Subgroups — C₄₇×#3068 #20240678

Order 6016.

466 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#575	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×AGL(1,ℤ/8)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×AGL(1,ℤ/8)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#472	C₂	1	4	yes	maximal, complemented
6	C₄₇×#471	C₂	1	4	yes	maximal, complemented
7	C₄₇×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#413	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#100×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
13	#137×C₄₇	—	2	1	no	—
14	#137×C₄₇	C₂²	1	2	yes	complemented
15	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
16	AGL(1,ℤ/8)×C₄₇	—	2	1	no	—
17	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
18	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
19	AGL(1,ℤ/8)×C₄₇	C₂²	1	4	yes	—
20	AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	complemented
21	#135×C₄₇	—	2	2	no	—
22	C₂×SD₁₆×C₄₇	—	2	1	no	—
23	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	complemented
24	C₂×D₈×C₄₇	—	2	1	no	—
25	C₂×D₈×C₄₇	C₂²	1	2	yes	complemented
26	#131×C₄₇	—	2	2	no	—
27	#131×C₄₇	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
33	#108×C₄₇	C₂²	1	4	yes	complemented
34	#101×C₄₇	C₂²	1	2	yes	complemented
35	#100×C₄₇	C₂²	1	4	yes	complemented
36	#100×C₄₇	C₂²	1	2	yes	complemented
37	C₂⁴×C₄₇	—	2	1	no	—
38	#41×C₄₇	—	2	1	no	—
39	#41×C₄₇	—	2	1	no	—
40	#41×C₄₇	—	2	1	no	—
41	#41×C₄₇	—	2	1	no	—
42	#41×C₄₇	—	2	1	no	—
43	#41×C₄₇	—	2	1	no	—
44	#41×C₄₇	D₄	1	2	yes	—
45	#41×C₄₇	C₂³	1	2	yes	—
46	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₄₇	—	2	1	no	—
49	C₂×D₄×C₄₇	—	2	1	no	—
50	C₂×D₄×C₄₇	—	2	1	no	—
51	C₂×D₄×C₄₇	—	2	1	no	—
52	C₂×D₄×C₄₇	—	2	1	no	—
53	C₂×D₄×C₄₇	—	2	1	no	—
54	C₂×D₄×C₄₇	—	2	1	no	—
55	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
56	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	Q₁₆×C₄₇	—	2	2	no	—
62	SD₁₆×C₄₇	—	2	4	no	—
63	SD₁₆×C₄₇	—	2	2	no	—
64	SD₁₆×C₄₇	—	2	2	no	—
65	SD₁₆×C₄₇	—	2	4	no	—
66	D₈×C₄₇	—	2	4	no	—
67	D₈×C₄₇	—	2	2	no	—
68	D₈×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	1	no	—
70	M₁₆×C₄₇	—	2	1	no	—
71	M₁₆×C₄₇	—	2	1	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	D₄	1	2	yes	—
74	M₁₆×C₄₇	D₄	1	2	yes	—
75	M₁₆×C₄₇	C₂³	1	2	yes	—
76	M₁₆×C₄₇	C₂³	1	2	yes	—
77	M₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	C₂³	1	2	yes	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	4	1	no	—
87	C₂³×C₄₇	—	4	1	no	—
88	C₂³×C₄₇	—	8	2	no	—
89	C₂³×C₄₇	#41	1	1	yes	characteristic
90	Q₈×C₄₇	—	2	1	no	—
91	Q₈×C₄₇	—	2	1	no	—
92	Q₈×C₄₇	—	2	1	no	—
93	Q₈×C₄₇	C₂×D₄	1	2	yes	—
94	D₄×C₄₇	—	2	2	no	—
95	D₄×C₄₇	—	2	2	no	—
96	D₄×C₄₇	—	2	1	no	—
97	D₄×C₄₇	—	2	1	no	—
98	D₄×C₄₇	—	2	2	no	—
99	D₄×C₄₇	—	2	2	no	—
100	D₄×C₄₇	—	2	1	no	—
101	D₄×C₄₇	—	4	1	no	—
102	D₄×C₄₇	—	4	1	no	—
103	D₄×C₄₇	—	4	1	no	—
104	D₄×C₄₇	—	4	1	no	—
105	D₄×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
118	C₈×C₄₇	—	2	2	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	C₂×D₄	1	4	yes	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	4	1	no	—
124	C₂²×C₄₇	—	4	1	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	8	1	no	—
127	C₂²×C₄₇	—	8	1	no	—
128	C₂²×C₄₇	—	8	2	no	—
129	C₂²×C₄₇	—	8	2	no	—
130	C₂²×C₄₇	—	8	2	no	—
131	C₂²×C₄₇	#121	1	1	yes	characteristic
132	C₂²×C₄₇	#121	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	4	1	no	—
136	C₄×C₄₇	#121	1	1	yes	characteristic
137	C₄×C₄₇	#121	1	1	yes	characteristic
138	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	#3068	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	—	2	1	no	—
143	C₂×C₄₇	—	2	1	no	—
144	C₂×C₄₇	—	4	2	no	—
145	C₂×C₄₇	—	8	1	no	—
146	C₂×C₄₇	—	8	2	no	—
147	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
148	#575	C₂×C₄₇	1	1	yes	characteristic, complemented
149	C₂×AGL(1,ℤ/8)	C₂×C₄₇	1	1	yes	characteristic, complemented
150	C₂×AGL(1,ℤ/8)	C₂×C₄₇	1	1	yes	characteristic, complemented
151	#568	C₂×C₄₇	1	1	yes	characteristic, complemented
152	#472	C₂×C₄₇	1	4	yes	complemented
153	#471	C₂×C₄₇	1	4	yes	complemented
154	#430	C₂×C₄₇	1	1	yes	characteristic, complemented
155	#413	C₂×C₄₇	1	1	yes	characteristic, complemented
156	C₂×#100	C₂×C₄₇	1	1	yes	characteristic, complemented
157	C₄₇	#3068	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#41	C₂²×C₄₇	1	1	yes	characteristic
159	C₂×#41	C₂²×C₄₇	1	1	yes	characteristic
160	C₂²×D₄	C₂²×C₄₇	1	1	yes	characteristic
161	#137	—	2	1	no	—
162	#137	C₂²×C₄₇	1	2	yes	complemented
163	AGL(1,ℤ/8)	—	2	2	no	—
164	AGL(1,ℤ/8)	—	2	1	no	—
165	AGL(1,ℤ/8)	—	2	2	no	—
166	AGL(1,ℤ/8)	—	2	2	no	—
167	AGL(1,ℤ/8)	C₂²×C₄₇	1	4	yes	—
168	AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	complemented
169	#135	—	2	2	no	—
170	C₂×SD₁₆	—	2	1	no	—
171	C₂×SD₁₆	C₂²×C₄₇	1	2	yes	complemented
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	C₂²×C₄₇	1	2	yes	complemented
174	#131	—	2	2	no	—
175	#131	C₂²×C₄₇	1	4	yes	complemented
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
178	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
179	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
180	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
181	#108	C₂²×C₄₇	1	4	yes	complemented
182	#101	C₂²×C₄₇	1	2	yes	complemented
183	#100	C₂²×C₄₇	1	4	yes	complemented
184	#100	C₂²×C₄₇	1	2	yes	complemented
185	C₂⁴	—	2	1	no	—
186	#41	—	2	1	no	—
187	#41	—	2	1	no	—
188	#41	—	2	1	no	—
189	#41	—	2	1	no	—
190	#41	—	2	1	no	—
191	#41	—	2	1	no	—
192	#41	D₄×C₄₇	1	2	yes	—
193	#41	C₂³×C₄₇	1	2	yes	—
194	C₂×Q₈	D₄×C₄₇	1	1	yes	characteristic
195	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	D₄×C₄₇	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₄₇	1	1	yes	characteristic
205	C₂×D₄	C₂³×C₄₇	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
209	Q₁₆	—	2	2	no	—
210	SD₁₆	—	2	4	no	—
211	SD₁₆	—	2	2	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	4	no	—
214	D₈	—	2	4	no	—
215	D₈	—	2	2	no	—
216	D₈	—	2	4	no	—
217	M₁₆	—	2	1	no	—
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	D₄×C₄₇	1	2	yes	—
222	M₁₆	D₄×C₄₇	1	2	yes	—
223	M₁₆	C₂³×C₄₇	1	2	yes	—
224	M₁₆	C₂³×C₄₇	1	2	yes	—
225	M₁₆	C₂³×C₄₇	1	2	yes	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	C₂³×C₄₇	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	4	1	no	—
235	C₂³	—	4	1	no	—
236	C₂³	—	8	2	no	—
237	C₂³	#41×C₄₇	1	1	yes	characteristic
238	Q₈	—	2	1	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	1	no	—
241	Q₈	C₂×D₄×C₄₇	1	2	yes	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	1	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	1	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	D₄	C₂×D₄×C₄₇	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
264	C₂×C₄	#41×C₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	C₂×D₄×C₄₇	1	4	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	1	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	#121×C₄₇	1	1	yes	characteristic
280	C₂²	#121×C₄₇	1	1	yes	characteristic
281	C₂²	C₂×#41×C₄₇	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	#121×C₄₇	1	1	yes	characteristic
285	C₄	#121×C₄₇	1	1	yes	characteristic
286	C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
287	C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	4	2	no	—
292	C₂	—	8	2	no	—
293	C₂	—	8	1	no	—
294	C₂	C₂×#121×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	C₂×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#575	C₂	semidirect
2. C₂×AGL(1,ℤ/8)×C₄₇	C₂	semidirect
3. C₂×AGL(1,ℤ/8)×C₄₇	C₂	semidirect
4. C₄₇×#568	C₂	semidirect
5. C₄₇×#472	C₂	semidirect
6. C₄₇×#471	C₂	semidirect
7. C₄₇×#430	C₂	semidirect
8. C₄₇×#413	C₂	semidirect
9. C₂×#100×C₄₇	C₂	semidirect
14. #137×C₄₇	C₂²	semidirect
20. AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
23. C₂×SD₁₆×C₄₇	C₂²	semidirect
25. C₂×D₈×C₄₇	C₂²	semidirect
27. #131×C₄₇	C₂²	semidirect
33. #108×C₄₇	C₂²	semidirect
34. #101×C₄₇	C₂²	semidirect
35. #100×C₄₇	C₂²	semidirect
36. #100×C₄₇	C₂²	semidirect
140. #3068	C₄₇	direct (N × H)
148. #575	C₂×C₄₇	semidirect
149. C₂×AGL(1,ℤ/8)	C₂×C₄₇	semidirect
150. C₂×AGL(1,ℤ/8)	C₂×C₄₇	semidirect
151. #568	C₂×C₄₇	semidirect
152. #472	C₂×C₄₇	semidirect
153. #471	C₂×C₄₇	semidirect
154. #430	C₂×C₄₇	semidirect
155. #413	C₂×C₄₇	semidirect
156. C₂×#100	C₂×C₄₇	semidirect
157. C₄₇	#3068	direct (N × H)
162. #137	C₂²×C₄₇	semidirect
168. AGL(1,ℤ/8)	C₂²×C₄₇	semidirect
171. C₂×SD₁₆	C₂²×C₄₇	semidirect
173. C₂×D₈	C₂²×C₄₇	semidirect
175. #131	C₂²×C₄₇	semidirect
181. #108	C₂²×C₄₇	semidirect
182. #101	C₂²×C₄₇	semidirect
183. #100	C₂²×C₄₇	semidirect
184. #100	C₂²×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#575	2
C₂×AGL(1,ℤ/8)×C₄₇	2
C₄₇×#568	2
C₄₇×#472	2
C₄₇×#471	2
C₄₇×#430	2
C₄₇×#413	2
C₂×#100×C₄₇	2
#3068	47

Subgroups — C47×#3068 #20240678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3068 #20240678