Subgroups — C₄₇×#19747 #26059445

Order 12032.

908 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3151	C₂	1	2	yes	maximal, complemented
4	C₄₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1721	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1717	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1714	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1706	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
11	C₄₇×#533	C₂²	1	1	yes	characteristic
12	C₄₇×#532	C₂²	1	2	yes	—
13	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#500	C₂²	1	2	yes	complemented
15	C₄₇×#498	C₂²	1	1	yes	characteristic, complemented
16	C₄₇×#496	—	2	4	no	—
17	C₄₇×#493	C₂²	1	2	yes	—
18	C₄₇×#493	C₂²	1	2	yes	—
19	C₄₇×#493	C₂²	1	1	yes	characteristic, complemented
20	C₄₇×#492	C₂²	1	2	yes	—
21	C₄₇×#492	C₂²	1	2	yes	—
22	C₄₇×#492	C₂²	1	2	yes	complemented
23	C₄₇×#490	—	2	2	no	—
24	C₄₇×#487	C₂²	1	2	yes	complemented
25	C₄₇×#486	C₂²	1	2	yes	—
26	C₄₇×#486	C₂²	1	2	yes	—
27	C₄₇×#486	C₂²	1	2	yes	complemented
28	C₄₇×#486	C₂²	1	2	yes	complemented
29	C₄₇×#469	—	2	2	no	—
30	C₄₇×#468	—	2	4	no	—
31	C₄₇×#442	—	2	2	no	—
32	C₄₇×#440	—	2	4	no	—
33	C₄₇×#418	—	2	2	no	—
34	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
36	C₄₇×#322	C₂²	1	2	yes	complemented
37	C₄₇×#322	C₂²	1	1	yes	characteristic, complemented
38	C₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₄₇	—	2	2	no	—
40	C₂²×D₄×C₄₇	—	2	2	no	—
41	C₂²×D₄×C₄₇	—	2	1	no	—
42	AGL(1,ℤ/8)×C₄₇	—	4	16	no	—
43	C₂×SD₁₆×C₄₇	—	2	4	no	—
44	C₂×SD₁₆×C₄₇	—	2	4	no	—
45	C₂×SD₁₆×C₄₇	—	2	4	no	—
46	C₂×D₈×C₄₇	—	2	2	no	—
47	C₂×D₈×C₄₇	—	2	4	no	—
48	C₂×D₈×C₄₇	—	2	4	no	—
49	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
50	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
51	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
53	#127×C₄₇	—	2	1	no	—
54	#127×C₄₇	—	2	2	no	—
55	#127×C₄₇	C₂³	1	2	yes	—
56	#127×C₄₇	C₂³	1	2	yes	—
57	#125×C₄₇	—	2	1	no	—
58	#124×C₄₇	—	2	2	no	—
59	C₂²⋊Q₈×C₄₇	—	2	2	no	—
60	#121×C₄₇	—	2	2	no	—
61	#121×C₄₇	—	2	2	no	—
62	C₄×Q₈×C₄₇	—	2	2	no	—
63	C₄×D₄×C₄₇	—	2	2	no	—
64	C₄×D₄×C₄₇	—	2	2	no	—
65	#117×C₄₇	—	2	1	no	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	#107×C₄₇	—	2	4	no	—
68	#106×C₄₇	—	2	2	no	—
69	#103×C₄₇	—	2	4	no	—
70	#102×C₄₇	—	2	4	no	—
71	#102×C₄₇	—	2	4	no	—
72	#102×C₄₇	—	2	4	no	—
73	#102×C₄₇	—	2	4	no	—
74	#97×C₄₇	D₄	1	2	yes	—
75	#97×C₄₇	D₄	1	4	yes	—
76	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	2	yes	—
79	C₄×C₈×C₄₇	C₂³	1	2	yes	—
80	C₂⁴×C₄₇	—	4	1	no	—
81	#41×C₄₇	—	4	8	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	2	2	no	—
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	4	8	no	—
91	C₂×D₄×C₄₇	—	4	4	no	—
92	C₂×D₄×C₄₇	—	4	2	no	—
93	C₂×D₄×C₄₇	—	4	1	no	—
94	C₂×D₄×C₄₇	—	4	2	no	—
95	C₂²×C₄×C₄₇	—	4	2	no	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
98	SD₁₆×C₄₇	—	4	8	no	—
99	SD₁₆×C₄₇	—	4	8	no	—
100	SD₁₆×C₄₇	—	4	8	no	—
101	D₈×C₄₇	—	4	4	no	—
102	D₈×C₄₇	—	4	8	no	—
103	D₈×C₄₇	—	4	8	no	—
104	M₁₆×C₄₇	—	2	8	no	—
105	M₁₆×C₄₇	—	4	2	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄⋊C₄×C₄₇	—	2	2	no	—
113	C₄⋊C₄×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	—	4	2	no	—
115	C₂²⋊C₄×C₄₇	—	4	1	no	—
116	C₂²⋊C₄×C₄₇	—	4	2	no	—
117	C₂²⋊C₄×C₄₇	—	4	2	no	—
118	C₄²×C₄₇	—	4	1	no	—
119	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	C₂×D₄	1	2	yes	—
121	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
122	C₂³×C₄₇	—	4	2	no	—
123	C₂³×C₄₇	—	4	2	no	—
124	C₂³×C₄₇	—	8	4	no	—
125	C₂³×C₄₇	—	8	2	no	—
126	C₂³×C₄₇	#127	1	1	yes	characteristic
127	Q₈×C₄₇	—	4	4	no	—
128	Q₈×C₄₇	—	4	4	no	—
129	D₄×C₄₇	—	4	4	no	—
130	D₄×C₄₇	—	4	4	no	—
131	D₄×C₄₇	—	4	4	no	—
132	D₄×C₄₇	—	4	4	no	—
133	D₄×C₄₇	—	4	4	no	—
134	D₄×C₄₇	—	4	4	no	—
135	D₄×C₄₇	—	8	4	no	—
136	D₄×C₄₇	—	8	2	no	—
137	D₄×C₄₇	—	8	4	no	—
138	C₂×C₄×C₄₇	—	2	4	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	8	4	no	—
143	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#127	1	2	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₄₇	—	2	4	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	—	2	2	no	—
150	C₈×C₄₇	AGL(1,ℤ/8)	1	8	yes	complemented
151	#19747	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	8	2	no	—
155	C₂²×C₄₇	—	8	2	no	—
156	C₂²×C₄₇	—	8	2	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	16	4	no	—
161	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
167	C₄×C₄₇	#575	1	2	yes	—
168	#3184	C₂×C₄₇	1	1	yes	characteristic, complemented
169	#3182	C₂×C₄₇	1	2	yes	complemented
170	#3151	C₂×C₄₇	1	2	yes	complemented
171	#3138	C₂×C₄₇	1	1	yes	characteristic, complemented
172	#1721	C₂×C₄₇	1	2	yes	complemented
173	#1717	C₂×C₄₇	1	2	yes	complemented
174	#1714	C₂×C₄₇	1	2	yes	complemented
175	#1706	C₂×C₄₇	1	2	yes	complemented
176	#1455	C₂×C₄₇	1	1	yes	characteristic, complemented
177	C₂×C₄₇	—	4	1	no	—
178	C₂×C₄₇	—	16	2	no	—
179	C₂×C₄₇	—	16	2	no	—
180	C₂×C₄₇	#3151	1	2	yes	—
181	C₂×C₄₇	#3153	1	1	yes	characteristic
182	C₂×AGL(1,ℤ/8)	—	2	4	no	—
183	#533	C₂²×C₄₇	1	1	yes	characteristic
184	#532	C₂²×C₄₇	1	2	yes	—
185	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
186	#500	C₂²×C₄₇	1	2	yes	complemented
187	#498	C₂²×C₄₇	1	1	yes	characteristic, complemented
188	#496	—	2	4	no	—
189	#493	C₂²×C₄₇	1	2	yes	—
190	#493	C₂²×C₄₇	1	2	yes	—
191	#493	C₂²×C₄₇	1	1	yes	characteristic, complemented
192	#492	C₂²×C₄₇	1	2	yes	—
193	#492	C₂²×C₄₇	1	2	yes	—
194	#492	C₂²×C₄₇	1	2	yes	complemented
195	#490	—	2	2	no	—
196	#487	C₂²×C₄₇	1	2	yes	complemented
197	#486	C₂²×C₄₇	1	2	yes	—
198	#486	C₂²×C₄₇	1	2	yes	—
199	#486	C₂²×C₄₇	1	2	yes	complemented
200	#486	C₂²×C₄₇	1	2	yes	complemented
201	#469	—	2	2	no	—
202	#468	—	2	4	no	—
203	#442	—	2	2	no	—
204	#440	—	2	4	no	—
205	#418	—	2	2	no	—
206	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
207	C₄×M₁₆	C₂²×C₄₇	1	2	yes	—
208	#322	C₂²×C₄₇	1	2	yes	complemented
209	#322	C₂²×C₄₇	1	1	yes	characteristic, complemented
210	C₈²	C₂²×C₄₇	1	1	yes	characteristic, complemented
211	C₄₇	#19747	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×#41	—	2	2	no	—
213	C₂²×D₄	—	2	2	no	—
214	C₂²×D₄	—	2	1	no	—
215	AGL(1,ℤ/8)	—	4	16	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₂×SD₁₆	—	2	4	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×D₈	—	2	4	no	—
220	C₂×D₈	—	2	4	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×M₁₆	D₄×C₄₇	1	4	yes	—
223	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
224	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
225	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
226	#127	—	2	1	no	—
227	#127	—	2	2	no	—
228	#127	C₂³×C₄₇	1	2	yes	—
229	#127	C₂³×C₄₇	1	2	yes	—
230	#125	—	2	1	no	—
231	#124	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	#121	—	2	2	no	—
234	#121	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	#117	—	2	1	no	—
239	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
240	#107	—	2	4	no	—
241	#106	—	2	2	no	—
242	#103	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	2	4	no	—
247	#97	D₄×C₄₇	1	4	yes	—
248	#97	D₄×C₄₇	1	2	yes	—
249	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₄₇	1	2	yes	—
251	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₄₇	1	2	yes	—
253	C₂⁴	—	4	1	no	—
254	#41	—	4	8	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	8	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
271	SD₁₆	—	4	8	no	—
272	SD₁₆	—	4	8	no	—
273	SD₁₆	—	4	8	no	—
274	D₈	—	4	4	no	—
275	D₈	—	4	8	no	—
276	D₈	—	4	8	no	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	4	2	no	—
279	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
282	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₄²	—	4	1	no	—
292	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₄₇	1	2	yes	—
294	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	#127×C₄₇	1	1	yes	characteristic
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	#127×C₄₇	1	1	yes	characteristic
317	C₂×C₄	#127×C₄₇	1	2	yes	—
318	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
319	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	AGL(1,ℤ/8)×C₄₇	1	8	yes	complemented
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	4	no	—
333	C₂²	C₂×#127×C₄₇	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	4	yes	—
339	C₄	C₄₇×#575	1	2	yes	—
340	C₂	—	4	1	no	—
341	C₂	—	16	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₄₇×#3151	1	2	yes	—
344	C₂	C₄₇×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
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×#3184	C₂	semidirect
2. C₄₇×#3182	C₂	semidirect
3. C₄₇×#3151	C₂	semidirect
4. C₄₇×#3138	C₂	semidirect
5. C₄₇×#1721	C₂	semidirect
6. C₄₇×#1717	C₂	semidirect
7. C₄₇×#1714	C₂	semidirect
8. C₄₇×#1706	C₂	semidirect
9. C₄₇×#1455	C₂	semidirect
14. C₄₇×#500	C₂²	semidirect
15. C₄₇×#498	C₂²	semidirect
19. C₄₇×#493	C₂²	semidirect
22. C₄₇×#492	C₂²	semidirect
24. C₄₇×#487	C₂²	semidirect
27. C₄₇×#486	C₂²	semidirect
28. C₄₇×#486	C₂²	semidirect
36. C₄₇×#322	C₂²	semidirect
37. C₄₇×#322	C₂²	semidirect
38. C₈²×C₄₇	C₂²	semidirect
150. C₈×C₄₇	AGL(1,ℤ/8)	semidirect
151. #19747	C₄₇	direct (N × H)
168. #3184	C₂×C₄₇	semidirect
169. #3182	C₂×C₄₇	semidirect
170. #3151	C₂×C₄₇	semidirect
171. #3138	C₂×C₄₇	semidirect
172. #1721	C₂×C₄₇	semidirect
173. #1717	C₂×C₄₇	semidirect
174. #1714	C₂×C₄₇	semidirect
175. #1706	C₂×C₄₇	semidirect
176. #1455	C₂×C₄₇	semidirect
186. #500	C₂²×C₄₇	semidirect
187. #498	C₂²×C₄₇	semidirect
191. #493	C₂²×C₄₇	semidirect
194. #492	C₂²×C₄₇	semidirect
196. #487	C₂²×C₄₇	semidirect
199. #486	C₂²×C₄₇	semidirect
200. #486	C₂²×C₄₇	semidirect
208. #322	C₂²×C₄₇	semidirect
209. #322	C₂²×C₄₇	semidirect
210. C₈²	C₂²×C₄₇	semidirect
211. C₄₇	#19747	direct (N × H)
323. C₈	AGL(1,ℤ/8)×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3184	2
C₄₇×#3182	2
C₄₇×#3151	2
C₄₇×#3138	2
C₄₇×#1721	2
C₄₇×#1717	2
C₄₇×#1714	2
C₄₇×#1706	2
C₄₇×#1455	2
#19747	47

Subgroups — C47×#19747 #26059445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#19747 #26059445