Subgroups — #14641473

Order 2816.

190 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622142	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₁×#366	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×C₁₁×#107	—	2	1	no	—
6	C₂×#66882	—	2	1	no	—
7	C₁₁×#366	—	2	1	no	—
8	C₁₁×#366	C₄	1	2	yes	complemented
9	#10622142	—	2	1	no	—
10	#10622142	C₄	1	2	yes	complemented
11	C₂×C₁₁⋊C₃₂	—	2	1	no	—
12	C₂×C₁₁⋊C₃₂	—	2	1	no	—
13	C₂×C₁₁⋊C₃₂	C₄	1	2	yes	complemented
14	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
15	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
16	C₂×C₄⋊Dic₁₁	—	4	1	no	—
17	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₁×C₁₆	Q₈	1	1	yes	characteristic
23	C₁₁×#107	—	2	2	no	—
24	C₁₁×#107	—	4	1	no	—
25	#66882	—	2	2	no	—
26	#66882	—	4	1	no	—
27	C₁₁⋊C₃₂	—	2	2	no	—
28	#7471	—	11	1	no	maximal, Sylow
29	C₂²×C₄×C₁₁	—	8	1	no	—
30	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
31	C₂²×Dic₁₁	—	8	1	no	—
32	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₁	Q₁₆	1	1	yes	characteristic
38	C₁₁×C₄⋊C₄	—	4	2	no	—
39	C₁₁×C₄⋊C₄	—	8	1	no	—
40	C₄⋊Dic₁₁	—	4	2	no	—
41	C₄⋊Dic₁₁	—	8	1	no	—
42	C₁₁×C₁₆	C₄²	1	1	yes	characteristic
43	C₁₁×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
44	C₁₁×C₁₆	C₄⋊C₄	1	1	yes	characteristic
45	C₁₁×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₃₂	—	11	1	no	—
47	C₂×#366	—	11	1	no	—
48	C₂×#366	D₁₁	1	1	yes	characteristic
49	C₂³×C₁₁	D₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₁	—	8	2	no	—
51	C₂×C₄×C₁₁	—	16	1	no	—
52	C₂×C₄×C₁₁	—	16	1	no	—
53	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
54	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₁	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₁	SD₃₂	1	1	yes	characteristic
57	C₂×C₄×C₁₁	SD₃₂	1	1	yes	characteristic
58	C₂×C₄×C₁₁	Q₃₂	1	1	yes	characteristic
59	C₂×Dic₁₁	—	8	2	no	—
60	C₂×Dic₁₁	—	16	1	no	—
61	C₂×Dic₁₁	—	16	1	no	—
62	C₈×C₁₁	#95	1	1	yes	characteristic
63	C₈×C₁₁	#103	1	1	yes	characteristic
64	C₈×C₁₁	#103	1	1	yes	characteristic
65	C₈×C₁₁	#106	1	1	yes	characteristic
66	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
67	C₂×#107	—	2	1	no	—
68	C₂×#107	—	22	1	no	—
69	C₂×C₃₂	—	11	2	no	—
70	C₂×C₃₂	—	22	1	no	—
71	C₂×C₃₂	—	22	1	no	—
72	#366	—	2	1	no	—
73	#366	—	11	2	no	—
74	#366	—	22	1	no	—
75	#366	Dic₁₁	1	2	yes	complemented
76	C₂²×C₁₁	#357	1	1	yes	characteristic
77	C₂²×C₁₁	#357	1	1	yes	characteristic
78	C₂²×C₁₁	#366	1	1	yes	characteristic
79	C₂²×C₁₁	D₃₂	1	1	yes	characteristic
80	C₂²×C₁₁	SD₆₄	1	1	yes	characteristic
81	C₂²×C₁₁	SD₆₄	1	1	yes	characteristic
82	C₂²×C₁₁	Q₆₄	1	1	yes	characteristic
83	C₄×C₁₁	—	16	2	no	—
84	C₄×C₁₁	#340	1	1	yes	characteristic
85	C₄×C₁₁	#358	1	1	yes	characteristic
86	C₄×C₁₁	#358	1	1	yes	characteristic
87	C₄×C₁₁	#367	1	1	yes	characteristic
88	Dic₁₁	—	16	2	no	—
89	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	4	1	no	—
91	C₂×C₄⋊C₄	—	44	1	no	—
92	C₂×C₁₆	Dic₂₂	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
94	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
95	C₂×C₁₆	D₄₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	4	1	no	—
100	#107	—	22	2	no	—
101	#107	—	44	1	no	—
102	C₃₂	—	22	2	no	—
103	C₂×C₁₁	#1380	1	1	yes	characteristic
104	C₂×C₁₁	#1415	1	1	yes	characteristic
105	C₂×C₁₁	#1415	1	1	yes	characteristic
106	C₂×C₁₁	#1416	1	1	yes	characteristic
107	C₂×C₁₁	#1416	1	1	yes	characteristic
108	C₂×C₁₁	#1423	1	1	yes	characteristic
109	C₂×C₁₁	#1424	1	1	yes	characteristic
110	C₂²×C₄	—	8	1	no	—
111	C₂²×C₄	—	88	1	no	—
112	C₂²×C₄	#4323	1	1	yes	characteristic
113	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
114	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
115	C₂×C₈	#4324	1	1	yes	characteristic
116	C₂×C₈	#4325	1	1	yes	characteristic
117	C₂×C₈	#4326	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	8	1	no	—
121	C₄⋊C₄	—	44	2	no	—
122	C₄⋊C₄	—	88	1	no	—
123	C₁₆	C₄×Dic₁₁	1	1	yes	characteristic
124	C₁₆	Dic₁₁⋊C₄	1	1	yes	characteristic
125	C₁₆	C₄⋊Dic₁₁	1	1	yes	characteristic
126	C₁₆	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
127	C₁₁	#7471	1	1	yes	characteristic, complemented, Sylow
128	C₂³	#66890	1	1	yes	characteristic
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	88	2	no	—
133	C₂×C₄	—	176	1	no	—
134	C₂×C₄	—	176	1	no	—
135	C₂×C₄	#66871	1	1	yes	characteristic
136	C₂×C₄	#66873	1	1	yes	characteristic
137	C₂×C₄	#66891	1	1	yes	characteristic
138	C₂×C₄	#66892	1	1	yes	characteristic
139	C₂×C₄	#66893	1	1	yes	characteristic
140	C₂×C₄	#66896	1	1	yes	characteristic
141	C₈	#66872	1	1	yes	characteristic
142	C₈	#66874	1	1	yes	characteristic
143	C₈	C₁₁⋊#95	1	1	yes	characteristic
144	C₈	#66899	1	1	yes	characteristic
145	C₂²	#10622127	1	1	yes	characteristic
146	C₂²	#10622129	1	1	yes	characteristic
147	C₂²	#10622157	1	1	yes	characteristic
148	C₂²	#10622158	1	1	yes	characteristic
149	C₂²	#10622159	1	1	yes	characteristic
150	C₂²	#10622160	1	1	yes	characteristic
151	C₂²	#10622200	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	176	2	no	—
154	C₄	#10622128	1	1	yes	characteristic
155	C₄	#10622130	1	1	yes	characteristic
156	C₄	#10622167	1	1	yes	characteristic
157	C₄	#10622202	1	1	yes	characteristic
158	C₂	#13066707	1	1	yes	characteristic
159	C₂	#13066793	1	1	yes	characteristic
160	C₂	#13066794	1	1	yes	characteristic
161	C₂	#13066796	1	1	yes	characteristic
162	C₂	#13066797	1	1	yes	characteristic
163	C₂	#13066815	1	1	yes	characteristic
164	C₂	#13066817	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#66890
commutator	C₁₁×C₁₆	C₄²
Frattini	C₂²×C₁₆	C₂×D₁₁
Fitting	C₂×C₁₁×#366	C₂
socle, prime radical	C₂³×C₁₁	D₁₆
cosocle	C₂²×C₁₁×C₁₆	C₂²
hypercenter	C₂×#366	D₁₁
hypocenter	C₁₁	#7471
LCS	G ▷ C₁₁×C₁₆ ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
8. C₁₁×#366	C₄	semidirect
10. #10622142	C₄	semidirect
13. C₂×C₁₁⋊C₃₂	C₄	semidirect
75. #366	Dic₁₁	semidirect
127. C₁₁	#7471	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₁⋊C₃₂	2
C₂×#10622142	2
C₂×C₁₁×#366	2
#7471	11