Subgroups — D₄₇×#1834 #26216459

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#384×D₄₇	C₂	1	2	yes	maximal
3	#20243516	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243515	C₂	1	2	yes	maximal, complemented
5	#20243514	C₂	1	4	yes	maximal, complemented
6	#340×D₄₇	C₂	1	4	yes	maximal
7	C₄₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×D₄₇	—	2	2	no	—
10	C₂×C₄⋊C₄×D₄₇	—	2	2	no	—
11	C₂×C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
15	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
16	#15688993	C₂²	1	2	yes	—
17	#15688992	—	2	2	no	—
18	#15688992	—	2	2	no	—
19	#15688992	C₂²	1	4	yes	—
20	#15688886	C₂²	1	4	yes	—
21	#15688885	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	2	yes	—
24	C₄₇×#340	C₂²	1	4	yes	—
25	C₂²×C₄×D₄₇	—	2	4	no	—
26	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₈×D₄₇	—	2	8	no	—
31	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	4	no	—
36	C₂×Dic₄₇⋊C₄	—	2	4	no	—
37	C₂×Dic₄₇⋊C₄	—	2	4	no	—
38	C₄⋊C₄×D₄₇	—	2	16	no	—
39	C₄⋊C₄×D₄₇	—	4	8	no	—
40	C₄⋊C₄×D₄₇	—	4	8	no	—
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₄²×D₄₇	C₂×C₄	1	4	yes	—
45	C₄²×D₄₇	Q₈	1	4	yes	—
46	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
47	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
54	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×D₄₇	—	4	8	no	—
64	C₂×C₄×D₄₇	—	4	8	no	—
65	C₂×C₄×D₄₇	—	4	8	no	—
66	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
67	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×D₄₇	D₈	1	4	yes	—
69	C₂×C₄×D₄₇	SD₁₆	1	8	yes	—
70	C₂×C₄×D₄₇	Q₁₆	1	4	yes	—
71	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
73	C₂×C₄×D₄₇	#41	1	8	yes	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
76	C₄⋊C₄×C₄₇	—	2	8	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	Dic₄₇⋊C₄	—	4	8	no	—
85	Dic₄₇⋊C₄	—	4	8	no	—
86	Dic₄₇⋊C₄	—	4	8	no	—
87	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
88	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
89	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
90	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
93	C₈×D₄₇	—	2	16	no	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂²×D₄₇	#124	1	4	yes	—
97	C₂²×D₄₇	C₂×D₈	1	2	yes	—
98	C₂²×D₄₇	C₂×SD₁₆	1	4	yes	—
99	C₂²×D₄₇	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
107	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×Dic₄₇	#124	1	4	yes	—
118	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
119	C₂×Dic₄₇	C₂×SD₁₆	1	4	yes	—
120	C₂×Dic₄₇	C₂×Q₁₆	1	2	yes	—
121	C₄×D₄₇	—	8	16	no	—
122	C₄×D₄₇	#106	1	16	yes	—
123	C₄×D₄₇	#107	1	16	yes	—
124	C₄×D₄₇	#117	1	8	yes	—
125	C₄×D₄₇	#125	1	8	yes	—
126	C₈×C₄₇	—	2	4	no	—
127	C₄₇⋊C₈	—	2	4	no	—
128	C₂×#1834	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#124	1	2	yes	—
131	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
133	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
134	C₂×D₄₇	#382	1	4	yes	—
135	C₂×D₄₇	C₂×#106	1	4	yes	—
136	C₂×D₄₇	C₂×#107	1	4	yes	—
137	C₂×D₄₇	#486	1	8	yes	—
138	C₂×D₄₇	#487	1	8	yes	—
139	C₄×C₄₇	—	8	4	no	—
140	C₄×C₄₇	C₂×#106	1	4	yes	—
141	C₄×C₄₇	C₂×#107	1	4	yes	—
142	C₄×C₄₇	C₂×#117	1	2	yes	—
143	C₄×C₄₇	C₂×#125	1	2	yes	—
144	Dic₄₇	—	8	4	no	—
145	Dic₄₇	#382	1	4	yes	—
146	Dic₄₇	C₂×#106	1	4	yes	—
147	Dic₄₇	C₂×#107	1	4	yes	—
148	C₂²×C₄×C₈	—	47	1	no	—
149	C₂×#384	—	47	2	no	—
150	#1834	—	47	1	no	—
151	#1834	—	47	2	no	—
152	#1834	—	47	4	no	—
153	#1834	D₄₇	1	1	yes	characteristic, direct product factor
154	C₂×#340	—	47	4	no	—
155	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
156	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
157	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#486	1	2	yes	—
159	C₂×C₄₇	C₂×#487	1	2	yes	—
160	D₄₇	#1834	1	8	yes	direct product factor
161	C₂³×C₈	—	47	2	no	—
162	C₂²×C₄⋊C₄	—	47	4	no	—
163	C₂²×C₄⋊C₄	—	94	2	no	—
164	C₂²×C₄⋊C₄	—	94	2	no	—
165	C₂²×C₄²	—	47	1	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	C₂×C₄×C₈	—	47	1	no	—
168	C₂×C₄×C₈	—	47	8	no	—
169	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
170	#384	—	47	4	no	—
171	#384	—	47	2	no	—
172	#384	—	94	2	no	—
173	#384	—	94	2	no	—
174	#384	C₂×D₄₇	1	2	yes	—
175	#340	—	47	4	no	—
176	#340	—	47	8	no	—
177	#340	C₂×D₄₇	1	4	yes	—
178	C₄₇	C₂×#1834	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	—	47	1	no	—
180	C₂³×C₄	—	47	1	no	—
181	C₂³×C₄	—	47	1	no	—
182	C₂³×C₄	—	94	4	no	—
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	—	47	8	no	—
185	C₂²×C₈	—	94	8	no	—
186	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	47	4	no	—
190	C₂×C₄⋊C₄	—	94	16	no	—
191	C₂×C₄⋊C₄	—	94	4	no	—
192	C₂×C₄⋊C₄	—	94	4	no	—
193	C₂×C₄⋊C₄	—	94	4	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	—	188	8	no	—
197	C₂×C₄⋊C₄	—	188	8	no	—
198	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	—	47	1	no	—
201	C₂×C₄²	—	47	1	no	—
202	C₂×C₄²	—	47	4	no	—
203	C₂×C₄²	—	47	4	no	—
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₄×C₈	—	47	8	no	—
206	C₄×C₈	—	47	4	no	—
207	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
208	C₂⁴	—	47	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	47	1	no	—
211	C₂²×C₄	—	47	1	no	—
212	C₂²×C₄	—	47	1	no	—
213	C₂²×C₄	—	47	4	no	—
214	C₂²×C₄	—	47	4	no	—
215	C₂²×C₄	—	47	4	no	—
216	C₂²×C₄	—	47	8	no	—
217	C₂²×C₄	—	47	4	no	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	47	2	no	—
220	C₂²×C₄	—	47	8	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	188	8	no	—
223	C₂²×C₄	—	188	8	no	—
224	C₂²×C₄	—	188	8	no	—
225	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	47	4	no	—
230	C₂×C₈	—	94	16	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	94	8	no	—
237	C₄⋊C₄	—	188	8	no	—
238	C₄⋊C₄	—	188	8	no	—
239	C₄⋊C₄	—	188	8	no	—
240	C₄⋊C₄	—	188	4	no	—
241	C₄⋊C₄	—	188	4	no	—
242	C₄²	—	47	4	no	—
243	C₄²	—	47	4	no	—
244	C₄²	—	47	2	no	—
245	C₄²	—	47	2	no	—
246	C₄²	C₂×C₄×D₄₇	1	2	yes	—
247	C₄²	Q₈×D₄₇	1	2	yes	—
248	C₂³	—	47	2	no	—
249	C₂³	—	47	2	no	—
250	C₂³	—	47	4	no	—
251	C₂³	—	47	4	no	—
252	C₂³	—	47	2	no	—
253	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	47	2	no	—
258	C₂×C₄	—	47	2	no	—
259	C₂×C₄	—	47	2	no	—
260	C₂×C₄	—	47	2	no	—
261	C₂×C₄	—	47	4	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	47	2	no	—
264	C₂×C₄	—	47	8	no	—
265	C₂×C₄	—	47	8	no	—
266	C₂×C₄	—	47	16	no	—
267	C₂×C₄	—	47	16	no	—
268	C₂×C₄	—	376	16	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
273	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
274	C₂×C₄	D₈×D₄₇	1	2	yes	—
275	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
276	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
277	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
279	C₂×C₄	#41×D₄₇	1	4	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	94	4	no	—
282	C₂²	—	47	8	no	—
283	C₂²	—	47	4	no	—
284	C₂²	—	47	4	no	—
285	C₂²	—	47	4	no	—
286	C₂²	—	47	8	no	—
287	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
288	C₂²	#124×D₄₇	1	2	yes	—
289	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
290	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
291	C₂²	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
292	C₄	—	8	4	no	—
293	C₄	—	47	4	no	—
294	C₄	—	47	4	no	—
295	C₄	—	47	4	no	—
296	C₄	—	376	4	no	—
297	C₄	#106×D₄₇	1	4	yes	—
298	C₄	#107×D₄₇	1	4	yes	—
299	C₄	#117×D₄₇	1	2	yes	—
300	C₄	#125×D₄₇	1	2	yes	—
301	C₂	—	47	8	no	—
302	C₂	#382×D₄₇	1	1	yes	characteristic
303	C₂	C₂×#106×D₄₇	1	1	yes	characteristic
304	C₂	C₂×#107×D₄₇	1	1	yes	characteristic
305	C₂	#486×D₄₇	1	2	yes	—
306	C₂	#487×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1834	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1834	D₄₇
hypocenter	C₄₇	C₂×#1834
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
3. #20243516	C₂	semidirect
4. #20243515	C₂	semidirect
5. #20243514	C₂	semidirect
7. C₄₇×#1834	C₂	semidirect
15. C₄×C₈×D₄₇	C₄	semidirect
46. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
47. C₈×Dic₄₇	C₂×C₄	semidirect
53. C₄×C₈×C₄₇	C₂×C₄	semidirect
153. #1834	D₄₇	direct (N × H)
160. D₄₇	#1834	direct (N × H)
178. C₄₇	C₂×#1834	semidirect
207. C₄×C₈	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₇	2
#384×D₄₇	2
#20243516	2
#20243515	2
#20243514	2
#340×D₄₇	2
C₄₇×#1834	2
C₂×#1834	47

Subgroups — D47×#1834 #26216459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1834 #26216459