Subgroups — C₄₇×#9145 #26048843

Order 12032.

694 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	C₄₇×#2855	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1817	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#328	C₂	1	2	yes	maximal
5	C₄₇×#1294	C₂	1	2	yes	maximal
6	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₄₇	—	2	1	no	—
8	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
10	C₄₇×#391	C₄	1	2	yes	complemented
11	C₄₇×#389	C₄	1	2	yes	complemented
12	C₄₇×#387	—	2	2	no	—
13	C₄₇×#385	—	2	1	no	—
14	C₄₇×#381	—	2	1	no	—
15	C₄₇×#380	C₂²	1	1	yes	characteristic
16	C₄₇×#376	—	2	4	no	—
17	C₂×#95×C₄₇	—	2	1	no	—
18	C₂×#95×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#336	—	2	2	no	—
20	C₄₇×#328	—	2	4	no	—
21	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	—	2	2	no	—
23	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂²⋊Q₈×C₄₇	—	2	2	no	—
28	C₂²⋊Q₈×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄²×C₄₇	—	2	8	no	—
36	C₂×C₄²×C₄₇	—	2	4	no	—
37	C₂×C₄²×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂²⋊C₈×C₄₇	—	2	2	no	—
42	C₂²⋊C₈×C₄₇	—	2	2	no	—
43	#95×C₄₇	—	2	4	no	—
44	#95×C₄₇	—	2	2	no	—
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	2	no	—
48	#95×C₄₇	—	2	1	no	—
49	#95×C₄₇	D₄	1	4	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₂×C₈×C₄₇	—	4	4	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	4	2	no	—
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₂²⋊C₄×C₄₇	—	4	1	no	—
82	C₂²⋊C₄×C₄₇	—	4	1	no	—
83	C₂²⋊C₄×C₄₇	—	4	2	no	—
84	C₄²×C₄₇	—	2	16	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	#120	1	1	yes	characteristic
91	C₂³×C₄₇	#123	1	1	yes	characteristic
92	C₂³×C₄₇	#135	1	2	yes	—
93	C₂³×C₄₇	#137	1	2	yes	—
94	Q₈×C₄₇	—	8	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	8	no	—
109	C₂×C₄×C₄₇	—	4	8	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	#104	1	4	yes	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	#121	1	2	yes	—
123	C₈×C₄₇	—	8	2	no	—
124	#9145	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	—	2	1	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	#386	1	1	yes	characteristic
132	C₂²×C₄₇	#410	1	2	yes	—
133	C₂²×C₄₇	#416	1	1	yes	characteristic
134	C₂²×C₄₇	#419	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#104	1	2	yes	—
136	C₂²×C₄₇	#452	1	2	yes	—
137	C₂²×C₄₇	#485	1	2	yes	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	4	8	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	—	8	1	no	—
144	C₄×C₄₇	—	8	1	no	—
145	C₄×C₄₇	—	8	2	no	—
146	#2855	C₂×C₄₇	1	1	yes	characteristic
147	C₂×#376	C₂×C₄₇	1	1	yes	characteristic
148	#1817	C₂×C₄₇	1	1	yes	characteristic
149	C₂×#328	C₂×C₄₇	1	2	yes	—
150	#1294	C₂×C₄₇	1	2	yes	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	#1500	1	2	yes	—
154	C₂×C₄₇	#1506	1	2	yes	—
155	C₂×C₄₇	#1873	1	1	yes	characteristic
156	C₂×C₄₇	#1896	1	1	yes	characteristic
157	C₂×C₄₇	#1899	1	1	yes	characteristic
158	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
159	C₂²×C₄²	—	2	1	no	—
160	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
161	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
162	#391	C₄×C₄₇	1	2	yes	complemented
163	#389	C₄×C₄₇	1	2	yes	complemented
164	#387	—	2	2	no	—
165	#385	—	2	1	no	—
166	#381	—	2	1	no	—
167	#380	C₂²×C₄₇	1	1	yes	characteristic
168	#376	—	2	4	no	—
169	C₂×#95	—	2	1	no	—
170	C₂×#95	C₂²×C₄₇	1	2	yes	—
171	#336	—	2	2	no	—
172	#328	—	2	4	no	—
173	C₄₇	#9145	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	2	no	—
180	C₂²⋊Q₈	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₂²⋊C₄	—	2	1	no	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	—	2	8	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
193	C₂×C₄²	D₄×C₄₇	1	2	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	#95	—	2	4	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	1	no	—
202	#95	D₄×C₄₇	1	4	yes	—
203	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
227	C₂²×C₄	#41×C₄₇	1	2	yes	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	—	2	16	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
243	C₂³	#120×C₄₇	1	1	yes	characteristic
244	C₂³	#123×C₄₇	1	1	yes	characteristic
245	C₂³	#135×C₄₇	1	2	yes	—
246	C₂³	#137×C₄₇	1	2	yes	—
247	Q₈	—	8	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	#104×C₄₇	1	4	yes	—
274	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
275	C₂×C₄	#121×C₄₇	1	2	yes	—
276	C₈	—	8	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₄₇×#386	1	1	yes	characteristic
284	C₂²	C₄₇×#410	1	2	yes	—
285	C₂²	C₄₇×#416	1	1	yes	characteristic
286	C₂²	C₄₇×#419	1	1	yes	characteristic
287	C₂²	C₂×#104×C₄₇	1	2	yes	—
288	C₂²	C₄₇×#452	1	2	yes	—
289	C₂²	C₄₇×#485	1	2	yes	—
290	C₄	—	2	4	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	8	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	1	no	—
297	C₄	—	8	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	2	no	—
300	C₂	C₄₇×#1500	1	2	yes	—
301	C₂	C₄₇×#1506	1	2	yes	—
302	C₂	C₄₇×#1873	1	1	yes	characteristic
303	C₂	C₄₇×#1896	1	1	yes	characteristic
304	C₂	C₄₇×#1899	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
prime radical	C₂⁴×C₄₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄₇×#391	C₄	semidirect
11. C₄₇×#389	C₄	semidirect
124. #9145	C₄₇	direct (N × H)
162. #391	C₄×C₄₇	semidirect
163. #389	C₄×C₄₇	semidirect
173. C₄₇	#9145	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#2855	2
C₂×C₄₇×#376	2
C₄₇×#1817	2
C₂×C₄₇×#328	2
C₄₇×#1294	2
#9145	47

Subgroups — C47×#9145 #26048843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#9145 #26048843