Subgroups — #14674256

Order 2816.

218 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×#10622137	C₂	1	2	yes	maximal
2	C₄⋊(C₁₁⋊C₃₂)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic
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	C₄×C₁₁×C₁₆	C₄	1	2	yes	—
8	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
9	#10622137	—	2	4	no	—
10	C₂×C₁₁⋊C₃₂	C₂²	1	4	yes	—
11	C₂²×C₈×C₁₁	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₁	C₈	1	1	yes	characteristic
14	C₂×C₁₁×C₁₆	—	2	2	no	—
15	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
16	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
18	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
19	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₁₁	C₈	1	2	yes	—
21	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
22	C₁₁⋊C₃₂	—	2	4	no	—
23	#13627	—	11	1	no	maximal, Sylow
24	C₂²×C₄×C₁₁	C₁₆	1	2	yes	—
25	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₁₁	—	2	2	no	—
27	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
31	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₁₁	C₁₆	1	1	yes	characteristic
33	C₄²×C₁₁	C₁₆	1	1	yes	characteristic
34	C₄²×C₁₁	C₂×C₈	1	2	yes	—
35	C₁₁×C₁₆	—	2	2	no	—
36	C₁₁×C₁₆	C₄⋊C₄	1	2	yes	—
37	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
38	C₁₁×C₁₆	C₂×Q₈	1	1	yes	characteristic
39	C₂×M₆₄	—	11	2	no	—
40	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic
41	C₄⋊C₃₂	—	11	4	no	—
42	C₂³×C₁₁	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₁₁	—	2	4	no	—
44	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
45	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₁₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
49	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₁₁	—	2	2	no	—
51	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
52	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₈×C₁₁	C₂×M₁₆	1	1	yes	characteristic
54	C₂²×C₁₆	Dic₁₁	1	1	yes	characteristic
55	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
56	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
57	M₆₄	—	22	4	no	—
58	C₂×C₃₂	—	11	4	no	—
59	C₄×C₁₆	Dic₁₁	1	2	yes	—
60	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
61	C₂²×C₁₁	—	2	2	no	—
62	C₂²×C₁₁	C₄⋊C₁₆	1	2	yes	—
63	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
64	C₄×C₁₁	—	2	2	no	—
65	C₄×C₁₁	C₄⋊C₁₆	1	2	yes	—
66	C₄×C₁₁	M₆₄	1	4	yes	complemented
67	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₄×C₁₁	C₂×M₃₂	1	1	yes	characteristic
69	C₂²×C₈	C₁₁⋊C₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
71	C₂×C₄²	C₁₁⋊C₈	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	Dic₂₂	1	2	yes	—
74	C₂×C₁₆	D₄₄	1	2	yes	—
75	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
76	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
77	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
78	C₄×C₈	C₁₁⋊C₈	1	2	yes	—
79	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
80	C₃₂	—	22	4	no	—
81	C₂×C₁₁	—	2	2	no	—
82	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
83	C₂×C₁₁	C₂×M₆₄	1	2	yes	—
84	C₂²×C₄	C₁₁⋊C₁₆	1	2	yes	—
85	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
88	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
89	C₂×C₈	#4318	1	2	yes	—
90	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
91	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
92	C₄²	C₁₁⋊C₁₆	1	1	yes	characteristic
93	C₄²	C₁₁⋊C₁₆	1	1	yes	characteristic
94	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
95	C₁₆	—	2	2	no	—
96	C₁₆	C₄⋊Dic₁₁	1	2	yes	—
97	C₁₆	C₂×Dic₂₂	1	1	yes	characteristic
98	C₁₆	C₂×D₄₄	1	1	yes	characteristic
99	C₁₁	#13627	1	1	yes	characteristic, complemented, Sylow
100	C₂³	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
103	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
104	C₂×C₄	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
105	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
106	C₂×C₄	#66876	1	2	yes	—
107	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
110	C₈	C₂×#4318	1	1	yes	characteristic
111	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	C₄⋊(C₁₁⋊C₁₆)	1	2	yes	—
114	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₄⋊(C₁₁⋊C₁₆)	1	2	yes	—
117	C₄	#10622137	1	4	yes	complemented
118	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
119	C₄	C₂×#66876	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
122	C₂	C₂×#10622137	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₁
commutator, socle	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
prime radical	C₂³×C₁₁	C₂×C₁₆
cosocle	C₂×C₁₁×C₁₆	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#13627
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₁⋊C₃₂)	C₂	semidirect
66. C₄×C₁₁	M₆₄	semidirect
99. C₁₁	#13627	semidirect
117. C₄	#10622137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622137	2
C₄⋊(C₁₁⋊C₃₂)	2
C₂×C₄×C₁₁×C₁₆	2
#13627	11