Subgroups — C₄₇×#19731 #26059429

Order 12032.

730 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	C₄₇×#3230	C₂	1	2	yes	maximal, complemented
2	C₄₇×#3228	C₂	1	2	yes	maximal
3	C₄₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3152	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1722	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1719	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#1708	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₄₇	—	2	2	no	—
13	C₄₇×#533	C₂²	1	2	yes	—
14	C₄₇×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#500	C₂²	1	2	yes	—
17	C₄₇×#500	C₂²	1	2	yes	—
18	C₄₇×#500	C₂²	1	2	yes	—
19	C₄₇×#498	C₂²	1	1	yes	characteristic
20	C₄₇×#498	C₂²	1	2	yes	—
21	C₄₇×#497	—	2	2	no	—
22	C₄₇×#494	C₂²	1	1	yes	characteristic
23	C₄₇×#494	C₂²	1	1	yes	characteristic
24	C₄₇×#494	C₂²	1	1	yes	characteristic
25	C₄₇×#492	C₂²	1	1	yes	characteristic
26	C₄₇×#492	C₂²	1	1	yes	characteristic
27	C₄₇×#487	C₂²	1	2	yes	—
28	C₄₇×#487	C₂²	1	2	yes	—
29	C₄₇×#487	C₂²	1	2	yes	—
30	C₄₇×#486	C₂²	1	2	yes	—
31	C₄₇×#486	C₂²	1	2	yes	—
32	C₄₇×#484	—	2	4	no	—
33	C₄₇×#482	—	2	4	no	—
34	C₄₇×#478	—	2	4	no	—
35	C₄₇×#477	—	2	4	no	—
36	C₄₇×#470	—	2	2	no	—
37	C₄₇×#441	—	2	2	no	—
38	C₄₇×#431	C₂²	1	2	yes	—
39	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#322	C₂²	1	1	yes	characteristic
41	C₄₇×#322	C₂²	1	2	yes	—
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#41×C₄₇	—	2	1	no	—
44	C₂²×Q₈×C₄₇	—	2	1	no	—
45	#137×C₄₇	—	4	8	no	—
46	C₂×Q₁₆×C₄₇	—	2	2	no	—
47	C₂×Q₁₆×C₄₇	—	2	2	no	—
48	C₂×Q₁₆×C₄₇	—	2	2	no	—
49	C₂×SD₁₆×C₄₇	—	2	2	no	—
50	C₂×SD₁₆×C₄₇	—	2	2	no	—
51	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
52	C₄⋊Q₈×C₄₇	—	2	2	no	—
53	C₄⋊Q₈×C₄₇	—	2	1	no	—
54	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
59	#127×C₄₇	C₂³	1	1	yes	characteristic
60	#125×C₄₇	—	2	2	no	—
61	#124×C₄₇	—	2	1	no	—
62	C₂²⋊Q₈×C₄₇	—	2	2	no	—
63	C₂²⋊Q₈×C₄₇	—	2	2	no	—
64	#121×C₄₇	—	2	2	no	—
65	C₄×Q₈×C₄₇	—	2	2	no	—
66	C₄×Q₈×C₄₇	—	2	2	no	—
67	C₄×D₄×C₄₇	—	2	2	no	—
68	#117×C₄₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	#107×C₄₇	—	2	4	no	—
72	#107×C₄₇	—	2	4	no	—
73	#107×C₄₇	—	2	4	no	—
74	#106×C₄₇	—	2	2	no	—
75	#106×C₄₇	—	2	4	no	—
76	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
77	#103×C₄₇	—	2	4	no	—
78	#103×C₄₇	—	2	4	no	—
79	#103×C₄₇	—	2	4	no	—
80	#102×C₄₇	—	2	4	no	—
81	#102×C₄₇	—	2	4	no	—
82	C₂²⋊C₈×C₄₇	D₄	1	4	yes	—
83	#97×C₄₇	D₄	1	2	yes	—
84	C₄×C₈×C₄₇	C₂³	1	2	yes	—
85	C₄×C₈×C₄₇	C₂³	1	2	yes	—
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	#41×C₄₇	—	4	4	no	—
89	C₂×Q₈×C₄₇	—	2	2	no	—
90	C₂×Q₈×C₄₇	—	2	2	no	—
91	C₂×Q₈×C₄₇	—	2	1	no	—
92	C₂×Q₈×C₄₇	—	2	1	no	—
93	C₂×Q₈×C₄₇	—	2	1	no	—
94	C₂×Q₈×C₄₇	—	4	4	no	—
95	C₂×Q₈×C₄₇	—	4	1	no	—
96	C₂×D₄×C₄₇	—	2	2	no	—
97	C₂×D₄×C₄₇	—	2	1	no	—
98	C₂×D₄×C₄₇	—	4	1	no	—
99	C₂²×C₄×C₄₇	—	4	1	no	—
100	C₂²×C₄×C₄₇	—	4	1	no	—
101	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₄₇	—	4	4	no	—
104	Q₁₆×C₄₇	—	4	4	no	—
105	Q₁₆×C₄₇	—	4	4	no	—
106	SD₁₆×C₄₇	—	4	4	no	—
107	SD₁₆×C₄₇	—	4	4	no	—
108	M₁₆×C₄₇	—	2	4	no	—
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
111	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	2	2	no	—
117	C₄⋊C₄×C₄₇	—	2	2	no	—
118	C₄⋊C₄×C₄₇	—	2	2	no	—
119	C₄⋊C₄×C₄₇	—	2	2	no	—
120	C₄⋊C₄×C₄₇	—	4	2	no	—
121	C₄⋊C₄×C₄₇	—	4	2	no	—
122	C₄⋊C₄×C₄₇	—	4	2	no	—
123	C₂²⋊C₄×C₄₇	—	4	2	no	—
124	C₂²⋊C₄×C₄₇	—	4	2	no	—
125	C₄²×C₄₇	—	4	2	no	—
126	C₄²×C₄₇	C₂×D₄	1	2	yes	—
127	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
129	C₂³×C₄₇	—	4	1	no	—
130	C₂³×C₄₇	#127	1	1	yes	characteristic
131	Q₈×C₄₇	—	4	2	no	—
132	Q₈×C₄₇	—	4	2	no	—
133	Q₈×C₄₇	—	4	2	no	—
134	Q₈×C₄₇	—	4	4	no	—
135	Q₈×C₄₇	—	4	4	no	—
136	Q₈×C₄₇	—	8	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	4	4	no	—
139	D₄×C₄₇	—	8	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	4	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	1	no	—
145	C₂×C₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	8	2	no	—
148	C₂×C₄×C₄₇	—	8	2	no	—
149	C₂×C₄×C₄₇	#127	1	2	yes	—
150	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	2	4	no	—
155	C₈×C₄₇	—	2	4	no	—
156	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
157	#19731	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	8	1	no	—
161	C₂²×C₄₇	—	8	2	no	—
162	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	8	1	no	—
166	C₄×C₄₇	—	8	1	no	—
167	C₄×C₄₇	—	8	1	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₄×C₄₇	—	8	2	no	—
170	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
171	C₄×C₄₇	#578	1	4	yes	—
172	#3230	C₂×C₄₇	1	2	yes	complemented
173	#3228	C₂×C₄₇	1	2	yes	—
174	#3185	C₂×C₄₇	1	1	yes	characteristic, complemented
175	#3152	C₂×C₄₇	1	1	yes	characteristic
176	#1723	C₂×C₄₇	1	2	yes	complemented
177	#1722	C₂×C₄₇	1	1	yes	characteristic, complemented
178	#1719	C₂×C₄₇	1	2	yes	complemented
179	#1716	C₂×C₄₇	1	1	yes	characteristic, complemented
180	#1713	C₂×C₄₇	1	1	yes	characteristic, complemented
181	#1708	C₂×C₄₇	1	1	yes	characteristic, complemented
182	#1453	C₂×C₄₇	1	1	yes	characteristic, complemented
183	C₂×C₄₇	—	4	1	no	—
184	C₂×C₄₇	—	16	1	no	—
185	C₂×C₄₇	#3151	1	1	yes	characteristic
186	C₂×C₄₇	#3156	1	2	yes	—
187	C₂×#137	—	2	2	no	—
188	#533	C₂²×C₄₇	1	2	yes	—
189	#532	C₂²×C₄₇	1	1	yes	characteristic
190	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
191	#500	C₂²×C₄₇	1	2	yes	—
192	#500	C₂²×C₄₇	1	2	yes	—
193	#500	C₂²×C₄₇	1	2	yes	—
194	#498	C₂²×C₄₇	1	1	yes	characteristic
195	#498	C₂²×C₄₇	1	2	yes	—
196	#497	—	2	2	no	—
197	#494	C₂²×C₄₇	1	1	yes	characteristic
198	#494	C₂²×C₄₇	1	1	yes	characteristic
199	#494	C₂²×C₄₇	1	1	yes	characteristic
200	#492	C₂²×C₄₇	1	1	yes	characteristic
201	#492	C₂²×C₄₇	1	1	yes	characteristic
202	#487	C₂²×C₄₇	1	2	yes	—
203	#487	C₂²×C₄₇	1	2	yes	—
204	#487	C₂²×C₄₇	1	2	yes	—
205	#486	C₂²×C₄₇	1	2	yes	—
206	#486	C₂²×C₄₇	1	2	yes	—
207	#484	—	2	4	no	—
208	#482	—	2	4	no	—
209	#478	—	2	4	no	—
210	#477	—	2	4	no	—
211	#470	—	2	2	no	—
212	#441	—	2	2	no	—
213	#431	C₂²×C₄₇	1	2	yes	—
214	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
215	#322	C₂²×C₄₇	1	1	yes	characteristic
216	#322	C₂²×C₄₇	1	2	yes	—
217	C₈²	C₂²×C₄₇	1	1	yes	characteristic
218	C₄₇	#19731	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×#41	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	#137	—	4	8	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
231	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
232	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
233	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
234	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
235	#127	C₂³×C₄₇	1	1	yes	characteristic
236	#125	—	2	2	no	—
237	#124	—	2	1	no	—
238	C₂²⋊Q₈	—	2	2	no	—
239	C₂²⋊Q₈	—	2	2	no	—
240	#121	—	2	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	#117	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
247	#107	—	2	4	no	—
248	#107	—	2	4	no	—
249	#107	—	2	4	no	—
250	#106	—	2	2	no	—
251	#106	—	2	4	no	—
252	C₄⋊C₈	D₄×C₄₇	1	4	yes	—
253	#103	—	2	4	no	—
254	#103	—	2	4	no	—
255	#103	—	2	4	no	—
256	#102	—	2	4	no	—
257	#102	—	2	4	no	—
258	C₂²⋊C₈	D₄×C₄₇	1	4	yes	—
259	#97	D₄×C₄₇	1	2	yes	—
260	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₄₇	1	2	yes	—
262	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₄₇	1	2	yes	—
264	#41	—	4	4	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
279	Q₁₆	—	4	4	no	—
280	Q₁₆	—	4	4	no	—
281	Q₁₆	—	4	4	no	—
282	SD₁₆	—	4	4	no	—
283	SD₁₆	—	4	4	no	—
284	M₁₆	—	2	4	no	—
285	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
287	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
288	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
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	—	2	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₄₇	1	2	yes	—
304	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
305	C₂³	—	4	1	no	—
306	C₂³	#127×C₄₇	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	4	no	—
315	D₄	—	8	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	#127×C₄₇	1	2	yes	—
326	C₂×C₄	#127×C₄₇	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
329	C₈	—	2	2	no	—
330	C₈	—	2	4	no	—
331	C₈	—	2	4	no	—
332	C₈	AGL(1,ℤ/8)×C₄₇	1	4	yes	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₂×#127×C₄₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	8	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	2	yes	—
346	C₄	C₄₇×#578	1	4	yes	—
347	C₂	—	4	1	no	—
348	C₂	—	16	1	no	—
349	C₂	C₄₇×#3151	1	1	yes	characteristic
350	C₂	C₄₇×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₄₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3230	C₂	semidirect
3. C₄₇×#3185	C₂	semidirect
5. C₄₇×#1723	C₂	semidirect
6. C₄₇×#1722	C₂	semidirect
7. C₄₇×#1719	C₂	semidirect
8. C₄₇×#1716	C₂	semidirect
9. C₄₇×#1713	C₂	semidirect
10. C₄₇×#1708	C₂	semidirect
11. C₄₇×#1453	C₂	semidirect
157. #19731	C₄₇	direct (N × H)
172. #3230	C₂×C₄₇	semidirect
174. #3185	C₂×C₄₇	semidirect
176. #1723	C₂×C₄₇	semidirect
177. #1722	C₂×C₄₇	semidirect
178. #1719	C₂×C₄₇	semidirect
179. #1716	C₂×C₄₇	semidirect
180. #1713	C₂×C₄₇	semidirect
181. #1708	C₂×C₄₇	semidirect
182. #1453	C₂×C₄₇	semidirect
218. C₄₇	#19731	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#3230	2
C₄₇×#3228	2
C₄₇×#3185	2
C₄₇×#3152	2
C₄₇×#1723	2
C₄₇×#1722	2
C₄₇×#1719	2
C₄₇×#1716	2
C₄₇×#1713	2
C₄₇×#1708	2
C₄₇×#1453	2
#19731	47

Subgroups — C47×#19731 #26059429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#19731 #26059429