Subgroups — D₄₇×#1744 #26213737

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 280 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	2	yes	maximal
2	C₄³×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243133	C₂	1	2	yes	maximal, complemented
5	#20243132	C₂	1	4	yes	maximal, complemented
6	#336×D₄₇	C₂	1	4	yes	maximal
7	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	4	yes	—
9	C₂×C₄²×D₄₇	C₄	1	2	yes	—
10	C₂×C₄²×D₄₇	C₄	1	2	yes	—
11	C₂×C₄²×D₄₇	C₂²	1	2	yes	—
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	C₄×C₈×D₄₇	C₄	1	16	yes	complemented
16	C₄²×Dic₄₇	C₄	1	2	yes	—
17	C₄²×Dic₄₇	C₄	1	2	yes	—
18	C₄²×Dic₄₇	C₂²	1	2	yes	—
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688878	C₂²	1	4	yes	—
21	#15688877	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#336	C₂²	1	4	yes	—
25	C₂²×C₄×D₄₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂×C₈×D₄₇	—	2	16	no	—
31	C₂×C₈×D₄₇	C₂×C₄	1	16	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₄²×D₄₇	—	2	8	no	—
42	C₄²×D₄₇	—	2	8	no	—
43	C₄²×D₄₇	C₈	1	16	yes	complemented
44	C₄²×D₄₇	C₂×C₄	1	8	yes	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
46	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
47	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
53	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
59	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×D₄₇	—	2	8	no	—
63	C₂×C₄×D₄₇	—	2	8	no	—
64	C₂×C₄×D₄₇	C₄²	1	4	yes	—
65	C₂×C₄×D₄₇	C₄²	1	4	yes	—
66	C₂×C₄×D₄₇	C₂×C₈	1	16	yes	—
67	C₂×C₄×D₄₇	C₂×C₈	1	8	yes	—
68	C₂×C₄×D₄₇	M₁₆	1	8	yes	—
69	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
70	C₂×C₄×D₄₇	#41	1	8	yes	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	C₂×C₈	1	8	yes	complemented
76	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
77	C₄×Dic₄₇	—	2	4	no	—
78	C₄×Dic₄₇	—	2	4	no	—
79	C₄×Dic₄₇	—	2	8	no	—
80	C₄×Dic₄₇	—	2	8	no	—
81	C₄×Dic₄₇	C₄²	1	4	yes	—
82	C₄×Dic₄₇	C₄²	1	4	yes	—
83	C₄×Dic₄₇	C₂×C₈	1	8	yes	complemented
84	C₄×Dic₄₇	C₂×C₈	1	16	yes	complemented
85	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	—	2	8	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
88	C₈×D₄₇	—	2	32	no	—
89	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
91	C₂²×D₄₇	#117	1	4	yes	—
92	C₂²×D₄₇	C₂²×C₈	1	4	yes	—
93	C₂²×D₄₇	C₂×M₁₆	1	4	yes	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
99	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
106	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
107	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
108	C₂×Dic₄₇	#117	1	4	yes	—
109	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
110	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
111	C₂×Dic₄₇	C₂×M₁₆	1	4	yes	—
112	C₄×D₄₇	—	2	32	no	—
113	C₄×D₄₇	C₄×C₈	1	16	yes	—
114	C₄×D₄₇	#97	1	16	yes	—
115	C₄×D₄₇	#117	1	16	yes	—
116	C₈×C₄₇	—	2	8	no	—
117	C₄₇⋊C₈	—	2	8	no	—
118	C₂×#1744	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#117	1	2	yes	—
121	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
122	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
123	C₂×D₄₇	#376	1	4	yes	—
124	C₂×D₄₇	C₂×C₄×C₈	1	4	yes	—
125	C₂×D₄₇	C₂×#97	1	4	yes	—
126	C₂×D₄₇	#431	1	16	yes	—
127	C₄×C₄₇	—	2	8	no	—
128	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₄₇	C₂×#97	1	4	yes	—
130	C₄×C₄₇	C₂×#117	1	4	yes	—
131	Dic₄₇	—	2	8	no	—
132	Dic₄₇	#376	1	4	yes	—
133	Dic₄₇	C₂×C₄×C₈	1	4	yes	—
134	Dic₄₇	C₂×#97	1	4	yes	—
135	C₂²×C₄×C₈	—	47	2	no	—
136	C₂×C₄³	—	47	1	no	—
137	#1744	—	47	2	no	—
138	#1744	—	47	1	no	—
139	#1744	—	47	4	no	—
140	#1744	D₄₇	1	1	yes	characteristic, direct product factor
141	C₂×#336	—	47	4	no	—
142	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
143	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
144	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#431	1	4	yes	—
146	D₄₇	#1744	1	8	yes	direct product factor
147	C₂³×C₈	—	47	4	no	—
148	C₂²×C₄²	—	47	2	no	—
149	C₂²×C₄²	—	47	1	no	—
150	C₂²×C₄²	—	47	2	no	—
151	C₂²×C₄²	—	47	2	no	—
152	C₂×C₄×C₈	—	47	4	no	—
153	C₂×C₄×C₈	—	47	2	no	—
154	C₂×C₄×C₈	—	47	16	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
156	C₄³	—	47	2	no	—
157	C₄³	—	47	2	no	—
158	C₄³	—	47	2	no	—
159	C₄³	—	47	1	no	—
160	C₄³	C₂×D₄₇	1	1	yes	characteristic
161	#336	—	47	8	no	—
162	#336	—	47	4	no	—
163	#336	C₂×D₄₇	1	4	yes	—
164	C₄₇	C₂×#1744	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	47	1	no	—
166	C₂³×C₄	—	47	1	no	—
167	C₂³×C₄	—	47	1	no	—
168	C₂³×C₄	—	47	4	no	—
169	C₂²×C₈	—	47	4	no	—
170	C₂²×C₈	—	47	16	no	—
171	C₂²×C₈	—	94	16	no	—
172	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	47	1	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	1	no	—
178	C₂×C₄²	—	47	16	no	—
179	C₂×C₄²	—	47	8	no	—
180	C₂×C₄²	—	47	4	no	—
181	C₂×C₄²	—	47	4	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	2	no	—
184	C₂×C₄²	—	94	8	no	—
185	C₂×C₄²	—	94	8	no	—
186	C₂×C₄²	C₄×D₄₇	1	2	yes	—
187	C₂×C₄²	C₄×D₄₇	1	2	yes	—
188	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	C₄×C₈	—	47	16	no	—
191	C₄×C₈	—	47	8	no	—
192	C₄×C₈	C₄×D₄₇	1	8	yes	complemented
193	C₂⁴	—	47	1	no	—
194	C₂²×C₄	—	47	1	no	—
195	C₂²×C₄	—	47	1	no	—
196	C₂²×C₄	—	47	1	no	—
197	C₂²×C₄	—	47	4	no	—
198	C₂²×C₄	—	47	4	no	—
199	C₂²×C₄	—	47	8	no	—
200	C₂²×C₄	—	47	8	no	—
201	C₂²×C₄	—	47	4	no	—
202	C₂²×C₄	—	47	8	no	—
203	C₂²×C₄	—	47	4	no	—
204	C₂²×C₄	—	47	16	no	—
205	C₂²×C₄	—	94	8	no	—
206	C₂²×C₄	—	94	8	no	—
207	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
208	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	47	8	no	—
213	C₂×C₈	—	94	32	no	—
214	C₂×C₈	—	94	8	no	—
215	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	47	4	no	—
219	C₄²	—	47	4	no	—
220	C₄²	—	47	16	no	—
221	C₄²	—	47	8	no	—
222	C₄²	—	47	4	no	—
223	C₄²	—	94	8	no	—
224	C₄²	—	94	8	no	—
225	C₄²	—	94	4	no	—
226	C₄²	—	94	4	no	—
227	C₄²	C₈×D₄₇	1	8	yes	complemented
228	C₄²	C₂×C₄×D₄₇	1	4	yes	—
229	C₂³	—	47	2	no	—
230	C₂³	—	47	4	no	—
231	C₂³	—	47	4	no	—
232	C₂³	—	47	4	no	—
233	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	47	2	no	—
237	C₂×C₄	—	47	2	no	—
238	C₂×C₄	—	47	2	no	—
239	C₂×C₄	—	47	4	no	—
240	C₂×C₄	—	47	4	no	—
241	C₂×C₄	—	47	4	no	—
242	C₂×C₄	—	47	16	no	—
243	C₂×C₄	—	47	16	no	—
244	C₂×C₄	—	47	16	no	—
245	C₂×C₄	—	47	8	no	—
246	C₂×C₄	—	94	32	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	C₄²×D₄₇	1	2	yes	—
250	C₂×C₄	C₄²×D₄₇	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₄₇	1	8	yes	—
252	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
253	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
254	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
255	C₂×C₄	#41×D₄₇	1	4	yes	—
256	C₈	—	2	8	no	—
257	C₈	—	94	8	no	—
258	C₂²	—	47	16	no	—
259	C₂²	—	47	4	no	—
260	C₂²	—	47	4	no	—
261	C₂²	—	47	4	no	—
262	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
263	C₂²	#117×D₄₇	1	2	yes	—
264	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
265	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
266	C₄	—	2	8	no	—
267	C₄	—	47	4	no	—
268	C₄	—	47	4	no	—
269	C₄	—	47	4	no	—
270	C₄	—	94	8	no	—
271	C₄	C₄×C₈×D₄₇	1	4	yes	—
272	C₄	#97×D₄₇	1	4	yes	—
273	C₄	#117×D₄₇	1	4	yes	—
274	C₂	—	47	8	no	—
275	C₂	#376×D₄₇	1	1	yes	characteristic
276	C₂	C₂×C₄×C₈×D₄₇	1	1	yes	characteristic
277	C₂	C₂×#97×D₄₇	1	1	yes	characteristic
278	C₂	#431×D₄₇	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1744	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1744	D₄₇
hypocenter	C₄₇	C₂×#1744
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
3. #20243134	C₂	semidirect
4. #20243133	C₂	semidirect
5. #20243132	C₂	semidirect
7. C₄₇×#1744	C₂	semidirect
15. C₄×C₈×D₄₇	C₄	semidirect
43. C₄²×D₄₇	C₈	semidirect
45. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
46. C₈×Dic₄₇	C₂×C₄	semidirect
52. C₄×C₈×C₄₇	C₂×C₄	semidirect
75. C₄²×C₄₇	C₂×C₈	semidirect
83. C₄×Dic₄₇	C₂×C₈	semidirect
84. C₄×Dic₄₇	C₂×C₈	semidirect
140. #1744	D₄₇	direct (N × H)
146. D₄₇	#1744	direct (N × H)
164. C₄₇	C₂×#1744	semidirect
192. C₄×C₈	C₄×D₄₇	semidirect
227. 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
C₂×C₄×C₈×D₄₇	2
C₄³×D₄₇	2
#20243134	2
#20243133	2
#20243132	2
#336×D₄₇	2
C₄₇×#1744	2
C₂×#1744	47

Subgroups — D47×#1744 #26213737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1744 #26213737