Subgroups — C₂²×C₁₆×#261 #16806708

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×#261	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#261	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×#261	C₄	1	1	yes	characteristic
6	C₂²×D₁₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₄	1	2	yes	complemented
9	C₂×C₈×#261	C₄	1	6	yes	—
10	C₂×C₈×#261	C₂²	1	6	yes	—
11	C₂×D₁₃⋊C₁₆	C₄	1	6	yes	complemented
12	C₂×D₁₃⋊C₁₆	C₂²	1	6	yes	—
13	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×D₁₃	C₄	1	6	yes	complemented
16	C₂×C₁₆×D₁₃	C₂²	1	6	yes	—
17	C₁₆×#261	C₂²	1	16	yes	direct product factor
18	C₂³×#261	C₈	1	2	yes	—
19	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×#1057	C₈	1	2	yes	—
21	C₂×C₄×#261	C₈	1	6	yes	—
22	C₂×C₄×#261	C₂×C₄	1	6	yes	—
23	C₂×D₁₃⋊C₈	C₈	1	6	yes	—
24	C₂×D₁₃⋊C₈	C₂×C₄	1	6	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
28	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₁₃	C₂³	1	3	yes	—
31	C₂×#6092	C₂×C₄	1	12	yes	complemented
32	C₈×#261	C₂×C₄	1	12	yes	—
33	C₈×#261	C₂³	1	4	yes	—
34	D₁₃⋊C₁₆	C₂×C₄	1	12	yes	complemented
35	D₁₃⋊C₁₆	C₂³	1	4	yes	—
36	C₂×C₁₃×C₁₆	C₂×C₄	1	6	yes	complemented
37	C₂×#6090	C₂×C₄	1	6	yes	complemented
38	C₁₆×D₁₃	C₂×C₄	1	12	yes	complemented
39	C₁₆×D₁₃	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₆	—	13	1	no	maximal, Sylow
41	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×#261	C₁₆	1	16	yes	direct product factor
43	C₂²×#261	C₂×C₈	1	12	yes	—
44	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₄²	1	3	yes	—
47	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₁₃	C₂²×C₄	1	3	yes	—
50	C₂×#1057	C₂×C₈	1	12	yes	—
51	C₄×#261	C₂×C₈	1	12	yes	—
52	C₄×#261	C₂²×C₄	1	4	yes	—
53	D₁₃⋊C₈	C₂×C₈	1	12	yes	—
54	D₁₃⋊C₈	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₃	C₄²	1	3	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂×#1055	C₄²	1	3	yes	—
58	C₂×#1055	C₂²×C₄	1	3	yes	—
59	C₈×D₁₃	C₄²	1	6	yes	—
60	C₈×D₁₃	C₂²×C₄	1	3	yes	—
61	C₈×D₁₃	C₂²×C₄	1	3	yes	—
62	C₈×D₁₃	C₂²×C₄	1	3	yes	—
63	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
64	#6092	C₂²×C₄	1	8	yes	complemented
65	C₁₃×C₁₆	C₂²×C₄	1	4	yes	complemented
66	#6090	C₂²×C₄	1	4	yes	complemented
67	C₂³×C₁₆	—	13	1	no	—
68	C₂³×C₁₆	—	13	1	no	—
69	C₂²×C₄×C₈	—	13	1	no	—
70	C₂×C₄×C₁₆	—	13	12	no	—
71	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
72	C₂²×D₁₃	C₄×C₈	1	3	yes	—
73	C₂²×D₁₃	C₂×C₁₆	1	8	yes	—
74	C₂²×D₁₃	C₂²×C₈	1	3	yes	—
75	C₂×#261	C₂×C₁₆	1	48	yes	direct product factor
76	C₂×#261	C₂²×C₈	1	8	yes	—
77	C₂×C₄×C₁₃	C₄×C₈	1	3	yes	—
78	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
79	C₂×Dic₁₃	C₄×C₈	1	3	yes	—
80	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
81	C₄×D₁₃	C₄×C₈	1	6	yes	—
82	C₄×D₁₃	C₂×C₄²	1	3	yes	—
83	C₄×D₁₃	C₂²×C₈	1	3	yes	—
84	C₄×D₁₃	C₂²×C₈	1	3	yes	—
85	C₄×D₁₃	C₂³×C₄	1	1	yes	characteristic
86	#1057	C₂²×C₈	1	8	yes	—
87	C₈×C₁₃	C₂×C₄²	1	3	yes	—
88	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	#1055	C₂×C₄²	1	3	yes	—
90	#1055	C₂³×C₄	1	1	yes	characteristic
91	C₂³×C₈	—	13	1	no	—
92	C₂³×C₈	—	13	1	no	—
93	C₂²×C₄²	—	13	1	no	—
94	C₂²×C₁₆	—	13	6	no	—
95	C₂²×C₁₆	—	13	6	no	—
96	C₂²×C₁₆	—	13	2	no	—
97	C₂²×C₁₆	—	13	1	no	—
98	C₂²×C₁₆	—	13	6	no	—
99	C₂²×C₁₆	—	13	6	no	—
100	C₂²×C₁₆	#261	1	1	yes	characteristic, direct product factor
101	C₂×C₄×C₈	—	13	6	no	—
102	C₂×C₄×C₈	—	13	6	no	—
103	C₄×C₁₆	—	13	16	no	—
104	C₂²×C₁₃	C₄×C₁₆	1	4	yes	complemented
105	C₂²×C₁₃	C₂×C₄×C₈	1	3	yes	—
106	C₂×D₁₃	C₄×C₁₆	1	12	yes	complemented
107	C₂×D₁₃	C₂×C₄×C₈	1	3	yes	—
108	C₂×D₁₃	C₂²×C₁₆	1	12	yes	—
109	C₂×D₁₃	C₂³×C₈	1	1	yes	characteristic
110	#261	C₂²×C₁₆	1	16	yes	direct product factor
111	C₄×C₁₃	C₂×C₄×C₈	1	3	yes	—
112	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
113	Dic₁₃	C₂×C₄×C₈	1	3	yes	—
114	Dic₁₃	C₂³×C₈	1	1	yes	characteristic
115	C₂³×C₄	—	13	2	no	—
116	C₂³×C₄	—	13	1	no	—
117	C₂²×C₈	—	13	6	no	—
118	C₂²×C₈	—	13	6	no	—
119	C₂²×C₈	—	13	2	no	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	—	13	3	no	—
122	C₂²×C₈	—	13	3	no	—
123	C₂²×C₈	—	13	3	no	—
124	C₂²×C₈	—	13	3	no	—
125	C₂²×C₈	C₂×#261	1	1	yes	characteristic
126	C₂×C₄²	—	13	6	no	—
127	C₂×C₄²	—	13	6	no	—
128	C₂×C₁₆	—	13	12	no	—
129	C₂×C₁₆	—	13	4	no	—
130	C₂×C₁₆	—	13	12	no	—
131	C₂×C₁₆	—	13	6	no	—
132	C₂×C₁₆	—	13	4	no	—
133	C₂×C₁₆	—	13	12	no	—
134	C₂×C₁₆	C₂×#261	1	6	yes	direct product factor
135	C₄×C₈	—	13	12	no	—
136	C₄×C₈	—	13	4	no	—
137	C₂×C₁₃	C₂×C₄×C₁₆	1	6	yes	complemented
138	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
139	D₁₃	C₂×C₄×C₁₆	1	6	yes	complemented
140	D₁₃	C₂³×C₁₆	1	2	yes	—
141	C₂⁴	—	13	1	no	—
142	C₂²×C₄	—	13	12	no	—
143	C₂²×C₄	—	13	16	no	—
144	C₂²×C₄	—	13	1	no	—
145	C₂²×C₄	—	13	3	no	—
146	C₂²×C₄	—	13	3	no	—
147	C₂²×C₄	—	13	3	no	—
148	C₂²×C₄	—	13	3	no	—
149	C₂²×C₄	C₄×#261	1	1	yes	characteristic
150	C₂×C₈	—	13	4	no	—
151	C₂×C₈	—	13	12	no	—
152	C₂×C₈	—	13	12	no	—
153	C₂×C₈	—	13	3	no	—
154	C₂×C₈	—	13	3	no	—
155	C₂×C₈	—	13	3	no	—
156	C₂×C₈	—	13	3	no	—
157	C₂×C₈	—	13	6	no	—
158	C₂×C₈	—	13	1	no	—
159	C₂×C₈	—	13	3	no	—
160	C₂×C₈	C₄×#261	1	3	yes	—
161	C₂×C₈	C₂²×#261	1	3	yes	—
162	C₄²	—	13	4	no	—
163	C₄²	—	13	12	no	—
164	C₁₆	—	13	8	no	—
165	C₁₆	—	13	4	no	—
166	C₁₆	C₂²×#261	1	4	yes	direct product factor
167	C₁₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
168	C₂³	—	13	8	no	—
169	C₂³	—	13	3	no	—
170	C₂³	—	13	3	no	—
171	C₂³	C₈×#261	1	1	yes	characteristic
172	C₂×C₄	—	13	8	no	—
173	C₂×C₄	—	13	48	no	—
174	C₂×C₄	—	13	3	no	—
175	C₂×C₄	—	13	3	no	—
176	C₂×C₄	—	13	3	no	—
177	C₂×C₄	—	13	3	no	—
178	C₂×C₄	—	13	6	no	—
179	C₂×C₄	—	13	1	no	—
180	C₂×C₄	—	13	3	no	—
181	C₂×C₄	C₈×#261	1	3	yes	—
182	C₂×C₄	C₂×C₄×#261	1	3	yes	—
183	C₈	—	13	8	no	—
184	C₈	—	13	1	no	—
185	C₈	—	13	3	no	—
186	C₈	C₂×C₄×#261	1	3	yes	—
187	C₈	C₂³×#261	1	1	yes	characteristic
188	C₂²	—	13	12	no	—
189	C₂²	—	13	12	no	—
190	C₂²	—	13	1	no	—
191	C₂²	—	13	3	no	—
192	C₂²	C₁₆×#261	1	4	yes	direct product factor
193	C₂²	C₂×C₈×#261	1	3	yes	—
194	C₄	—	13	16	no	—
195	C₄	—	13	1	no	—
196	C₄	—	13	3	no	—
197	C₄	C₂×C₈×#261	1	3	yes	—
198	C₄	C₂²×C₄×#261	1	1	yes	characteristic
199	C₂	—	13	2	no	—
200	C₂	—	13	6	no	—
201	C₂	C₂×C₁₆×#261	1	6	yes	direct product factor
202	C₂	C₂²×C₈×#261	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	#261
commutator, hypocenter	C₁₃	C₂²×C₄×C₁₆
Frattini	C₈	C₂³×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂³×C₁₃	C₄×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₈×D₁₃	C₂⁴
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×#261	C₂	direct (N × H)
8. C₂²×#6092	C₄	semidirect
11. C₂×D₁₃⋊C₁₆	C₄	semidirect
13. C₂²×C₁₃×C₁₆	C₄	semidirect
14. C₂²×#6090	C₄	semidirect
15. C₂×C₁₆×D₁₃	C₄	semidirect
17. C₁₆×#261	C₂²	direct (N × H)
31. C₂×#6092	C₂×C₄	semidirect
34. D₁₃⋊C₁₆	C₂×C₄	semidirect
36. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
37. C₂×#6090	C₂×C₄	semidirect
38. C₁₆×D₁₃	C₂×C₄	semidirect
42. C₂²×#261	C₁₆	direct (N × H)
64. #6092	C₂²×C₄	semidirect
65. C₁₃×C₁₆	C₂²×C₄	semidirect
66. #6090	C₂²×C₄	semidirect
75. C₂×#261	C₂×C₁₆	direct (N × H)
100. C₂²×C₁₆	#261	direct (N × H)
104. C₂²×C₁₃	C₄×C₁₆	semidirect
106. C₂×D₁₃	C₄×C₁₆	semidirect
110. #261	C₂²×C₁₆	direct (N × H)
134. C₂×C₁₆	C₂×#261	direct (N × H)
137. C₂×C₁₃	C₂×C₄×C₁₆	semidirect
139. D₁₃	C₂×C₄×C₁₆	semidirect
166. C₁₆	C₂²×#261	direct (N × H)
167. C₁₃	C₂²×C₄×C₁₆	semidirect
192. C₂²	C₁₆×#261	direct (N × H)
201. C₂	C₂×C₁₆×#261	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₃⋊C₁₆	2
C₂²×C₈×#261	2
C₂×C₁₆×#261	2
C₂²×C₁₆×D₁₃	2
C₂²×C₄×C₁₆	13

Subgroups — C22×C16×#261 #16806708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×#261 #16806708