Subgroups — C₄₇×#9146 #26048844

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2848	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1779	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#328	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1294	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	complemented
8	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₂²⋊C₄×C₄₇	—	2	1	no	—
10	C₂²×C₄²×C₄₇	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
12	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#385	—	2	2	no	—
14	C₄₇×#385	—	2	1	no	—
15	C₄₇×#380	C₄	1	2	yes	complemented
16	C₄₇×#380	C₂²	1	1	yes	characteristic
17	C₄₇×#379	—	2	1	no	—
18	C₄×C₂²⋊C₄×C₄₇	—	2	4	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#328	—	2	4	no	—
21	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	—	2	2	no	—
24	C₂³×C₄×C₄₇	—	2	2	no	—
25	C₂³×C₄×C₄₇	—	2	2	no	—
26	C₂³×C₄×C₄₇	—	4	4	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	D₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	2	2	no	—
32	C₂²⋊Q₈×C₄₇	—	2	2	no	—
33	C₂²⋊Q₈×C₄₇	—	2	2	no	—
34	C₂²⋊Q₈×C₄₇	—	4	4	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	—	2	4	no	—
46	C₂×C₄²×C₄₇	—	4	4	no	—
47	C₂²⋊C₈×C₄₇	—	2	2	no	—
48	C₂²⋊C₈×C₄₇	—	2	2	no	—
49	C₂²⋊C₈×C₄₇	—	4	4	no	—
50	#95×C₄₇	—	2	2	no	—
51	#95×C₄₇	—	2	2	no	—
52	#95×C₄₇	—	2	1	no	—
53	#95×C₄₇	D₄	1	4	yes	—
54	C₂⁴×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	—	2	1	no	—
56	C₂⁴×C₄₇	—	2	1	no	—
57	C₂⁴×C₄₇	—	2	2	no	—
58	C₂⁴×C₄₇	—	2	2	no	—
59	C₂⁴×C₄₇	—	4	2	no	—
60	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂²×C₄×C₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	2	1	no	—
74	C₂²×C₄×C₄₇	—	2	1	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	—	4	2	no	—
78	C₂²×C₄×C₄₇	—	4	2	no	—
79	C₂²×C₄×C₄₇	—	4	8	no	—
80	C₂²×C₄×C₄₇	—	4	8	no	—
81	C₂²×C₄×C₄₇	—	4	4	no	—
82	C₂²×C₄×C₄₇	—	4	2	no	—
83	C₂²×C₄×C₄₇	—	4	2	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	2	yes	—
88	C₂²×C₄×C₄₇	#41	1	2	yes	—
89	C₂×C₈×C₄₇	—	4	4	no	—
90	C₂×C₈×C₄₇	—	4	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	4	2	no	—
93	C₄⋊C₄×C₄₇	—	4	4	no	—
94	C₄⋊C₄×C₄₇	—	4	2	no	—
95	C₂²⋊C₄×C₄₇	—	2	16	no	—
96	C₂²⋊C₄×C₄₇	—	4	4	no	—
97	C₂²⋊C₄×C₄₇	—	4	4	no	—
98	C₂²⋊C₄×C₄₇	—	4	1	no	—
99	C₂²⋊C₄×C₄₇	—	4	1	no	—
100	C₂²⋊C₄×C₄₇	—	4	2	no	—
101	C₄²×C₄₇	—	4	8	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	4	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	—	4	8	no	—
113	C₂³×C₄₇	—	4	8	no	—
114	C₂³×C₄₇	—	4	2	no	—
115	C₂³×C₄₇	—	4	2	no	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	#104	1	4	yes	—
119	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₄₇	#120	1	1	yes	characteristic
121	C₂³×C₄₇	#123	1	1	yes	characteristic
122	C₂³×C₄₇	#135	1	2	yes	—
123	C₂³×C₄₇	#137	1	2	yes	—
124	Q₈×C₄₇	—	8	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	8	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	8	no	—
135	C₂×C₄×C₄₇	—	4	8	no	—
136	C₂×C₄×C₄₇	—	4	4	no	—
137	C₂×C₄×C₄₇	—	4	8	no	—
138	C₂×C₄×C₄₇	—	4	8	no	—
139	C₂×C₄×C₄₇	—	4	4	no	—
140	C₂×C₄×C₄₇	—	4	1	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	#99	1	4	yes	—
148	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₄₇	#121	1	2	yes	—
150	C₈×C₄₇	—	8	2	no	—
151	#9146	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	2	1	no	—
155	C₂²×C₄₇	—	2	1	no	—
156	C₂²×C₄₇	—	2	4	no	—
157	C₂²×C₄₇	—	2	4	no	—
158	C₂²×C₄₇	—	2	4	no	—
159	C₂²×C₄₇	—	2	4	no	—
160	C₂²×C₄₇	—	2	4	no	—
161	C₂²×C₄₇	—	4	8	no	—
162	C₂²×C₄₇	—	4	8	no	—
163	C₂²×C₄₇	—	4	2	no	—
164	C₂²×C₄₇	—	4	2	no	—
165	C₂²×C₄₇	—	4	2	no	—
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	#386	1	1	yes	characteristic
168	C₂²×C₄₇	C₂×#99	1	2	yes	—
169	C₂²×C₄₇	#416	1	1	yes	characteristic
170	C₂²×C₄₇	#419	1	1	yes	characteristic
171	C₂²×C₄₇	C₂×#104	1	2	yes	—
172	C₂²×C₄₇	#452	1	2	yes	—
173	C₂²×C₄₇	#485	1	2	yes	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	2	4	no	—
176	C₄×C₄₇	—	2	2	no	—
177	C₄×C₄₇	—	4	8	no	—
178	C₄×C₄₇	—	8	1	no	—
179	C₄×C₄₇	—	8	1	no	—
180	C₄×C₄₇	—	8	2	no	—
181	#2848	C₂×C₄₇	1	1	yes	characteristic
182	C₂×C₄×C₂²⋊C₄	C₂×C₄₇	1	1	yes	characteristic
183	#1779	C₂×C₄₇	1	1	yes	characteristic
184	C₂×#328	C₂×C₄₇	1	2	yes	complemented
185	#1294	C₂×C₄₇	1	2	yes	complemented
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	—	2	2	no	—
188	C₂×C₄₇	—	4	4	no	—
189	C₂×C₄₇	#1501	1	2	yes	—
190	C₂×C₄₇	#1508	1	2	yes	—
191	C₂×C₄₇	#1873	1	1	yes	characteristic
192	C₂×C₄₇	#1896	1	1	yes	characteristic
193	C₂×C₄₇	#1898	1	1	yes	characteristic
194	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
195	C₂×C₂²⋊Q₈	C₄×C₄₇	1	2	yes	complemented
196	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	—	2	1	no	—
198	C₂²×C₄²	—	2	1	no	—
199	C₂×C₂²⋊C₈	—	2	2	no	—
200	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
201	#385	—	2	1	no	—
202	#385	—	2	2	no	—
203	#380	C₄×C₄₇	1	2	yes	complemented
204	#380	C₂²×C₄₇	1	1	yes	characteristic
205	#379	—	2	1	no	—
206	C₄×C₂²⋊C₄	—	2	4	no	—
207	C₂×#95	C₂²×C₄₇	1	2	yes	—
208	#328	—	2	4	no	—
209	C₄₇	#9146	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁵	D₄×C₄₇	1	1	yes	characteristic
211	C₂²×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
212	C₂³×C₄	—	2	2	no	—
213	C₂³×C₄	—	2	2	no	—
214	C₂³×C₄	—	2	2	no	—
215	C₂³×C₄	—	4	4	no	—
216	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
217	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
218	C₂³×C₄	D₄×C₄₇	1	2	yes	—
219	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
220	C₂²×C₈	—	2	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₂²⋊Q₈	—	4	4	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
228	C₂×C₂²⋊C₄	—	2	8	no	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	—	2	1	no	—
233	C₂×C₂²⋊C₄	—	2	1	no	—
234	C₂×C₄²	—	2	4	no	—
235	C₂×C₄²	—	4	4	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	2	2	no	—
238	C₂²⋊C₈	—	4	4	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	1	no	—
242	#95	D₄×C₄₇	1	4	yes	—
243	C₂⁴	—	2	4	no	—
244	C₂⁴	—	2	1	no	—
245	C₂⁴	—	2	1	no	—
246	C₂⁴	—	2	2	no	—
247	C₂⁴	—	2	2	no	—
248	C₂⁴	—	4	2	no	—
249	C₂⁴	C₂²⋊C₄×C₄₇	1	2	yes	—
250	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	8	no	—
269	C₂²×C₄	—	4	8	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
277	C₂²×C₄	#41×C₄₇	1	2	yes	—
278	C₂×C₈	—	4	4	no	—
279	C₂×C₈	—	4	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	2	16	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	4	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	8	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	2	4	no	—
294	C₂³	—	2	4	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	4	8	no	—
302	C₂³	—	4	8	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	#104×C₄₇	1	4	yes	—
308	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
309	C₂³	#120×C₄₇	1	1	yes	characteristic
310	C₂³	#123×C₄₇	1	1	yes	characteristic
311	C₂³	#135×C₄₇	1	2	yes	—
312	C₂³	#137×C₄₇	1	2	yes	—
313	Q₈	—	8	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	8	no	—
324	C₂×C₄	—	4	8	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	8	no	—
327	C₂×C₄	—	4	8	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	#99×C₄₇	1	4	yes	—
337	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
338	C₂×C₄	#121×C₄₇	1	2	yes	—
339	C₈	—	8	2	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	4	no	—
347	C₂²	—	2	4	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	4	8	no	—
350	C₂²	—	4	8	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	C₄₇×#386	1	1	yes	characteristic
356	C₂²	C₂×#99×C₄₇	1	2	yes	—
357	C₂²	C₄₇×#416	1	1	yes	characteristic
358	C₂²	C₄₇×#419	1	1	yes	characteristic
359	C₂²	C₂×#104×C₄₇	1	2	yes	—
360	C₂²	C₄₇×#452	1	2	yes	—
361	C₂²	C₄₇×#485	1	2	yes	—
362	C₄	—	2	2	no	—
363	C₄	—	2	4	no	—
364	C₄	—	2	2	no	—
365	C₄	—	4	8	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	—	8	1	no	—
369	C₂	—	2	2	no	—
370	C₂	—	2	2	no	—
371	C₂	—	4	4	no	—
372	C₂	C₄₇×#1501	1	2	yes	—
373	C₂	C₄₇×#1508	1	2	yes	—
374	C₂	C₄₇×#1873	1	1	yes	characteristic
375	C₂	C₄₇×#1896	1	1	yes	characteristic
376	C₂	C₄₇×#1898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
prime radical	C₂⁵×C₄₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂⁴×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₄₇×#328	C₂	semidirect
5. C₄₇×#1294	C₂	semidirect
7. C₂×C₂²⋊Q₈×C₄₇	C₄	semidirect
15. C₄₇×#380	C₄	semidirect
151. #9146	C₄₇	direct (N × H)
184. C₂×#328	C₂×C₄₇	semidirect
185. #1294	C₂×C₄₇	semidirect
195. C₂×C₂²⋊Q₈	C₄×C₄₇	semidirect
203. #380	C₄×C₄₇	semidirect
209. C₄₇	#9146	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×#2848	2
C₂×C₄×C₂²⋊C₄×C₄₇	2
C₄₇×#1779	2
C₂×C₄₇×#328	2
C₄₇×#1294	2
#9146	47

Subgroups — C47×#9146 #26048844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#9146 #26048844