Subgroups — C₄₇×#10893 #26050591

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2873	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1740	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1733	C₂	1	2	yes	maximal
5	C₂×C₄₇×#339	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
7	C₂×#117×C₄₇	C₂²	1	2	yes	—
8	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#405	—	2	4	no	—
12	C₄×M₁₆×C₄₇	—	2	2	no	—
13	C₄₇×#384	C₄	1	2	yes	complemented
14	C₄₇×#382	C₄	1	2	yes	complemented
15	C₄₇×#380	—	2	2	no	—
16	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
17	C₄₇×#342	—	2	2	no	—
18	C₄₇×#339	—	2	4	no	—
19	C₄₇×#339	C₄	1	8	yes	complemented
20	C₂³×C₄×C₄₇	—	2	1	no	—
21	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂×M₁₆×C₄₇	—	2	2	no	—
24	C₂×M₁₆×C₄₇	—	2	2	no	—
25	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
27	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	#117×C₄₇	—	2	2	no	—
30	#117×C₄₇	—	2	2	no	—
31	#117×C₄₇	—	2	2	no	—
32	#117×C₄₇	—	2	2	no	—
33	#117×C₄₇	—	2	2	no	—
34	#117×C₄₇	—	2	2	no	—
35	#117×C₄₇	—	2	2	no	—
36	#117×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	—	2	1	no	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	#97×C₄₇	—	2	4	no	—
53	C₄×C₈×C₄₇	—	2	4	no	—
54	#95×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	C₄⋊C₄	1	8	yes	complemented
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	4	8	no	—
82	C₄⋊C₄×C₄₇	—	4	2	no	—
83	C₂²⋊C₄×C₄₇	—	2	4	no	—
84	C₂²⋊C₄×C₄₇	—	2	4	no	—
85	C₂²⋊C₄×C₄₇	—	4	2	no	—
86	C₂²⋊C₄×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	—	2	1	no	—
88	C₄²×C₄₇	—	2	1	no	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	4	1	no	—
94	C₄²×C₄₇	—	4	1	no	—
95	C₄²×C₄₇	—	4	2	no	—
96	C₄²×C₄₇	C₄²	1	2	yes	—
97	C₄²×C₄₇	C₄²	1	2	yes	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₄×D₄	1	2	yes	—
101	C₂³×C₄₇	#121	1	2	yes	—
102	C₂³×C₄₇	#124	1	1	yes	characteristic
103	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	8	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	8	4	no	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	#117	1	2	yes	—
118	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	#123	1	2	yes	—
128	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#125	1	2	yes	—
130	C₂×C₄×C₄₇	#126	1	2	yes	—
131	C₈×C₄₇	—	2	8	no	—
132	#10893	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	4	2	no	—
136	C₂²×C₄₇	—	4	1	no	—
137	C₂²×C₄₇	—	4	1	no	—
138	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	#386	1	1	yes	characteristic
140	C₂²×C₄₇	#388	1	2	yes	—
141	C₂²×C₄₇	#389	1	2	yes	—
142	C₂²×C₄₇	#391	1	1	yes	characteristic
143	C₄×C₄₇	—	2	4	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	—	4	4	no	—
146	C₄×C₄₇	—	8	4	no	—
147	C₄×C₄₇	—	8	2	no	—
148	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
149	C₄×C₄₇	#385	1	2	yes	—
150	C₄×C₄₇	#387	1	2	yes	—
151	C₄×C₄₇	#387	1	2	yes	—
152	#2873	C₂×C₄₇	1	1	yes	characteristic
153	#2285	C₂×C₄₇	1	1	yes	characteristic
154	#1740	C₂×C₄₇	1	1	yes	characteristic
155	#1733	C₂×C₄₇	1	2	yes	—
156	C₂×#339	C₂×C₄₇	1	2	yes	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	—	2	2	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	#1436	1	1	yes	characteristic
162	C₂×C₄₇	#1884	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
164	C₂×#117	C₂²×C₄₇	1	2	yes	—
165	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
166	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
167	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
168	#405	—	2	4	no	—
169	C₄×M₁₆	—	2	2	no	—
170	#384	C₄×C₄₇	1	2	yes	complemented
171	#382	C₄×C₄₇	1	2	yes	complemented
172	#380	—	2	2	no	—
173	C₄×C₄⋊C₄	—	2	2	no	—
174	#342	—	2	2	no	—
175	#339	—	2	4	no	—
176	#339	C₄×C₄₇	1	8	yes	complemented
177	C₄₇	#10893	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³×C₄	—	2	1	no	—
179	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂×C₄×C₄₇	1	4	yes	—
184	C₂×M₁₆	D₄×C₄₇	1	4	yes	—
185	C₂×M₁₆	Q₈×C₄₇	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	C₂×C₄×C₄₇	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
207	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₄₇	1	2	yes	—
210	#97	—	2	4	no	—
211	C₄×C₈	—	2	4	no	—
212	#95	—	2	4	no	—
213	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
226	C₂²×C₄	#41×C₄₇	1	2	yes	—
227	C₂²×C₄	#41×C₄₇	1	2	yes	—
228	C₂²×C₄	#41×C₄₇	1	2	yes	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	C₄⋊C₄×C₄₇	1	8	yes	complemented
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	8	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	2	no	—
254	C₄²	C₄²×C₄₇	1	2	yes	—
255	C₄²	C₄²×C₄₇	1	2	yes	—
256	C₂³	—	4	2	no	—
257	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
258	C₂³	C₄×D₄×C₄₇	1	2	yes	—
259	C₂³	#121×C₄₇	1	2	yes	—
260	C₂³	#124×C₄₇	1	1	yes	characteristic
261	C₂³	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	C₂×C₄²×C₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
275	C₂×C₄	#117×C₄₇	1	2	yes	—
276	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
277	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
278	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
280	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
281	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
282	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
284	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
285	C₂×C₄	#123×C₄₇	1	2	yes	—
286	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
287	C₂×C₄	#125×C₄₇	1	2	yes	—
288	C₂×C₄	#126×C₄₇	1	2	yes	—
289	C₈	—	2	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	C₄×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
296	C₂²	C₄₇×#386	1	1	yes	characteristic
297	C₂²	C₄₇×#388	1	2	yes	—
298	C₂²	C₄₇×#389	1	2	yes	—
299	C₂²	C₄₇×#391	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	4	no	—
304	C₄	—	8	2	no	—
305	C₄	C₄×C₄⋊C₄×C₄₇	1	2	yes	—
306	C₄	C₄₇×#385	1	2	yes	—
307	C₄	C₄₇×#387	1	2	yes	—
308	C₄	C₄₇×#387	1	2	yes	—
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	—	2	1	no	—
313	C₂	C₄₇×#1436	1	1	yes	characteristic
314	C₂	C₄₇×#1884	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₇	#120
commutator	C₂×C₄	C₂×C₄²×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄₇×#384	C₄	semidirect
14. C₄₇×#382	C₄	semidirect
19. C₄₇×#339	C₄	semidirect
74. M₁₆×C₄₇	C₄⋊C₄	semidirect
132. #10893	C₄₇	direct (N × H)
170. #384	C₄×C₄₇	semidirect
171. #382	C₄×C₄₇	semidirect
176. #339	C₄×C₄₇	semidirect
177. C₄₇	#10893	direct (N × H)
232. M₁₆	C₄⋊C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#2873	2
C₄₇×#2285	2
C₄₇×#1740	2
C₄₇×#1733	2
C₂×C₄₇×#339	2
#10893	47

Subgroups — C47×#10893 #26050591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10893 #26050591