Subgroups — C₄₇×#19724 #26059422

Order 12032.

712 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3227	C₂	1	2	yes	maximal, complemented
2	C₄₇×#3210	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3184	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1720	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1717	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1716	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1708	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#533	C₂²	1	1	yes	characteristic
11	C₄₇×#532	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#498	C₂²	1	1	yes	characteristic
14	C₄₇×#498	C₂²	1	2	yes	—
15	C₄₇×#498	C₂²	1	2	yes	—
16	C₄₇×#494	C₂²	1	2	yes	—
17	C₄₇×#493	C₂²	1	1	yes	characteristic
18	C₄₇×#492	C₂²	1	2	yes	—
19	C₄₇×#491	—	2	2	no	—
20	C₄₇×#487	C₂²	1	2	yes	—
21	C₄₇×#487	C₂²	1	2	yes	—
22	C₄₇×#487	C₂²	1	2	yes	—
23	C₄₇×#486	C₂²	1	2	yes	—
24	C₄₇×#486	C₂²	1	2	yes	—
25	C₄₇×#486	C₂²	1	2	yes	—
26	C₄₇×#486	C₂²	1	2	yes	—
27	C₄₇×#481	—	2	4	no	—
28	C₄₇×#476	—	2	4	no	—
29	C₄₇×#469	—	2	2	no	—
30	C₄₇×#464	—	2	4	no	—
31	C₄₇×#452	—	2	4	no	—
32	C₄₇×#442	—	2	2	no	—
33	C₄₇×#431	C₂²	1	2	yes	—
34	C₄₇×#419	—	2	2	no	—
35	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#322	C₂²	1	2	yes	—
37	C₄₇×#322	C₂²	1	1	yes	characteristic
38	C₈²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₄₇	—	2	2	no	—
40	C₂²×Q₈×C₄₇	—	2	1	no	—
41	C₂×Q₁₆×C₄₇	—	2	4	no	—
42	C₂×SD₁₆×C₄₇	—	2	4	no	—
43	C₂×D₈×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₄₇	—	2	2	no	—
46	C₄⋊Q₈×C₄₇	—	2	1	no	—
47	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
50	#127×C₄₇	C₂³	1	2	yes	—
51	#125×C₄₇	—	2	1	no	—
52	#124×C₄₇	—	2	2	no	—
53	C₂²⋊Q₈×C₄₇	—	2	2	no	—
54	#121×C₄₇	—	2	2	no	—
55	#121×C₄₇	—	2	2	no	—
56	C₄×Q₈×C₄₇	—	2	2	no	—
57	C₄×D₄×C₄₇	—	2	2	no	—
58	C₄×D₄×C₄₇	—	2	2	no	—
59	#117×C₄₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#106×C₄₇	—	2	2	no	—
63	#106×C₄₇	—	2	4	no	—
64	#106×C₄₇	—	2	4	no	—
65	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
66	#103×C₄₇	—	2	4	no	—
67	#103×C₄₇	—	2	4	no	—
68	#103×C₄₇	—	2	4	no	—
69	#102×C₄₇	—	2	4	no	—
70	#102×C₄₇	—	2	4	no	—
71	#102×C₄₇	—	2	4	no	—
72	#102×C₄₇	—	2	4	no	—
73	C₂²⋊C₈×C₄₇	D₄	1	4	yes	—
74	#97×C₄₇	D₄	1	2	yes	—
75	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₄₇	C₂³	1	2	yes	—
77	C₄×C₈×C₄₇	C₂³	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	#41×C₄₇	—	8	4	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	—	4	4	no	—
84	C₂×Q₈×C₄₇	—	4	1	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	—	4	2	no	—
89	C₂²×C₄×C₄₇	—	4	1	no	—
90	C₂²×C₄×C₄₇	—	4	2	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₄₇	—	8	4	no	—
94	SD₁₆×C₄₇	—	8	4	no	—
95	D₈×C₄₇	—	4	4	no	—
96	M₁₆×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	4	2	no	—
108	C₄⋊C₄×C₄₇	—	4	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	1	no	—
110	C₂²⋊C₄×C₄₇	—	4	2	no	—
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₄²×C₄₇	—	4	1	no	—
113	C₄²×C₄₇	C₂×D₄	1	2	yes	—
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	#127	1	1	yes	characteristic
118	Q₈×C₄₇	—	4	4	no	—
119	Q₈×C₄₇	—	4	4	no	—
120	Q₈×C₄₇	—	4	4	no	—
121	Q₈×C₄₇	—	8	2	no	—
122	D₄×C₄₇	—	4	4	no	—
123	D₄×C₄₇	—	4	4	no	—
124	D₄×C₄₇	—	4	4	no	—
125	D₄×C₄₇	—	8	4	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	8	4	no	—
132	C₂×C₄×C₄₇	—	8	2	no	—
133	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#127	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₈×C₄₇	—	2	4	no	—
141	#19724	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	8	2	no	—
145	C₂²×C₄₇	—	8	2	no	—
146	C₂²×C₄₇	—	8	2	no	—
147	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	8	2	no	—
151	C₄×C₄₇	—	8	2	no	—
152	C₄×C₄₇	—	8	2	no	—
153	C₄×C₄₇	#575	1	2	yes	—
154	C₄×C₄₇	#577	1	4	yes	—
155	#3227	C₂×C₄₇	1	2	yes	complemented
156	#3210	C₂×C₄₇	1	2	yes	complemented
157	#3184	C₂×C₄₇	1	1	yes	characteristic
158	#3139	C₂×C₄₇	1	1	yes	characteristic, complemented
159	#1720	C₂×C₄₇	1	2	yes	complemented
160	#1717	C₂×C₄₇	1	2	yes	complemented
161	#1716	C₂×C₄₇	1	2	yes	complemented
162	#1708	C₂×C₄₇	1	2	yes	complemented
163	#1453	C₂×C₄₇	1	1	yes	characteristic, complemented
164	C₂×C₄₇	—	4	1	no	—
165	C₂×C₄₇	—	16	2	no	—
166	C₂×C₄₇	#3153	1	1	yes	characteristic
167	C₂×C₄₇	#3154	1	2	yes	—
168	#533	C₂²×C₄₇	1	1	yes	characteristic
169	#532	C₂²×C₄₇	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
171	#498	C₂²×C₄₇	1	1	yes	characteristic
172	#498	C₂²×C₄₇	1	2	yes	—
173	#498	C₂²×C₄₇	1	2	yes	—
174	#494	C₂²×C₄₇	1	2	yes	—
175	#493	C₂²×C₄₇	1	1	yes	characteristic
176	#492	C₂²×C₄₇	1	2	yes	—
177	#491	—	2	2	no	—
178	#487	C₂²×C₄₇	1	2	yes	—
179	#487	C₂²×C₄₇	1	2	yes	—
180	#487	C₂²×C₄₇	1	2	yes	—
181	#486	C₂²×C₄₇	1	2	yes	—
182	#486	C₂²×C₄₇	1	2	yes	—
183	#486	C₂²×C₄₇	1	2	yes	—
184	#486	C₂²×C₄₇	1	2	yes	—
185	#481	—	2	4	no	—
186	#476	—	2	4	no	—
187	#469	—	2	2	no	—
188	#464	—	2	4	no	—
189	#452	—	2	4	no	—
190	#442	—	2	2	no	—
191	#431	C₂²×C₄₇	1	2	yes	—
192	#419	—	2	2	no	—
193	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
194	#322	C₂²×C₄₇	1	2	yes	—
195	#322	C₂²×C₄₇	1	1	yes	characteristic
196	C₈²	C₂²×C₄₇	1	1	yes	characteristic
197	C₄₇	#19724	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×#41	—	2	2	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×SD₁₆	—	2	4	no	—
202	C₂×D₈	—	2	2	no	—
203	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
208	C₄⋊Q₈	C₂³×C₄₇	1	2	yes	—
209	#127	C₂³×C₄₇	1	2	yes	—
210	#125	—	2	1	no	—
211	#124	—	2	2	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	#117	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
221	#106	—	2	2	no	—
222	#106	—	2	4	no	—
223	#106	—	2	4	no	—
224	C₄⋊C₈	D₄×C₄₇	1	4	yes	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	C₂²⋊C₈	D₄×C₄₇	1	4	yes	—
233	#97	D₄×C₄₇	1	2	yes	—
234	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₄₇	1	2	yes	—
236	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₄₇	1	2	yes	—
238	#41	—	8	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
252	Q₁₆	—	8	4	no	—
253	SD₁₆	—	8	4	no	—
254	D₈	—	4	4	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
259	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₄₇	1	2	yes	—
274	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
275	C₂³	—	4	2	no	—
276	C₂³	#127×C₄₇	1	1	yes	characteristic
277	Q₈	—	4	4	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	4	no	—
280	Q₈	—	8	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	#127×C₄₇	1	1	yes	characteristic
293	C₂×C₄	#127×C₄₇	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂×#127×C₄₇	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₄₇×#575	1	2	yes	—
312	C₄	C₄₇×#577	1	4	yes	—
313	C₂	—	4	1	no	—
314	C₂	—	16	2	no	—
315	C₂	C₄₇×#3153	1	1	yes	characteristic
316	C₂	C₄₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₄₇×#3184	C₂
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×#3227	C₂	semidirect
2. C₄₇×#3210	C₂	semidirect
4. C₄₇×#3139	C₂	semidirect
5. C₄₇×#1720	C₂	semidirect
6. C₄₇×#1717	C₂	semidirect
7. C₄₇×#1716	C₂	semidirect
8. C₄₇×#1708	C₂	semidirect
9. C₄₇×#1453	C₂	semidirect
141. #19724	C₄₇	direct (N × H)
155. #3227	C₂×C₄₇	semidirect
156. #3210	C₂×C₄₇	semidirect
158. #3139	C₂×C₄₇	semidirect
159. #1720	C₂×C₄₇	semidirect
160. #1717	C₂×C₄₇	semidirect
161. #1716	C₂×C₄₇	semidirect
162. #1708	C₂×C₄₇	semidirect
163. #1453	C₂×C₄₇	semidirect
197. C₄₇	#19724	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×#3227	2
C₄₇×#3210	2
C₄₇×#3184	2
C₄₇×#3139	2
C₄₇×#1720	2
C₄₇×#1717	2
C₄₇×#1716	2
C₄₇×#1708	2
C₄₇×#1453	2
#19724	47

Subgroups — C47×#19724 #26059422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#19724 #26059422