Subgroups — Dic₁₃×C₂²⋊C₁₆ #15777948

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal, complemented
4	#13112913	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₄	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₂×C₈×Dic₁₃	—	2	4	no	—
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	2	yes	—
16	C₁₃×C₂²⋊C₁₆	C₄	1	4	yes	complemented
17	C₂²⋊#6090	C₄	1	4	yes	complemented
18	C₁₆×Dic₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×Dic₁₃	C₈	1	2	yes	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	—	2	2	no	—
28	C₂×C₄×Dic₁₃	—	2	2	no	—
29	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	—	2	2	no	—
33	C₂²×#1055	—	2	2	no	—
34	C₂²×#1055	C₈	1	4	yes	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₂×C₁₃×C₁₆	—	2	2	no	—
39	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
40	C₈×Dic₁₃	—	2	2	no	—
41	C₈×Dic₁₃	—	2	2	no	—
42	C₈×Dic₁₃	—	2	2	no	—
43	C₈×Dic₁₃	D₄	1	4	yes	—
44	C₂×#6090	—	2	2	no	—
45	C₂×#6090	—	2	2	no	—
46	C₂×#6090	C₂×C₄	1	4	yes	complemented
47	C₄×C₂²⋊C₁₆	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	2	4	no	—
58	C₂²×Dic₁₃	—	2	4	no	—
59	C₂²×Dic₁₃	C₁₆	1	8	yes	complemented
60	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
61	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	C₄²	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₄×Dic₁₃	—	2	1	no	—
72	C₄×Dic₁₃	—	2	1	no	—
73	C₄×Dic₁₃	—	2	4	no	—
74	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
75	C₄×Dic₁₃	M₁₆	1	2	yes	—
76	C₂×#1055	—	2	1	no	—
77	C₂×#1055	—	2	1	no	—
78	C₂×#1055	—	2	8	no	—
79	C₂×#1055	C₂²⋊C₄	1	2	yes	—
80	C₂×#1055	C₂×C₈	1	2	yes	—
81	C₂×#1055	C₂×C₈	1	2	yes	—
82	C₂×#1055	M₁₆	1	2	yes	—
83	C₁₃×C₁₆	—	2	4	no	—
84	#6090	—	2	4	no	—
85	C₂²×C₄×C₈	—	13	1	no	—
86	C₂×C₂²⋊C₁₆	—	13	1	no	—
87	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
88	C₂×C₄×C₁₆	—	13	2	no	—
89	#1374	—	13	2	no	—
90	C₂³×C₁₃	—	2	1	no	—
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	C₄×C₈	1	2	yes	—
94	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
95	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	2	16	no	—
112	C₂×Dic₁₃	C₂²⋊C₈	1	4	yes	—
113	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
114	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
115	C₂×Dic₁₃	M₃₂	1	4	yes	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	C₄×D₄	1	2	yes	—
118	C₈×C₁₃	C₄×D₄	1	2	yes	—
119	#1055	—	2	2	no	—
120	#1055	C₂²⋊C₈	1	4	yes	—
121	C₂³×C₈	—	13	1	no	—
122	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₄²	—	13	1	no	—
124	C₂²×C₁₆	—	13	2	no	—
125	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
126	C₂×C₄×C₈	—	13	1	no	—
127	C₂×C₄×C₈	—	13	1	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂×C₄×C₈	—	13	1	no	—
130	C₂×C₄×C₈	—	26	4	no	—
131	C₂²⋊C₁₆	—	13	4	no	—
132	C₂²⋊C₁₆	Dic₁₃	1	4	yes	direct product factor
133	C₄×C₁₆	—	26	4	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	—	2	4	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	C₄×C₁₆	1	4	yes	complemented
139	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₁₃	#431	1	1	yes	characteristic
142	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
143	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₁₃	#431	1	1	yes	characteristic
149	Dic₁₃	—	2	4	no	—
150	Dic₁₃	C₂²⋊C₁₆	1	8	yes	direct product factor
151	C₂³×C₄	—	13	2	no	—
152	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₂²×C₈	—	13	4	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	26	2	no	—
159	C₂²×C₈	—	26	2	no	—
160	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₁₃	1	2	yes	—
162	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
163	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	13	2	no	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	26	2	no	—
168	C₂×C₄²	—	26	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	13	4	no	—
172	C₂×C₁₆	—	26	2	no	—
173	C₂×C₁₆	—	26	2	no	—
174	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	complemented
175	C₄×C₈	—	13	4	no	—
176	C₄×C₈	—	26	2	no	—
177	C₄×C₈	—	26	2	no	—
178	C₄×C₈	—	26	2	no	—
179	C₂×C₁₃	—	2	4	no	—
180	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
181	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
182	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
183	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	2	yes	—
184	C₂×C₁₃	#2162	1	2	yes	—
185	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	8	no	—
193	C₂²×C₄	—	26	4	no	—
194	C₂²×C₄	—	26	4	no	—
195	C₂²×C₄	C₈×D₁₃	1	2	yes	—
196	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	26	8	no	—
208	C₂×C₈	—	26	1	no	—
209	C₂×C₈	—	26	1	no	—
210	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₈	D₄×D₁₃	1	2	yes	—
213	C₂×C₈	#6129	1	2	yes	—
214	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
215	C₄²	—	13	2	no	—
216	C₄²	—	13	2	no	—
217	C₄²	—	26	4	no	—
218	C₄²	—	26	1	no	—
219	C₄²	—	26	1	no	—
220	C₁₆	—	2	4	no	—
221	C₁₆	—	26	4	no	—
222	C₁₃	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	C₁₆×D₁₃	1	4	yes	—
227	C₂³	C₈×Dic₁₃	1	2	yes	—
228	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	13	4	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	26	16	no	—
240	C₂×C₄	—	26	2	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
243	C₂×C₄	#88131	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
246	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	13	4	no	—
251	C₈	—	26	2	no	—
252	C₈	#88135	1	2	yes	—
253	C₈	D₄×Dic₁₃	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	C₁₆×Dic₁₃	1	4	yes	complemented
259	C₂²	#11718379	1	1	yes	characteristic
260	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
261	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
262	C₂²	D₁₃×M₃₂	1	2	yes	—
263	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	13	8	no	—
266	C₄	—	26	4	no	—
267	C₄	#11718379	1	1	yes	characteristic
268	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
269	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
270	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	C₂²⋊C₈×Dic₁₃	1	1	yes	characteristic
273	C₂	C₂×C₁₆×Dic₁₃	1	1	yes	characteristic
274	C₂	M₃₂×Dic₁₃	1	1	yes	characteristic
275	C₂	D₁₃×C₂²⋊C₁₆	1	2	yes	—
276	C₂	#13116632	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×Dic₁₃	C₂	semidirect
4. #13112913	C₂	semidirect
16. C₁₃×C₂²⋊C₁₆	C₄	semidirect
17. C₂²⋊#6090	C₄	semidirect
39. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
46. C₂×#6090	C₂×C₄	semidirect
59. C₂²×Dic₁₃	C₁₆	semidirect
132. C₂²⋊C₁₆	Dic₁₃	direct (N × H)
138. C₂²×C₁₃	C₄×C₁₆	semidirect
150. Dic₁₃	C₂²⋊C₁₆	direct (N × H)
174. C₂×C₁₆	C₂×Dic₁₃	semidirect
222. C₁₃	C₄×C₂²⋊C₁₆	semidirect
258. C₂²	C₁₆×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₁₃	2
C₂×C₂²⋊#6090	2
C₂×C₁₆×Dic₁₃	2
#13112913	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₄×C₂²⋊C₁₆	13

Subgroups — Dic13×C22⋊C16 #15777948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃×C₂²⋊C₁₆ #15777948