Subgroups — D₄₇×#1375 #26131123

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₇	C₂	1	2	yes	maximal
2	C₂×#97×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241468	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241467	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1375	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₄₇	C₄	1	1	yes	characteristic
9	C₂×#13093964	C₂²	1	2	yes	—
10	C₂×#13093963	C₄	1	4	yes	complemented
11	M₃₂×D₄₇	—	2	4	no	—
12	M₃₂×D₄₇	—	2	4	no	—
13	C₂×C₁₆×D₄₇	C₄	1	4	yes	complemented
14	C₂×#13093935	C₄	1	1	yes	characteristic
15	C₂×#13093934	C₄	1	1	yes	characteristic
16	C₂×#13093934	C₂²	1	1	yes	characteristic
17	#97×D₄₇	—	2	4	no	—
18	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
19	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₄₇	—	2	1	no	—
21	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₄₇	—	2	2	no	—
25	C₂×C₈×D₄₇	—	2	1	no	—
26	C₂×C₈×D₄₇	—	2	1	no	—
27	C₂×C₈×D₄₇	C₈	1	4	yes	—
28	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₄₇	D₄	1	2	yes	—
31	C₂×C₈×D₄₇	D₄	1	4	yes	—
32	C₂×C₈×D₄₇	Q₈	1	2	yes	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₄₇	—	2	2	no	—
36	C₄²×D₄₇	C₈	1	4	yes	—
37	#13093964	—	2	2	no	—
38	#13093964	—	2	2	no	—
39	#13093963	—	2	8	no	—
40	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
41	C₁₆×D₄₇	—	2	8	no	—
42	#13093935	—	2	2	no	—
43	#13093934	—	2	2	no	—
44	#13093934	—	2	2	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₄₇	—	2	2	no	—
49	M₃₂×C₄₇	—	2	2	no	—
50	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
51	#97×C₄₇	—	2	2	no	—
52	C₂³×D₄₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	1	no	—
56	C₂²×Dic₄₇	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₄₇	—	2	4	no	—
58	C₂×C₄×D₄₇	—	4	4	no	—
59	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
60	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
61	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
66	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
67	C₂×C₈×C₄₇	—	2	1	no	—
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	C₂×C₈	1	2	yes	—
76	C₄×Dic₄₇	—	2	1	no	—
77	C₄×Dic₄₇	—	2	4	no	—
78	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
79	C₂×C₄₇⋊C₈	—	2	1	no	—
80	C₂×C₄₇⋊C₈	—	2	1	no	—
81	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
82	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
85	C₈×D₄₇	—	2	4	no	—
86	C₈×D₄₇	—	2	4	no	—
87	C₈×D₄₇	C₂²⋊C₄	1	4	yes	—
88	C₈×D₄₇	C₂²⋊C₄	1	4	yes	—
89	C₈×D₄₇	C₄⋊C₄	1	4	yes	—
90	C₈×D₄₇	C₄⋊C₄	1	4	yes	—
91	C₁₆×C₄₇	—	2	4	no	—
92	C₄₇⋊C₁₆	—	2	4	no	—
93	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	4	no	—
95	C₂²×D₄₇	#97	1	1	yes	characteristic
96	C₂²×D₄₇	#97	1	1	yes	characteristic
97	C₂²×D₄₇	C₂²⋊C₈	1	2	yes	—
98	C₂²×D₄₇	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₂×Dic₄₇	#97	1	1	yes	characteristic
110	C₂×Dic₄₇	#97	1	1	yes	characteristic
111	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
112	C₂×Dic₄₇	C₄⋊C₈	1	2	yes	—
113	C₄×D₄₇	—	4	8	no	—
114	C₄×D₄₇	#95	1	2	yes	—
115	C₄×D₄₇	#95	1	2	yes	—
116	C₄×D₄₇	C₄×C₈	1	2	yes	—
117	C₄×D₄₇	C₄×C₈	1	2	yes	—
118	C₄×D₄₇	C₂²⋊C₈	1	2	yes	—
119	C₄×D₄₇	C₂²⋊C₈	1	2	yes	—
120	C₄×D₄₇	C₄⋊C₈	1	2	yes	—
121	C₄×D₄₇	C₄⋊C₈	1	2	yes	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
124	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	#95	1	4	yes	—
127	C₂×#1375	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	2	4	no	—
134	C₂×D₄₇	—	2	4	no	—
135	C₂×D₄₇	#336	1	2	yes	—
136	C₂×D₄₇	#336	1	2	yes	—
137	C₂×D₄₇	#347	1	4	yes	—
138	C₄×C₄₇	—	4	2	no	—
139	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
140	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
143	Dic₄₇	—	4	2	no	—
144	Dic₄₇	#336	1	2	yes	—
145	Dic₄₇	#336	1	2	yes	—
146	C₂²×M₃₂	—	47	2	no	—
147	C₂²×#97	—	47	1	no	—
148	#1375	—	47	2	no	—
149	#1375	—	47	1	no	—
150	#1375	D₄₇	1	1	yes	characteristic, direct product factor
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#347	1	2	yes	—
155	D₄₇	—	2	2	no	—
156	D₄₇	#1375	1	4	yes	direct product factor
157	C₂³×C₈	—	47	1	no	—
158	C₂³×C₈	—	47	1	no	—
159	C₂²×C₄²	—	47	1	no	—
160	C₂×M₃₂	—	47	2	no	—
161	C₂×M₃₂	—	47	4	no	—
162	C₂×M₃₂	—	94	4	no	—
163	C₂×M₃₂	—	94	4	no	—
164	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
165	C₂²×C₁₆	—	47	4	no	—
166	C₂×#97	—	47	1	no	—
167	C₂×#97	—	47	1	no	—
168	C₂×#97	—	47	1	no	—
169	C₂×#97	—	94	4	no	—
170	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
171	C₄₇	C₂×#1375	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	47	1	no	—
173	C₂³×C₄	—	94	1	no	—
174	C₂²×C₈	—	47	1	no	—
175	C₂²×C₈	—	47	2	no	—
176	C₂²×C₈	—	47	2	no	—
177	C₂²×C₈	—	47	4	no	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	—	47	1	no	—
181	C₂²×C₈	—	47	4	no	—
182	C₂²×C₈	—	94	1	no	—
183	C₂²×C₈	—	94	1	no	—
184	C₂²×C₈	—	94	2	no	—
185	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×C₄²	—	47	1	no	—
188	C₂×C₄²	—	47	4	no	—
189	C₂×C₄²	—	94	2	no	—
190	C₂×C₄²	—	94	1	no	—
191	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
192	M₃₂	—	2	2	no	—
193	M₃₂	—	2	2	no	—
194	M₃₂	—	94	8	no	—
195	M₃₂	—	94	2	no	—
196	M₃₂	—	94	2	no	—
197	C₂×C₁₆	—	47	4	no	—
198	C₂×C₁₆	—	94	8	no	—
199	C₂×C₁₆	C₄×D₄₇	1	4	yes	complemented
200	#97	—	2	2	no	—
201	#97	—	94	2	no	—
202	#97	—	94	2	no	—
203	#97	—	94	2	no	—
204	C₂⁴	—	47	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	47	1	no	—
207	C₂²×C₄	—	47	1	no	—
208	C₂²×C₄	—	47	1	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	1	no	—
214	C₂²×C₄	—	47	1	no	—
215	C₂²×C₄	—	94	4	no	—
216	C₂²×C₄	—	94	1	no	—
217	C₂²×C₄	—	188	4	no	—
218	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	47	2	no	—
222	C₂×C₈	—	47	2	no	—
223	C₂×C₈	—	47	2	no	—
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	47	4	no	—
226	C₂×C₈	—	47	4	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	47	2	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	94	4	no	—
231	C₂×C₈	—	94	1	no	—
232	C₂×C₈	—	94	1	no	—
233	C₂×C₈	C₈×D₄₇	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
235	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
236	C₂×C₈	D₄×D₄₇	1	2	yes	—
237	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
238	C₄²	—	2	1	no	—
239	C₄²	—	47	2	no	—
240	C₄²	—	94	4	no	—
241	C₄²	—	94	1	no	—
242	C₄²	C₈×D₄₇	1	2	yes	—
243	C₁₆	—	2	4	no	—
244	C₁₆	—	94	4	no	—
245	C₂³	—	47	1	no	—
246	C₂³	—	47	1	no	—
247	C₂³	—	47	2	no	—
248	C₂³	—	47	2	no	—
249	C₂³	—	94	4	no	—
250	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	47	1	no	—
254	C₂×C₄	—	47	1	no	—
255	C₂×C₄	—	47	2	no	—
256	C₂×C₄	—	47	2	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	2	no	—
262	C₂×C₄	—	47	2	no	—
263	C₂×C₄	—	47	2	no	—
264	C₂×C₄	—	47	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	188	8	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	47	4	no	—
276	C₈	—	94	2	no	—
277	C₈	C₂²⋊C₄×D₄₇	1	2	yes	—
278	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	47	2	no	—
281	C₂²	—	47	4	no	—
282	C₂²	—	47	2	no	—
283	C₂²	—	94	4	no	—
284	C₂²	—	94	4	no	—
285	C₂²	—	94	2	no	—
286	C₂²	#97×D₄₇	1	1	yes	characteristic
287	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
288	C₂²	C₄⋊C₈×D₄₇	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	47	2	no	—
291	C₄	—	47	2	no	—
292	C₄	—	188	2	no	—
293	C₄	#95×D₄₇	1	1	yes	characteristic
294	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
295	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
296	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	47	4	no	—
300	C₂	—	94	2	no	—
301	C₂	#336×D₄₇	1	1	yes	characteristic
302	C₂	#347×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1375	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×D₄₇	M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1375	D₄₇
hypocenter	C₄₇	C₂×#1375
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #1375

Complements

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

N	H ≅ G/N	type
3. #20241468	C₂	semidirect
4. #20241467	C₂	semidirect
5. C₄₇×#1375	C₂	semidirect
10. C₂×#13093963	C₄	semidirect
13. C₂×C₁₆×D₄₇	C₄	semidirect
40. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
50. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
150. #1375	D₄₇	direct (N × H)
156. D₄₇	#1375	direct (N × H)
171. C₄₇	C₂×#1375	semidirect
199. C₂×C₁₆	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×M₃₂×D₄₇	2
C₂×#97×D₄₇	2
#20241468	2
#20241467	2
C₄₇×#1375	2
C₂×#1375	47

Subgroups — D47×#1375 #26131123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1375 #26131123