Subgroups — C₄₇×#8391 #26048089

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1800	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1548	C₂	1	2	yes	maximal, complemented
4	C₄₇×#1487	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1460	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₄₇	C₄	1	2	yes	—
7	D₄×Q₈×C₄₇	—	2	4	no	—
8	D₄×Q₈×C₄₇	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
12	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₇	—	2	1	no	—
15	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#431	C₂²	1	2	yes	—
17	C₄₇×#423	C₂²	1	2	yes	—
18	C₂×#100×C₄₇	—	2	1	no	—
19	C₄₇×#343	—	2	1	no	—
20	C₄₇×#333	—	2	4	no	—
21	C₄₇×Q₈⋊C₈	—	2	4	no	—
22	C₄₇×#323	—	2	2	no	—
23	C₂²×Q₈×C₄₇	—	2	4	no	—
24	C₂²×Q₈×C₄₇	—	2	4	no	—
25	C₂²×Q₈×C₄₇	—	2	2	no	—
26	C₂²×Q₈×C₄₇	—	4	8	no	—
27	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	—	2	2	no	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₄₇	—	2	2	no	—
33	C₄⋊Q₈×C₄₇	—	2	8	no	—
34	C₄⋊Q₈×C₄₇	—	2	2	no	—
35	C₄⋊Q₈×C₄₇	D₄	1	4	yes	—
36	C₂²⋊Q₈×C₄₇	—	2	16	no	—
37	C₂²⋊Q₈×C₄₇	—	2	4	no	—
38	C₂²⋊Q₈×C₄₇	—	2	4	no	—
39	C₄×Q₈×C₄₇	—	2	2	no	—
40	C₄×Q₈×C₄₇	D₄	1	4	yes	—
41	C₄×D₄×C₄₇	—	2	8	no	—
42	C₄×D₄×C₄₇	—	2	2	no	—
43	C₄×D₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	—	2	2	no	—
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	C₄⋊C₈×C₄₇	—	2	2	no	—
57	#100×C₄₇	—	4	2	no	—
58	C₂²⋊C₈×C₄₇	—	2	2	no	—
59	C₄×C₈×C₄₇	—	2	2	no	—
60	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂×Q₈×C₄₇	—	2	4	no	—
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂×Q₈×C₄₇	—	2	1	no	—
64	C₂×Q₈×C₄₇	—	2	2	no	—
65	C₂×Q₈×C₄₇	—	4	8	no	—
66	C₂×Q₈×C₄₇	—	4	4	no	—
67	C₂×Q₈×C₄₇	—	4	16	no	—
68	C₂×Q₈×C₄₇	—	4	8	no	—
69	C₂×Q₈×C₄₇	—	4	8	no	—
70	C₂×Q₈×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	SD₁₆	1	2	yes	—
75	C₂×D₄×C₄₇	Q₁₆	1	2	yes	—
76	C₂²×C₄×C₄₇	—	2	1	no	—
77	C₂²×C₄×C₄₇	—	2	8	no	—
78	C₂²×C₄×C₄₇	—	2	1	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	4	8	no	—
83	C₂²×C₄×C₄₇	—	4	4	no	—
84	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	M₁₆×C₄₇	—	4	2	no	—
89	M₁₆×C₄₇	—	4	2	no	—
90	C₂×C₈×C₄₇	—	4	2	no	—
91	C₂×C₈×C₄₇	—	4	2	no	—
92	C₄⋊C₄×C₄₇	—	2	8	no	—
93	C₄⋊C₄×C₄₇	—	2	1	no	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	2	4	no	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
99	C₂²⋊C₄×C₄₇	—	2	8	no	—
100	C₂²⋊C₄×C₄₇	—	2	4	no	—
101	C₄²×C₄₇	—	2	4	no	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	#103	1	4	yes	—
109	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	Q₈×C₄₇	—	2	4	no	—
111	Q₈×C₄₇	—	2	2	no	—
112	Q₈×C₄₇	—	4	4	no	—
113	Q₈×C₄₇	—	4	16	no	—
114	Q₈×C₄₇	—	4	4	no	—
115	Q₈×C₄₇	#100	1	4	yes	complemented
116	D₄×C₄₇	—	2	4	no	—
117	D₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	4	1	no	—
125	C₂×C₄×C₄₇	—	4	16	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#120	1	2	yes	—
134	C₂×C₄×C₄₇	#120	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#137	1	2	yes	—
138	C₈×C₄₇	—	8	2	no	—
139	C₈×C₄₇	—	8	2	no	—
140	#8391	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	—	4	4	no	—
146	C₂²×C₄₇	—	4	2	no	—
147	C₂²×C₄₇	—	4	2	no	—
148	C₂²×C₄₇	#379	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
150	C₂²×C₄₇	#419	1	1	yes	characteristic
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	—	4	2	no	—
157	C₄×C₄₇	C₂×#100	1	2	yes	—
158	C₄×C₄₇	#448	1	2	yes	—
159	C₄×C₄₇	#451	1	2	yes	—
160	C₂×D₄×Q₈	C₂×C₄₇	1	1	yes	characteristic
161	#1800	C₂×C₄₇	1	1	yes	characteristic
162	#1548	C₂×C₄₇	1	2	yes	complemented
163	#1487	C₂×C₄₇	1	2	yes	complemented
164	#1460	C₂×C₄₇	1	1	yes	characteristic
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	4	4	no	—
168	C₂×C₄₇	#1783	1	1	yes	characteristic
169	C₂×C₄₇	#1787	1	1	yes	characteristic
170	C₂×C₄₇	#1796	1	1	yes	characteristic
171	C₂³×Q₈	C₄×C₄₇	1	2	yes	—
172	D₄×Q₈	—	2	4	no	—
173	D₄×Q₈	—	2	4	no	—
174	C₂×C₄⋊Q₈	—	2	1	no	—
175	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
176	C₂×C₂²⋊Q₈	—	2	2	no	—
177	C₂×C₂²⋊Q₈	C₄×C₄₇	1	2	yes	—
178	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
179	C₂×C₄×D₄	—	2	1	no	—
180	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
181	#431	C₂²×C₄₇	1	2	yes	—
182	#423	C₂²×C₄₇	1	2	yes	—
183	C₂×#100	—	2	1	no	—
184	#343	—	2	1	no	—
185	#333	—	2	4	no	—
186	Q₈⋊C₈	—	2	4	no	—
187	#323	—	2	2	no	—
188	C₄₇	#8391	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×Q₈	—	2	4	no	—
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×Q₈	—	4	8	no	—
193	C₂²×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
194	C₂²×Q₈	C₂×C₄×C₄₇	1	2	yes	—
195	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
196	C₂³×C₄	—	2	2	no	—
197	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
198	C₂×M₁₆	—	2	2	no	—
199	C₄⋊Q₈	—	2	8	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	D₄×C₄₇	1	4	yes	—
202	C₂²⋊Q₈	—	2	16	no	—
203	C₂²⋊Q₈	—	2	4	no	—
204	C₂²⋊Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	D₄×C₄₇	1	4	yes	—
207	C₄×D₄	—	2	8	no	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	#100	—	4	2	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₄×C₈	—	2	2	no	—
226	C₂⁴	C₂²⋊C₄×C₄₇	1	2	yes	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	16	no	—
234	C₂×Q₈	—	4	8	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	C₂²⋊C₄×C₄₇	1	2	yes	—
237	C₂×Q₈	C₂²⋊C₄×C₄₇	1	4	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	SD₁₆×C₄₇	1	2	yes	—
241	C₂×D₄	Q₁₆×C₄₇	1	2	yes	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	8	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
254	M₁₆	—	4	2	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	C₂×D₄×C₄₇	1	2	yes	—
269	C₄²	C₂×D₄×C₄₇	1	2	yes	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	#103×C₄₇	1	4	yes	—
275	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
276	Q₈	—	2	4	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	16	no	—
280	Q₈	—	4	4	no	—
281	Q₈	#100×C₄₇	1	4	yes	complemented
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
299	C₂×C₄	#120×C₄₇	1	2	yes	—
300	C₂×C₄	#120×C₄₇	1	2	yes	—
301	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₁₆×C₄₇	1	1	yes	characteristic
303	C₂×C₄	#137×C₄₇	1	2	yes	—
304	C₈	—	8	2	no	—
305	C₈	—	8	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₄₇×#379	1	1	yes	characteristic
314	C₂²	C₂×#103×C₄₇	1	1	yes	characteristic
315	C₂²	C₄₇×#419	1	1	yes	characteristic
316	C₄	—	2	4	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	C₂×#100×C₄₇	1	2	yes	—
323	C₄	C₄₇×#448	1	2	yes	—
324	C₄	C₄₇×#451	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	4	4	no	—
328	C₂	C₄₇×#1783	1	1	yes	characteristic
329	C₂	C₄₇×#1787	1	1	yes	characteristic
330	C₂	C₄₇×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#379
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
prime radical	C₂²×D₄×C₄₇	D₄
LCS	G ▷ C₂²×C₄ ▷ 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
3. C₄₇×#1548	C₂	semidirect
4. C₄₇×#1487	C₂	semidirect
115. Q₈×C₄₇	#100	semidirect
140. #8391	C₄₇	direct (N × H)
162. #1548	C₂×C₄₇	semidirect
163. #1487	C₂×C₄₇	semidirect
188. C₄₇	#8391	direct (N × H)
281. Q₈	#100×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₄₇	2
C₄₇×#1800	2
C₄₇×#1548	2
C₄₇×#1487	2
C₄₇×#1460	2
#8391	47

Subgroups — C47×#8391 #26048089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#8391 #26048089