Subgroups — C₂×C₄₇×#2238 #26073352

Order 12032.

604 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2238	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#361	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#349	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₇	—	2	1	no	—
9	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×#121×C₄₇	—	2	1	no	—
11	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
12	C₄₇×#469	—	2	2	no	—
13	C₄₇×#469	C₂²	1	4	yes	—
14	C₄₇×#430	—	2	2	no	—
15	C₄₇×#430	C₂²	1	4	yes	complemented
16	C₂×#108×C₄₇	—	2	1	no	—
17	C₂×#108×C₄₇	C₂²	1	2	yes	—
18	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#102×C₄₇	—	2	1	no	—
20	C₄₇×#365	C₂²	1	8	yes	complemented
21	C₄₇×#361	C₂²	1	8	yes	complemented
22	C₄₇×#349	C₂²	1	4	yes	complemented
23	C₂²×D₄×C₄₇	—	2	1	no	—
24	C₂²×D₄×C₄₇	—	4	1	no	—
25	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂×D₈×C₄₇	—	2	1	no	—
27	C₂×D₈×C₄₇	—	2	2	no	—
28	C₂×D₈×C₄₇	—	2	2	no	—
29	C₂×D₈×C₄₇	C₂³	1	2	yes	—
30	C₂×M₁₆×C₄₇	—	2	2	no	—
31	C₂×M₁₆×C₄₇	—	2	2	no	—
32	C₂×M₁₆×C₄₇	—	2	4	no	—
33	C₂×M₁₆×C₄₇	—	2	2	no	—
34	C₂×M₁₆×C₄₇	—	4	2	no	—
35	C₂×M₁₆×C₄₇	—	4	1	no	—
36	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
37	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₇	—	4	1	no	—
39	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	#121×C₄₇	—	2	4	no	—
42	#121×C₄₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
45	M₃₂×C₄₇	—	2	2	no	—
46	M₃₂×C₄₇	C₂³	1	4	yes	—
47	C₂×C₁₆×C₄₇	—	2	2	no	—
48	#108×C₄₇	—	2	4	no	—
49	#108×C₄₇	—	2	2	no	—
50	#108×C₄₇	C₂³	1	4	yes	—
51	#106×C₄₇	—	2	1	no	—
52	#106×C₄₇	C₂³	1	2	yes	—
53	#102×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	—	4	1	no	—
55	C₂⁴×C₄₇	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₄₇	—	2	2	no	—
57	C₂×D₄×C₄₇	—	4	4	no	—
58	C₂×D₄×C₄₇	—	4	1	no	—
59	C₂×D₄×C₄₇	—	4	1	no	—
60	C₂×D₄×C₄₇	—	4	1	no	—
61	C₂×D₄×C₄₇	—	4	1	no	—
62	C₂×D₄×C₄₇	—	4	1	no	—
63	C₂×D₄×C₄₇	—	8	4	no	—
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂²×C₄×C₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	—	2	1	no	—
67	C₂²×C₄×C₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	D₈×C₄₇	—	2	4	no	—
74	D₈×C₄₇	—	4	2	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	2	4	no	—
78	M₁₆×C₄₇	—	4	4	no	—
79	M₁₆×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	1	no	—
82	C₂×C₈×C₄₇	—	4	4	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	#41	1	2	yes	—
87	C₂×C₈×C₄₇	#41	1	2	yes	—
88	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	4	1	no	—
91	C₂²⋊C₄×C₄₇	—	4	4	no	—
92	C₁₆×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	—	8	2	no	—
98	C₂³×C₄₇	—	8	2	no	—
99	C₂³×C₄₇	—	8	1	no	—
100	C₂³×C₄₇	—	8	1	no	—
101	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
103	D₄×C₄₇	—	4	2	no	—
104	D₄×C₄₇	—	4	2	no	—
105	D₄×C₄₇	—	8	8	no	—
106	D₄×C₄₇	—	8	1	no	—
107	D₄×C₄₇	—	8	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	—	4	4	no	—
125	C₈×C₄₇	#123	1	2	yes	—
126	C₈×C₄₇	C₂×#41	1	2	yes	—
127	C₂×#2238	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	8	1	no	—
133	C₂²×C₄₇	—	8	2	no	—
134	C₂²×C₄₇	—	8	2	no	—
135	C₂²×C₄₇	—	8	2	no	—
136	C₂²×C₄₇	—	16	2	no	—
137	C₂²×C₄₇	—	16	2	no	—
138	C₂²×C₄₇	#481	1	1	yes	characteristic
139	C₂²×C₄₇	#481	1	1	yes	characteristic
140	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	8	2	no	—
143	C₄×C₄₇	#481	1	1	yes	characteristic
144	C₄×C₄₇	#481	1	1	yes	characteristic
145	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
146	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×#469	C₂×C₄₇	1	1	yes	characteristic
148	C₂×#430	C₂×C₄₇	1	1	yes	characteristic, complemented
149	#2238	C₂×C₄₇	1	8	yes	direct product factor
150	C₂×#365	C₂×C₄₇	1	2	yes	complemented
151	C₂×#361	C₂×C₄₇	1	2	yes	complemented
152	C₂×#349	C₂×C₄₇	1	1	yes	characteristic, complemented
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	4	2	no	—
156	C₂×C₄₇	—	16	2	no	—
157	C₂×C₄₇	#2238	1	2	yes	direct product factor
158	C₂×C₄₇	C₂×#481	1	1	yes	characteristic
159	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
160	C₂²×M₁₆	—	2	1	no	—
161	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
162	C₂×#121	—	2	1	no	—
163	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
164	#469	—	2	2	no	—
165	#469	C₂²×C₄₇	1	4	yes	—
166	#430	—	2	2	no	—
167	#430	C₂²×C₄₇	1	4	yes	complemented
168	C₂×#108	—	2	1	no	—
169	C₂×#108	C₂²×C₄₇	1	2	yes	—
170	C₂×#106	C₂²×C₄₇	1	1	yes	characteristic
171	C₂×#102	—	2	1	no	—
172	#365	C₂²×C₄₇	1	8	yes	complemented
173	#361	C₂²×C₄₇	1	8	yes	complemented
174	#349	C₂²×C₄₇	1	4	yes	complemented
175	C₄₇	C₂×#2238	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	4	1	no	—
178	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	C₂³×C₄₇	1	2	yes	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	4	2	no	—
188	C₂×M₁₆	—	4	1	no	—
189	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
190	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
191	C₂²×C₈	—	4	1	no	—
192	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
194	#121	—	2	4	no	—
195	#121	—	4	2	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	M₃₂	—	2	2	no	—
199	M₃₂	C₂³×C₄₇	1	4	yes	—
200	C₂×C₁₆	—	2	2	no	—
201	#108	—	2	4	no	—
202	#108	—	2	2	no	—
203	#108	C₂³×C₄₇	1	4	yes	—
204	#106	—	2	1	no	—
205	#106	C₂³×C₄₇	1	2	yes	—
206	#102	—	2	4	no	—
207	C₂⁴	—	4	1	no	—
208	C₂⁴	SD₁₆×C₄₇	1	1	yes	characteristic
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	8	4	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	SD₁₆×C₄₇	1	1	yes	characteristic
223	C₂²×C₄	SD₁₆×C₄₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
226	D₈	—	2	4	no	—
227	D₈	—	4	2	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	4	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
239	C₂×C₈	#41×C₄₇	1	2	yes	—
240	C₂×C₈	#41×C₄₇	1	2	yes	—
241	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₁₆	—	2	4	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	1	no	—
254	C₂³	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
255	C₂³	C₂×SD₁₆×C₄₇	1	2	yes	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	8	8	no	—
259	D₄	—	8	1	no	—
260	D₄	—	8	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
270	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
272	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
273	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
274	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	4	no	—
278	C₈	#123×C₄₇	1	2	yes	—
279	C₈	C₂×#41×C₄₇	1	2	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	1	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	16	2	no	—
289	C₂²	—	16	2	no	—
290	C₂²	C₄₇×#481	1	1	yes	characteristic
291	C₂²	C₄₇×#481	1	1	yes	characteristic
292	C₂²	C₂²×SD₁₆×C₄₇	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	8	2	no	—
295	C₄	C₄₇×#481	1	1	yes	characteristic
296	C₄	C₄₇×#481	1	1	yes	characteristic
297	C₄	C₂×#123×C₄₇	1	1	yes	characteristic
298	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	2	no	—
302	C₂	—	16	2	no	—
303	C₂	C₄₇×#2238	1	2	yes	direct product factor
304	C₂	C₂×C₄₇×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₂×C₄₇×#469	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂²×M₁₆×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#430	C₂	semidirect
3. C₄₇×#2238	C₂	direct (N × H)
4. C₂×C₄₇×#365	C₂	semidirect
5. C₂×C₄₇×#361	C₂	semidirect
6. C₂×C₄₇×#349	C₂	semidirect
15. C₄₇×#430	C₂²	semidirect
20. C₄₇×#365	C₂²	semidirect
21. C₄₇×#361	C₂²	semidirect
22. C₄₇×#349	C₂²	semidirect
127. C₂×#2238	C₄₇	direct (N × H)
148. C₂×#430	C₂×C₄₇	semidirect
149. #2238	C₂×C₄₇	direct (N × H)
150. C₂×#365	C₂×C₄₇	semidirect
151. C₂×#361	C₂×C₄₇	semidirect
152. C₂×#349	C₂×C₄₇	semidirect
157. C₂×C₄₇	#2238	direct (N × H)
167. #430	C₂²×C₄₇	semidirect
172. #365	C₂²×C₄₇	semidirect
173. #361	C₂²×C₄₇	semidirect
174. #349	C₂²×C₄₇	semidirect
175. C₄₇	C₂×#2238	direct (N × H)
303. C₂	C₄₇×#2238	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#469	2
C₂×C₄₇×#430	2
C₄₇×#2238	2
C₂×C₄₇×#365	2
C₂×C₄₇×#361	2
C₂×C₄₇×#349	2
C₂×#2238	47

Subgroups — C2×C47×#2238 #26073352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2238 #26073352