Subgroups — D₄₇×#2242 #26371569

Order 12032.

610 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245622	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245621	C₂	1	2	yes	maximal, complemented
5	#20245620	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245619	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245618	C₂	1	2	yes	maximal, complemented
8	#358×D₄₇	C₂	1	2	yes	maximal, complemented
9	#357×D₄₇	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2242	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×D₄₇	—	2	1	no	—
14	C₂×D₈×D₄₇	C₂²	1	1	yes	characteristic
15	#125×D₄₇	—	2	1	no	—
16	#124×D₄₇	—	2	1	no	—
17	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
18	#15689525	C₂²	1	1	yes	characteristic, complemented
19	#15689524	C₂²	1	2	yes	complemented
20	#15689523	—	2	1	no	—
21	#15689522	—	2	1	no	—
22	#15689508	C₂²	1	1	yes	characteristic
23	#15689507	C₂²	1	1	yes	characteristic
24	#15689506	C₂²	1	1	yes	characteristic
25	#15689505	—	2	1	no	—
26	#15689504	—	2	1	no	—
27	#107×D₄₇	C₂²	1	2	yes	—
28	#106×D₄₇	—	2	1	no	—
29	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
30	#15688928	C₂²	1	2	yes	complemented
31	#15688927	C₂²	1	2	yes	complemented
32	#15688926	C₂²	1	2	yes	complemented
33	#15688925	C₂²	1	2	yes	—
34	#15688924	C₂²	1	2	yes	—
35	#15688923	C₂²	1	2	yes	—
36	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
38	C₄₇×#499	C₂²	1	1	yes	characteristic, complemented
39	C₄₇×#495	C₂²	1	1	yes	characteristic
40	C₄₇×#358	C₂²	1	2	yes	complemented
41	C₄₇×#357	C₂²	1	2	yes	—
42	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₄₇	—	2	1	no	—
44	C₂×D₄×D₄₇	—	2	1	no	—
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625337	—	2	1	no	—
47	Q₁₆×D₄₇	—	2	4	no	—
48	C₂×#10625325	—	2	1	no	—
49	C₂×#10625335	—	2	1	no	—
50	C₂×#10625336	—	2	1	no	—
51	SD₁₆×D₄₇	—	4	4	no	—
52	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
53	C₂×#10625334	—	2	1	no	—
54	D₈×D₄₇	—	2	4	no	—
55	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
56	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
57	#13094016	—	2	1	no	—
58	#13094015	—	2	2	no	—
59	#13094014	—	4	2	no	—
60	#13094013	—	2	1	no	—
61	#13094012	—	2	1	no	—
62	#13094011	—	2	1	no	—
63	#13094010	—	4	2	no	—
64	C₄⋊C₄×D₄₇	—	2	2	no	—
65	C₄⋊C₄×D₄₇	—	4	2	no	—
66	C₂²⋊C₄×D₄₇	—	4	2	no	—
67	C₄²×D₄₇	D₄	1	1	yes	characteristic
68	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
69	C₁₆×D₄₇	—	2	4	no	—
70	#13093958	C₂³	1	2	yes	—
71	#13093957	—	2	2	no	—
72	#13093956	—	2	1	no	—
73	#13093955	—	2	2	no	—
74	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
75	C₈×Dic₄₇	D₄	1	1	yes	characteristic
76	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₄₇	—	2	1	no	—
79	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
80	#125×C₄₇	—	2	1	no	—
81	#124×C₄₇	—	2	1	no	—
82	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
83	#107×C₄₇	C₂³	1	2	yes	—
84	#106×C₄₇	—	2	1	no	—
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₂³×D₄₇	—	4	1	no	—
87	C₂×Q₈×C₄₇	—	2	1	no	—
88	C₂×D₄×C₄₇	—	2	1	no	—
89	C₂×C₂²⋊D₄₇	—	4	1	no	—
90	Q₈×D₄₇	—	4	2	no	—
91	Q₈×D₄₇	—	4	2	no	—
92	D₄×D₄₇	—	4	2	no	—
93	D₄×D₄₇	—	4	2	no	—
94	C₂×D₁₈₈	—	2	1	no	—
95	C₂×C₄×D₄₇	—	4	1	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×C₄×D₄₇	D₈	1	2	yes	—
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	4	1	no	—
101	Q₁₆×C₄₇	—	2	2	no	—
102	SD₁₆×C₄₇	—	4	2	no	—
103	D₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	4	2	no	—
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂²⋊Dic₄₇	—	4	2	no	—
111	#10625337	—	4	2	no	—
112	#10625336	—	4	2	no	—
113	#10625335	—	4	2	no	—
114	#10625334	—	4	2	no	—
115	(C₂×D₄₇)⋊C₄	—	4	2	no	—
116	(C₂×D₄₇)⋊C₄	—	4	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	4	2	no	—
119	Dic₄₇⋊C₄	—	4	2	no	—
120	Dic₄₇⋊C₄	—	4	2	no	—
121	Dic₄₇⋊C₄	—	4	2	no	—
122	C₄×Dic₄₇	D₈	1	2	yes	—
123	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
126	Dic₁₈₈	—	2	2	no	—
127	D₃₇₆	—	2	2	no	—
128	#10625325	—	4	2	no	—
129	C₈×D₄₇	SD₁₆	1	4	yes	—
130	C₈×D₄₇	#41	1	4	yes	—
131	C₁₆×C₄₇	—	2	2	no	—
132	C₄₇⋊C₁₆	—	2	2	no	—
133	C₂³×C₄₇	—	4	1	no	—
134	C₂²×D₄₇	—	4	1	no	—
135	C₂²×D₄₇	—	8	2	no	—
136	C₂²×D₄₇	—	8	4	no	—
137	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₄₇	—	4	1	no	—
139	Q₈×C₄₇	—	4	1	no	—
140	D₄×C₄₇	—	4	1	no	—
141	D₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	1	no	—
144	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₄₇	—	8	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	1	no	—
149	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
150	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
151	D₁₈₈	—	4	1	no	—
152	D₁₈₈	—	4	1	no	—
153	C₄×D₄₇	—	2	4	no	—
154	C₄×D₄₇	—	8	2	no	—
155	C₄×D₄₇	—	8	4	no	—
156	C₄×D₄₇	#124	1	2	yes	—
157	C₄×D₄₇	C₂×SD₁₆	1	2	yes	—
158	Dic₉₄	—	4	1	no	—
159	Dic₉₄	—	4	1	no	—
160	Dic₉₄	—	8	2	no	—
161	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
162	C₈×C₄₇	C₂×#41	1	2	yes	—
163	C₄₇⋊C₈	#124	1	2	yes	—
164	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
165	C₂×#2242	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	8	1	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₄₇	—	8	1	no	—
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	16	4	no	—
172	C₂×D₄₇	#486	1	2	yes	—
173	C₂×D₄₇	#508	1	4	yes	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	8	1	no	—
176	C₄×C₄₇	—	8	2	no	—
177	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
178	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
179	Dic₄₇	—	2	2	no	—
180	Dic₄₇	—	8	1	no	—
181	Dic₄₇	—	8	2	no	—
182	Dic₄₇	#486	1	2	yes	—
183	C₂×#499	—	47	1	no	—
184	C₂×#495	—	47	1	no	—
185	#2242	—	47	1	no	—
186	#2242	—	47	1	no	—
187	#2242	—	47	2	no	—
188	#2242	—	47	2	no	—
189	#2242	—	47	1	no	—
190	#2242	D₄₇	1	1	yes	characteristic, direct product factor
191	C₂×#358	—	47	2	no	—
192	C₂×#357	—	47	2	no	—
193	C₂×C₄×C₁₆	—	47	1	no	—
194	C₂×C₄₇	—	16	1	no	—
195	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
196	C₂×C₄₇	C₂×#508	1	2	yes	—
197	D₄₇	—	16	1	no	—
198	D₄₇	#2242	1	4	yes	direct product factor
199	C₂²×Q₁₆	—	47	1	no	—
200	C₂²×SD₁₆	—	94	1	no	—
201	C₂²×D₈	—	47	1	no	—
202	C₂×#125	—	94	1	no	—
203	C₂×#124	—	94	1	no	—
204	C₂²×C₁₆	—	47	2	no	—
205	#499	—	47	2	no	—
206	#499	—	47	1	no	—
207	#499	—	94	1	no	—
208	#499	—	94	1	no	—
209	#499	C₂×D₄₇	1	1	yes	characteristic, complemented
210	#495	—	47	1	no	—
211	#495	—	47	1	no	—
212	#495	—	47	1	no	—
213	#495	—	94	1	no	—
214	#495	—	94	1	no	—
215	#495	C₂×D₄₇	1	1	yes	characteristic
216	C₂×#107	—	47	2	no	—
217	C₂×#106	—	94	1	no	—
218	C₂×C₄×C₈	—	47	1	no	—
219	#358	—	47	2	no	—
220	#358	—	47	2	no	—
221	#358	—	47	2	no	—
222	#358	C₂×D₄₇	1	2	yes	complemented
223	#357	—	47	2	no	—
224	#357	—	47	2	no	—
225	#357	—	47	2	no	—
226	#357	C₂×D₄₇	1	2	yes	—
227	C₄×C₁₆	—	47	2	no	—
228	C₄×C₁₆	—	47	1	no	—
229	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
230	C₄₇	C₂×#2242	1	1	yes	characteristic, complemented, Sylow
231	C₂²×Q₈	—	94	1	no	—
232	C₂²×D₄	—	94	1	no	—
233	C₂×Q₁₆	—	47	1	no	—
234	C₂×Q₁₆	—	94	4	no	—
235	C₂×Q₁₆	—	94	1	no	—
236	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×SD₁₆	—	94	1	no	—
239	C₂×SD₁₆	—	94	1	no	—
240	C₂×SD₁₆	—	94	1	no	—
241	C₂×SD₁₆	—	188	4	no	—
242	C₂×D₈	—	47	1	no	—
243	C₂×D₈	—	94	4	no	—
244	C₂×D₈	—	94	1	no	—
245	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₂²×C₈	—	47	1	no	—
247	C₂²×C₈	—	47	1	no	—
248	#125	—	2	1	no	—
249	#125	—	94	2	no	—
250	#125	—	94	1	no	—
251	#125	—	188	2	no	—
252	#124	—	2	1	no	—
253	#124	—	94	1	no	—
254	#124	—	94	1	no	—
255	#124	—	94	1	no	—
256	#124	—	188	2	no	—
257	C₂×C₄⋊C₄	—	94	2	no	—
258	C₂×C₄⋊C₄	—	188	2	no	—
259	C₂×C₂²⋊C₄	—	188	2	no	—
260	C₂×C₄²	—	47	1	no	—
261	C₂×C₁₆	—	47	2	no	—
262	C₂×C₁₆	—	94	4	no	—
263	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
264	#107	—	47	2	no	—
265	#107	—	94	2	no	—
266	#107	C₂²×D₄₇	1	2	yes	—
267	#106	—	2	1	no	—
268	#106	—	94	2	no	—
269	#106	—	94	1	no	—
270	C₄×C₈	—	47	1	no	—
271	C₄×C₈	—	47	1	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
274	C₂⁴	—	188	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	94	1	no	—
277	C₂×Q₈	—	188	1	no	—
278	C₂×Q₈	—	188	2	no	—
279	C₂×Q₈	—	188	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	94	1	no	—
282	C₂×D₄	—	188	2	no	—
283	C₂×D₄	—	188	2	no	—
284	C₂×D₄	—	188	1	no	—
285	C₂²×C₄	—	47	1	no	—
286	C₂²×C₄	—	47	2	no	—
287	C₂²×C₄	—	188	1	no	—
288	C₂²×C₄	—	188	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	94	2	no	—
291	Q₁₆	—	188	2	no	—
292	SD₁₆	—	4	2	no	—
293	SD₁₆	—	188	2	no	—
294	SD₁₆	—	188	2	no	—
295	SD₁₆	—	188	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	94	2	no	—
298	D₈	—	188	2	no	—
299	C₂×C₈	—	47	1	no	—
300	C₂×C₈	—	47	1	no	—
301	C₂×C₈	—	47	4	no	—
302	C₂×C₈	—	47	4	no	—
303	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	188	2	no	—
309	C₄⋊C₄	—	188	2	no	—
310	C₄⋊C₄	—	188	2	no	—
311	C₄⋊C₄	—	188	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	188	2	no	—
314	C₂²⋊C₄	—	188	2	no	—
315	C₂²⋊C₄	—	188	2	no	—
316	C₄²	—	47	2	no	—
317	C₄²	—	47	1	no	—
318	C₄²	D₄×D₄₇	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	94	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	47	1	no	—
323	C₂³	—	376	2	no	—
324	C₂³	—	376	4	no	—
325	C₂³	—	188	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	376	2	no	—
329	Q₈	—	188	1	no	—
330	Q₈	—	188	1	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	376	2	no	—
334	D₄	—	188	1	no	—
335	D₄	—	188	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	47	1	no	—
339	C₂×C₄	—	47	2	no	—
340	C₂×C₄	—	47	2	no	—
341	C₂×C₄	—	47	2	no	—
342	C₂×C₄	—	94	4	no	—
343	C₂×C₄	—	376	4	no	—
344	C₂×C₄	—	376	2	no	—
345	C₂×C₄	—	188	1	no	—
346	C₂×C₄	—	188	2	no	—
347	C₂×C₄	D₈×D₄₇	1	2	yes	—
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₈	—	47	2	no	—
350	C₈	—	47	2	no	—
351	C₈	SD₁₆×D₄₇	1	2	yes	—
352	C₈	#41×D₄₇	1	2	yes	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	47	4	no	—
356	C₂²	—	47	2	no	—
357	C₂²	—	376	2	no	—
358	C₂²	—	376	1	no	—
359	C₂²	—	752	4	no	—
360	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	47	2	no	—
365	C₄	—	94	2	no	—
366	C₄	—	376	2	no	—
367	C₄	—	376	1	no	—
368	C₄	#124×D₄₇	1	1	yes	characteristic
369	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
370	C₂	—	16	1	no	—
371	C₂	—	47	4	no	—
372	C₂	—	752	1	no	—
373	C₂	#486×D₄₇	1	1	yes	characteristic
374	C₂	#508×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2242	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×D₈×D₄₇	C₂²
hypercenter	#2242	D₄₇
hypocenter	C₄₇	C₂×#2242
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

Complements

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

N	H ≅ G/N	type
1. #499×D₄₇	C₂	semidirect
3. #20245622	C₂	semidirect
4. #20245621	C₂	semidirect
5. #20245620	C₂	semidirect
6. #20245619	C₂	semidirect
7. #20245618	C₂	semidirect
8. #358×D₄₇	C₂	semidirect
10. C₄×C₁₆×D₄₇	C₂	semidirect
11. C₄₇×#2242	C₂	semidirect
18. #15689525	C₂²	semidirect
19. #15689524	C₂²	semidirect
30. #15688928	C₂²	semidirect
31. #15688927	C₂²	semidirect
32. #15688926	C₂²	semidirect
36. C₄×C₄₇⋊C₁₆	C₂²	semidirect
37. C₁₆×Dic₄₇	C₂²	semidirect
38. C₄₇×#499	C₂²	semidirect
40. C₄₇×#358	C₂²	semidirect
42. C₄×C₁₆×C₄₇	C₂²	semidirect
190. #2242	D₄₇	direct (N × H)
198. D₄₇	#2242	direct (N × H)
209. #499	C₂×D₄₇	semidirect
222. #358	C₂×D₄₇	semidirect
229. C₄×C₁₆	C₂×D₄₇	semidirect
230. C₄₇	C₂×#2242	semidirect

Maximal subgroups

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

subgroup	index
#499×D₄₇	2
#495×D₄₇	2
#20245622	2
#20245621	2
#20245620	2
#20245619	2
#20245618	2
#358×D₄₇	2
#357×D₄₇	2
C₄×C₁₆×D₄₇	2
C₄₇×#2242	2
C₂×#2242	47

Subgroups — D47×#2242 #26371569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#2242 #26371569