Subgroups — Q₁₆×#11761612 #30827292

Order 17152.

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×#11761612	C₂	1	2	yes	maximal
2	#22559106	C₂	1	2	yes	maximal, complemented
3	#22559060	C₂	1	4	yes	maximal, complemented
4	Q₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	#22558968	C₂	1	1	yes	maximal, characteristic
6	#22558638	C₂	1	2	yes	maximal
7	C₈×#11761612	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	4	no	—
10	#16863527	C₂²	1	4	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863420	—	2	2	no	—
14	#16863420	C₂²	1	2	yes	—
15	C₄×#11761612	—	2	2	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863149	C₂²	1	2	yes	—
18	#16863149	C₂²	1	2	yes	—
19	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	#16863118	C₂²	1	2	yes	—
22	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
27	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
28	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	2	no	—
35	C₂×#11761612	D₄	1	1	yes	characteristic
36	C₂×#11761612	D₄	1	1	yes	characteristic
37	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
41	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
43	#14523954	—	2	4	no	—
44	#14523954	D₄	1	2	yes	—
45	C₄×C₆₇⋊C₈	—	2	4	no	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₆₇	—	2	4	no	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#107×C₆₇	C₂×C₄	1	2	yes	—
66	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
68	#103×C₆₇	C₂×C₄	1	2	yes	—
69	#103×C₆₇	C₂×C₄	1	2	yes	—
70	#103×C₆₇	C₂×C₄	1	2	yes	—
71	#103×C₆₇	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	#11761612	—	2	2	no	—
77	#11761612	—	4	4	no	—
78	#11761612	Q₁₆	1	4	yes	direct product factor
79	C₂×C₆₇⋊C₈	—	2	4	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
82	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	4	no	—
85	C₂×Q₈×C₆₇	—	4	2	no	—
86	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	Q₁₆×C₆₇	—	2	4	no	—
94	Q₁₆×C₆₇	—	2	2	no	—
95	Q₁₆×C₆₇	M₁₆	1	4	yes	complemented
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	4	1	no	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	#41	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	2	2	no	—
120	Q₈×C₆₇	—	2	2	no	—
121	Q₈×C₆₇	—	4	4	no	—
122	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	8	2	no	—
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₆₇	#135	1	2	yes	—
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
144	C₆₇⋊C₈	—	2	2	no	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	—	4	4	no	—
147	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	complemented
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
151	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
160	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₆₇	#568	1	1	yes	characteristic
162	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
164	M₁₆×Q₁₆	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₆₇	#2957	1	1	yes	characteristic
169	Q₈×M₁₆	—	67	2	no	—
170	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
171	#1591	—	67	2	no	—
172	#1584	—	67	4	no	—
173	C₈×Q₁₆	—	67	2	no	—
174	#1567	—	67	1	no	—
175	#1487	—	67	2	no	—
176	C₈×M₁₆	—	67	1	no	—
177	C₆₇	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
180	#446	—	67	4	no	—
181	#446	—	134	4	no	—
182	C₈×Q₈	—	67	4	no	—
183	C₄×Q₁₆	Dic₆₇	1	2	yes	—
184	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
185	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
186	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
187	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
188	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
189	#431	—	67	1	no	—
190	C₂×#107	Dic₆₇	1	1	yes	characteristic
191	#423	—	67	2	no	—
192	#423	—	134	2	no	—
193	C₂×#103	Dic₆₇	1	2	yes	—
194	C₄×M₁₆	—	67	1	no	—
195	C₄×M₁₆	—	134	2	no	—
196	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
197	#335	—	67	2	no	—
198	#335	—	67	2	no	—
199	Q₈⋊C₈	—	67	4	no	—
200	Q₈⋊C₈	—	67	4	no	—
201	#322	—	67	2	no	—
202	C₈²	—	67	2	no	—
203	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
206	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
208	C₂×M₁₆	—	67	1	no	—
209	C₂×M₁₆	—	67	1	no	—
210	C₂×M₁₆	—	134	2	no	—
211	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
216	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
217	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
218	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	#107	C₂×Dic₆₇	1	2	yes	—
224	#107	C₂×Dic₆₇	1	1	yes	characteristic
225	#107	C₂×Dic₆₇	1	1	yes	characteristic
226	C₄⋊C₈	—	67	4	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	4	no	—
229	C₄⋊C₈	—	134	2	no	—
230	C₄⋊C₈	—	134	1	no	—
231	#103	C₂×Dic₆₇	1	2	yes	—
232	#103	C₂×Dic₆₇	1	2	yes	—
233	#103	C₂×Dic₆₇	1	2	yes	—
234	#103	C₂×Dic₆₇	1	2	yes	—
235	C₂²⋊C₈	—	134	1	no	—
236	#97	—	67	2	no	—
237	#97	—	134	4	no	—
238	C₄×C₈	—	67	2	no	—
239	C₄×C₈	—	67	2	no	—
240	C₄×C₈	—	134	4	no	—
241	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
242	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
249	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
254	C₂²×C₄	#11761627	1	1	yes	characteristic
255	Q₁₆	—	2	4	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	#11761612	1	4	yes	direct product factor
258	M₁₆	—	268	4	no	—
259	M₁₆	—	67	4	no	—
260	M₁₆	—	134	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	67	2	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	134	4	no	—
266	C₂×C₈	—	134	2	no	—
267	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
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	—	2	2	no	—
276	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
277	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₄²	#11761627	1	1	yes	characteristic
285	C₄²	C₂³×D₆₇	1	1	yes	characteristic
286	C₂³	D₄×Dic₆₇	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×#11761612	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	D₄×Dic₆₇	1	1	yes	characteristic
302	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
303	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
305	C₂×C₄	#14524066	1	2	yes	—
306	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	268	2	no	—
313	C₈	—	268	4	no	—
314	C₈	—	67	4	no	—
315	C₈	—	134	2	no	—
316	C₈	C₂×#11761612	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	Q₁₆×Dic₆₇	1	2	yes	—
320	C₂²	C₂×D₄×Dic₆₇	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×Dic₆₇	1	2	yes	—
329	C₄	C₂²×#11761612	1	1	yes	characteristic
330	C₄	#16864163	1	1	yes	characteristic
331	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
332	C₄	C₂×#14524066	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	D₄×#11761612	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×Dic₆₇	1	1	yes	characteristic
336	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
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. #22559106	C₂	semidirect
3. #22559060	C₂	semidirect
4. Q₁₆×C₆₇⋊C₈	C₂	semidirect
78. #11761612	Q₁₆	direct (N × H)
95. Q₁₆×C₆₇	M₁₆	semidirect
147. C₆₇⋊C₈	C₂×Q₁₆	semidirect
177. C₆₇	M₁₆×Q₁₆	semidirect
257. Q₁₆	#11761612	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#11761612	2
#22559106	2
#22559060	2
Q₁₆×C₆₇⋊C₈	2
#22558968	2
#22558638	2
C₈×#11761612	2
C₂×C₄×Q₁₆×C₆₇	2
M₁₆×Q₁₆	67

Subgroups — Q16×#11761612 #30827292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#11761612 #30827292