Subgroups — #261×C₄⋊C₁₆ #16789557

Order 3328.

438 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#261	C₂	1	2	yes	maximal
3	C₄×C₈×#261	C₂	1	1	yes	maximal, characteristic
4	(C₂×#261)⋊C₁₆	C₂	1	2	yes	maximal
5	D₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#261	C₄	1	1	yes	characteristic
7	C₂×C₈×#261	C₄	1	1	yes	characteristic
8	C₂×C₈×#261	C₂²	1	1	yes	characteristic
9	C₂×C₈×#261	C₂²	1	1	yes	characteristic
10	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
11	C₄²×#261	C₄	1	1	yes	characteristic
12	C₄×D₁₃⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
14	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
15	#11718260	C₄	1	2	yes	complemented
16	C₄⋊#6092	C₄	1	2	yes	complemented
17	C₁₆×#261	—	2	4	no	—
18	C₁₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
19	#11718093	C₄	1	2	yes	complemented
20	C₄⋊#6090	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×#261	C₈	1	2	yes	—
22	C₂×C₄×#261	C₈	1	2	yes	—
23	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
25	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×#1057	C₈	1	2	yes	—
31	C₄×#1057	C₈	1	2	yes	—
32	C₂×#6092	C₂×C₄	1	4	yes	—
33	C₈×#261	—	2	2	no	—
34	C₈×#261	—	2	2	no	—
35	C₈×#261	—	2	1	no	—
36	C₈×#261	—	2	1	no	—
37	C₈×#261	D₄	1	2	yes	—
38	C₈×#261	Q₈	1	2	yes	—
39	D₁₃⋊C₁₆	—	2	2	no	—
40	D₁₃⋊C₁₆	—	2	2	no	—
41	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#6090	C₂×C₄	1	2	yes	—
46	C₄×#1055	C₂×C₄	1	1	yes	characteristic
47	C₁₆×D₁₃	—	2	2	no	—
48	C₁₆×D₁₃	—	2	2	no	—
49	C₄×C₄⋊C₁₆	—	13	1	no	maximal, Sylow
50	C₂²×#261	C₂×C₈	1	2	yes	—
51	C₂²×#261	C₂×C₈	1	2	yes	—
52	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×#1057	C₂×C₈	1	2	yes	—
56	C₂×#1057	C₂×C₈	1	2	yes	—
57	C₄×#261	—	2	4	no	—
58	C₄×#261	—	2	1	no	—
59	C₄×#261	—	2	1	no	—
60	C₄×#261	C₁₆	1	8	yes	complemented
61	C₄×#261	C₄⋊C₄	1	2	yes	—
62	C₄×#261	M₁₆	1	2	yes	—
63	D₁₃⋊C₈	—	2	1	no	—
64	D₁₃⋊C₈	—	2	1	no	—
65	D₁₃⋊C₈	C₄⋊C₄	1	2	yes	—
66	D₁₃⋊C₈	M₁₆	1	2	yes	—
67	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₄²	1	1	yes	characteristic
70	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
71	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂×#1055	C₄²	1	1	yes	characteristic
74	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
75	C₈×D₁₃	—	2	1	no	—
76	C₈×D₁₃	—	2	1	no	—
77	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
78	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₈×D₁₃	#41	1	1	yes	characteristic
80	C₈×D₁₃	#41	1	1	yes	characteristic
81	#6092	—	2	4	no	—
82	C₁₃×C₁₆	—	2	2	no	—
83	#6090	—	2	2	no	—
84	C₂×C₄⋊C₁₆	—	13	1	no	—
85	C₂×C₄⋊C₁₆	—	13	1	no	—
86	C₂×C₄×C₁₆	—	13	2	no	—
87	C₄²×C₈	—	13	1	no	—
88	#1374	—	13	2	no	—
89	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×#261	—	2	2	no	—
91	C₂×#261	—	2	2	no	—
92	C₂×#261	C₄⋊C₈	1	4	yes	—
93	C₂×#261	C₂×C₁₆	1	4	yes	—
94	C₂×#261	C₂×C₁₆	1	4	yes	—
95	C₂×#261	M₃₂	1	4	yes	—
96	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
100	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
101	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	C₂×C₁₆	1	4	yes	—
104	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₁₃	#117	1	1	yes	characteristic
106	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
108	#1057	—	2	2	no	—
109	#1057	C₄⋊C₈	1	4	yes	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
113	#1055	—	2	1	no	—
114	#1055	C₄×D₄	1	1	yes	characteristic
115	#1055	C₄×Q₈	1	1	yes	characteristic
116	C₂²×C₁₆	—	13	2	no	—
117	C₂²×C₁₆	—	13	2	no	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₂×C₄×C₈	—	13	1	no	—
122	C₂×C₄×C₈	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₄³	—	13	1	no	—
125	C₄⋊C₁₆	—	13	2	no	—
126	C₄⋊C₁₆	—	13	2	no	—
127	C₄⋊C₁₆	—	13	2	no	—
128	C₄⋊C₁₆	—	13	1	no	—
129	C₄⋊C₁₆	#261	1	1	yes	characteristic, direct product factor
130	C₄×C₁₆	—	26	4	no	—
131	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂×D₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂×D₁₃	#431	1	1	yes	characteristic
134	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
135	C₂×D₁₃	C₂×M₃₂	1	2	yes	—
136	#261	—	2	4	no	—
137	#261	C₄⋊C₁₆	1	8	yes	direct product factor
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	C₄×C₁₆	1	2	yes	complemented
140	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
142	Dic₁₃	—	2	1	no	—
143	Dic₁₃	C₄×C₁₆	1	2	yes	complemented
144	Dic₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
145	Dic₁₃	#431	1	1	yes	characteristic
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	1	no	—
148	C₂²×C₈	—	13	1	no	—
149	C₂²×C₈	—	13	1	no	—
150	C₂×C₄²	—	13	1	no	—
151	C₂×C₄²	—	13	2	no	—
152	C₂×C₄²	—	13	1	no	—
153	C₂×C₄²	—	13	1	no	—
154	C₂×C₄²	—	13	2	no	—
155	C₂×C₁₆	—	13	4	no	—
156	C₂×C₁₆	—	13	2	no	—
157	C₂×C₁₆	—	26	2	no	—
158	C₂×C₁₆	—	26	2	no	—
159	C₂×C₁₆	—	26	2	no	—
160	C₂×C₁₆	—	26	2	no	—
161	C₂×C₁₆	C₂×#261	1	2	yes	—
162	C₄×C₈	—	13	2	no	—
163	C₄×C₈	—	13	2	no	—
164	C₄×C₈	—	13	2	no	—
165	C₄×C₈	—	13	2	no	—
166	C₄×C₈	—	13	1	no	—
167	C₄×C₈	—	13	1	no	—
168	C₄×C₈	—	13	1	no	—
169	C₄×C₈	—	26	2	no	—
170	C₄×C₈	—	26	2	no	—
171	C₄×C₈	—	26	1	no	—
172	C₄×C₈	—	26	1	no	—
173	C₄×C₈	C₂×#261	1	1	yes	characteristic
174	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
175	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
176	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
177	D₁₃	C₂×C₄⋊C₁₆	1	2	yes	—
178	D₁₃	#2162	1	2	yes	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	—	13	2	no	—
183	C₂²×C₄	—	13	1	no	—
184	C₂×C₈	—	13	2	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	13	2	no	—
187	C₂×C₈	—	13	1	no	—
188	C₂×C₈	—	13	1	no	—
189	C₂×C₈	—	13	1	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	1	no	—
194	C₂×C₈	—	26	1	no	—
195	C₂×C₈	—	26	1	no	—
196	C₂×C₈	—	26	1	no	—
197	C₂×C₈	—	26	1	no	—
198	C₂×C₈	C₄×#261	1	1	yes	characteristic
199	C₂×C₈	C₂²×#261	1	1	yes	characteristic
200	C₄²	—	13	2	no	—
201	C₄²	—	13	2	no	—
202	C₄²	—	13	8	no	—
203	C₄²	—	13	1	no	—
204	C₄²	—	13	1	no	—
205	C₄²	—	13	1	no	—
206	C₄²	—	26	1	no	—
207	C₄²	—	26	1	no	—
208	C₄²	—	26	4	no	—
209	C₄²	C₄×#261	1	1	yes	characteristic
210	C₁₆	—	2	2	no	—
211	C₁₆	—	26	4	no	—
212	C₁₆	—	26	2	no	—
213	C₁₃	C₄×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	13	1	no	—
215	C₂×C₄	—	13	4	no	—
216	C₂×C₄	—	13	4	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	13	1	no	—
219	C₂×C₄	—	13	1	no	—
220	C₂×C₄	—	13	4	no	—
221	C₂×C₄	—	13	1	no	—
222	C₂×C₄	—	13	1	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	4	no	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	C₈×#261	1	1	yes	characteristic
231	C₂×C₄	C₈×#261	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	13	4	no	—
235	C₈	—	13	1	no	—
236	C₈	—	13	1	no	—
237	C₈	—	26	2	no	—
238	C₈	—	26	1	no	—
239	C₈	D₄×#261	1	1	yes	characteristic
240	C₈	Q₈×#261	1	1	yes	characteristic
241	C₂²	—	13	2	no	—
242	C₂²	—	13	2	no	—
243	C₂²	—	13	1	no	—
244	C₂²	—	13	1	no	—
245	C₂²	C₂×C₈×#261	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	13	8	no	—
248	C₄	—	13	2	no	—
249	C₄	—	13	1	no	—
250	C₄	—	13	1	no	—
251	C₄	—	26	4	no	—
252	C₄	—	26	1	no	—
253	C₄	C₁₆×#261	1	2	yes	complemented
254	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
255	C₄	M₁₆×#261	1	1	yes	characteristic
256	C₂	—	13	2	no	—
257	C₂	—	13	2	no	—
258	C₂	C₄⋊C₈×#261	1	1	yes	characteristic
259	C₂	C₂×C₁₆×#261	1	1	yes	characteristic
260	C₂	M₃₂×#261	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
15. #11718260	C₄	semidirect
16. C₄⋊#6092	C₄	semidirect
18. C₁₃×C₄⋊C₁₆	C₄	semidirect
19. #11718093	C₄	semidirect
20. C₄⋊#6090	C₄	semidirect
60. C₄×#261	C₁₆	semidirect
129. C₄⋊C₁₆	#261	direct (N × H)
137. #261	C₄⋊C₁₆	direct (N × H)
139. C₄×C₁₃	C₄×C₁₆	semidirect
143. Dic₁₃	C₄×C₁₆	semidirect
213. C₁₃	C₄×C₄⋊C₁₆	semidirect
253. C₄	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⋊(C₄⋊C₁₆)	2
C₂×C₁₆×#261	2
C₄×C₈×#261	2
(C₂×#261)⋊C₁₆	2
D₁₃×C₄⋊C₁₆	2
C₄×C₄⋊C₁₆	13

Subgroups — #261×C4⋊C16 #16789557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #261×C₄⋊C₁₆ #16789557