Subgroups — C₁₆×#10625350 #26369795

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#10625350	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₉₄	C₂	1	2	yes	maximal, complemented
3	C₁₆×D₁₈₈	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₂²⋊D₄₇	C₂	1	4	yes	maximal, complemented
5	#20245187	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245185	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₆×D₄₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10625350	C₄	1	1	yes	characteristic
10	C₂×C₁₆×D₄₇	C₂²	1	4	yes	—
11	C₈×Dic₉₄	C₄	1	1	yes	characteristic
12	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₈×D₁₈₈	C₄	1	1	yes	characteristic
14	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₄₇	C₄	1	2	yes	—
16	C₈×C₂²⋊D₄₇	C₂²	1	2	yes	—
17	#15689176	C₄	1	1	yes	characteristic
18	#15689174	C₄	1	2	yes	—
19	C₄×C₈×D₄₇	C₂²	1	2	yes	—
20	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
21	Dic₄₇⋊C₁₆	C₂²	1	4	yes	—
22	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
23	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
24	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
25	C₁₆×Dic₄₇	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
29	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#10625350	C₈	1	1	yes	characteristic
32	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₄₇	C₂³	1	2	yes	—
34	C₄×Dic₉₄	C₈	1	1	yes	characteristic
35	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
36	C₄×D₁₈₈	C₈	1	1	yes	characteristic
37	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₄₇	C₈	1	2	yes	—
39	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
40	#13093983	C₈	1	1	yes	characteristic
41	#13093981	C₈	1	2	yes	—
42	C₄²×D₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
44	C₁₆×D₄₇	—	2	8	no	—
45	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
48	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
50	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
51	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
54	C₈×Dic₄₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₄₇	—	2	4	no	—
59	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
60	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
61	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
69	#10625350	C₁₆	1	8	yes	direct product factor
70	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
74	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
86	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
87	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
90	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
92	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
93	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
94	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
97	C₈×D₄₇	—	2	4	no	—
98	C₈×D₄₇	—	2	4	no	—
99	C₁₆×C₄₇	—	2	2	no	—
100	C₁₆×C₄₇	#41	1	4	yes	complemented
101	C₄₇⋊C₁₆	—	2	4	no	—
102	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	complemented
108	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂²⋊D₄₇	C₂×C₁₆	1	8	yes	complemented
115	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
116	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
117	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
118	D₁₈₈	C₂×C₁₆	1	4	yes	complemented
119	C₄×D₄₇	—	2	2	no	—
120	C₄×D₄₇	—	2	2	no	—
121	C₄×D₄₇	—	2	4	no	—
122	C₄×D₄₇	C₂×C₁₆	1	8	yes	complemented
123	Dic₉₄	C₂×C₁₆	1	4	yes	complemented
124	C₈×C₄₇	—	2	1	no	—
125	C₈×C₄₇	—	2	1	no	—
126	C₈×C₄₇	#131	1	2	yes	—
127	C₈×C₄₇	C₂×#41	1	2	yes	—
128	C₄₇⋊C₈	—	2	2	no	—
129	C₄₇⋊C₈	—	2	2	no	—
130	C₁₆×#41	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	C₂²×C₁₆	1	2	yes	—
134	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₄₇	—	2	2	no	—
136	C₂×D₄₇	—	2	2	no	—
137	C₂×D₄₇	C₂²×C₁₆	1	4	yes	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	C₂²×C₁₆	1	2	yes	—
141	C₄×C₄₇	C₂²×C₁₆	1	2	yes	—
142	C₄×C₄₇	#504	1	2	yes	—
143	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
145	Dic₄₇	—	2	2	no	—
146	Dic₄₇	—	2	2	no	—
147	Dic₄₇	C₂²×C₁₆	1	4	yes	—
148	C₈×#41	—	47	1	no	—
149	Q₈×C₁₆	—	47	2	no	—
150	D₄×C₁₆	—	47	4	no	—
151	D₄×C₁₆	—	47	2	no	—
152	#2162	—	47	1	no	—
153	#2162	—	47	2	no	—
154	C₂×C₄×C₁₆	—	47	2	no	—
155	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic, complemented
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₈×#41	1	1	yes	characteristic
158	C₂×C₄₇	C₂³×C₁₆	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
160	D₄₇	—	2	4	no	—
161	C₄×#41	—	47	1	no	—
162	C₂²×C₁₆	—	47	4	no	—
163	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
164	C₈×Q₈	—	47	1	no	—
165	C₈×Q₈	—	47	1	no	—
166	C₈×D₄	—	47	2	no	—
167	C₈×D₄	—	47	2	no	—
168	C₈×D₄	—	47	1	no	—
169	C₈×D₄	—	47	1	no	—
170	#431	—	47	1	no	—
171	#431	—	47	2	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
174	C₄⋊C₁₆	—	47	4	no	—
175	C₄⋊C₁₆	—	47	2	no	—
176	C₂²⋊C₁₆	—	47	2	no	—
177	C₂²⋊C₁₆	—	47	4	no	—
178	C₄×C₁₆	—	47	4	no	—
179	C₄×C₁₆	—	47	2	no	—
180	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
181	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
182	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
183	C₄₇	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	47	1	no	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₄×Q₈	—	47	1	no	—
190	C₄×Q₈	—	47	1	no	—
191	C₄×D₄	—	47	2	no	—
192	C₄×D₄	—	47	2	no	—
193	C₄×D₄	—	47	1	no	—
194	C₄×D₄	—	47	1	no	—
195	#117	—	47	1	no	—
196	#117	—	47	2	no	—
197	C₂×C₄²	—	47	2	no	—
198	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₁₆	—	2	4	no	—
200	C₂×C₁₆	—	47	4	no	—
201	C₂×C₁₆	—	94	8	no	—
202	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
203	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
204	C₄⋊C₈	—	47	2	no	—
205	C₄⋊C₈	—	47	2	no	—
206	C₄⋊C₈	—	47	1	no	—
207	C₄⋊C₈	—	47	1	no	—
208	C₂²⋊C₈	—	47	2	no	—
209	C₂²⋊C₈	—	47	2	no	—
210	C₂²⋊C₈	—	47	2	no	—
211	C₄×C₈	—	47	2	no	—
212	C₄×C₈	—	47	2	no	—
213	C₄×C₈	—	47	2	no	—
214	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
215	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	#41	—	47	8	no	—
219	C₂×Q₈	—	47	1	no	—
220	C₂×D₄	—	47	2	no	—
221	C₂×D₄	—	47	1	no	—
222	C₂²×C₄	—	47	2	no	—
223	C₂²×C₄	—	47	2	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	47	2	no	—
231	C₂×C₈	—	47	2	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
238	C₄⋊C₄	—	47	2	no	—
239	C₄⋊C₄	—	47	2	no	—
240	C₄⋊C₄	—	47	1	no	—
241	C₄⋊C₄	—	47	1	no	—
242	C₂²⋊C₄	—	47	2	no	—
243	C₂²⋊C₄	—	47	2	no	—
244	C₂²⋊C₄	—	47	2	no	—
245	C₄²	—	47	2	no	—
246	C₄²	—	47	2	no	—
247	C₄²	—	47	2	no	—
248	C₄²	C₈×D₄₇	1	1	yes	characteristic
249	C₄²	C₈×D₄₇	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₁₆	—	2	2	no	—
253	C₁₆	—	94	4	no	—
254	C₁₆	#10625350	1	4	yes	direct product factor
255	C₂³	—	47	2	no	—
256	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
257	Q₈	—	47	4	no	—
258	D₄	—	47	8	no	—
259	D₄	—	47	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	47	2	no	—
264	C₂×C₄	—	47	8	no	—
265	C₂×C₄	—	47	2	no	—
266	C₂×C₄	—	47	2	no	—
267	C₂×C₄	—	94	4	no	—
268	C₂×C₄	—	94	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	C₁₆×D₄₇	1	4	yes	complemented
271	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	94	2	no	—
280	C₈	—	94	2	no	—
281	C₈	#13094030	1	2	yes	—
282	C₈	C₂×#10625350	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	47	4	no	—
286	C₂²	—	94	2	no	—
287	C₂²	—	94	2	no	—
288	C₂²	C₂×C₁₆×D₄₇	1	2	yes	—
289	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	47	4	no	—
293	C₄	—	94	2	no	—
294	C₄	—	94	2	no	—
295	C₄	C₂×C₁₆×D₄₇	1	2	yes	—
296	C₄	C₂×C₁₆×D₄₇	1	2	yes	—
297	C₄	#15689539	1	2	yes	—
298	C₄	C₄×#10625350	1	1	yes	characteristic
299	C₄	C₂×#13094030	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	94	4	no	—
302	C₂	C₈×#10625350	1	1	yes	characteristic
303	C₂	C₂²×C₁₆×D₄₇	1	1	yes	characteristic
304	C₂	C₂×#15689539	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×C₈
prime radical	C₂×#10625350	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	C₁₆×#41
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₉₄	C₂	semidirect
3. C₁₆×D₁₈₈	C₂	semidirect
4. C₁₆×C₂²⋊D₄₇	C₂	semidirect
5. #20245187	C₂	semidirect
6. #20245185	C₂	semidirect
7. C₄×C₁₆×D₄₇	C₂	semidirect
8. C₂×C₄×C₁₆×C₄₇	C₂	semidirect
69. #10625350	C₁₆	direct (N × H)
100. C₁₆×C₄₇	#41	semidirect
107. C₂×C₄×C₄₇	C₂×C₁₆	semidirect
114. C₂²⋊D₄₇	C₂×C₁₆	semidirect
118. D₁₈₈	C₂×C₁₆	semidirect
122. C₄×D₄₇	C₂×C₁₆	semidirect
123. Dic₉₄	C₂×C₁₆	semidirect
155. C₂×C₄×C₁₆	D₄₇	semidirect
183. C₄₇	C₁₆×#41	semidirect
254. C₁₆	#10625350	direct (N × H)
270. C₂×C₄	C₁₆×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#10625350	2
C₁₆×Dic₉₄	2
C₁₆×D₁₈₈	2
C₁₆×C₂²⋊D₄₇	2
#20245187	2
#20245185	2
C₄×C₁₆×D₄₇	2
C₂×C₄×C₁₆×C₄₇	2
C₁₆×#41	47

Subgroups — C16×#10625350 #26369795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#10625350 #26369795