Subgroups — Q₁₆×#6099 #15834380

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6099	C₂	1	2	yes	maximal
2	#13113733	C₂	1	2	yes	maximal, complemented
3	#13113687	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	#13113595	C₂	1	1	yes	maximal, characteristic
6	#13113265	C₂	1	2	yes	maximal
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
12	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
14	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₃×#103	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718685	—	2	4	no	—
21	#11718685	C₂²	1	4	yes	—
22	Q₈×#1055	C₂²	1	4	yes	—
23	#11718586	C₂²	1	1	yes	characteristic
24	#11718585	—	2	2	no	—
25	#11718585	C₂²	1	2	yes	—
26	C₄×#6099	—	2	2	no	—
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	Q₈⋊#1055	C₂²	1	4	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718050	C₂²	1	2	yes	—
32	#11718046	C₂²	1	2	yes	—
33	C₈×#1055	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×Q₁₆	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	—	2	2	no	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	D₄	1	1	yes	characteristic
53	C₂×#6099	D₄	1	1	yes	characteristic
54	C₁₃×#107	C₂×C₄	1	2	yes	—
55	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#103	C₂×C₄	1	2	yes	—
59	C₁₃×#103	C₂×C₄	1	2	yes	—
60	C₁₃×#103	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²⋊#1055	—	2	1	no	—
66	C₄⋊#1055	—	2	2	no	—
67	C₄⋊#1055	—	2	1	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	—	2	4	no	—
70	C₄⋊#1055	C₂³	1	4	yes	—
71	#88043	—	2	4	no	—
72	#88043	D₄	1	2	yes	—
73	C₄×#1055	—	2	4	no	—
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	M₁₆×Q₁₆	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	4	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
87	C₁₃×Q₁₆	—	2	4	no	—
88	C₁₃×Q₁₆	—	2	2	no	—
89	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
102	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	#41	1	1	yes	characteristic
110	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
111	#6099	—	2	2	no	—
112	#6099	—	4	4	no	—
113	#6099	Q₁₆	1	4	yes	direct product factor
114	C₂×#1055	—	2	4	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	C₂×#1055	C₂×D₄	1	2	yes	—
118	Q₈×M₁₆	—	13	2	no	—
119	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
120	#1591	—	13	2	no	—
121	#1584	—	13	4	no	—
122	C₈×Q₁₆	—	13	2	no	—
123	#1567	—	13	1	no	—
124	#1487	—	13	2	no	—
125	C₈×M₁₆	—	13	1	no	—
126	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	2	no	—
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	4	4	no	—
131	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	4	no	—
140	C₂×C₄×C₁₃	—	8	2	no	—
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
145	C₂×C₄×C₁₃	#135	1	2	yes	—
146	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
153	#1055	—	2	2	no	—
154	#1055	—	4	2	no	—
155	#1055	—	4	4	no	—
156	#1055	C₂×Q₁₆	1	4	yes	complemented
157	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
159	#446	—	13	4	no	—
160	#446	—	26	4	no	—
161	C₈×Q₈	—	13	4	no	—
162	C₄×Q₁₆	Dic₁₃	1	2	yes	—
163	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
164	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
165	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
166	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
167	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
168	#431	—	13	1	no	—
169	C₂×#107	Dic₁₃	1	1	yes	characteristic
170	#423	—	13	2	no	—
171	#423	—	26	2	no	—
172	C₂×#103	Dic₁₃	1	2	yes	—
173	C₄×M₁₆	—	13	1	no	—
174	C₄×M₁₆	—	26	2	no	—
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#335	—	13	2	no	—
177	#335	—	13	2	no	—
178	Q₈⋊C₈	—	13	4	no	—
179	Q₈⋊C₈	—	13	4	no	—
180	#322	—	13	2	no	—
181	C₈²	—	13	2	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
185	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	4	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	—	4	2	no	—
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
194	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
195	C₄×C₁₃	#568	1	1	yes	characteristic
196	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
197	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
198	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
201	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
202	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
203	C₂×M₁₆	—	13	1	no	—
204	C₂×M₁₆	—	13	1	no	—
205	C₂×M₁₆	—	26	2	no	—
206	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
211	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
212	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
213	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
218	#107	C₂×Dic₁₃	1	2	yes	—
219	#107	C₂×Dic₁₃	1	1	yes	characteristic
220	#107	C₂×Dic₁₃	1	1	yes	characteristic
221	C₄⋊C₈	—	13	4	no	—
222	C₄⋊C₈	—	26	4	no	—
223	C₄⋊C₈	—	26	4	no	—
224	C₄⋊C₈	—	26	2	no	—
225	C₄⋊C₈	—	26	1	no	—
226	#103	C₂×Dic₁₃	1	2	yes	—
227	#103	C₂×Dic₁₃	1	2	yes	—
228	#103	C₂×Dic₁₃	1	2	yes	—
229	#103	C₂×Dic₁₃	1	2	yes	—
230	C₂²⋊C₈	—	26	1	no	—
231	#97	—	13	2	no	—
232	#97	—	26	4	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	2	no	—
235	C₄×C₈	—	26	4	no	—
236	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
242	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₂×C₁₃	#2957	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
248	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
252	C₂²×C₄	#6129	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	#6099	1	4	yes	direct product factor
257	M₁₆	—	13	4	no	—
258	M₁₆	—	26	2	no	—
259	M₁₆	—	52	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	26	4	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
276	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	D₄×D₁₃	1	1	yes	characteristic
282	C₄²	#6129	1	1	yes	characteristic
283	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
284	C₄²	C₂³×D₁₃	1	1	yes	characteristic
285	C₁₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
286	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	4	no	—
291	Q₈	C₂×#6099	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
302	C₂×C₄	#88173	1	2	yes	—
303	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂×#6129	1	1	yes	characteristic
308	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	13	4	no	—
313	C₈	—	26	2	no	—
314	C₈	—	52	2	no	—
315	C₈	—	52	4	no	—
316	C₈	C₂×#6099	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	Q₁₆×Dic₁₃	1	2	yes	—
320	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	Q₁₆×Dic₁₃	1	2	yes	—
329	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
330	C₄	C₂×#88173	1	1	yes	characteristic
331	C₄	C₂²×#6099	1	1	yes	characteristic
332	C₄	#11719524	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	D₄×#6099	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×Dic₁₃	1	1	yes	characteristic
336	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13113733	C₂	semidirect
3. #13113687	C₂	semidirect
4. Q₁₆×#1055	C₂	semidirect
89. C₁₃×Q₁₆	M₁₆	semidirect
113. #6099	Q₁₆	direct (N × H)
156. #1055	C₂×Q₁₆	semidirect
256. Q₁₆	#6099	direct (N × H)
285. C₁₃	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6099	2
#13113733	2
#13113687	2
Q₁₆×#1055	2
#13113595	2
#13113265	2
C₈×#6099	2
C₂×C₄×C₁₃×Q₁₆	2
M₁₆×Q₁₆	13

Subgroups — Q16×#6099 #15834380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#6099 #15834380