Subgroups — C₄×M₁₆×#261 #16796726

Order 3328.

1734 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×#261	C₂	1	4	yes	maximal
3	C₂×C₄²×#261	C₂	1	1	yes	maximal, characteristic
4	C₄×#88100	C₂	1	2	yes	maximal, complemented
5	#97×#261	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×#261	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×#261	C₄	1	4	yes	—
9	C₂²×C₄×#261	C₂²	1	2	yes	—
10	C₂×D₁₃⋊M₁₆	C₂²	1	2	yes	—
11	C₂×M₁₆×D₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
13	C₄×#6122	C₄	1	4	yes	complemented
14	M₁₆×#261	C₄	1	32	yes	direct product factor
15	C₂×#88100	C₂²	1	8	yes	—
16	C₂×C₈×#261	C₂²	1	8	yes	—
17	C₄²×#261	C₄	1	4	yes	—
18	C₄²×#261	C₂²	1	1	yes	characteristic
19	C₄²×#261	C₂²	1	2	yes	—
20	C₄²×#261	C₂²	1	1	yes	characteristic
21	C₄×#6118	C₄	1	4	yes	complemented
22	#11719047	C₂²	1	2	yes	—
23	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
24	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic, complemented
25	M₁₆×Dic₁₃	C₄	1	2	yes	complemented
26	C₄×#6099	C₄	1	1	yes	characteristic, complemented
27	#11718368	C₄	1	2	yes	complemented
28	D₁₃×#97	C₂²	1	2	yes	—
29	C₄×#6094	C₄	1	2	yes	complemented
30	C₄×C₈×D₁₃	C₂²	1	2	yes	—
31	C₂³×#261	C₂×C₄	1	4	yes	—
32	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6122	C₂×C₄	1	4	yes	—
35	C₂×C₄×#261	—	2	16	no	—
36	C₂×C₄×#261	C₂×C₄	1	8	yes	—
37	C₂×C₄×#261	C₂×C₄	1	4	yes	—
38	C₂×C₄×#261	C₂×C₄	1	4	yes	—
39	C₂×C₄×#261	C₂×C₄	1	16	yes	—
40	C₂×C₄×#261	C₂×C₄	1	4	yes	—
41	C₂×C₄×#261	C₂³	1	2	yes	—
42	C₂×C₄×#261	C₂³	1	2	yes	—
43	D₁₃⋊M₁₆	C₂×C₄	1	16	yes	—
44	C₂×#6118	C₂×C₄	1	4	yes	—
45	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
46	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×#6099	C₂×C₄	1	2	yes	—
51	M₁₆×D₁₃	C₂×C₄	1	16	yes	—
52	C₂×#6094	C₂×C₄	1	4	yes	—
53	C₂×C₈×D₁₃	C₂³	1	4	yes	—
54	C₄²×D₁₃	C₂×C₄	1	2	yes	—
55	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₁₃	C₂×C₄	1	2	yes	—
58	C₄²×D₁₃	C₂³	1	1	yes	characteristic
59	C₄²×D₁₃	C₂³	1	1	yes	characteristic
60	#88110	C₂×C₄	1	8	yes	complemented
61	C₄×#1057	C₂×C₄	1	8	yes	complemented
62	#88100	C₂×C₄	1	32	yes	complemented
63	C₈×#261	C₂×C₄	1	32	yes	complemented
64	C₁₃×#97	C₂×C₄	1	2	yes	complemented
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	complemented
66	#88055	C₂×C₄	1	4	yes	complemented
67	C₈×Dic₁₃	C₂×C₄	1	4	yes	complemented
68	#88043	C₂×C₄	1	2	yes	complemented
69	C₄×#1055	C₂×C₄	1	2	yes	complemented
70	C₄²×M₁₆	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×#261	—	2	16	no	—
73	C₂²×#261	C₄²	1	16	yes	—
74	C₂²×#261	C₂²×C₄	1	4	yes	—
75	C₂²×#261	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₄²	1	2	yes	—
79	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	—	2	8	no	—
81	C₂×C₄×D₁₃	C₄²	1	2	yes	—
82	C₂×C₄×D₁₃	C₄²	1	2	yes	—
83	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	8	yes	—
88	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
90	#6122	C₄²	1	16	yes	complemented
91	C₂×#1057	C₂²×C₄	1	8	yes	—
92	C₄×#261	—	2	16	no	—
93	C₄×#261	—	2	8	no	—
94	C₄×#261	C₄²	1	16	yes	—
95	C₄×#261	M₁₆	1	16	yes	direct product factor
96	C₄×#261	C₂²×C₄	1	8	yes	—
97	C₄×#261	C₂²×C₄	1	8	yes	—
98	#6118	C₄²	1	16	yes	complemented
99	D₁₃⋊C₈	C₂²×C₄	1	16	yes	—
100	C₁₃×M₁₆	C₄²	1	8	yes	complemented
101	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₁₃	C₄²	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₄²	1	2	yes	—
106	C₄×Dic₁₃	C₄²	1	2	yes	—
107	C₄×Dic₁₃	C₄²	1	2	yes	—
108	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
111	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
112	#6099	C₄²	1	8	yes	complemented
113	C₂×#1055	C₂²×C₄	1	4	yes	—
114	#6094	C₄²	1	16	yes	complemented
115	C₈×D₁₃	C₂²×C₄	1	16	yes	—
116	C₂×C₄×M₁₆	—	13	4	no	—
117	C₂×C₄×M₁₆	—	13	1	no	—
118	C₂×C₄×M₁₆	—	13	1	no	—
119	C₂×C₄³	—	13	1	no	—
120	C₄×#97	—	13	4	no	—
121	C₄×#97	—	13	2	no	—
122	C₄²×C₈	—	13	2	no	—
123	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
124	C₂²×D₁₃	—	2	4	no	—
125	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₂²×D₁₃	C₂×C₄²	1	4	yes	—
127	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×#261	—	2	8	no	—
129	C₂×#261	—	2	32	no	—
130	C₂×#261	C₂×C₄²	1	8	yes	—
131	C₂×#261	C₂×C₄²	1	8	yes	—
132	C₂×#261	C₂×M₁₆	1	16	yes	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
137	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	4	no	—
140	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
141	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
143	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
144	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
145	C₄×D₁₃	—	2	4	no	—
146	C₄×D₁₃	—	2	4	no	—
147	C₄×D₁₃	—	2	4	no	—
148	C₄×D₁₃	C₂×C₄²	1	4	yes	—
149	C₄×D₁₃	C₂×C₄²	1	4	yes	—
150	C₄×D₁₃	C₂×C₄²	1	4	yes	—
151	C₄×D₁₃	C₂×M₁₆	1	8	yes	—
152	C₄×D₁₃	C₂³×C₄	1	4	yes	—
153	#1057	C₂×C₄²	1	16	yes	complemented
154	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
155	#1055	C₂×C₄²	1	8	yes	complemented
156	C₂²×M₁₆	—	13	2	no	—
157	C₂²×M₁₆	—	13	2	no	—
158	C₂²×C₄²	—	13	2	no	—
159	C₂²×C₄²	—	13	1	no	—
160	C₂²×C₄²	—	13	4	no	—
161	C₄×M₁₆	—	13	4	no	—
162	C₄×M₁₆	—	13	32	no	—
163	C₄×M₁₆	—	13	4	no	—
164	C₄×M₁₆	—	13	2	no	—
165	C₄×M₁₆	—	13	1	no	—
166	C₄×M₁₆	—	13	2	no	—
167	C₄×M₁₆	—	13	2	no	—
168	C₄×M₁₆	#261	1	1	yes	characteristic, direct product factor
169	C₂×#97	—	13	2	no	—
170	C₂×#97	—	13	8	no	—
171	C₂×#97	—	13	2	no	—
172	C₂×C₄×C₈	—	13	2	no	—
173	C₂×C₄×C₈	—	13	8	no	—
174	C₂×C₄×C₈	—	13	2	no	—
175	C₄³	—	13	4	no	—
176	C₄³	—	13	1	no	—
177	C₄³	—	13	2	no	—
178	C₄³	—	13	1	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	C₄³	1	2	yes	—
181	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
182	C₂×D₁₃	—	2	2	no	—
183	C₂×D₁₃	—	2	8	no	—
184	C₂×D₁₃	C₄³	1	2	yes	—
185	C₂×D₁₃	C₂²×C₄²	1	2	yes	—
186	C₂×D₁₃	C₂²×M₁₆	1	4	yes	—
187	#261	—	2	8	no	—
188	#261	C₄×M₁₆	1	16	yes	direct product factor
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	C₄³	1	2	yes	—
191	C₄×C₁₃	C₄×M₁₆	1	4	yes	complemented
192	C₄×C₁₃	C₂²×C₄²	1	2	yes	—
193	Dic₁₃	—	2	2	no	—
194	Dic₁₃	C₄³	1	2	yes	—
195	Dic₁₃	C₄×M₁₆	1	4	yes	complemented
196	Dic₁₃	C₂²×C₄²	1	2	yes	—
197	C₂³×C₄	—	13	1	no	—
198	C₂³×C₄	—	13	4	no	—
199	C₂³×C₄	—	13	2	no	—
200	C₂×M₁₆	—	13	4	no	—
201	C₂×M₁₆	—	13	16	no	—
202	C₂×M₁₆	—	13	4	no	—
203	C₂×M₁₆	—	13	2	no	—
204	C₂×M₁₆	—	13	16	no	—
205	C₂×M₁₆	—	13	4	no	—
206	C₂×M₁₆	C₂×#261	1	2	yes	—
207	C₂²×C₈	—	13	4	no	—
208	C₂²×C₈	—	13	4	no	—
209	C₂×C₄²	—	13	2	no	—
210	C₂×C₄²	—	13	2	no	—
211	C₂×C₄²	—	13	4	no	—
212	C₂×C₄²	—	13	4	no	—
213	C₂×C₄²	—	13	16	no	—
214	C₂×C₄²	—	13	2	no	—
215	C₂×C₄²	—	13	1	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	—	13	2	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	1	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	8	no	—
223	C₂×C₄²	—	13	4	no	—
224	C₂×C₄²	—	26	16	no	—
225	C₂×C₄²	C₂×#261	1	1	yes	characteristic
226	#97	—	13	32	no	—
227	#97	—	13	8	no	—
228	#97	—	13	4	no	—
229	#97	—	13	2	no	—
230	#97	C₂×#261	1	2	yes	complemented
231	C₄×C₈	—	13	32	no	—
232	C₄×C₈	—	13	8	no	—
233	C₄×C₈	—	13	4	no	—
234	C₄×C₈	—	13	2	no	—
235	C₄×C₈	C₂×#261	1	2	yes	complemented
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
238	C₂×C₁₃	C₂×C₄×M₁₆	1	2	yes	—
239	D₁₃	—	2	2	no	—
240	D₁₃	C₂×C₄×M₁₆	1	4	yes	—
241	C₂⁴	—	13	1	no	—
242	C₂²×C₄	—	13	4	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	8	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	13	4	no	—
251	C₂²×C₄	—	13	16	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	13	2	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	26	8	no	—
257	C₂²×C₄	—	26	16	no	—
258	C₂²×C₄	C₄×#261	1	2	yes	—
259	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
260	M₁₆	—	13	16	no	—
261	M₁₆	—	13	16	no	—
262	M₁₆	—	13	8	no	—
263	M₁₆	—	13	16	no	—
264	M₁₆	C₄×#261	1	8	yes	direct product factor
265	C₂×C₈	—	13	16	no	—
266	C₂×C₈	—	13	8	no	—
267	C₂×C₈	—	13	4	no	—
268	C₂×C₈	—	13	16	no	—
269	C₂×C₈	C₂²×#261	1	4	yes	—
270	C₄²	—	13	8	no	—
271	C₄²	—	13	8	no	—
272	C₄²	—	13	16	no	—
273	C₄²	—	13	16	no	—
274	C₄²	—	13	2	no	—
275	C₄²	—	13	2	no	—
276	C₄²	—	13	2	no	—
277	C₄²	—	13	2	no	—
278	C₄²	—	13	2	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	26	16	no	—
282	C₄²	—	26	8	no	—
283	C₄²	C₄×#261	1	2	yes	—
284	C₄²	C₂²×#261	1	1	yes	characteristic
285	C₄²	C₂²×#261	1	1	yes	characteristic
286	C₁₃	C₄²×M₁₆	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	13	1	no	—
288	C₂³	—	13	4	no	—
289	C₂³	—	13	1	no	—
290	C₂³	—	26	4	no	—
291	C₂³	C₂×C₄×#261	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	13	8	no	—
294	C₂×C₄	—	13	16	no	—
295	C₂×C₄	—	13	4	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	4	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	13	8	no	—
302	C₂×C₄	—	13	4	no	—
303	C₂×C₄	—	13	4	no	—
304	C₂×C₄	—	13	8	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	26	4	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	32	no	—
310	C₂×C₄	—	26	8	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	C₂×C₄×#261	1	2	yes	—
313	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×#261	1	4	yes	—
315	C₂×C₄	C₂×C₄×#261	1	2	yes	—
316	C₂×C₄	C₂³×#261	1	1	yes	characteristic
317	C₈	—	13	16	no	—
318	C₈	—	13	8	no	—
319	C₈	C₂×C₄×#261	1	8	yes	complemented
320	C₂²	—	2	2	no	—
321	C₂²	—	13	4	no	—
322	C₂²	—	13	2	no	—
323	C₂²	—	13	2	no	—
324	C₂²	—	26	8	no	—
325	C₂²	—	26	2	no	—
326	C₂²	C₄²×#261	1	2	yes	—
327	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	13	16	no	—
330	C₄	—	13	4	no	—
331	C₄	—	13	2	no	—
332	C₄	—	13	2	no	—
333	C₄	—	26	8	no	—
334	C₄	—	26	2	no	—
335	C₄	C₄²×#261	1	2	yes	—
336	C₄	M₁₆×#261	1	4	yes	direct product factor
337	C₄	C₂²×C₄×#261	1	2	yes	—
338	C₂	—	2	2	no	—
339	C₂	—	13	4	no	—
340	C₂	—	26	2	no	—
341	C₂	C₂×C₄²×#261	1	1	yes	characteristic
342	C₂	C₂×M₁₆×#261	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	C₄²×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₄×#88100	C₂	semidirect
5. #97×#261	C₂	semidirect
6. C₄×C₈×#261	C₂	semidirect
13. C₄×#6122	C₄	semidirect
14. M₁₆×#261	C₄	direct (N × H)
21. C₄×#6118	C₄	semidirect
24. C₄×C₁₃×M₁₆	C₄	semidirect
25. M₁₆×Dic₁₃	C₄	semidirect
26. C₄×#6099	C₄	semidirect
27. #11718368	C₄	semidirect
29. C₄×#6094	C₄	semidirect
60. #88110	C₂×C₄	semidirect
61. C₄×#1057	C₂×C₄	semidirect
62. #88100	C₂×C₄	semidirect
63. C₈×#261	C₂×C₄	semidirect
64. C₁₃×#97	C₂×C₄	semidirect
65. C₄×C₈×C₁₃	C₂×C₄	semidirect
66. #88055	C₂×C₄	semidirect
67. C₈×Dic₁₃	C₂×C₄	semidirect
68. #88043	C₂×C₄	semidirect
69. C₄×#1055	C₂×C₄	semidirect
90. #6122	C₄²	semidirect
95. C₄×#261	M₁₆	direct (N × H)
98. #6118	C₄²	semidirect
100. C₁₃×M₁₆	C₄²	semidirect
112. #6099	C₄²	semidirect
114. #6094	C₄²	semidirect
153. #1057	C₂×C₄²	semidirect
154. C₈×C₁₃	C₂×C₄²	semidirect
155. #1055	C₂×C₄²	semidirect
168. C₄×M₁₆	#261	direct (N × H)
188. #261	C₄×M₁₆	direct (N × H)
191. C₄×C₁₃	C₄×M₁₆	semidirect
195. Dic₁₃	C₄×M₁₆	semidirect
230. #97	C₂×#261	semidirect
235. C₄×C₈	C₂×#261	semidirect
264. M₁₆	C₄×#261	direct (N × H)
286. C₁₃	C₄²×M₁₆	semidirect
319. C₈	C₂×C₄×#261	semidirect
336. C₄	M₁₆×#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
C₄×D₁₃⋊M₁₆	2
C₂×M₁₆×#261	2
C₂×C₄²×#261	2
C₄×#88100	2
#97×#261	2
C₄×C₈×#261	2
C₄×M₁₆×D₁₃	2
C₄²×M₁₆	13

Subgroups — C4×M16×#261 #16796726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₁₆×#261 #16796726