Subgroups — D₄₇×#1716 #26204001

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498×D₄₇	C₂	1	2	yes	maximal
2	#498×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#494×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#487×D₄₇	C₂	1	2	yes	maximal
5	#20243023	C₂	1	2	yes	maximal, complemented
6	#20243022	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243021	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243020	C₂	1	2	yes	maximal, complemented
9	#20243019	C₂	1	1	yes	maximal, characteristic, complemented
10	#322×D₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	—	2	2	no	—
13	C₄⋊Q₈×D₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×D₄₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×D₄₇	C₂²	1	1	yes	characteristic
16	#15689521	C₂²	1	1	yes	characteristic
17	#15689521	C₂²	1	2	yes	—
18	#15689520	—	2	4	no	—
19	#15689520	—	2	2	no	—
20	#15689519	C₂²	1	2	yes	—
21	#15689519	C₂²	1	2	yes	—
22	#15689519	C₂²	1	2	yes	—
23	#15689503	C₂²	1	1	yes	characteristic
24	#15689502	C₂²	1	1	yes	characteristic
25	#15689502	C₂²	1	1	yes	characteristic
26	#15689501	—	2	2	no	—
27	#15689475	C₂²	1	2	yes	—
28	#15689474	C₂²	1	2	yes	—
29	#15689473	C₂²	1	2	yes	—
30	#15689472	—	2	4	no	—
31	#106×D₄₇	—	2	4	no	—
32	#106×D₄₇	—	2	2	no	—
33	#103×D₄₇	—	2	4	no	—
34	C₄×C₈×D₄₇	C₂²	1	2	yes	—
35	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
36	#15688845	C₂²	1	2	yes	—
37	#15688844	C₂²	1	1	yes	characteristic
38	C₄₇×#498	C₂²	1	2	yes	—
39	C₄₇×#498	C₂²	1	1	yes	characteristic
40	C₄₇×#494	C₂²	1	1	yes	characteristic
41	C₄₇×#487	C₂²	1	2	yes	—
42	C₄₇×#322	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₄₇	—	2	2	no	—
44	C₂×Q₈×D₄₇	—	2	1	no	—
45	C₂×Q₈×D₄₇	—	2	1	no	—
46	C₂×Dic₁₈₈	—	2	2	no	—
47	C₂×#10625337	—	2	2	no	—
48	C₂×#10625337	—	2	2	no	—
49	Q₁₆×D₄₇	—	4	8	no	—
50	C₂×C₈×D₄₇	D₄	1	4	yes	—
51	C₂×C₈×D₄₇	D₄	1	2	yes	—
52	#13094024	C₂³	1	1	yes	characteristic
53	#13094024	C₂³	1	2	yes	—
54	#13094024	C₂³	1	1	yes	characteristic
55	#13094023	—	2	2	no	—
56	#13094023	—	2	2	no	—
57	#13094023	—	2	2	no	—
58	#13094023	—	2	2	no	—
59	#13094022	—	2	2	no	—
60	#13094022	—	2	1	no	—
61	#13094022	—	2	1	no	—
62	C₄⋊C₄×D₄₇	—	2	2	no	—
63	C₄⋊C₄×D₄₇	—	2	2	no	—
64	C₄⋊C₄×D₄₇	—	2	2	no	—
65	C₄⋊C₄×D₄₇	—	2	2	no	—
66	C₄²×D₄₇	C₂³	1	1	yes	characteristic
67	#13093956	—	2	4	no	—
68	#13093956	—	2	2	no	—
69	#13093955	—	2	4	no	—
70	#13093955	—	2	4	no	—
71	#13093955	—	2	4	no	—
72	#13093949	—	2	4	no	—
73	#13093948	—	2	4	no	—
74	#13093947	—	2	4	no	—
75	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₇	D₄	1	4	yes	—
78	C₈×Dic₄₇	D₄	1	2	yes	—
79	C₂×Q₁₆×C₄₇	—	2	2	no	—
80	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
83	#106×C₄₇	—	2	4	no	—
84	#106×C₄₇	—	2	2	no	—
85	#103×C₄₇	—	2	4	no	—
86	C₄×C₈×C₄₇	C₂³	1	2	yes	—
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₄₇	—	2	2	no	—
89	C₂×Q₈×C₄₇	—	2	1	no	—
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	Q₈×D₄₇	—	4	4	no	—
92	Q₈×D₄₇	—	4	4	no	—
93	Q₈×D₄₇	—	4	8	no	—
94	C₂×C₄×D₄₇	—	4	1	no	—
95	C₂×C₄×D₄₇	—	4	1	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
99	C₂×Dic₉₄	—	2	2	no	—
100	C₂×Dic₉₄	—	2	1	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	C₂×Dic₉₄	—	4	1	no	—
103	C₂×Dic₉₄	—	4	1	no	—
104	C₂×Dic₉₄	—	4	2	no	—
105	Q₁₆×C₄₇	—	4	4	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
113	#10625337	—	4	4	no	—
114	#10625337	—	4	4	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	Dic₄₇⋊C₄	—	4	2	no	—
120	Dic₄₇⋊C₄	—	4	2	no	—
121	Dic₄₇⋊C₄	—	4	2	no	—
122	Dic₄₇⋊C₄	—	4	2	no	—
123	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₁₈₈	—	4	4	no	—
128	C₈×D₄₇	—	2	8	no	—
129	C₈×D₄₇	SD₁₆	1	8	yes	—
130	C₂²×D₄₇	#127	1	1	yes	characteristic
131	Q₈×C₄₇	—	4	2	no	—
132	Q₈×C₄₇	—	4	2	no	—
133	Q₈×C₄₇	—	4	4	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	4	1	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	—	4	1	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	#127	1	1	yes	characteristic
143	C₂×Dic₄₇	#127	1	2	yes	—
144	C₄×D₄₇	—	8	2	no	—
145	C₄×D₄₇	—	8	2	no	—
146	C₄×D₄₇	—	8	4	no	—
147	C₄×D₄₇	C₂×SD₁₆	1	4	yes	—
148	C₄×D₄₇	#137	1	8	yes	—
149	Dic₉₄	—	4	2	no	—
150	Dic₉₄	—	4	2	no	—
151	Dic₉₄	—	4	4	no	—
152	Dic₉₄	—	8	2	no	—
153	Dic₉₄	—	8	2	no	—
154	Dic₉₄	—	8	4	no	—
155	C₈×C₄₇	—	2	4	no	—
156	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
157	C₄₇⋊C₈	—	2	4	no	—
158	C₄₇⋊C₈	C₂×SD₁₆	1	4	yes	—
159	C₂×#1716	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
161	C₂×D₄₇	#492	1	2	yes	—
162	C₂×D₄₇	#497	1	4	yes	—
163	C₄×C₄₇	—	8	1	no	—
164	C₄×C₄₇	—	8	1	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
167	C₄×C₄₇	C₂×#137	1	4	yes	—
168	Dic₄₇	—	8	1	no	—
169	Dic₄₇	—	8	1	no	—
170	Dic₄₇	—	8	2	no	—
171	Dic₄₇	#492	1	2	yes	—
172	Dic₄₇	#497	1	4	yes	—
173	C₂×#498	—	47	1	no	—
174	C₂×#498	—	47	2	no	—
175	C₂×#494	—	47	1	no	—
176	C₂×#487	—	47	2	no	—
177	#1716	—	47	2	no	—
178	#1716	—	47	1	no	—
179	#1716	—	47	1	no	—
180	#1716	—	47	2	no	—
181	#1716	—	47	1	no	—
182	#1716	D₄₇	1	1	yes	characteristic, direct product factor
183	C₂×#322	—	47	1	no	—
184	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#497	1	2	yes	—
186	D₄₇	#1716	1	4	yes	direct product factor
187	C₂²×Q₁₆	—	94	2	no	—
188	C₂×C₄⋊Q₈	—	47	1	no	—
189	C₂×C₄⋊Q₈	—	47	2	no	—
190	C₂×C₄⋊Q₈	—	47	1	no	—
191	#498	—	47	2	no	—
192	#498	—	47	2	no	—
193	#498	—	47	2	no	—
194	#498	—	47	2	no	—
195	#498	—	47	1	no	—
196	#498	—	94	4	no	—
197	#498	—	94	2	no	—
198	#498	C₂×D₄₇	1	2	yes	—
199	#498	C₂×D₄₇	1	1	yes	characteristic
200	#494	—	47	1	no	—
201	#494	—	47	1	no	—
202	#494	—	47	1	no	—
203	#494	—	94	2	no	—
204	#494	C₂×D₄₇	1	1	yes	characteristic
205	#487	—	47	2	no	—
206	#487	—	47	2	no	—
207	#487	—	47	2	no	—
208	#487	—	94	4	no	—
209	#487	C₂×D₄₇	1	2	yes	—
210	C₂×#106	—	94	2	no	—
211	C₂×#106	—	94	4	no	—
212	C₂×#103	—	94	4	no	—
213	C₂×C₄×C₈	—	47	2	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	#322	—	47	1	no	—
216	#322	—	47	2	no	—
217	#322	C₂×D₄₇	1	1	yes	characteristic
218	C₄₇	C₂×#1716	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	94	1	no	—
220	C₂²×Q₈	—	94	1	no	—
221	C₂²×Q₈	—	94	2	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	94	2	no	—
224	C₂×Q₁₆	—	94	2	no	—
225	C₂×Q₁₆	—	94	2	no	—
226	C₂×Q₁₆	—	188	8	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	—	47	4	no	—
229	C₄⋊Q₈	—	47	1	no	—
230	C₄⋊Q₈	—	47	2	no	—
231	C₄⋊Q₈	—	47	1	no	—
232	C₄⋊Q₈	—	94	2	no	—
233	C₄⋊Q₈	—	94	2	no	—
234	C₄⋊Q₈	—	94	2	no	—
235	C₄⋊Q₈	—	94	2	no	—
236	C₄⋊Q₈	—	94	2	no	—
237	C₄⋊Q₈	—	94	1	no	—
238	C₄⋊Q₈	—	94	1	no	—
239	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
240	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
241	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×C₄⋊C₄	—	94	2	no	—
243	C₂×C₄⋊C₄	—	94	2	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄⋊C₄	—	94	2	no	—
246	C₂×C₄²	—	47	1	no	—
247	#106	—	2	4	no	—
248	#106	—	2	2	no	—
249	#106	—	94	4	no	—
250	#106	—	94	4	no	—
251	#106	—	94	4	no	—
252	#106	—	94	4	no	—
253	#106	—	94	2	no	—
254	#103	—	2	4	no	—
255	#103	—	94	4	no	—
256	#103	—	94	4	no	—
257	#103	—	94	4	no	—
258	C₄×C₈	—	47	4	no	—
259	C₄×C₈	—	47	2	no	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	—	47	1	no	—
262	C₄×C₈	C₂²×D₄₇	1	2	yes	—
263	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	94	2	no	—
268	C₂×Q₈	—	94	1	no	—
269	C₂×Q₈	—	94	1	no	—
270	C₂×Q₈	—	188	2	no	—
271	C₂×Q₈	—	188	1	no	—
272	C₂×Q₈	—	188	1	no	—
273	C₂×Q₈	—	188	4	no	—
274	C₂×Q₈	—	188	4	no	—
275	C₂×Q₈	—	188	8	no	—
276	C₂²×C₄	—	47	2	no	—
277	C₂²×C₄	—	47	1	no	—
278	C₂²×C₄	—	188	1	no	—
279	C₂²×C₄	—	188	1	no	—
280	C₂²×C₄	—	188	2	no	—
281	Q₁₆	—	4	4	no	—
282	Q₁₆	—	188	4	no	—
283	Q₁₆	—	188	4	no	—
284	Q₁₆	—	188	4	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	47	8	no	—
287	C₂×C₈	—	47	4	no	—
288	C₂×C₈	—	94	8	no	—
289	C₂×C₈	D₄×D₄₇	1	4	yes	—
290	C₂×C₈	D₄×D₄₇	1	2	yes	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	188	2	no	—
300	C₄⋊C₄	—	188	2	no	—
301	C₄⋊C₄	—	188	2	no	—
302	C₄⋊C₄	—	188	2	no	—
303	C₄²	—	47	1	no	—
304	C₄²	—	47	2	no	—
305	C₄²	C₂³×D₄₇	1	1	yes	characteristic
306	C₂³	—	47	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	376	2	no	—
311	Q₈	—	376	2	no	—
312	Q₈	—	376	4	no	—
313	Q₈	—	188	2	no	—
314	Q₈	—	188	2	no	—
315	Q₈	—	188	4	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	47	2	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	4	no	—
322	C₂×C₄	—	47	8	no	—
323	C₂×C₄	—	376	2	no	—
324	C₂×C₄	—	376	2	no	—
325	C₂×C₄	—	376	4	no	—
326	C₂×C₄	—	188	1	no	—
327	C₂×C₄	—	188	1	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
331	C₈	—	2	4	no	—
332	C₈	—	47	4	no	—
333	C₈	—	94	4	no	—
334	C₈	SD₁₆×D₄₇	1	4	yes	—
335	C₂²	—	47	2	no	—
336	C₂²	—	47	4	no	—
337	C₂²	#127×D₄₇	1	1	yes	characteristic
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	47	2	no	—
342	C₄	—	47	4	no	—
343	C₄	—	376	1	no	—
344	C₄	—	376	1	no	—
345	C₄	—	376	2	no	—
346	C₄	C₂×SD₁₆×D₄₇	1	2	yes	—
347	C₄	#137×D₄₇	1	4	yes	—
348	C₂	—	47	4	no	—
349	C₂	#492×D₄₇	1	1	yes	characteristic
350	C₂	#497×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1716	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	C₂²×D₄₇	#127
hypercenter	#1716	D₄₇
hypocenter	C₄₇	C₂×#1716
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1716

Complements

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

N	H ≅ G/N	type
5. #20243023	C₂	semidirect
6. #20243022	C₂	semidirect
7. #20243021	C₂	semidirect
8. #20243020	C₂	semidirect
9. #20243019	C₂	semidirect
11. C₄₇×#1716	C₂	semidirect
182. #1716	D₄₇	direct (N × H)
186. D₄₇	#1716	direct (N × H)
218. C₄₇	C₂×#1716	semidirect

Maximal subgroups

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

subgroup	index
#498×D₄₇	2
#494×D₄₇	2
#487×D₄₇	2
#20243023	2
#20243022	2
#20243021	2
#20243020	2
#20243019	2
#322×D₄₇	2
C₄₇×#1716	2
C₂×#1716	47

Subgroups — D47×#1716 #26204001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1716 #26204001