Subgroups — #14673953

Order 2816.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 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	1	yes	maximal, characteristic
2	C₂²×C₁₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#13066750	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₁⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₁₁×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
8	C₁₁×#405	C₄	1	1	yes	characteristic
9	C₁₁×#346	C₄	1	2	yes	—
10	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
11	#10622137	C₄	1	4	yes	complemented
12	C₂×C₁₁⋊C₃₂	C₄	1	4	yes	complemented
13	C₂×C₁₁⋊C₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₁⋊C₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₁⋊C₃₂	C₂²	1	2	yes	—
16	C₂×C₁₁×M₁₆	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₁₁×#117	C₈	1	1	yes	characteristic
19	C₁₁×M₃₂	C₈	1	4	yes	—
20	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
25	C₁₁×#97	C₈	1	2	yes	—
26	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
27	C₁₁⋊C₃₂	C₂×C₄	1	2	yes	—
28	C₁₁⋊C₃₂	C₂×C₄	1	2	yes	—
29	C₁₁⋊C₃₂	C₂×C₄	1	4	yes	—
30	#13549	—	11	1	no	maximal, Sylow
31	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
32	C₁₁×M₁₆	C₁₆	1	4	yes	—
33	C₂×C₈×C₁₁	C₄²	1	2	yes	—
34	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₁₁×C₄⋊C₄	C₁₆	1	2	yes	—
39	C₁₁×C₂²⋊C₄	C₁₆	1	2	yes	—
40	C₄²×C₁₁	C₂×C₈	1	2	yes	—
41	C₁₁×C₁₆	C₄²	1	2	yes	—
42	C₁₁×C₁₆	C₂×C₈	1	4	yes	—
43	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
45	C₂×M₆₄	—	11	1	no	—
46	C₂²×C₃₂	—	11	1	no	—
47	#2108	D₁₁	1	1	yes	characteristic
48	#1397	—	11	2	no	—
49	C₄×C₃₂	—	11	2	no	—
50	C₂³×C₁₁	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₁	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₁₁	C₂×C₁₆	1	4	yes	—
53	C₂×C₄×C₁₁	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₁₁	C₄×C₈	1	2	yes	—
57	C₈×C₁₁	C₂×C₁₆	1	4	yes	—
58	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₁₁	C₂²×C₈	1	1	yes	characteristic
60	C₂×M₃₂	Dic₁₁	1	1	yes	characteristic
61	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
62	#405	Dic₁₁	1	1	yes	characteristic
63	M₆₄	—	11	4	no	—
64	C₂×C₃₂	—	11	1	no	—
65	C₂×C₃₂	—	11	2	no	—
66	C₂×C₃₂	—	11	1	no	—
67	C₂×C₃₂	—	11	4	no	—
68	#346	Dic₁₁	1	2	yes	—
69	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
70	C₂²×C₁₁	—	2	2	no	—
71	C₂²×C₁₁	C₄×C₁₆	1	2	yes	—
72	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
73	C₄×C₁₁	—	2	4	no	—
74	C₄×C₁₁	C₄×C₁₆	1	2	yes	—
75	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
77	C₂×M₁₆	C₁₁⋊C₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
79	#117	C₁₁⋊C₈	1	1	yes	characteristic
80	M₃₂	C₁₁⋊C₈	1	4	yes	—
81	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
82	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
84	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
85	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
86	#97	C₁₁⋊C₈	1	2	yes	—
87	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
88	C₃₂	—	11	2	no	—
89	C₃₂	—	11	2	no	—
90	C₃₂	—	11	4	no	—
91	C₂×C₁₁	—	2	2	no	—
92	C₂×C₁₁	C₂×C₄×C₁₆	1	1	yes	characteristic
93	C₂×C₁₁	#2258	1	2	yes	—
94	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
95	M₁₆	C₁₁⋊C₁₆	1	4	yes	—
96	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
97	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
98	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
99	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
100	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
101	C₄⋊C₄	C₁₁⋊C₁₆	1	2	yes	—
102	C₂²⋊C₄	C₁₁⋊C₁₆	1	2	yes	—
103	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
104	C₁₆	C₂×C₁₁⋊C₈	1	4	yes	—
105	C₁₆	C₄×Dic₁₁	1	2	yes	—
106	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
107	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
108	C₁₁	#13549	1	1	yes	characteristic, complemented, Sylow
109	C₂³	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
110	C₂×C₄	C₄×C₁₁⋊C₈	1	2	yes	—
111	C₂×C₄	C₂×C₁₁⋊C₁₆	1	4	yes	—
112	C₂×C₄	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
113	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
114	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
115	C₈	C₄×C₁₁⋊C₈	1	2	yes	—
116	C₈	C₂×C₁₁⋊C₁₆	1	4	yes	—
117	C₈	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
118	C₈	C₂×C₄×Dic₁₁	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×C₁₁⋊C₁₆	1	2	yes	—
121	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×C₁₁⋊C₁₆	1	2	yes	—
124	C₄	C₂×C₄×C₁₁⋊C₈	1	1	yes	characteristic
125	C₄	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×C₁₁⋊C₁₆	1	1	yes	characteristic
128	C₂	#13070910	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₁
commutator	C₂×C₁₁	C₂×C₄×C₁₆
Fitting	C₁₁×#2108	C₂
socle	C₂²×C₁₁	C₂²×C₁₆
prime radical	C₂³×C₁₁	C₂×C₁₆
cosocle	C₂×C₁₁×C₁₆	C₂³
hypercenter	#2108	D₁₁
hypocenter	C₁₁	#13549
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #13066750	C₂	semidirect
4. C₄×C₁₁⋊C₃₂	C₂	semidirect
11. #10622137	C₄	semidirect
12. C₂×C₁₁⋊C₃₂	C₄	semidirect
108. C₁₁	#13549	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
C₂²×C₁₁⋊C₃₂	2
#13066750	2
C₄×C₁₁⋊C₃₂	2
C₁₁×#2108	2
#13549	11