Subgroups — D₄₇×#1276 #26105712

Order 12032.

974 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	#431×D₄₇	C₂	1	2	yes	maximal
2	C₄³×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241221	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241220	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₄₇	—	2	2	no	—
7	C₂×C₄²×D₄₇	C₄	1	2	yes	—
8	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
9	#15689176	C₂²	1	2	yes	—
10	#15689175	C₄	1	4	yes	complemented
11	#15689174	—	2	2	no	—
12	#15689173	—	2	2	no	—
13	C₄⋊C₈×D₄₇	C₄	1	4	yes	complemented
14	C₂²⋊C₈×D₄₇	—	2	2	no	—
15	C₄×C₈×D₄₇	—	2	2	no	—
16	C₄²×Dic₄₇	—	2	2	no	—
17	C₄²×Dic₄₇	C₄	1	2	yes	—
18	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#431	C₂²	1	2	yes	—
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₄₇	—	2	2	no	—
22	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×D₄₇	D₄	1	2	yes	—
24	C₂×C₈×D₄₇	—	2	4	no	—
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	4	no	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
31	C₄²×D₄₇	—	2	16	no	—
32	C₄²×D₄₇	—	2	4	no	—
33	C₄²×D₄₇	C₈	1	8	yes	complemented
34	C₄²×D₄₇	C₂×C₄	1	2	yes	—
35	C₄²×D₄₇	C₂×C₄	1	2	yes	—
36	C₄²×D₄₇	D₄	1	1	yes	characteristic
37	C₄²×D₄₇	D₄	1	1	yes	characteristic
38	C₄²×D₄₇	Q₈	1	1	yes	characteristic
39	C₄²×D₄₇	Q₈	1	1	yes	characteristic
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	complemented
41	Dic₄₇⋊C₈	—	2	4	no	—
42	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
43	(C₂×D₄₇)⋊C₈	—	2	4	no	—
44	C₄×C₄₇⋊C₈	—	2	2	no	—
45	C₈×Dic₄₇	—	2	4	no	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	complemented
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	C₄×C₈×C₄₇	—	2	2	no	—
52	C₂³×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×Dic₄₇	—	2	2	no	—
57	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₄₇	—	2	16	no	—
60	C₂×C₄×D₄₇	—	2	8	no	—
61	C₂×C₄×D₄₇	—	2	8	no	—
62	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
63	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
64	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₄₇	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₄₇	M₁₆	1	4	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	8	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	C₂×C₈	1	4	yes	complemented
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₄×Dic₄₇	—	2	8	no	—
80	C₄×Dic₄₇	—	2	16	no	—
81	C₄×Dic₄₇	—	2	2	no	—
82	C₄×Dic₄₇	—	2	8	no	—
83	C₄×Dic₄₇	C₄²	1	2	yes	—
84	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
85	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
86	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
89	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
90	C₄×Dic₄₇	C₂×C₈	1	4	yes	complemented
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₈×D₄₇	—	4	8	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	4	no	—
95	C₂²×D₄₇	#95	1	1	yes	characteristic
96	C₂²×D₄₇	#95	1	1	yes	characteristic
97	C₂²×D₄₇	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
107	C₂×Dic₄₇	—	2	8	no	—
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	#95	1	2	yes	—
111	C₂×Dic₄₇	#95	1	1	yes	characteristic
112	C₂×Dic₄₇	#95	1	1	yes	characteristic
113	C₂×Dic₄₇	#95	1	2	yes	—
114	C₂×Dic₄₇	C₂²⋊C₈	1	4	yes	—
115	C₄×D₄₇	—	2	32	no	—
116	C₄×D₄₇	—	2	4	no	—
117	C₄×D₄₇	—	2	4	no	—
118	C₄×D₄₇	—	2	4	no	—
119	C₄×D₄₇	C₄×C₈	1	2	yes	—
120	C₄×D₄₇	C₄×C₈	1	2	yes	—
121	C₄×D₄₇	#97	1	2	yes	—
122	C₄×D₄₇	#97	1	2	yes	—
123	C₄×D₄₇	#104	1	8	yes	—
124	C₄×D₄₇	C₄⋊C₈	1	4	yes	—
125	C₄×D₄₇	C₄⋊C₈	1	4	yes	—
126	C₈×C₄₇	—	4	4	no	—
127	C₄₇⋊C₈	—	4	4	no	—
128	C₂×#1276	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	2	8	no	—
134	C₂×D₄₇	#336	1	2	yes	—
135	C₂×D₄₇	#336	1	2	yes	—
136	C₂×D₄₇	#339	1	4	yes	—
137	C₄×C₄₇	—	2	8	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
141	C₄×C₄₇	C₂×#104	1	4	yes	—
142	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
143	Dic₄₇	—	2	8	no	—
144	Dic₄₇	—	2	2	no	—
145	Dic₄₇	#336	1	2	yes	—
146	Dic₄₇	#336	1	2	yes	—
147	Dic₄₇	#339	1	4	yes	—
148	C₂×#431	—	47	2	no	—
149	C₂×C₄³	—	47	1	no	—
150	#1276	—	47	2	no	—
151	#1276	—	47	1	no	—
152	#1276	D₄₇	1	1	yes	characteristic, direct product factor
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#339	1	2	yes	—
156	D₄₇	—	2	2	no	—
157	D₄₇	#1276	1	4	yes	direct product factor
158	C₂²×C₄²	—	47	1	no	—
159	C₂²×C₄²	—	47	2	no	—
160	C₂²×C₄²	—	94	2	no	—
161	#431	—	47	2	no	—
162	#431	—	47	4	no	—
163	#431	—	94	2	no	—
164	#431	—	94	2	no	—
165	#431	C₂×D₄₇	1	2	yes	—
166	C₂×C₄⋊C₈	—	47	4	no	—
167	C₂×C₂²⋊C₈	—	94	2	no	—
168	C₂×C₄×C₈	—	94	2	no	—
169	C₄³	—	47	2	no	—
170	C₄³	—	47	1	no	—
171	C₄³	—	94	2	no	—
172	C₄³	C₂×D₄₇	1	1	yes	characteristic
173	C₄₇	C₂×#1276	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	47	1	no	—
175	C₂³×C₄	—	47	2	no	—
176	C₂³×C₄	—	94	2	no	—
177	C₂²×C₈	—	94	4	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	47	1	no	—
180	C₂×C₄²	—	47	2	no	—
181	C₂×C₄²	—	47	2	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	1	no	—
184	C₂×C₄²	—	47	1	no	—
185	C₂×C₄²	—	47	1	no	—
186	C₂×C₄²	—	47	1	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	—	47	8	no	—
189	C₂×C₄²	—	94	4	no	—
190	C₂×C₄²	—	94	16	no	—
191	C₂×C₄²	—	94	4	no	—
192	C₂×C₄²	—	94	2	no	—
193	C₂×C₄²	—	94	2	no	—
194	C₂×C₄²	C₄×D₄₇	1	2	yes	—
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊C₈	—	47	4	no	—
197	C₄⋊C₈	—	94	4	no	—
198	C₄⋊C₈	C₄×D₄₇	1	4	yes	complemented
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	94	4	no	—
201	C₂²⋊C₈	—	94	2	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	—	94	4	no	—
204	C₄×C₈	—	94	2	no	—
205	C₂⁴	—	47	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	47	1	no	—
208	C₂²×C₄	—	47	1	no	—
209	C₂²×C₄	—	47	1	no	—
210	C₂²×C₄	—	47	2	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	2	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	—	47	2	no	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	94	8	no	—
220	C₂²×C₄	—	94	8	no	—
221	C₂²×C₄	—	94	16	no	—
222	C₂²×C₄	—	94	2	no	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₄₇	1	2	yes	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	188	8	no	—
228	C₄²	—	2	8	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	47	2	no	—
231	C₄²	—	47	2	no	—
232	C₄²	—	47	2	no	—
233	C₄²	—	47	2	no	—
234	C₄²	—	47	1	no	—
235	C₄²	—	47	2	no	—
236	C₄²	—	47	1	no	—
237	C₄²	—	47	4	no	—
238	C₄²	—	94	8	no	—
239	C₄²	—	94	16	no	—
240	C₄²	—	94	8	no	—
241	C₄²	—	94	2	no	—
242	C₄²	C₈×D₄₇	1	4	yes	complemented
243	C₄²	C₂×C₄×D₄₇	1	2	yes	—
244	C₄²	D₄×D₄₇	1	1	yes	characteristic
245	C₄²	Q₈×D₄₇	1	1	yes	characteristic
246	C₂³	—	47	1	no	—
247	C₂³	—	47	1	no	—
248	C₂³	—	47	4	no	—
249	C₂³	—	94	4	no	—
250	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	47	1	no	—
255	C₂×C₄	—	47	1	no	—
256	C₂×C₄	—	47	2	no	—
257	C₂×C₄	—	47	4	no	—
258	C₂×C₄	—	47	2	no	—
259	C₂×C₄	—	47	2	no	—
260	C₂×C₄	—	47	2	no	—
261	C₂×C₄	—	47	2	no	—
262	C₂×C₄	—	47	8	no	—
263	C₂×C₄	—	47	4	no	—
264	C₂×C₄	—	47	4	no	—
265	C₂×C₄	—	47	2	no	—
266	C₂×C₄	—	94	4	no	—
267	C₂×C₄	—	94	4	no	—
268	C₂×C₄	—	94	32	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	8	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
276	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
278	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
279	C₈	—	4	4	no	—
280	C₈	—	188	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	47	2	no	—
283	C₂²	—	47	4	no	—
284	C₂²	—	47	2	no	—
285	C₂²	—	94	8	no	—
286	C₂²	—	94	2	no	—
287	C₂²	#95×D₄₇	1	1	yes	characteristic
288	C₂²	C₂²⋊C₈×D₄₇	1	2	yes	—
289	C₄	—	2	8	no	—
290	C₄	—	2	2	no	—
291	C₄	—	47	2	no	—
292	C₄	—	47	2	no	—
293	C₄	—	47	4	no	—
294	C₄	—	94	8	no	—
295	C₄	—	94	2	no	—
296	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
297	C₄	#97×D₄₇	1	1	yes	characteristic
298	C₄	#104×D₄₇	1	4	yes	—
299	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	47	4	no	—
302	C₂	—	94	2	no	—
303	C₂	#336×D₄₇	1	1	yes	characteristic
304	C₂	#339×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1276	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂³×D₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1276	D₄₇
hypocenter	C₄₇	C₂×#1276
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. #20241221	C₂	semidirect
4. #20241220	C₂	semidirect
5. C₄₇×#1276	C₂	semidirect
10. #15689175	C₄	semidirect
13. C₄⋊C₈×D₄₇	C₄	semidirect
33. C₄²×D₄₇	C₈	semidirect
40. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
49. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
75. C₄²×C₄₇	C₂×C₈	semidirect
90. C₄×Dic₄₇	C₂×C₈	semidirect
152. #1276	D₄₇	direct (N × H)
157. D₄₇	#1276	direct (N × H)
173. C₄₇	C₂×#1276	semidirect
198. C₄⋊C₈	C₄×D₄₇	semidirect
242. C₄²	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#431×D₄₇	2
C₄³×D₄₇	2
#20241221	2
#20241220	2
C₄₇×#1276	2
C₂×#1276	47

Subgroups — D47×#1276 #26105712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1276 #26105712