Subgroups — C₂×D₁₆×#261 #16807187

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×#261	C₂	1	2	yes	maximal, complemented
2	D₁₆×#261	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718221	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718276	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#261	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×#261	—	2	2	no	—
8	C₂×D₈×D₁₃	C₂²	1	2	yes	—
9	C₂×#88115	—	2	2	no	—
10	D₈×#261	—	2	4	no	—
11	D₈×#261	C₂²	1	8	yes	complemented
12	C₂×#88102	C₂²	1	2	yes	—
13	C₂×C₈×#261	C₂²	1	1	yes	characteristic
14	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×#88066	C₄	1	2	yes	complemented
17	D₁₃×D₁₆	C₄	1	4	yes	complemented
18	D₁₃×D₁₆	C₂²	1	4	yes	—
19	C₂×D₂₀₈	C₄	1	1	yes	characteristic, complemented
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	#11718276	C₂²	1	8	yes	complemented
22	#11718221	C₂²	1	4	yes	complemented
23	C₁₆×#261	C₂²	1	4	yes	complemented
24	C₂³×#261	—	4	2	no	—
25	C₂×D₄×D₁₃	—	2	2	no	—
26	C₂×C₂²⋊#261	—	4	2	no	—
27	D₄×#261	—	2	8	no	—
28	D₄×#261	—	4	4	no	—
29	C₂×C₄⋊#261	—	2	2	no	—
30	C₂×C₄×#261	D₄	1	1	yes	characteristic
31	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	complemented
33	C₂×#6104	—	2	2	no	—
34	D₈×D₁₃	—	2	2	no	—
35	D₈×D₁₃	—	2	2	no	—
36	D₈×D₁₃	C₂×C₄	1	4	yes	complemented
37	D₈×D₁₃	C₂³	1	4	yes	—
38	C₂×D₁₀₄	C₂×C₄	1	2	yes	complemented
39	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
40	#88115	—	2	8	no	—
41	C₂×#6092	—	2	1	no	—
42	#88102	—	2	2	no	—
43	#88102	C₂³	1	4	yes	—
44	C₈×#261	D₄	1	2	yes	—
45	C₈×#261	C₂³	1	2	yes	—
46	D₁₃⋊C₁₆	—	2	1	no	—
47	D₁₃⋊C₁₆	C₂³	1	2	yes	—
48	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic, complemented
50	#88066	C₂×C₄	1	8	yes	complemented
51	C₂×#6090	C₂×C₄	1	1	yes	characteristic, complemented
52	D₂₀₈	C₂×C₄	1	4	yes	complemented
53	C₁₆×D₁₃	C₂×C₄	1	2	yes	complemented
54	C₁₆×D₁₃	C₂³	1	2	yes	—
55	C₂×C₄×D₁₆	—	13	1	no	maximal, Sylow
56	C₂³×D₁₃	—	4	2	no	—
57	C₂²×#261	—	4	2	no	—
58	C₂²×#261	—	4	8	no	—
59	C₂²×#261	—	8	8	no	—
60	C₂²×#261	D₈	1	1	yes	characteristic
61	C₂²×#261	D₈	1	1	yes	characteristic
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×C₂²⋊D₁₃	—	4	2	no	—
64	D₄×D₁₃	—	2	4	no	—
65	D₄×D₁₃	—	2	4	no	—
66	D₄×D₁₃	—	4	2	no	—
67	D₄×D₁₃	—	4	2	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²⋊#261	—	4	8	no	—
71	C₂×#1057	—	2	1	no	—
72	C₄⋊#261	—	2	4	no	—
73	C₄⋊#261	—	4	2	no	—
74	C₄×#261	D₈	1	2	yes	—
75	C₄×#261	C₂×D₄	1	2	yes	—
76	D₁₃⋊C₈	—	2	1	no	—
77	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
78	C₁₃×D₈	—	2	2	no	—
79	C₁₃×D₈	C₂²×C₄	1	4	yes	complemented
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	#6104	—	2	4	no	—
82	#6104	—	2	4	no	—
83	C₂×#1055	#41	1	1	yes	characteristic
84	D₁₀₄	—	2	2	no	—
85	D₁₀₄	C₂²×C₄	1	4	yes	complemented
86	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
88	C₈×D₁₃	#41	1	1	yes	characteristic
89	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
90	#6092	—	2	2	no	—
91	C₁₃×C₁₆	C₂²×C₄	1	2	yes	complemented
92	#6090	C₂²×C₄	1	2	yes	complemented
93	C₂²×D₁₆	—	13	1	no	—
94	C₂×C₄×D₈	—	13	2	no	—
95	C₄×D₁₆	—	13	8	no	—
96	C₂×#366	—	13	1	no	—
97	C₂×#357	—	13	2	no	—
98	C₂×C₄×C₁₆	—	13	1	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂²×D₁₃	—	4	2	no	—
101	C₂²×D₁₃	—	4	4	no	—
102	C₂²×D₁₃	—	4	4	no	—
103	C₂²×D₁₃	—	8	4	no	—
104	C₂²×D₁₃	—	8	4	no	—
105	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
106	C₂×#261	—	2	2	no	—
107	C₂×#261	—	4	4	no	—
108	C₂×#261	—	8	4	no	—
109	C₂×#261	—	8	16	no	—
110	C₂×#261	D₁₆	1	4	yes	direct product factor
111	C₂×#261	C₂×D₈	1	2	yes	—
112	C₂×#261	C₂×D₈	1	2	yes	—
113	D₄×C₁₃	—	2	4	no	—
114	D₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂²⋊D₁₃	—	4	4	no	—
117	C₂²⋊D₁₃	—	4	4	no	—
118	C₂×Dic₁₃	#135	1	1	yes	characteristic
119	D₅₂	—	2	4	no	—
120	D₅₂	—	4	2	no	—
121	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
122	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
123	C₄×D₁₃	#135	1	1	yes	characteristic
124	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
125	#1057	—	2	2	no	—
126	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	#1055	C₄×D₄	1	1	yes	characteristic
129	#1055	C₂×#41	1	1	yes	characteristic
130	C₂²×D₈	—	13	2	no	—
131	C₂×C₄×D₄	—	26	2	no	—
132	C₂×D₁₆	—	13	2	no	—
133	C₂×D₁₆	—	13	4	no	—
134	C₂×D₁₆	—	13	4	no	—
135	C₂×D₁₆	—	13	1	no	—
136	C₂×D₁₆	#261	1	1	yes	characteristic, direct product factor
137	C₂²×C₁₆	—	13	1	no	—
138	C₂²×C₁₆	—	13	1	no	—
139	C₄×D₈	—	13	8	no	—
140	C₄×D₈	—	26	4	no	—
141	C₂×#107	—	13	2	no	—
142	C₂×#102	—	26	2	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	#366	—	13	4	no	—
145	#357	—	13	8	no	—
146	C₄×C₁₆	—	13	4	no	—
147	C₂²×C₁₃	—	4	4	no	—
148	C₂²×C₁₃	—	8	4	no	—
149	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
150	C₂×D₁₃	—	4	4	no	—
151	C₂×D₁₃	—	8	4	no	—
152	C₂×D₁₃	—	8	4	no	—
153	C₂×D₁₃	—	8	4	no	—
154	C₂×D₁₃	—	8	8	no	—
155	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
156	C₂×D₁₃	C₂×D₁₆	1	2	yes	—
157	C₂×D₁₃	#508	1	2	yes	—
158	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
159	#261	—	2	2	no	—
160	#261	—	8	4	no	—
161	#261	C₂×D₁₆	1	4	yes	direct product factor
162	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
164	Dic₁₃	C₄×D₈	1	1	yes	characteristic
165	Dic₁₃	C₂×#135	1	1	yes	characteristic
166	C₂²×D₄	—	26	2	no	—
167	C₂³×C₄	—	52	2	no	—
168	C₂×D₈	—	13	4	no	—
169	C₂×D₈	—	13	4	no	—
170	C₂×D₈	—	13	2	no	—
171	C₂×D₈	—	26	2	no	—
172	C₂×D₈	—	26	2	no	—
173	C₂×D₈	—	26	2	no	—
174	C₂×D₈	C₂×#261	1	2	yes	complemented
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₄×D₄	—	26	8	no	—
178	C₄×D₄	—	52	4	no	—
179	C₂×C₄⋊C₄	—	26	2	no	—
180	C₂×C₂²⋊C₄	—	52	2	no	—
181	C₂×C₄²	—	13	1	no	—
182	D₁₆	—	13	8	no	—
183	D₁₆	—	13	4	no	—
184	D₁₆	C₂×#261	1	4	yes	direct product factor
185	C₂×C₁₆	—	13	1	no	—
186	C₂×C₁₆	—	13	2	no	—
187	C₂×C₁₆	—	13	2	no	—
188	C₂×C₁₆	—	13	2	no	—
189	C₂×C₁₆	—	26	1	no	—
190	C₂×C₁₆	—	26	1	no	—
191	C₂×C₁₆	C₂×#261	1	1	yes	characteristic, complemented
192	#107	—	13	4	no	—
193	#107	—	26	2	no	—
194	#102	—	26	8	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	—	13	2	no	—
197	C₂×C₁₃	—	8	4	no	—
198	C₂×C₁₃	C₄×D₁₆	1	2	yes	complemented
199	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
200	D₁₃	—	8	4	no	—
201	D₁₃	C₄×D₁₆	1	2	yes	complemented
202	D₁₃	C₂²×D₁₆	1	1	yes	characteristic
203	D₁₃	C₂×#508	1	1	yes	characteristic
204	C₂⁴	—	52	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	26	4	no	—
207	C₂×D₄	—	26	4	no	—
208	C₂×D₄	—	26	2	no	—
209	C₂×D₄	—	52	2	no	—
210	C₂×D₄	—	52	2	no	—
211	C₂×D₄	—	52	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	52	8	no	—
216	C₂²×C₄	—	52	2	no	—
217	C₂²×C₄	—	104	8	no	—
218	D₈	—	2	2	no	—
219	D₈	—	13	4	no	—
220	D₈	—	26	4	no	—
221	D₈	—	26	4	no	—
222	D₈	—	26	2	no	—
223	D₈	C₂²×#261	1	4	yes	complemented
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	1	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	C₂²×#261	1	1	yes	characteristic
233	C₄⋊C₄	—	26	4	no	—
234	C₄⋊C₄	—	52	2	no	—
235	C₂²⋊C₄	—	52	8	no	—
236	C₄²	—	13	2	no	—
237	C₄²	—	13	2	no	—
238	C₁₆	—	13	2	no	—
239	C₁₆	—	26	2	no	—
240	C₁₆	C₂²×#261	1	2	yes	complemented
241	C₁₃	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	4	2	no	—
243	C₂³	—	13	1	no	—
244	C₂³	—	52	4	no	—
245	C₂³	—	52	4	no	—
246	C₂³	—	52	2	no	—
247	C₂³	—	104	4	no	—
248	C₂³	—	104	4	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	26	4	no	—
252	D₄	—	52	4	no	—
253	D₄	—	52	4	no	—
254	D₄	—	52	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	4	no	—
257	C₂×C₄	—	13	1	no	—
258	C₂×C₄	—	13	1	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	104	16	no	—
266	C₂×C₄	—	104	4	no	—
267	C₂×C₄	D₄×#261	1	1	yes	characteristic
268	C₈	—	13	1	no	—
269	C₈	—	13	1	no	—
270	C₈	—	26	2	no	—
271	C₈	D₄×#261	1	1	yes	characteristic
272	C₈	C₂³×#261	1	1	yes	characteristic
273	C₂²	—	4	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	13	1	no	—
276	C₂²	—	13	1	no	—
277	C₂²	—	13	2	no	—
278	C₂²	—	13	2	no	—
279	C₂²	—	52	4	no	—
280	C₂²	—	104	4	no	—
281	C₂²	—	104	4	no	—
282	C₂²	—	104	8	no	—
283	C₂²	—	104	4	no	—
284	C₂²	D₈×#261	1	1	yes	characteristic
285	C₄	—	13	4	no	—
286	C₄	—	13	1	no	—
287	C₄	—	13	1	no	—
288	C₄	—	26	2	no	—
289	C₄	—	104	4	no	—
290	C₄	D₈×#261	1	1	yes	characteristic
291	C₄	C₂×D₄×#261	1	1	yes	characteristic
292	C₂	—	8	4	no	—
293	C₂	—	13	1	no	—
294	C₂	—	13	1	no	—
295	C₂	—	13	2	no	—
296	C₂	—	104	4	no	—
297	C₂	D₁₆×#261	1	2	yes	direct product factor
298	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×D₁₆	C₄
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂×D₁₃×D₁₆	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×D₁₆	#261
hypocenter	C₁₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×#261	C₂	semidirect
2. D₁₆×#261	C₂	direct (N × H)
3. C₂×#11718221	C₂	semidirect
4. C₂×#11718276	C₂	semidirect
5. C₂×C₁₆×#261	C₂	semidirect
11. D₈×#261	C₂²	semidirect
15. C₂×C₁₃×D₁₆	C₄	semidirect
16. C₂×#88066	C₄	semidirect
17. D₁₃×D₁₆	C₄	semidirect
19. C₂×D₂₀₈	C₄	semidirect
21. #11718276	C₂²	semidirect
22. #11718221	C₂²	semidirect
23. C₁₆×#261	C₂²	semidirect
32. C₂×C₁₃×D₈	C₂×C₄	semidirect
36. D₈×D₁₃	C₂×C₄	semidirect
38. C₂×D₁₀₄	C₂×C₄	semidirect
48. C₁₃×D₁₆	C₂×C₄	semidirect
49. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
50. #88066	C₂×C₄	semidirect
51. C₂×#6090	C₂×C₄	semidirect
52. D₂₀₈	C₂×C₄	semidirect
53. C₁₆×D₁₃	C₂×C₄	semidirect
79. C₁₃×D₈	C₂²×C₄	semidirect
85. D₁₀₄	C₂²×C₄	semidirect
91. C₁₃×C₁₆	C₂²×C₄	semidirect
92. #6090	C₂²×C₄	semidirect
110. C₂×#261	D₁₆	direct (N × H)
136. C₂×D₁₆	#261	direct (N × H)
161. #261	C₂×D₁₆	direct (N × H)
174. C₂×D₈	C₂×#261	semidirect
184. D₁₆	C₂×#261	direct (N × H)
191. C₂×C₁₆	C₂×#261	semidirect
198. C₂×C₁₃	C₄×D₁₆	semidirect
201. D₁₃	C₄×D₁₆	semidirect
223. D₈	C₂²×#261	semidirect
240. C₁₆	C₂²×#261	semidirect
241. C₁₃	C₂×C₄×D₁₆	semidirect
297. C₂	D₁₆×#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
C₂×D₈×#261	2
D₁₆×#261	2
C₂×#11718221	2
C₂×#11718276	2
C₂×C₁₆×#261	2
C₂×D₁₃×D₁₆	2
C₂×C₄×D₁₆	13

Subgroups — C2×D16×#261 #16807187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×#261 #16807187