Subgroups — D₄₇×#1274 #26105672

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 298 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	#376×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241217	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241216	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₄₇	C₄	1	2	yes	—
7	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
8	#15689176	C₂²	1	2	yes	—
9	#15689175	—	2	2	no	—
10	#15689174	C₄	1	4	yes	complemented
11	#15689173	—	2	2	no	—
12	C₄⋊C₈×D₄₇	—	2	2	no	—
13	C₂²⋊C₈×D₄₇	—	2	2	no	—
14	C₄×C₈×D₄₇	C₄	1	4	yes	complemented
15	#15688969	C₄	1	2	yes	—
16	#15688969	C₂²	1	1	yes	characteristic
17	#15688968	—	2	2	no	—
18	#95×D₄₇	—	2	2	no	—
19	C₄₇×#431	C₂²	1	2	yes	—
20	C₄₇×#376	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	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
29	C₄²×D₄₇	—	2	4	no	—
30	C₄²×D₄₇	C₈	1	8	yes	complemented
31	C₄²×D₄₇	C₂×C₄	1	2	yes	—
32	C₄²×D₄₇	C₂×C₄	1	2	yes	—
33	C₄²×D₄₇	D₄	1	1	yes	characteristic
34	C₄²×D₄₇	D₄	1	1	yes	characteristic
35	C₄²×D₄₇	Q₈	1	1	yes	characteristic
36	C₄²×D₄₇	Q₈	1	1	yes	characteristic
37	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
38	Dic₄₇⋊C₈	—	2	4	no	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
40	(C₂×D₄₇)⋊C₈	—	2	4	no	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
42	C₈×Dic₄₇	—	2	4	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₄₇⋊#95	—	2	4	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₇	—	2	2	no	—
48	C₂²⋊C₈×C₄₇	—	2	2	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
50	#95×C₄₇	—	2	2	no	—
51	C₂³×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×Dic₄₇	—	2	2	no	—
56	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₄₇	—	2	8	no	—
59	C₂×C₄×D₄₇	—	4	4	no	—
60	C₂×C₄×D₄₇	—	4	8	no	—
61	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
62	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
63	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₄₇	C₂×C₈	1	4	yes	—
70	C₂×C₄×D₄₇	M₁₆	1	4	yes	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	C₂×C₈	1	4	yes	complemented
74	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
77	C₄×Dic₄₇	—	2	2	no	—
78	C₄×Dic₄₇	—	2	8	no	—
79	C₄×Dic₄₇	C₄²	1	2	yes	—
80	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
81	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
83	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
84	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
85	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
86	C₄×Dic₄₇	C₂×C₈	1	4	yes	complemented
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₈×D₄₇	—	4	8	no	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×D₄₇	—	2	4	no	—
91	C₂²×D₄₇	#95	1	1	yes	characteristic
92	C₂²×D₄₇	#95	1	1	yes	characteristic
93	C₂²×D₄₇	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	#95	1	2	yes	—
107	C₂×Dic₄₇	#95	1	2	yes	—
108	C₂×Dic₄₇	#95	1	1	yes	characteristic
109	C₂×Dic₄₇	#95	1	1	yes	characteristic
110	C₂×Dic₄₇	C₂²⋊C₈	1	4	yes	—
111	C₄×D₄₇	—	2	4	no	—
112	C₄×D₄₇	—	2	4	no	—
113	C₄×D₄₇	—	4	2	no	—
114	C₄×D₄₇	—	4	2	no	—
115	C₄×D₄₇	—	4	16	no	—
116	C₄×D₄₇	C₄×C₈	1	2	yes	—
117	C₄×D₄₇	C₄×C₈	1	2	yes	—
118	C₄×D₄₇	#97	1	2	yes	—
119	C₄×D₄₇	#97	1	2	yes	—
120	C₄×D₄₇	C₄⋊C₈	1	4	yes	—
121	C₄×D₄₇	C₄⋊C₈	1	4	yes	—
122	C₈×C₄₇	—	4	4	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	C₂×#1274	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
128	C₂×D₄₇	—	2	2	no	—
129	C₂×D₄₇	—	2	8	no	—
130	C₂×D₄₇	#336	1	2	yes	—
131	C₂×D₄₇	#336	1	2	yes	—
132	C₂×D₄₇	H(3,ℤ/4)	1	4	yes	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
139	Dic₄₇	—	2	2	no	—
140	Dic₄₇	—	2	2	no	—
141	Dic₄₇	—	4	4	no	—
142	Dic₄₇	#336	1	2	yes	—
143	Dic₄₇	#336	1	2	yes	—
144	C₂×#431	—	47	2	no	—
145	C₂×#376	—	47	1	no	—
146	#1274	—	47	2	no	—
147	#1274	—	47	1	no	—
148	#1274	D₄₇	1	1	yes	characteristic, direct product factor
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
151	C₂×C₄₇	C₂×H(3,ℤ/4)	1	2	yes	—
152	D₄₇	—	2	2	no	—
153	D₄₇	#1274	1	4	yes	direct product factor
154	C₂²×C₄²	—	47	1	no	—
155	C₂²×C₄²	—	47	2	no	—
156	#431	—	47	2	no	—
157	#431	—	47	4	no	—
158	#431	—	94	2	no	—
159	#431	—	94	2	no	—
160	#431	C₂×D₄₇	1	2	yes	—
161	C₂×C₄⋊C₈	—	94	2	no	—
162	C₂×C₂²⋊C₈	—	94	2	no	—
163	C₂×C₄×C₈	—	47	4	no	—
164	#376	—	47	2	no	—
165	#376	—	47	1	no	—
166	#376	—	94	2	no	—
167	#376	C₂×D₄₇	1	1	yes	characteristic
168	C₂×#95	—	94	2	no	—
169	C₄₇	C₂×#1274	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	47	1	no	—
171	C₂³×C₄	—	47	2	no	—
172	C₂³×C₄	—	94	2	no	—
173	C₂²×C₈	—	94	4	no	—
174	C₂×C₄²	—	47	1	no	—
175	C₂×C₄²	—	47	2	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	1	no	—
179	C₂×C₄²	—	47	1	no	—
180	C₂×C₄²	—	47	1	no	—
181	C₂×C₄²	—	47	1	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	8	no	—
184	C₂×C₄²	—	94	4	no	—
185	C₂×C₄²	—	94	2	no	—
186	C₂×C₄²	C₄×D₄₇	1	2	yes	—
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	94	4	no	—
190	C₄⋊C₈	—	94	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	94	4	no	—
193	C₂²⋊C₈	—	94	2	no	—
194	C₄×C₈	—	47	4	no	—
195	C₄×C₈	—	94	4	no	—
196	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
197	#95	—	2	2	no	—
198	#95	—	94	4	no	—
199	#95	—	94	2	no	—
200	C₂⁴	—	47	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	47	1	no	—
203	C₂²×C₄	—	47	1	no	—
204	C₂²×C₄	—	47	1	no	—
205	C₂²×C₄	—	47	1	no	—
206	C₂²×C₄	—	47	1	no	—
207	C₂²×C₄	—	47	4	no	—
208	C₂²×C₄	—	47	4	no	—
209	C₂²×C₄	—	47	2	no	—
210	C₂²×C₄	—	47	2	no	—
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	47	2	no	—
213	C₂²×C₄	—	47	2	no	—
214	C₂²×C₄	—	94	8	no	—
215	C₂²×C₄	—	94	2	no	—
216	C₂²×C₄	—	188	4	no	—
217	C₂²×C₄	—	188	8	no	—
218	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₄₇	1	2	yes	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	188	8	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	47	2	no	—
225	C₄²	—	47	2	no	—
226	C₄²	—	47	2	no	—
227	C₄²	—	47	2	no	—
228	C₄²	—	47	2	no	—
229	C₄²	—	47	1	no	—
230	C₄²	—	47	1	no	—
231	C₄²	—	47	4	no	—
232	C₄²	—	94	8	no	—
233	C₄²	—	94	2	no	—
234	C₄²	C₈×D₄₇	1	4	yes	complemented
235	C₄²	C₂×C₄×D₄₇	1	2	yes	—
236	C₄²	D₄×D₄₇	1	1	yes	characteristic
237	C₄²	Q₈×D₄₇	1	1	yes	characteristic
238	C₂³	—	47	1	no	—
239	C₂³	—	47	1	no	—
240	C₂³	—	47	4	no	—
241	C₂³	—	94	4	no	—
242	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	47	1	no	—
247	C₂×C₄	—	47	1	no	—
248	C₂×C₄	—	47	4	no	—
249	C₂×C₄	—	47	2	no	—
250	C₂×C₄	—	47	2	no	—
251	C₂×C₄	—	47	2	no	—
252	C₂×C₄	—	47	2	no	—
253	C₂×C₄	—	47	4	no	—
254	C₂×C₄	—	47	4	no	—
255	C₂×C₄	—	47	2	no	—
256	C₂×C₄	—	47	2	no	—
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	—	94	4	no	—
259	C₂×C₄	—	94	4	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	—	188	2	no	—
262	C₂×C₄	—	188	16	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
267	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
270	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₈	—	188	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	47	2	no	—
275	C₂²	—	47	4	no	—
276	C₂²	—	47	2	no	—
277	C₂²	—	94	8	no	—
278	C₂²	—	94	2	no	—
279	C₂²	#95×D₄₇	1	1	yes	characteristic
280	C₂²	C₂²⋊C₈×D₄₇	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	4	no	—
284	C₄	—	47	2	no	—
285	C₄	—	47	2	no	—
286	C₄	—	94	2	no	—
287	C₄	—	94	2	no	—
288	C₄	—	188	4	no	—
289	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
290	C₄	#97×D₄₇	1	1	yes	characteristic
291	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	47	4	no	—
294	C₂	—	94	2	no	—
295	C₂	#336×D₄₇	1	1	yes	characteristic
296	C₂	H(3,ℤ/4)×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂³×D₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1274	D₄₇
hypocenter	C₄₇	C₂×#1274
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #1274

Complements

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

N	H ≅ G/N	type
3. #20241217	C₂	semidirect
4. #20241216	C₂	semidirect
5. C₄₇×#1274	C₂	semidirect
10. #15689174	C₄	semidirect
14. C₄×C₈×D₄₇	C₄	semidirect
30. C₄²×D₄₇	C₈	semidirect
41. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
49. C₄×C₈×C₄₇	C₂×C₄	semidirect
73. C₄²×C₄₇	C₂×C₈	semidirect
86. C₄×Dic₄₇	C₂×C₈	semidirect
148. #1274	D₄₇	direct (N × H)
153. D₄₇	#1274	direct (N × H)
169. C₄₇	C₂×#1274	semidirect
196. C₄×C₈	C₄×D₄₇	semidirect
234. 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
#376×D₄₇	2
#20241217	2
#20241216	2
C₄₇×#1274	2
C₂×#1274	47

Subgroups — D47×#1274 #26105672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1274 #26105672