Subgroups — C₄₇×#10964 #26050662

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₄₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#336	C₂	1	2	yes	maximal, complemented
4	C₂³×D₄×C₄₇	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
6	C₂³×C₈×C₄₇	C₂²	1	4	yes	—
7	C₂×C₄×D₄×C₄₇	—	2	8	no	—
8	C₂×C₄×D₄×C₄₇	—	2	4	no	—
9	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
10	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₄₇	—	2	8	no	—
13	C₂×C₂²⋊C₈×C₄₇	—	2	16	no	—
14	C₄₇×#336	—	2	4	no	—
15	C₂⁵×C₄₇	C₂×C₄	1	2	yes	—
16	C₂²×D₄×C₄₇	—	2	4	no	—
17	C₂²×D₄×C₄₇	—	2	4	no	—
18	C₂²×D₄×C₄₇	—	2	2	no	—
19	C₂²×D₄×C₄₇	—	2	2	no	—
20	C₂²×D₄×C₄₇	C₈	1	4	yes	complemented
21	C₂³×C₄×C₄₇	—	2	4	no	—
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	—	2	8	no	—
24	C₂³×C₄×C₄₇	—	2	8	no	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	D₄	1	8	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	—	2	4	no	—
31	C₂²×C₈×C₄₇	—	2	8	no	—
32	C₂²×C₈×C₄₇	—	2	8	no	—
33	C₂²×C₈×C₄₇	—	2	4	no	—
34	C₂²×C₈×C₄₇	D₄	1	8	yes	complemented
35	C₄×D₄×C₄₇	—	2	32	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄²×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	D₄	1	4	yes	—
43	C₂²⋊C₈×C₄₇	—	2	32	no	—
44	C₂⁴×C₄₇	—	2	2	no	—
45	C₂⁴×C₄₇	—	2	2	no	—
46	C₂⁴×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	—	2	2	no	—
48	C₂⁴×C₄₇	—	2	2	no	—
49	C₂⁴×C₄₇	—	2	8	no	—
50	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₂⁴×C₄₇	C₂×C₈	1	4	yes	—
52	C₂⁴×C₄₇	M₁₆	1	4	yes	—
53	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×D₄×C₄₇	—	2	8	no	—
55	C₂×D₄×C₄₇	—	2	8	no	—
56	C₂×D₄×C₄₇	—	2	16	no	—
57	C₂×D₄×C₄₇	—	2	16	no	—
58	C₂×D₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	16	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	16	no	—
69	C₂²×C₄×C₄₇	—	2	8	no	—
70	C₂²×C₄×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	—	4	8	no	—
72	C₂²×C₄×C₄₇	—	4	8	no	—
73	C₂²×C₄×C₄₇	—	4	4	no	—
74	C₂²×C₄×C₄₇	—	4	4	no	—
75	C₂²×C₄×C₄₇	—	4	4	no	—
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	4	yes	—
85	C₂×C₈×C₄₇	—	2	16	no	—
86	C₂×C₈×C₄₇	—	2	8	no	—
87	C₂×C₈×C₄₇	—	4	4	no	—
88	C₂×C₈×C₄₇	—	4	8	no	—
89	C₂×C₈×C₄₇	—	4	4	no	—
90	C₄⋊C₄×C₄₇	—	2	8	no	—
91	C₂²⋊C₄×C₄₇	—	2	16	no	—
92	C₄²×C₄₇	—	2	8	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	8	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	4	no	—
99	C₂³×C₄₇	—	2	4	no	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	2	16	no	—
102	C₂³×C₄₇	—	2	16	no	—
103	C₂³×C₄₇	—	4	4	no	—
104	C₂³×C₄₇	—	4	4	no	—
105	C₂³×C₄₇	—	4	4	no	—
106	C₂³×C₄₇	—	4	4	no	—
107	C₂³×C₄₇	—	4	8	no	—
108	C₂³×C₄₇	—	4	8	no	—
109	C₂³×C₄₇	C₂²⋊C₈	1	8	yes	complemented
110	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₄₇	C₄×D₄	1	4	yes	—
113	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
114	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
115	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
116	C₂³×C₄₇	#131	1	4	yes	—
117	D₄×C₄₇	—	2	32	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	—	4	16	no	—
130	C₂×C₄×C₄₇	—	4	16	no	—
131	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	complemented
132	C₂×C₄×C₄₇	#120	1	4	yes	—
133	C₂×C₄×C₄₇	#120	1	4	yes	—
134	C₂×C₄×C₄₇	#121	1	8	yes	—
135	C₂×C₄×C₄₇	#121	1	4	yes	—
136	C₂×C₄×C₄₇	#123	1	4	yes	—
137	C₂×C₄×C₄₇	#124	1	2	yes	—
138	C₂×C₄×C₄₇	#127	1	2	yes	—
139	C₈×C₄₇	—	4	8	no	—
140	#10964	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₄₇	—	2	16	no	—
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	4	8	no	—
144	C₂²×C₄₇	—	4	8	no	—
145	C₂²×C₄₇	—	4	8	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	—	4	8	no	—
148	C₂²×C₄₇	—	4	8	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	#379	1	1	yes	characteristic
151	C₂²×C₄₇	#386	1	4	yes	—
152	C₂²×C₄₇	#390	1	2	yes	—
153	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
154	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₄₇	#407	1	2	yes	—
156	C₂²×C₄₇	#407	1	1	yes	characteristic
157	C₂²×C₄₇	#408	1	4	yes	—
158	C₂²×C₄₇	#408	1	2	yes	—
159	C₂²×C₄₇	C₈×D₄	1	4	yes	—
160	C₂²×C₄₇	#435	1	8	yes	—
161	C₂²×C₄₇	#436	1	4	yes	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	4	4	no	—
164	C₄×C₄₇	—	4	2	no	—
165	C₄×C₄₇	#392	1	4	yes	—
166	C₄×C₄₇	#394	1	4	yes	—
167	C₂²×C₄×D₄	C₂×C₄₇	1	1	yes	characteristic
168	C₂²×C₂²⋊C₈	C₂×C₄₇	1	4	yes	complemented
169	C₂×#336	C₂×C₄₇	1	2	yes	complemented
170	C₂×C₄₇	—	4	8	no	—
171	C₂×C₄₇	#1438	1	1	yes	characteristic
172	C₂×C₄₇	#1779	1	1	yes	characteristic
173	C₂×C₄₇	#1780	1	1	yes	characteristic
174	C₂×C₄₇	#1869	1	4	yes	—
175	C₂×C₄₇	#1870	1	4	yes	—
176	C₂×C₄₇	#1954	1	2	yes	—
177	C₂×C₄₇	#1955	1	2	yes	—
178	C₂³×D₄	C₄×C₄₇	1	1	yes	characteristic
179	C₂⁴×C₄	C₂²×C₄₇	1	2	yes	—
180	C₂³×C₈	C₂²×C₄₇	1	4	yes	—
181	C₂×C₄×D₄	—	2	8	no	—
182	C₂×C₄×D₄	—	2	4	no	—
183	C₂²×C₄⋊C₄	C₄×C₄₇	1	1	yes	characteristic
184	C₂²×C₂²⋊C₄	C₄×C₄₇	1	2	yes	—
185	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
186	C₂×C₂²⋊C₈	—	2	8	no	—
187	C₂×C₂²⋊C₈	—	2	16	no	—
188	#336	—	2	4	no	—
189	C₄₇	#10964	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁵	C₂×C₄×C₄₇	1	2	yes	—
191	C₂²×D₄	—	2	4	no	—
192	C₂²×D₄	—	2	4	no	—
193	C₂²×D₄	—	2	2	no	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	C₈×C₄₇	1	4	yes	complemented
196	C₂³×C₄	—	2	4	no	—
197	C₂³×C₄	—	2	4	no	—
198	C₂³×C₄	—	2	8	no	—
199	C₂³×C₄	—	2	8	no	—
200	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
201	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
202	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
203	C₂³×C₄	D₄×C₄₇	1	8	yes	—
204	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	—	2	8	no	—
207	C₂²×C₈	—	2	8	no	—
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	D₄×C₄₇	1	8	yes	complemented
210	C₄×D₄	—	2	32	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₂²⋊C₄	—	2	4	no	—
215	C₂×C₂²⋊C₄	—	2	4	no	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	D₄×C₄₇	1	4	yes	—
218	C₂²⋊C₈	—	2	32	no	—
219	C₂⁴	—	2	2	no	—
220	C₂⁴	—	2	2	no	—
221	C₂⁴	—	2	8	no	—
222	C₂⁴	—	2	2	no	—
223	C₂⁴	—	2	2	no	—
224	C₂⁴	—	2	8	no	—
225	C₂⁴	C₂²⋊C₄×C₄₇	1	4	yes	—
226	C₂⁴	C₂×C₈×C₄₇	1	4	yes	—
227	C₂⁴	M₁₆×C₄₇	1	4	yes	—
228	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
229	C₂×D₄	—	2	8	no	—
230	C₂×D₄	—	2	8	no	—
231	C₂×D₄	—	2	16	no	—
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	16	no	—
243	C₂²×C₄	—	2	16	no	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	8	no	—
247	C₂²×C₄	—	4	8	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
252	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
253	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
254	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
255	C₂²×C₄	M₁₆×C₄₇	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
257	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
258	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
259	C₂²×C₄	#41×C₄₇	1	4	yes	—
260	C₂×C₈	—	2	16	no	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	—	4	4	no	—
263	C₂×C₈	—	4	8	no	—
264	C₂×C₈	—	4	4	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	2	16	no	—
267	C₄²	—	2	8	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	8	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	4	no	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	4	no	—
275	C₂³	—	2	8	no	—
276	C₂³	—	2	16	no	—
277	C₂³	—	2	16	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	8	no	—
283	C₂³	—	4	8	no	—
284	C₂³	C₂²⋊C₈×C₄₇	1	8	yes	complemented
285	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
286	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
287	C₂³	C₄×D₄×C₄₇	1	4	yes	—
288	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
289	C₂³	C₂×M₁₆×C₄₇	1	2	yes	—
290	C₂³	C₂×M₁₆×C₄₇	1	1	yes	characteristic
291	C₂³	#131×C₄₇	1	4	yes	—
292	D₄	—	2	32	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	16	no	—
305	C₂×C₄	—	4	16	no	—
306	C₂×C₄	C₂²⋊C₈×C₄₇	1	4	yes	complemented
307	C₂×C₄	#120×C₄₇	1	4	yes	—
308	C₂×C₄	#120×C₄₇	1	4	yes	—
309	C₂×C₄	#121×C₄₇	1	8	yes	—
310	C₂×C₄	#121×C₄₇	1	4	yes	—
311	C₂×C₄	#123×C₄₇	1	4	yes	—
312	C₂×C₄	#124×C₄₇	1	2	yes	—
313	C₂×C₄	#127×C₄₇	1	2	yes	—
314	C₈	—	4	8	no	—
315	C₂²	—	2	8	no	—
316	C₂²	—	2	16	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	8	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	8	no	—
321	C₂²	—	4	4	no	—
322	C₂²	—	4	8	no	—
323	C₂²	—	4	8	no	—
324	C₂²	C₄₇×#379	1	1	yes	characteristic
325	C₂²	C₄₇×#386	1	4	yes	—
326	C₂²	C₄₇×#390	1	2	yes	—
327	C₂²	C₂×C₂²⋊C₈×C₄₇	1	2	yes	—
328	C₂²	C₂×C₂²⋊C₈×C₄₇	1	1	yes	characteristic
329	C₂²	C₄₇×#407	1	2	yes	—
330	C₂²	C₄₇×#407	1	1	yes	characteristic
331	C₂²	C₄₇×#408	1	4	yes	—
332	C₂²	C₄₇×#408	1	2	yes	—
333	C₂²	C₈×D₄×C₄₇	1	4	yes	—
334	C₂²	C₄₇×#435	1	8	yes	—
335	C₂²	C₄₇×#436	1	4	yes	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	4	no	—
339	C₄	C₄₇×#392	1	4	yes	—
340	C₄	C₄₇×#394	1	4	yes	—
341	C₂	—	4	8	no	—
342	C₂	C₄₇×#1438	1	1	yes	characteristic
343	C₂	C₄₇×#1779	1	1	yes	characteristic
344	C₂	C₄₇×#1780	1	1	yes	characteristic
345	C₂	C₄₇×#1869	1	4	yes	—
346	C₂	C₄₇×#1870	1	4	yes	—
347	C₂	C₄₇×#1954	1	2	yes	—
348	C₂	C₄₇×#1955	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₇	C₂³
commutator	C₂³	C₂²×C₈×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂³×D₄×C₄₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊C₈×C₄₇	C₂	semidirect
3. C₂×C₄₇×#336	C₂	semidirect
20. C₂²×D₄×C₄₇	C₈	semidirect
34. C₂²×C₈×C₄₇	D₄	semidirect
109. C₂³×C₄₇	C₂²⋊C₈	semidirect
131. C₂×C₄×C₄₇	C₂²⋊C₈	semidirect
140. #10964	C₄₇	direct (N × H)
168. C₂²×C₂²⋊C₈	C₂×C₄₇	semidirect
169. C₂×#336	C₂×C₄₇	semidirect
189. C₄₇	#10964	direct (N × H)
195. C₂²×D₄	C₈×C₄₇	semidirect
209. C₂²×C₈	D₄×C₄₇	semidirect
284. C₂³	C₂²⋊C₈×C₄₇	semidirect
306. C₂×C₄	C₂²⋊C₈×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₄₇	2
C₂²×C₂²⋊C₈×C₄₇	2
C₂×C₄₇×#336	2
#10964	47

Subgroups — C47×#10964 #26050662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10964 #26050662