Subgroups — C₁₆×#6129 #16016434

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#6129	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116738	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
6	#13116653	C₂	1	2	yes	maximal, complemented
7	#13116634	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116632	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#6129	C₄	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₁₃	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
17	C₂²×#6090	C₂²	1	2	yes	—
18	D₄×#1055	C₄	1	1	yes	characteristic
19	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	#11718494	C₄	1	1	yes	characteristic
22	#11718491	C₄	1	1	yes	characteristic
23	#11718386	C₄	1	2	yes	—
24	#11718379	C₄	1	2	yes	—
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊#6090	C₂²	1	2	yes	—
31	#11718099	C₂²	1	2	yes	—
32	#11718093	C₂²	1	2	yes	—
33	#11718093	C₂²	1	2	yes	—
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	2	yes	—
37	C₁₆×Dic₁₃	C₂²	1	2	yes	—
38	C₁₆×Dic₁₃	C₂²	1	2	yes	—
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6129	C₈	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
43	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	D₄×Dic₁₃	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
47	C₂²×#1055	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	#88146	C₈	1	1	yes	characteristic
51	#88141	C₈	1	1	yes	characteristic
52	#88135	C₈	1	2	yes	—
53	#88131	C₈	1	2	yes	—
54	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	—	2	4	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊#1055	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88054	C₂×C₄	1	2	yes	—
66	#88054	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂³	1	2	yes	—
72	C₂×#6090	—	2	4	no	—
73	C₂×#6090	C₂³	1	2	yes	—
74	C₂×#6090	C₂³	1	1	yes	characteristic
75	C₂×#6090	C₂³	1	1	yes	characteristic
76	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
77	C₄×#1055	C₂×C₄	1	1	yes	characteristic
78	C₁₆×D₁₃	—	2	2	no	—
79	C₁₆×D₁₃	—	2	2	no	—
80	C₁₆×#41	—	13	1	no	maximal, Sylow
81	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
84	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
85	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
86	#6129	C₁₆	1	8	yes	direct product factor
87	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
100	#6103	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
102	#6101	C₂×C₈	1	2	yes	—
103	#6101	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
107	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
108	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×#1055	—	2	4	no	—
111	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
112	C₂×#1055	C₂²×C₄	1	2	yes	—
113	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
114	C₈×D₁₃	—	2	1	no	—
115	C₈×D₁₃	—	2	1	no	—
116	C₈×D₁₃	—	2	2	no	—
117	C₁₃×C₁₆	—	2	2	no	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	#41	1	2	yes	complemented
120	#6090	—	2	1	no	—
121	#6090	—	2	2	no	—
122	#6090	#41	1	2	yes	complemented
123	C₈×#41	—	13	1	no	—
124	Q₈×C₁₆	—	13	1	no	—
125	Q₈×C₁₆	—	13	1	no	—
126	D₄×C₁₆	—	13	2	no	—
127	D₄×C₁₆	—	13	1	no	—
128	D₄×C₁₆	—	13	2	no	—
129	D₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
130	#2162	—	13	1	no	—
131	#2162	—	13	2	no	—
132	C₂×C₄×C₁₆	—	13	2	no	—
133	C₂×C₄×C₁₆	—	13	1	no	—
134	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
136	D₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂²⋊D₁₃	C₂×C₁₆	1	8	yes	complemented
144	C₂×Dic₁₃	—	2	4	no	—
145	C₂×Dic₁₃	C₂×C₁₆	1	8	yes	complemented
146	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
148	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
150	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
151	C₄×D₁₃	—	2	1	no	—
152	C₄×D₁₃	—	2	1	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	C₂×C₁₆	1	4	yes	complemented
155	Dic₂₆	C₂×C₁₆	1	4	yes	complemented
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
160	#1055	—	2	1	no	—
161	#1055	—	2	2	no	—
162	#1055	#131	1	2	yes	—
163	C₄×#41	—	13	1	no	—
164	C₂²×C₁₆	—	13	2	no	—
165	C₂²×C₁₆	—	13	1	no	—
166	C₂²×C₁₆	—	13	1	no	—
167	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
168	C₈×Q₈	—	13	1	no	—
169	C₈×Q₈	—	13	1	no	—
170	C₈×D₄	—	13	2	no	—
171	C₈×D₄	—	13	1	no	—
172	C₈×D₄	—	13	2	no	—
173	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
174	#431	—	13	1	no	—
175	#431	—	13	2	no	—
176	C₂×C₄×C₈	—	13	2	no	—
177	C₂×C₄×C₈	—	13	1	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₄⋊C₁₆	—	13	2	no	—
180	C₄⋊C₁₆	—	13	1	no	—
181	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
182	C₂²⋊C₁₆	—	13	2	no	—
183	C₂²⋊C₁₆	—	13	2	no	—
184	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
185	C₄×C₁₆	—	13	2	no	—
186	C₄×C₁₆	—	13	1	no	—
187	C₄×C₁₆	—	13	2	no	—
188	C₄×C₁₆	—	13	1	no	—
189	C₄×C₁₆	—	13	2	no	—
190	C₄×C₁₆	—	13	1	no	—
191	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
192	C₂²×C₁₃	—	2	2	no	—
193	C₂²×C₁₃	—	2	2	no	—
194	C₂²×C₁₃	C₂²×C₁₆	1	4	yes	—
195	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
196	C₂×D₁₃	—	2	1	no	—
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
202	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
203	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
204	Dic₁₃	—	2	2	no	—
205	Dic₁₃	—	2	1	no	—
206	Dic₁₃	C₂²×C₁₆	1	4	yes	—
207	Dic₁₃	C₂²×C₁₆	1	2	yes	—
208	Dic₁₃	#504	1	2	yes	—
209	C₂×#41	—	13	1	no	—
210	C₂²×C₈	—	13	2	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
214	C₄×Q₈	—	13	1	no	—
215	C₄×Q₈	—	13	1	no	—
216	C₄×D₄	—	13	1	no	—
217	C₄×D₄	—	13	2	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	C₄×D₁₃	1	1	yes	characteristic
220	#117	—	13	1	no	—
221	#117	—	13	2	no	—
222	C₂×C₄²	—	13	2	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	1	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	26	2	no	—
229	C₂×C₁₆	—	26	4	no	—
230	C₂×C₁₆	—	26	2	no	—
231	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
232	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	C₄⋊C₈	—	13	2	no	—
235	C₄⋊C₈	—	13	2	no	—
236	C₄⋊C₈	—	13	1	no	—
237	C₄⋊C₈	C₄×D₁₃	1	1	yes	characteristic
238	C₂²⋊C₈	—	13	2	no	—
239	C₂²⋊C₈	—	13	2	no	—
240	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
241	C₄×C₈	—	13	2	no	—
242	C₄×C₈	—	13	1	no	—
243	C₄×C₈	—	13	2	no	—
244	C₄×C₈	—	13	1	no	—
245	C₄×C₈	—	13	2	no	—
246	C₄×C₈	—	13	1	no	—
247	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
248	C₂×C₁₃	—	2	4	no	—
249	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
250	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
251	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
252	D₁₃	—	2	2	no	—
253	#41	—	13	8	no	—
254	C₂×Q₈	—	13	1	no	—
255	C₂×D₄	—	13	2	no	—
256	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	2	no	—
260	C₂²×C₄	—	13	1	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	26	2	no	—
269	C₂×C₈	—	26	4	no	—
270	C₂×C₈	—	26	1	no	—
271	C₂×C₈	—	26	1	no	—
272	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
273	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
275	C₄⋊C₄	—	13	2	no	—
276	C₄⋊C₄	—	13	2	no	—
277	C₄⋊C₄	—	13	1	no	—
278	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	13	2	no	—
280	C₂²⋊C₄	—	13	2	no	—
281	C₂²⋊C₄	C₈×D₁₃	1	2	yes	—
282	C₄²	—	13	2	no	—
283	C₄²	—	13	1	no	—
284	C₄²	—	13	2	no	—
285	C₄²	—	13	1	no	—
286	C₄²	—	13	1	no	—
287	C₄²	—	13	2	no	—
288	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	13	2	no	—
292	C₁₆	—	26	1	no	—
293	C₁₆	—	26	2	no	—
294	C₁₆	#6129	1	2	yes	direct product factor
295	C₁₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	13	1	no	—
297	C₂³	C₂×C₈×D₁₃	1	2	yes	—
298	Q₈	—	13	4	no	—
299	D₄	—	13	8	no	—
300	D₄	C₁₆×D₁₃	1	4	yes	complemented
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	8	no	—
307	C₂×C₄	—	13	4	no	—
308	C₂×C₄	—	13	1	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	—	26	1	no	—
313	C₂×C₄	—	26	1	no	—
314	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	13	2	no	—
321	C₈	—	26	2	no	—
322	C₈	—	26	1	no	—
323	C₈	C₂×#6129	1	1	yes	characteristic
324	C₈	#41×D₁₃	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	13	2	no	—
328	C₂²	—	26	1	no	—
329	C₂²	—	26	1	no	—
330	C₂²	C₂×C₁₆×D₁₃	1	4	yes	—
331	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	13	2	no	—
335	C₄	—	13	4	no	—
336	C₄	—	13	2	no	—
337	C₄	—	26	1	no	—
338	C₄	—	26	2	no	—
339	C₄	C₂×C₁₆×D₁₃	1	2	yes	—
340	C₄	C₄×#6129	1	1	yes	characteristic
341	C₄	D₁₃×#131	1	1	yes	characteristic
342	C₂	—	2	4	no	—
343	C₂	—	26	2	no	—
344	C₂	C₈×#6129	1	1	yes	characteristic
345	C₂	C₂²×C₁₆×D₁₃	1	1	yes	characteristic
346	C₂	D₁₃×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	D₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×#6129	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	D₄×C₁₆	D₁₃
hypocenter	C₁₃	C₁₆×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₂₆	C₂	semidirect
3. #13116738	C₂	semidirect
4. D₄×#6090	C₂	semidirect
5. C₁₆×C₂²⋊D₁₃	C₂	semidirect
6. #13116653	C₂	semidirect
7. #13116634	C₂	semidirect
8. #13116632	C₂	semidirect
9. C₂×C₁₆×Dic₁₃	C₂	semidirect
10. C₄×C₁₆×D₁₃	C₂	semidirect
11. D₄×C₁₃×C₁₆	C₂	semidirect
86. #6129	C₁₆	direct (N × H)
119. C₁₃×C₁₆	#41	semidirect
122. #6090	#41	semidirect
129. D₄×C₁₆	D₁₃	semidirect
136. D₄×C₁₃	C₂×C₁₆	semidirect
143. C₂²⋊D₁₃	C₂×C₁₆	semidirect
145. C₂×Dic₁₃	C₂×C₁₆	semidirect
154. C₄×D₁₃	C₂×C₁₆	semidirect
155. Dic₂₆	C₂×C₁₆	semidirect
294. C₁₆	#6129	direct (N × H)
295. C₁₃	C₁₆×#41	semidirect
300. D₄	C₁₆×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#6129	2
C₁₆×Dic₂₆	2
#13116738	2
D₄×#6090	2
C₁₆×C₂²⋊D₁₃	2
#13116653	2
#13116634	2
#13116632	2
C₂×C₁₆×Dic₁₃	2
C₄×C₁₆×D₁₃	2
D₄×C₁₃×C₁₆	2
C₁₆×#41	13

Subgroups — C16×#6129 #16016434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#6129 #16016434