Subgroups — M₁₆×Dic₂₂ #13077569

Order 1408.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#10622731	C₂	1	2	yes	maximal
4	M₁₆×Dic₁₁	C₂	1	2	yes	maximal
5	#10622537	C₂	1	1	yes	maximal, characteristic
6	#10622314	C₂	1	4	yes	maximal, complemented
7	#10622292	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₂	C₂	1	2	yes	maximal, complemented
9	C₂²×Dic₂₂	C₄	1	1	yes	characteristic
10	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₁	C₄	1	1	yes	characteristic
14	C₂×Dic₁₁⋊C₄	C₄	1	2	yes	—
15	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
16	C₂×#4318	C₂²	1	2	yes	—
17	C₄×Dic₂₂	C₄	1	1	yes	characteristic
18	C₄×Dic₂₂	C₄	1	2	yes	—
19	C₄×Dic₂₂	C₄	1	1	yes	characteristic
20	C₄×Dic₂₂	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₂	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₂	C₂²	1	2	yes	—
23	C₁₁×#97	C₂²	1	2	yes	—
24	C₄×C₈×C₁₁	C₂²	1	2	yes	—
25	#66879	C₂²	1	4	yes	—
26	Dic₁₁⋊C₈	C₂²	1	4	yes	—
27	Dic₁₁⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₁₁	C₂²	1	4	yes	—
29	C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
30	C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₁	C₂×C₄	1	2	yes	—
35	C₂²×Dic₁₁	C₂×C₄	1	2	yes	—
36	C₂×Dic₂₂	—	2	4	no	—
37	C₂×Dic₂₂	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₂	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₂₂	C₂×C₄	1	1	yes	characteristic
40	C₁₁×M₁₆	—	2	2	no	—
41	C₁₁×M₁₆	Q₈	1	4	yes	complemented
42	C₂×C₈×C₁₁	C₂³	1	2	yes	—
43	C₂×C₈×C₁₁	C₂³	1	2	yes	—
44	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₁₁	C₂³	1	1	yes	characteristic
47	C₄²×C₁₁	C₂³	1	1	yes	characteristic
48	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
51	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₁	C₂³	1	2	yes	—
58	C₄×Dic₁₁	C₂³	1	2	yes	—
59	#4318	—	2	4	no	—
60	C₂×C₁₁⋊C₈	C₂³	1	4	yes	—
61	Q₈×M₁₆	—	11	1	no	maximal, Sylow
62	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁	—	2	2	no	—
64	C₂×C₄×C₁₁	—	4	1	no	—
65	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₁₁	#41	1	2	yes	—
72	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₁₁	—	2	4	no	—
74	C₂×Dic₁₁	—	4	2	no	—
75	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
79	Dic₂₂	—	2	2	no	—
80	Dic₂₂	—	2	4	no	—
81	Dic₂₂	M₁₆	1	4	yes	direct product factor
82	C₈×C₁₁	—	2	2	no	—
83	C₈×C₁₁	C₂×Q₈	1	4	yes	complemented
84	C₁₁⋊C₈	—	2	4	no	—
85	C₂×C₄×Q₈	—	11	1	no	—
86	#446	—	11	4	no	—
87	#446	—	11	2	no	—
88	C₈×Q₈	—	11	2	no	—
89	#423	—	11	2	no	—
90	#423	—	11	1	no	—
91	C₄×M₁₆	—	11	2	no	—
92	C₄×M₁₆	D₁₁	1	1	yes	characteristic
93	C₂²×C₁₁	—	2	1	no	—
94	C₂²×C₁₁	—	2	1	no	—
95	C₂²×C₁₁	C₄×Q₈	1	2	yes	—
96	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₁	—	2	1	no	—
98	C₄×C₁₁	—	2	1	no	—
99	C₄×C₁₁	—	2	1	no	—
100	C₄×C₁₁	C₄×Q₈	1	2	yes	—
101	C₄×C₁₁	C₂×M₁₆	1	2	yes	—
102	C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
103	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
104	Dic₁₁	—	2	2	no	—
105	Dic₁₁	—	2	2	no	—
106	Dic₁₁	—	2	2	no	—
107	Dic₁₁	C₂×M₁₆	1	4	yes	—
108	C₂²×Q₈	—	11	1	no	—
109	C₂×M₁₆	—	11	2	no	—
110	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
111	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
112	C₄×Q₈	—	11	1	no	—
113	C₄×Q₈	—	11	2	no	—
114	C₄×Q₈	—	11	1	no	—
115	C₄×Q₈	—	11	1	no	—
116	C₄×Q₈	—	11	2	no	—
117	C₄×Q₈	—	11	1	no	—
118	C₂×C₄⋊C₄	—	11	2	no	—
119	C₂×C₄⋊C₄	—	11	1	no	—
120	C₂×C₄²	—	11	2	no	—
121	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
122	C₄⋊C₈	—	11	4	no	—
123	C₄⋊C₈	—	11	4	no	—
124	C₄⋊C₈	—	11	2	no	—
125	C₄⋊C₈	—	11	2	no	—
126	#97	—	11	4	no	—
127	#97	C₂×D₁₁	1	2	yes	—
128	C₄×C₈	—	11	4	no	—
129	C₄×C₈	C₂×D₁₁	1	2	yes	—
130	C₂×C₁₁	—	2	2	no	—
131	C₂×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
133	C₂×C₁₁	#568	1	1	yes	characteristic
134	C₂×Q₈	—	11	1	no	—
135	C₂×Q₈	—	11	2	no	—
136	C₂×Q₈	—	11	1	no	—
137	C₂×Q₈	—	22	4	no	—
138	C₂²×C₄	—	11	2	no	—
139	C₂²×C₄	—	11	2	no	—
140	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
141	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
143	M₁₆	—	2	2	no	—
144	M₁₆	—	22	4	no	—
145	M₁₆	Dic₂₂	1	4	yes	direct product factor
146	C₂×C₈	—	11	4	no	—
147	C₂×C₈	C₂²×D₁₁	1	2	yes	—
148	C₂×C₈	C₂²×D₁₁	1	2	yes	—
149	C₄⋊C₄	—	11	2	no	—
150	C₄⋊C₄	—	11	2	no	—
151	C₄⋊C₄	—	11	2	no	—
152	C₄⋊C₄	—	11	2	no	—
153	C₄⋊C₄	—	11	2	no	—
154	C₄⋊C₄	—	11	1	no	—
155	C₄⋊C₄	—	11	1	no	—
156	C₄²	—	11	2	no	—
157	C₄²	—	11	2	no	—
158	C₄²	—	11	2	no	—
159	C₄²	—	11	2	no	—
160	C₄²	C₄×D₁₁	1	1	yes	characteristic
161	C₄²	C₄×D₁₁	1	1	yes	characteristic
162	C₄²	C₂²×D₁₁	1	1	yes	characteristic
163	C₄²	C₂²×D₁₁	1	1	yes	characteristic
164	C₁₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
166	Q₈	—	11	4	no	—
167	Q₈	—	22	4	no	—
168	Q₈	—	22	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	11	2	no	—
172	C₂×C₄	—	11	2	no	—
173	C₂×C₄	—	11	2	no	—
174	C₂×C₄	—	11	2	no	—
175	C₂×C₄	—	22	4	no	—
176	C₂×C₄	—	44	2	no	—
177	C₂×C₄	C₂×Dic₂₂	1	2	yes	—
178	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
183	C₂×C₄	#4339	1	2	yes	—
184	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	22	4	no	—
187	C₈	C₂×Dic₂₂	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₄×Dic₂₂	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	11	4	no	—
196	C₄	—	22	2	no	—
197	C₄	—	22	2	no	—
198	C₄	—	22	2	no	—
199	C₄	C₄×Dic₂₂	1	2	yes	—
200	C₄	M₁₆×D₁₁	1	2	yes	—
201	C₄	C₂²×Dic₂₂	1	1	yes	characteristic
202	C₄	C₂×#4339	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×C₄×Dic₂₂	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₁₁	1	1	yes	characteristic
206	C₂	#10623276	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₄×C₁₁×M₁₆	C₂
prime radical	C₂³×C₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	C₄×M₁₆	D₁₁
hypocenter	C₁₁	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
6. #10622314	C₂	semidirect
7. #10622292	C₂	semidirect
8. C₈×Dic₂₂	C₂	semidirect
41. C₁₁×M₁₆	Q₈	semidirect
81. Dic₂₂	M₁₆	direct (N × H)
83. C₈×C₁₁	C₂×Q₈	semidirect
145. M₁₆	Dic₂₂	direct (N × H)
164. C₁₁	Q₈×M₁₆	semidirect
187. C₈	C₂×Dic₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₂	2
C₄×C₁₁×M₁₆	2
#10622731	2
M₁₆×Dic₁₁	2
#10622537	2
#10622314	2
#10622292	2
C₈×Dic₂₂	2
Q₈×M₁₆	11

Subgroups — M16×Dic22 #13077569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₂₂ #13077569