Subgroups — D₄₇×#1401 #26136483

Order 12032.

294 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₄₇	C₂	1	1	yes	maximal, characteristic
2	M₆₄×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241531	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241530	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241529	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1401	C₂	1	1	yes	maximal, characteristic, complemented
8	#131×D₄₇	C₄	1	1	yes	characteristic
9	#15689541	—	2	1	no	—
10	#15689541	C₄	1	1	yes	characteristic
11	#15689540	C₄	1	1	yes	characteristic
12	#15689539	—	2	1	no	—
13	#15689539	C₂²	1	1	yes	characteristic
14	M₃₂×D₄₇	—	2	1	no	—
15	M₃₂×D₄₇	C₄	1	1	yes	characteristic
16	C₂×C₁₆×D₄₇	—	2	1	no	—
17	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
18	#15688957	C₂²	1	1	yes	characteristic, complemented
19	#15688956	—	2	1	no	—
20	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic, complemented
21	C₃₂×D₄₇	—	2	1	no	—
22	C₃₂×D₄₇	—	2	1	no	—
23	C₃₂×D₄₇	—	2	1	no	—
24	C₄₇×#504	C₂²	1	1	yes	characteristic
25	C₄₇×M₆₄	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic, complemented
27	#41×D₄₇	C₈	1	1	yes	characteristic
28	#13094032	—	2	1	no	—
29	#13094032	C₈	1	1	yes	characteristic
30	#13094031	C₈	1	1	yes	characteristic
31	#13094030	—	2	1	no	—
32	#13094030	C₂×C₄	1	1	yes	characteristic
33	M₁₆×D₄₇	—	2	1	no	—
34	M₁₆×D₄₇	C₈	1	1	yes	characteristic
35	C₂×C₈×D₄₇	—	2	1	no	—
36	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
37	#13093964	—	2	1	no	—
38	#13093964	C₂×C₄	1	1	yes	characteristic
39	#13093963	—	2	1	no	—
40	#13093963	—	2	1	no	—
41	#13093963	—	2	1	no	—
42	C₂×C₄₇⋊C₁₆	—	2	1	no	—
43	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₄₇	—	2	1	no	—
45	C₁₆×D₄₇	—	2	1	no	—
46	C₁₆×D₄₇	—	2	1	no	—
47	C₁₆×D₄₇	D₄	1	1	yes	characteristic
48	C₁₆×D₄₇	D₄	1	1	yes	characteristic
49	C₁₆×D₄₇	D₄	1	1	yes	characteristic
50	C₁₆×D₄₇	D₄	1	1	yes	characteristic
51	C₄₇⋊C₃₂	—	2	1	no	—
52	C₄₇⋊C₃₂	—	2	1	no	—
53	#131×C₄₇	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₄₇	—	2	1	no	—
55	M₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₄₇	—	2	1	no	—
57	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
58	C₃₂×C₄₇	—	2	1	no	—
59	C₃₂×C₄₇	—	2	1	no	—
60	#41×C₄₇	C₂×C₈	1	1	yes	characteristic
61	#10625354	C₁₆	1	1	yes	characteristic
62	Q₈×D₄₇	C₁₆	1	1	yes	characteristic
63	#10625352	—	2	1	no	—
64	#10625352	C₁₆	1	1	yes	characteristic
65	D₄×D₄₇	—	2	1	no	—
66	D₄×D₄₇	C₁₆	1	1	yes	characteristic
67	#10625350	—	2	1	no	—
68	#10625350	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₄₇	—	2	1	no	—
70	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
71	M₁₆×C₄₇	—	2	1	no	—
72	M₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	#10625329	—	2	1	no	—
76	#10625329	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄₇⋊C₈	—	2	1	no	—
78	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
79	#10625324	—	2	1	no	—
80	#10625324	—	2	1	no	—
81	#10625324	—	2	1	no	—
82	C₈×D₄₇	—	2	1	no	—
83	C₈×D₄₇	—	2	1	no	—
84	C₈×D₄₇	—	2	1	no	—
85	C₈×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₈×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₈×D₄₇	M₁₆	1	1	yes	characteristic
88	C₈×D₄₇	M₁₆	1	1	yes	characteristic
89	C₁₆×C₄₇	—	2	1	no	—
90	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₄₇⋊C₁₆	—	2	1	no	—
93	C₄₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
94	C₄₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×D₄₇	—	2	1	no	—
96	C₂²×D₄₇	M₃₂	1	1	yes	characteristic
97	Q₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
98	D₄×C₄₇	—	2	1	no	—
99	D₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂²⋊D₄₇	—	2	1	no	—
103	C₂²⋊D₄₇	—	2	1	no	—
104	C₂²⋊D₄₇	—	2	1	no	—
105	C₂×Dic₄₇	—	2	1	no	—
106	C₂×Dic₄₇	M₃₂	1	1	yes	characteristic
107	D₁₈₈	—	2	1	no	—
108	D₁₈₈	C₂×C₁₆	1	1	yes	characteristic
109	C₄×D₄₇	—	2	1	no	—
110	C₄×D₄₇	—	2	1	no	—
111	C₄×D₄₇	—	2	1	no	—
112	C₄×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
113	C₄×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
114	C₄×D₄₇	C₂×C₁₆	1	1	yes	characteristic
115	C₄×D₄₇	C₂×C₁₆	1	1	yes	characteristic
116	Dic₉₄	—	2	1	no	—
117	Dic₉₄	C₂×C₁₆	1	1	yes	characteristic
118	C₈×C₄₇	—	2	1	no	—
119	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
121	C₄₇⋊C₈	—	2	1	no	—
122	C₄₇⋊C₈	C₂²⋊C₈	1	1	yes	characteristic
123	C₄₇⋊C₈	C₂²⋊C₈	1	1	yes	characteristic
124	C₂×#1401	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
127	C₂×D₄₇	—	2	2	no	—
128	C₂×D₄₇	—	2	1	no	—
129	C₂×D₄₇	—	4	2	no	—
130	C₂×D₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
131	C₂×D₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
135	Dic₄₇	—	2	1	no	—
136	Dic₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
137	Dic₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
138	C₂×#504	—	47	1	no	—
139	C₂×M₆₄	—	47	1	no	—
140	C₂²×C₃₂	—	47	1	no	—
141	#1401	—	47	1	no	—
142	#1401	—	47	1	no	—
143	#1401	—	47	1	no	—
144	#1401	D₄₇	1	1	yes	characteristic, direct product factor
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	4	1	no	—
147	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
148	D₄₇	—	2	1	no	—
149	D₄₇	—	4	1	no	—
150	D₄₇	#1401	1	2	yes	direct product factor
151	C₂×#131	—	47	1	no	—
152	#504	—	47	1	no	—
153	#504	—	47	1	no	—
154	#504	—	47	1	no	—
155	#504	—	94	1	no	—
156	#504	—	94	1	no	—
157	#504	C₂×D₄₇	1	1	yes	characteristic
158	C₂×M₃₂	—	47	1	no	—
159	C₂×M₃₂	—	94	1	no	—
160	C₂²×C₁₆	—	47	1	no	—
161	C₂²×C₁₆	—	94	1	no	—
162	M₆₄	—	47	1	no	—
163	M₆₄	—	94	1	no	—
164	M₆₄	C₂×D₄₇	1	1	yes	characteristic, complemented
165	C₂×C₃₂	—	47	1	no	—
166	C₂×C₃₂	—	94	1	no	—
167	C₂×C₃₂	—	94	1	no	—
168	C₂×C₃₂	—	94	1	no	—
169	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic, complemented
170	C₄₇	C₂×#1401	1	1	yes	characteristic, complemented, Sylow
171	C₂×#41	—	47	1	no	—
172	#131	—	47	1	no	—
173	#131	—	47	1	no	—
174	#131	—	47	1	no	—
175	#131	—	94	1	no	—
176	#131	—	94	1	no	—
177	#131	C₄×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	47	1	no	—
179	C₂×M₁₆	—	94	1	no	—
180	C₂²×C₈	—	47	1	no	—
181	C₂²×C₈	—	94	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	47	1	no	—
184	M₃₂	—	94	1	no	—
185	M₃₂	—	94	1	no	—
186	M₃₂	—	94	1	no	—
187	M₃₂	—	94	1	no	—
188	M₃₂	C₄×D₄₇	1	1	yes	characteristic
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	47	1	no	—
191	C₂×C₁₆	—	47	1	no	—
192	C₂×C₁₆	—	47	1	no	—
193	C₂×C₁₆	—	47	1	no	—
194	C₂×C₁₆	—	47	1	no	—
195	C₂×C₁₆	—	94	1	no	—
196	C₂×C₁₆	—	94	1	no	—
197	C₂×C₁₆	—	94	1	no	—
198	C₂×C₁₆	—	94	1	no	—
199	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
200	C₃₂	—	2	1	no	—
201	C₃₂	—	2	1	no	—
202	C₃₂	—	94	1	no	—
203	C₃₂	—	94	1	no	—
204	#41	—	47	1	no	—
205	#41	—	47	1	no	—
206	#41	—	47	1	no	—
207	#41	—	94	1	no	—
208	#41	—	94	1	no	—
209	#41	C₈×D₄₇	1	1	yes	characteristic
210	C₂×Q₈	—	47	1	no	—
211	C₂×D₄	—	47	1	no	—
212	C₂×D₄	—	94	1	no	—
213	C₂²×C₄	—	47	1	no	—
214	C₂²×C₄	—	94	1	no	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	47	1	no	—
217	M₁₆	—	94	1	no	—
218	M₁₆	—	94	1	no	—
219	M₁₆	—	94	1	no	—
220	M₁₆	—	94	1	no	—
221	M₁₆	C₈×D₄₇	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	47	1	no	—
224	C₂×C₈	—	47	1	no	—
225	C₂×C₈	—	47	1	no	—
226	C₂×C₈	—	47	1	no	—
227	C₂×C₈	—	47	1	no	—
228	C₂×C₈	—	94	1	no	—
229	C₂×C₈	—	94	1	no	—
230	C₂×C₈	—	94	1	no	—
231	C₂×C₈	—	94	1	no	—
232	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₁₆	—	2	1	no	—
234	C₁₆	—	47	1	no	—
235	C₁₆	—	47	1	no	—
236	C₁₆	—	94	1	no	—
237	C₁₆	D₄×D₄₇	1	1	yes	characteristic
238	C₁₆	D₄×D₄₇	1	1	yes	characteristic
239	C₂³	—	47	1	no	—
240	C₂³	—	94	1	no	—
241	Q₈	—	47	1	no	—
242	Q₈	—	94	1	no	—
243	Q₈	C₁₆×D₄₇	1	1	yes	characteristic
244	D₄	—	2	1	no	—
245	D₄	—	47	1	no	—
246	D₄	—	94	1	no	—
247	D₄	—	94	1	no	—
248	D₄	—	94	1	no	—
249	D₄	—	94	1	no	—
250	D₄	C₁₆×D₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	47	1	no	—
253	C₂×C₄	—	47	1	no	—
254	C₂×C₄	—	47	1	no	—
255	C₂×C₄	—	47	1	no	—
256	C₂×C₄	—	47	1	no	—
257	C₂×C₄	—	94	1	no	—
258	C₂×C₄	—	94	1	no	—
259	C₂×C₄	—	94	1	no	—
260	C₂×C₄	—	94	1	no	—
261	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	47	1	no	—
264	C₈	—	47	1	no	—
265	C₈	—	94	1	no	—
266	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
267	C₈	M₁₆×D₄₇	1	1	yes	characteristic
268	C₂²	—	2	1	no	—
269	C₂²	—	47	1	no	—
270	C₂²	—	47	1	no	—
271	C₂²	—	94	2	no	—
272	C₂²	—	94	1	no	—
273	C₂²	—	188	2	no	—
274	C₂²	M₃₂×D₄₇	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	47	1	no	—
277	C₄	—	47	1	no	—
278	C₄	—	94	1	no	—
279	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
280	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
281	C₂	—	2	1	no	—
282	C₂	—	4	1	no	—
283	C₂	—	47	2	no	—
284	C₂	—	94	1	no	—
285	C₂	—	188	1	no	—
286	C₂	C₂²⋊C₁₆×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1401	C₂
socle	C₂×C₄₇	C₂×C₂²⋊C₁₆
prime radical	#41×D₄₇	C₈
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	#1401	D₄₇
hypocenter	C₄₇	C₂×#1401
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1401

Complements

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

N	H ≅ G/N	type
2. M₆₄×D₄₇	C₂	semidirect
3. C₂×C₃₂×D₄₇	C₂	semidirect
4. #20241531	C₂	semidirect
5. #20241530	C₂	semidirect
6. #20241529	C₂	semidirect
7. C₄₇×#1401	C₂	semidirect
18. #15688957	C₂²	semidirect
20. C₂×C₄₇⋊C₃₂	C₂²	semidirect
25. C₄₇×M₆₄	C₂²	semidirect
26. C₂×C₃₂×C₄₇	C₂²	semidirect
144. #1401	D₄₇	direct (N × H)
150. D₄₇	#1401	direct (N × H)
164. M₆₄	C₂×D₄₇	semidirect
169. C₂×C₃₂	C₂×D₄₇	semidirect
170. C₄₇	C₂×#1401	semidirect

Maximal subgroups

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

subgroup	index
#504×D₄₇	2
M₆₄×D₄₇	2
C₂×C₃₂×D₄₇	2
#20241531	2
#20241530	2
#20241529	2
C₄₇×#1401	2
C₂×#1401	47

Subgroups — D47×#1401 #26136483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1401 #26136483