Subgroups — C₄₇×#3005 #20240615

Order 6016.

758 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#137×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×AGL(1,ℤ/8)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#452	C₂	1	4	yes	maximal, complemented
6	C₄₇×#449	C₂	1	4	yes	maximal, complemented
7	C₄₇×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#418	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₄₇	—	2	4	no	—
11	C₂×#41×C₄₇	—	2	4	no	—
12	C₂×#41×C₄₇	—	2	2	no	—
13	C₂×#41×C₄₇	C₂²	1	4	yes	—
14	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₄₇	—	2	1	no	—
17	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	—	2	1	no	—
19	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	#137×C₄₇	—	2	4	no	—
21	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
22	C₂×Q₁₆×C₄₇	C₂²	1	2	yes	complemented
23	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	complemented
24	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	complemented
25	C₂×D₈×C₄₇	C₂²	1	2	yes	complemented
26	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	#123×C₄₇	—	2	1	no	—
29	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	complemented
30	#121×C₄₇	C₂²	1	2	yes	complemented
31	#120×C₄₇	—	2	1	no	—
32	C₄×D₄×C₄₇	—	2	2	no	—
33	#117×C₄₇	C₂²	1	1	yes	characteristic
34	#103×C₄₇	C₂²	1	4	yes	complemented
35	#102×C₄₇	C₂²	1	4	yes	complemented
36	C₂²⋊C₈×C₄₇	C₂²	1	4	yes	complemented
37	C₂⁴×C₄₇	—	2	1	no	—
38	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
39	#41×C₄₇	—	2	16	no	—
40	#41×C₄₇	—	2	8	no	—
41	#41×C₄₇	—	2	8	no	—
42	#41×C₄₇	—	2	2	no	—
43	#41×C₄₇	—	2	2	no	—
44	C₂×Q₈×C₄₇	—	2	2	no	—
45	C₂×Q₈×C₄₇	—	2	1	no	—
46	C₂×Q₈×C₄₇	D₄	1	4	yes	complemented
47	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
49	C₂×D₄×C₄₇	—	2	8	no	—
50	C₂×D₄×C₄₇	—	2	4	no	—
51	C₂×D₄×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	—	2	1	no	—
53	C₂×D₄×C₄₇	—	2	1	no	—
54	C₂×D₄×C₄₇	—	2	2	no	—
55	C₂×D₄×C₄₇	D₄	1	4	yes	complemented
56	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
68	Q₁₆×C₄₇	—	2	4	no	—
69	SD₁₆×C₄₇	—	2	4	no	—
70	SD₁₆×C₄₇	—	2	4	no	—
71	D₈×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	2	no	—
73	M₁₆×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	C₂³	1	2	yes	—
75	C₂×C₈×C₄₇	C₂³	1	2	yes	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₄₇	—	2	1	no	—
79	C₂²⋊C₄×C₄₇	—	2	2	no	—
80	C₂²⋊C₄×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	—	4	2	no	—
87	C₂³×C₄₇	—	4	4	no	—
88	C₂³×C₄₇	C₂×D₄	1	2	yes	—
89	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₄₇	—	2	4	no	—
91	Q₈×C₄₇	—	2	2	no	—
92	Q₈×C₄₇	—	2	1	no	—
93	Q₈×C₄₇	—	2	1	no	—
94	Q₈×C₄₇	C₂×D₄	1	4	yes	—
95	D₄×C₄₇	—	2	16	no	—
96	D₄×C₄₇	—	2	4	no	—
97	D₄×C₄₇	—	2	2	no	—
98	D₄×C₄₇	—	2	1	no	—
99	D₄×C₄₇	—	2	1	no	—
100	D₄×C₄₇	—	4	4	no	—
101	D₄×C₄₇	—	4	2	no	—
102	D₄×C₄₇	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₄₇	—	2	16	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
118	C₈×C₄₇	—	2	2	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	4	no	—
126	C₂²×C₄₇	—	4	8	no	—
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	#120	1	2	yes	—
131	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	—	4	1	no	—
135	C₄×C₄₇	—	4	2	no	—
136	C₄×C₄₇	#120	1	2	yes	—
137	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	#3005	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	—	4	2	no	—
142	C₂×C₄₇	—	4	4	no	—
143	C₂×C₄₇	—	8	1	no	—
144	C₂×C₄₇	C₂×#120	1	1	yes	characteristic
145	C₂×C₄₇	#575	1	2	yes	—
146	C₂²×#41	C₂×C₄₇	1	1	yes	characteristic, complemented
147	C₂×#137	C₂×C₄₇	1	1	yes	characteristic, complemented
148	C₂×AGL(1,ℤ/8)	C₂×C₄₇	1	1	yes	characteristic, complemented
149	#525	C₂×C₄₇	1	1	yes	characteristic, complemented
150	#452	C₂×C₄₇	1	4	yes	complemented
151	#449	C₂×C₄₇	1	4	yes	complemented
152	#419	C₂×C₄₇	1	1	yes	characteristic, complemented
153	#418	C₂×C₄₇	1	1	yes	characteristic, complemented
154	#407	C₂×C₄₇	1	1	yes	characteristic, complemented
155	C₄₇	#3005	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#41	—	2	4	no	—
157	C₂×#41	—	2	4	no	—
158	C₂×#41	—	2	2	no	—
159	C₂×#41	C₂²×C₄₇	1	1	yes	characteristic
160	C₂×#41	C₂²×C₄₇	1	4	yes	—
161	C₂²×Q₈	C₂²×C₄₇	1	1	yes	characteristic
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	C₂²×C₄₇	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
166	#137	—	2	4	no	—
167	AGL(1,ℤ/8)	—	2	4	no	—
168	C₂×Q₁₆	C₂²×C₄₇	1	2	yes	complemented
169	C₂×SD₁₆	C₂²×C₄₇	1	2	yes	complemented
170	C₂×SD₁₆	C₂²×C₄₇	1	2	yes	complemented
171	C₂×D₈	C₂²×C₄₇	1	2	yes	complemented
172	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
173	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
174	#123	—	2	1	no	—
175	C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	complemented
176	#121	C₂²×C₄₇	1	2	yes	complemented
177	#120	—	2	1	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	C₂²×C₄₇	1	1	yes	characteristic
180	#103	C₂²×C₄₇	1	4	yes	complemented
181	#102	C₂²×C₄₇	1	4	yes	complemented
182	C₂²⋊C₈	C₂²×C₄₇	1	4	yes	complemented
183	C₂⁴	—	2	1	no	—
184	C₂⁴	D₄×C₄₇	1	1	yes	characteristic
185	#41	—	2	16	no	—
186	#41	—	2	8	no	—
187	#41	—	2	8	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	D₄×C₄₇	1	4	yes	complemented
193	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
194	C₂×Q₈	C₂³×C₄₇	1	2	yes	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	D₄×C₄₇	1	4	yes	complemented
202	C₂×D₄	C₂³×C₄₇	1	1	yes	characteristic
203	C₂×D₄	C₂³×C₄₇	1	2	yes	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	D₄×C₄₇	1	2	yes	—
211	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
213	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	SD₁₆	—	2	4	no	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	4	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	C₂×C₈	C₂³×C₄₇	1	2	yes	—
221	C₂×C₈	C₂³×C₄₇	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	C₂×D₄×C₄₇	1	2	yes	—
235	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	2	2	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	2	1	no	—
240	Q₈	C₂×D₄×C₄₇	1	4	yes	—
241	D₄	—	2	16	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	1	no	—
245	D₄	—	2	1	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	2	no	—
248	D₄	C₂×D₄×C₄₇	1	4	yes	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
263	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	8	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	#120×C₄₇	1	2	yes	—
277	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	2	no	—
282	C₄	#120×C₄₇	1	2	yes	—
283	C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
284	C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	4	no	—
288	C₂	—	8	1	no	—
289	C₂	C₂×#120×C₄₇	1	1	yes	characteristic
290	C₂	C₄₇×#575	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₄₇	C₂	semidirect
2. C₂×#137×C₄₇	C₂	semidirect
3. C₂×AGL(1,ℤ/8)×C₄₇	C₂	semidirect
4. C₄₇×#525	C₂	semidirect
5. C₄₇×#452	C₂	semidirect
6. C₄₇×#449	C₂	semidirect
7. C₄₇×#419	C₂	semidirect
8. C₄₇×#418	C₂	semidirect
9. C₄₇×#407	C₂	semidirect
22. C₂×Q₁₆×C₄₇	C₂²	semidirect
23. C₂×SD₁₆×C₄₇	C₂²	semidirect
24. C₂×SD₁₆×C₄₇	C₂²	semidirect
25. C₂×D₈×C₄₇	C₂²	semidirect
29. C₂²⋊Q₈×C₄₇	C₂²	semidirect
30. #121×C₄₇	C₂²	semidirect
34. #103×C₄₇	C₂²	semidirect
35. #102×C₄₇	C₂²	semidirect
36. C₂²⋊C₈×C₄₇	C₂²	semidirect
46. C₂×Q₈×C₄₇	D₄	semidirect
55. C₂×D₄×C₄₇	D₄	semidirect
139. #3005	C₄₇	direct (N × H)
146. C₂²×#41	C₂×C₄₇	semidirect
147. C₂×#137	C₂×C₄₇	semidirect
148. C₂×AGL(1,ℤ/8)	C₂×C₄₇	semidirect
149. #525	C₂×C₄₇	semidirect
150. #452	C₂×C₄₇	semidirect
151. #449	C₂×C₄₇	semidirect
152. #419	C₂×C₄₇	semidirect
153. #418	C₂×C₄₇	semidirect
154. #407	C₂×C₄₇	semidirect
155. C₄₇	#3005	direct (N × H)
168. C₂×Q₁₆	C₂²×C₄₇	semidirect
169. C₂×SD₁₆	C₂²×C₄₇	semidirect
170. C₂×SD₁₆	C₂²×C₄₇	semidirect
171. C₂×D₈	C₂²×C₄₇	semidirect
175. C₂²⋊Q₈	C₂²×C₄₇	semidirect
176. #121	C₂²×C₄₇	semidirect
180. #103	C₂²×C₄₇	semidirect
181. #102	C₂²×C₄₇	semidirect
182. C₂²⋊C₈	C₂²×C₄₇	semidirect
192. C₂×Q₈	D₄×C₄₇	semidirect
201. C₂×D₄	D₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₄₇	2
C₂×#137×C₄₇	2
C₂×AGL(1,ℤ/8)×C₄₇	2
C₄₇×#525	2
C₄₇×#452	2
C₄₇×#449	2
C₄₇×#419	2
C₄₇×#418	2
C₄₇×#407	2
#3005	47

Subgroups — C47×#3005 #20240615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3005 #20240615