Subgroups — C₁₆×C₂²⋊#261 #16789346

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊(C₂²⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#261	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊#261	C₂	1	1	yes	maximal, characteristic
4	(C₂×#261)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₄×C₂²⋊#261	C₄	1	1	yes	characteristic
8	D₁₃⋊(C₂²⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×#261	C₂²	1	2	yes	—
10	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
13	C₂×C₁₆×D₁₃	—	2	4	no	—
14	C₂×C₁₆×D₁₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×D₁₃	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
18	C₂²⋊#6092	C₄	1	2	yes	complemented
19	(C₂×#261)⋊C₈	C₄	1	2	yes	—
20	#11718259	C₄	1	2	yes	complemented
21	C₁₆×#261	—	2	4	no	—
22	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₂²⋊#261	C₈	1	2	yes	—
24	C₂×C₄×#261	C₂×C₄	1	2	yes	—
25	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
34	C₂²⋊#1057	C₈	1	2	yes	—
35	(C₂×#261)⋊C₄	C₈	1	2	yes	—
36	#88111	C₈	1	2	yes	—
37	C₂×#6092	C₂×C₄	1	4	yes	—
38	C₈×#261	—	2	2	no	—
39	C₈×#261	—	2	2	no	—
40	D₁₃⋊C₁₆	—	2	2	no	—
41	D₁₃⋊C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#6090	—	2	2	no	—
46	C₂×#6090	C₂×C₄	1	1	yes	characteristic
47	C₂×#6090	C₂×C₄	1	1	yes	characteristic
48	C₁₆×D₁₃	—	2	2	no	—
49	C₁₆×D₁₃	—	2	2	no	—
50	C₁₆×D₁₃	—	2	2	no	—
51	C₁₆×D₁₃	D₄	1	4	yes	—
52	C₁₆×C₂²⋊C₄	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×#261	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄×D₁₃	—	2	2	no	—
58	C₂×C₄×D₁₃	—	2	2	no	—
59	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
60	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²⋊#261	C₁₆	1	8	yes	direct product factor
64	C₂×#1057	C₂×C₈	1	4	yes	—
65	C₄×#261	—	2	2	no	—
66	C₄×#261	—	2	2	no	—
67	D₁₃⋊C₈	—	2	2	no	—
68	D₁₃⋊C₈	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	C₄²	1	1	yes	characteristic
75	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
76	C₈×D₁₃	—	2	2	no	—
77	C₈×D₁₃	—	2	1	no	—
78	C₈×D₁₃	—	2	1	no	—
79	C₈×D₁₃	—	2	2	no	—
80	C₈×D₁₃	C₂×D₄	1	2	yes	—
81	C₈×D₁₃	#41	1	2	yes	—
82	#6092	—	2	4	no	—
83	C₁₃×C₁₆	—	2	1	no	—
84	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	complemented
85	#6090	—	2	1	no	—
86	#6090	C₂²⋊C₄	1	2	yes	complemented
87	C₂³×C₁₆	—	13	1	no	—
88	C₂×C₂²⋊C₁₆	—	13	1	no	—
89	C₂×C₄×C₁₆	—	13	2	no	—
90	C₈×C₂²⋊C₄	—	13	1	no	—
91	#1374	—	13	2	no	—
92	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
93	C₂²×D₁₃	—	2	2	no	—
94	C₂²×D₁₃	—	2	2	no	—
95	C₂²×D₁₃	C₄×C₈	1	1	yes	characteristic
96	C₂²×D₁₃	C₂×C₁₆	1	4	yes	—
97	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×#261	—	2	4	no	—
99	C₂×#261	—	2	4	no	—
100	C₂×#261	C₂×C₁₆	1	8	yes	complemented
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
107	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₄×D₁₃	—	2	2	no	—
109	C₄×D₁₃	—	2	1	no	—
110	C₄×D₁₃	—	2	1	no	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	C₄×D₄	1	2	yes	—
113	C₄×D₁₃	#131	1	2	yes	—
114	#1057	—	2	4	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₈×C₁₃	#117	1	1	yes	characteristic
118	#1055	—	2	1	no	—
119	#1055	C₄×D₄	1	2	yes	—
120	C₂³×C₈	—	13	1	no	—
121	C₂²×C₁₆	—	13	1	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	C₂²×C₁₆	—	13	1	no	—
124	C₂²×C₁₆	—	13	1	no	—
125	C₂²×C₁₆	—	13	1	no	—
126	C₂²×C₁₆	—	13	1	no	—
127	C₂²×C₁₆	—	26	4	no	—
128	C₂²×C₁₆	#261	1	1	yes	characteristic, complemented
129	C₂×C₂²⋊C₈	—	13	1	no	—
130	C₂×C₄×C₈	—	13	2	no	—
131	C₄×C₂²⋊C₄	—	13	1	no	—
132	C₂²⋊C₁₆	—	13	2	no	—
133	C₂²⋊C₁₆	—	13	2	no	—
134	C₄×C₁₆	—	26	4	no	—
135	#336	—	13	2	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	C₄×C₁₆	1	2	yes	complemented
139	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
140	C₂×D₁₃	—	2	1	no	—
141	C₂×D₁₃	—	2	1	no	—
142	C₂×D₁₃	—	2	8	no	—
143	C₂×D₁₃	C₄×C₁₆	1	2	yes	complemented
144	C₂×D₁₃	C₈×D₄	1	2	yes	—
145	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
146	C₂×D₁₃	#504	1	2	yes	—
147	#261	—	2	8	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₁₃	#405	1	1	yes	characteristic
151	Dic₁₃	—	2	1	no	—
152	Dic₁₃	C₈×D₄	1	2	yes	—
153	C₂³×C₄	—	13	1	no	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	2	no	—
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	1	no	—
160	C₂²×C₈	—	26	2	no	—
161	C₂²×C₈	—	26	2	no	—
162	C₂²×C₈	C₂×#261	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	—	13	2	no	—
164	C₂×C₄²	—	13	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	13	1	no	—
167	C₂×C₁₆	—	13	4	no	—
168	C₂×C₁₆	—	13	4	no	—
169	C₂×C₁₆	—	13	1	no	—
170	C₂×C₁₆	—	26	2	no	—
171	C₂×C₁₆	—	26	2	no	—
172	C₂×C₁₆	—	26	2	no	—
173	C₂×C₁₆	—	26	2	no	—
174	C₂×C₁₆	—	26	2	no	—
175	C₂×C₁₆	—	26	2	no	—
176	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
177	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
178	C₂²⋊C₈	—	13	2	no	—
179	C₂²⋊C₈	—	13	2	no	—
180	C₄×C₈	—	26	2	no	—
181	C₄×C₈	—	26	2	no	—
182	#95	—	13	2	no	—
183	C₂×C₁₃	—	2	2	no	—
184	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
186	C₂×C₁₃	#2108	1	1	yes	characteristic
187	D₁₃	—	2	2	no	—
188	D₁₃	D₄×C₁₆	1	4	yes	—
189	C₂⁴	—	13	1	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	4	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	26	2	no	—
197	C₂²×C₄	—	26	2	no	—
198	C₂²×C₄	C₄×#261	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	2	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	13	4	no	—
205	C₂×C₈	—	13	1	no	—
206	C₂×C₈	—	26	2	no	—
207	C₂×C₈	—	26	2	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	—	26	2	no	—
210	C₂×C₈	—	26	2	no	—
211	C₂×C₈	—	26	1	no	—
212	C₂×C₈	—	26	1	no	—
213	C₂×C₈	C₄×#261	1	1	yes	characteristic
214	C₂×C₈	C₂²×#261	1	1	yes	characteristic
215	C₂²⋊C₄	—	13	8	no	—
216	C₄²	—	26	2	no	—
217	C₄²	—	26	2	no	—
218	C₁₆	—	2	1	no	—
219	C₁₆	—	13	2	no	—
220	C₁₆	—	26	4	no	—
221	C₁₆	—	26	1	no	—
222	C₁₆	C₂²⋊#261	1	2	yes	direct product factor
223	C₁₃	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	13	1	no	—
225	C₂³	—	13	4	no	—
226	C₂³	—	13	1	no	—
227	C₂³	—	26	2	no	—
228	C₂³	—	26	2	no	—
229	C₂³	C₈×#261	1	1	yes	characteristic
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	13	1	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	13	8	no	—
236	C₂×C₄	—	13	1	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	4	no	—
241	C₂×C₄	—	26	4	no	—
242	C₂×C₄	—	26	1	no	—
243	C₂×C₄	—	26	1	no	—
244	C₂×C₄	C₈×#261	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	13	2	no	—
248	C₈	—	26	4	no	—
249	C₈	—	26	1	no	—
250	C₈	#88237	1	1	yes	characteristic
251	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	13	2	no	—
255	C₂²	—	13	2	no	—
256	C₂²	—	13	2	no	—
257	C₂²	—	13	2	no	—
258	C₂²	—	26	8	no	—
259	C₂²	—	26	1	no	—
260	C₂²	—	26	1	no	—
261	C₂²	C₁₆×#261	1	2	yes	complemented
262	C₂²	C₂×C₈×#261	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	13	2	no	—
265	C₄	—	26	8	no	—
266	C₄	—	26	1	no	—
267	C₄	#11719089	1	1	yes	characteristic
268	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	13	4	no	—
271	C₂	—	26	2	no	—
272	C₂	C₈×C₂²⋊#261	1	1	yes	characteristic
273	C₂	C₂×C₁₆×#261	1	1	yes	characteristic
274	C₂	#13131044	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×#261	C₂	semidirect
4. (C₂×#261)⋊C₁₆	C₂	semidirect
11. C₂²×C₁₃×C₁₆	C₄	semidirect
12. C₂²×#6090	C₄	semidirect
14. C₂×C₁₆×D₁₃	C₄	semidirect
15. C₂×C₁₆×D₁₃	C₄	semidirect
18. C₂²⋊#6092	C₄	semidirect
20. #11718259	C₄	semidirect
63. C₂²⋊#261	C₁₆	direct (N × H)
84. C₁₃×C₁₆	C₂²⋊C₄	semidirect
86. #6090	C₂²⋊C₄	semidirect
100. C₂×#261	C₂×C₁₆	semidirect
128. C₂²×C₁₆	#261	semidirect
138. C₂²×C₁₃	C₄×C₁₆	semidirect
143. C₂×D₁₃	C₄×C₁₆	semidirect
222. C₁₆	C₂²⋊#261	direct (N × H)
223. C₁₃	C₁₆×C₂²⋊C₄	semidirect
261. C₂²	C₁₆×#261	semidirect

Maximal subgroups

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

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

Subgroups — C16×C22⋊#261 #16789346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊#261 #16789346