#346	1	1	yes	characteristic
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂×C₈×C₁₁	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₄²×C₁₁	1	1	yes	characteristic
98	C₂×C₈	C₄²×C₁₁	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₁₁	1	2	yes	—
100	C₂×C₈	C₂×C₈×C₁₁	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₁₁	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₁₁	1	1	yes	characteristic
103	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₂×C₈×C₁₁	1	2	yes	—
106	C₁₆	C₄²×C₁₁	1	2	yes	—
107	C₁₆	C₂×C₈×C₁₁	1	4	yes	—
108	C₁₆	C₂²×C₄×C₁₁	1	2	yes	—
109	C₁₁	C₂×#1398	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³	C₁₁×M₃₂	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₈×C₁₁	1	1	yes	characteristic
114	C₂×C₄	C₄×C₈×C₁₁	1	1	yes	characteristic
115	C₂×C₄	C₁₁×M₃₂	1	1	yes	characteristic
116	C₂×C₄	C₁₁×M₃₂	1	1	yes	characteristic
117	C₂×C₄	C₁₁×M₃₂	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₁₁	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	C₄×C₈×C₁₁	1	1	yes	characteristic
121	C₈	C₄×C₈×C₁₁	1	1	yes	characteristic
122	C₈	C₂×C₄²×C₁₁	1	1	yes	characteristic
123	C₈	C₂²×C₈×C₁₁	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	C₁₁×#346	1	1	yes	characteristic
126	C₂²	C₁₁×#346	1	1	yes	characteristic
127	C₂²	C₂×C₁₁×M₃₂	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	C₁₁×#346	1	1	yes	characteristic
130	C₄	C₁₁×#346	1	1	yes	characteristic
131	C₄	C₂×C₄×C₈×C₁₁	1	1	yes	characteristic
132	C₄	C₂×C₁₁×M₃₂	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	C₁₁×#1398	1	2	yes	direct product factor
136	C₂	C₂×C₁₁×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₁	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₁
socle	C₂²×C₁₁	#346
prime radical	C₂³×C₁₁	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₁×#1398	C₂	direct (N × H)
7. C₁₁×M₆₄	C₄	semidirect
9. C₂×C₁₁×C₃₂	C₄	semidirect
24. C₁₁×C₃₂	C₂×C₄	semidirect
25. C₂×#1398	C₁₁	direct (N × H)
44. #1398	C₂×C₁₁	direct (N × H)
62. M₆₄	C₄×C₁₁	semidirect
64. C₂×C₃₂	C₄×C₁₁	semidirect
88. C₃₂	C₂×C₄×C₁₁	semidirect
91. C₂×C₁₁	#1398	direct (N × H)
109. C₁₁	C₂×#1398	direct (N × H)
135. C₂	C₁₁×#1398	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×C₁₁×M₆₄	2
C₂×C₁₁×#346	2
C₁₁×#1398	2
C₂×#1398	11

Subgroups — C2×C11×#1398 #14590743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#1398 #14590743