Subgroups — Dic₄₇⋊#338 #26122084

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689146	C₂	1	2	yes	maximal
2	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇⋊#338	C₂	1	2	yes	maximal
4	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689146	—	2	2	no	—
10	#15689146	—	2	2	no	—
11	#15689146	—	2	4	no	—
12	C₂×Dic₄₇⋊C₈	C₄	1	8	yes	—
13	M₁₆×Dic₄₇	—	2	2	no	—
14	M₁₆×Dic₄₇	—	2	2	no	—
15	C₂×#13093934	C₄	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
17	C₄₇⋊#338	—	2	4	no	—
18	C₄₇⋊#338	—	2	2	no	—
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#338	—	2	2	no	—
22	C₂³×Dic₄₇	Q₈	1	2	yes	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	D₄	1	2	yes	—
32	C₂×#10625329	Q₈	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
41	Dic₄₇⋊C₈	—	2	8	no	—
42	Dic₄₇⋊C₈	—	2	8	no	—
43	#13093934	—	2	4	no	—
44	C₈×Dic₄₇	—	2	2	no	—
45	C₈×Dic₄₇	—	2	2	no	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	—	2	2	no	—
50	C₂×M₁₆×C₄₇	—	2	2	no	—
51	C₂×M₁₆×C₄₇	—	2	2	no	—
52	C₂×M₁₆×C₄₇	—	2	2	no	—
53	C₂×M₁₆×C₄₇	—	2	2	no	—
54	C₂×M₁₆×C₄₇	—	2	2	no	—
55	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
56	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
57	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
72	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
73	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	4	2	no	—
78	M₁₆×C₄₇	—	4	2	no	—
79	M₁₆×C₄₇	—	4	2	no	—
80	M₁₆×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄×Dic₄₇	C₄²	1	2	yes	—
88	C₄×Dic₄₇	C₄²	1	2	yes	—
89	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
92	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
93	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
94	#10625329	—	2	4	no	—
95	#10625329	—	2	4	no	—
96	#10625329	—	4	2	no	—
97	#10625329	—	4	2	no	—
98	#10625329	—	4	2	no	—
99	#10625329	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₄₇	#125	1	2	yes	—
110	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
111	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#121	1	2	yes	—
124	C₂×C₄×C₄₇	#121	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	#124	1	2	yes	—
128	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	#95	1	2	yes	—
135	C₂×Dic₄₇	#95	1	4	yes	—
136	C₂×Dic₄₇	#95	1	2	yes	—
137	C₈×C₄₇	—	2	4	no	—
138	C₈×C₄₇	—	4	4	no	—
139	C₄₇⋊C₈	—	2	4	no	—
140	C₄₇⋊C₈	—	4	4	no	—
141	C₄⋊#338	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	4	2	no	—
145	C₂²×C₄₇	—	4	1	no	—
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₄₇	#382	1	1	yes	characteristic
149	C₂²×C₄₇	#384	1	1	yes	characteristic
150	C₂²×C₄₇	#389	1	2	yes	—
151	C₂²×C₄₇	#389	1	2	yes	—
152	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	#380	1	1	yes	characteristic
155	C₄×C₄₇	#380	1	1	yes	characteristic
156	C₄×C₄₇	#390	1	2	yes	—
157	C₄×C₄₇	#391	1	2	yes	—
158	Dic₄₇	—	2	2	no	—
159	Dic₄₇	—	2	4	no	—
160	Dic₄₇	#338	1	4	yes	complemented
161	C₂×#423	—	47	2	no	—
162	C₂×C₄×M₁₆	—	47	1	no	—
163	C₂×#338	—	47	2	no	—
164	C₂×#338	—	47	1	no	—
165	C₂×#338	D₄₇	1	1	yes	characteristic
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#338	1	2	yes	—
172	C₂²×M₁₆	—	47	1	no	—
173	C₂²×M₁₆	—	47	2	no	—
174	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
175	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
176	C₂²×C₄²	—	47	1	no	—
177	#423	—	94	2	no	—
178	#423	—	94	2	no	—
179	#423	—	94	4	no	—
180	C₂×C₄⋊C₈	—	47	8	no	—
181	C₄×M₁₆	—	94	2	no	—
182	C₄×M₁₆	—	94	2	no	—
183	C₂×#97	—	47	2	no	—
184	C₂×C₄×C₈	—	47	2	no	—
185	#338	—	2	2	no	—
186	#338	—	94	4	no	—
187	#338	—	94	2	no	—
188	C₄₇	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	—	47	2	no	—
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	47	2	no	—
200	C₂×M₁₆	—	47	2	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂×M₁₆	—	94	2	no	—
203	C₂×M₁₆	—	94	2	no	—
204	C₂×M₁₆	—	94	2	no	—
205	C₂×M₁₆	—	94	2	no	—
206	C₂×M₁₆	—	94	2	no	—
207	C₂×M₁₆	—	94	2	no	—
208	C₂×M₁₆	—	94	2	no	—
209	C₂×M₁₆	Dic₉₄	1	2	yes	—
210	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	47	4	no	—
213	C₂²×C₈	C₄×D₄₇	1	4	yes	—
214	C₂²×C₈	C₄×D₄₇	1	2	yes	—
215	C₂×C₄²	—	47	1	no	—
216	C₂×C₄²	—	47	1	no	—
217	C₂×C₄²	—	47	4	no	—
218	C₂×C₄²	—	47	2	no	—
219	C₂×C₄²	—	47	2	no	—
220	C₂×C₄²	—	47	2	no	—
221	C₄⋊C₈	—	94	8	no	—
222	C₄⋊C₈	—	94	8	no	—
223	#97	—	94	4	no	—
224	C₄×C₈	—	94	2	no	—
225	C₄×C₈	—	94	2	no	—
226	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	—	47	2	no	—
228	C₂²×C₄	—	47	4	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	47	4	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
235	C₂²×C₄	#10625350	1	2	yes	—
236	C₂²×C₄	D₄×D₄₇	1	2	yes	—
237	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	#10625352	1	2	yes	—
240	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	94	4	no	—
249	M₁₆	—	94	4	no	—
250	M₁₆	—	188	2	no	—
251	M₁₆	—	188	2	no	—
252	M₁₆	—	188	2	no	—
253	M₁₆	—	188	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	94	2	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	—	94	4	no	—
263	C₂×C₈	—	94	4	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
266	C₄²	—	47	2	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	1	no	—
269	C₄²	—	47	1	no	—
270	C₄²	—	47	2	no	—
271	C₄²	—	47	4	no	—
272	C₄²	—	47	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
275	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
276	C₂³	#13094014	1	2	yes	—
277	C₂³	#13094022	1	1	yes	characteristic
278	C₂³	#13094023	1	1	yes	characteristic
279	C₂×C₄	—	47	2	no	—
280	C₂×C₄	—	47	2	no	—
281	C₂×C₄	—	47	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	8	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
289	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
291	C₂×C₄	#13093980	1	1	yes	characteristic
292	C₂×C₄	#13093981	1	1	yes	characteristic
293	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
294	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
295	C₂×C₄	#13093989	1	2	yes	—
296	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
297	C₂×C₄	#13093996	1	2	yes	—
298	C₂×C₄	#13093997	1	2	yes	—
299	C₂×C₄	#13094001	1	2	yes	—
300	C₂×C₄	#13094002	1	2	yes	—
301	C₂×C₄	#13094010	1	2	yes	—
302	C₂×C₄	#13094020	1	1	yes	characteristic
303	C₂×C₄	#13094023	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	4	4	no	—
306	C₈	—	94	4	no	—
307	C₈	—	188	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	#95×D₄₇	1	1	yes	characteristic
314	C₂²	#15688987	1	1	yes	characteristic
315	C₂²	#15688992	1	1	yes	characteristic
316	C₂²	#15689018	1	2	yes	—
317	C₂²	#15689020	1	2	yes	—
318	C₄	—	47	4	no	—
319	C₄	—	94	2	no	—
320	C₄	—	94	4	no	—
321	C₄	#15688974	1	1	yes	characteristic
322	C₄	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
323	C₄	#15688981	1	1	yes	characteristic
324	C₄	#15688982	1	1	yes	characteristic
325	C₄	#15689023	1	2	yes	—
326	C₄	#15689027	1	2	yes	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	2	no	—
330	C₂	—	2	2	no	—
331	C₂	Dic₄₇⋊#95	1	1	yes	characteristic
332	C₂	#338×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#338	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#338	D₄₇
hypocenter	C₄₇	C₄⋊#338
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
160. Dic₄₇	#338	semidirect
188. C₄₇	C₄⋊#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689146	2
C₂×M₁₆×Dic₄₇	2
C₂×C₄₇⋊#338	2
C₂×C₄₇×#338	2
C₄⋊#338	47

Subgroups — Dic47⋊#338 #26122084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₄₇⋊#338 #26122084