Finite Group Databasebeta

← #26143046

Subgroups — #26143046

Order 12032.

1830 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689034C211yesmaximal, characteristic
2C2×#15689014C213yesmaximal
3#20241659C211yesmaximal, characteristic, complemented
4#20241651C213yesmaximal, complemented
5#20241630C213yesmaximal
6#20241615C213yesmaximal, complemented
7C22×C4×Q8×C47C211yesmaximal, characteristic, complemented
8C22×Dic47⋊C4C2213yes—
9C22×(C2×D47)⋊C4C2211yescharacteristic
10C22×(C2×D47)⋊C4C2213yes—
11C22×(C2×D47)⋊C4C2213yes—
12#15689034C418yescomplemented
13#15689014—212no—
14C2×C47⋊#95C2213yes—
15C2×C47⋊#95C2213yes—
16C2×C47⋊#95C2213yes—
17C2×C47⋊#95C2216yes—
18C2×C47⋊#95C2213yes—
19C23×Q8×C47C2211yescharacteristic
20C2×C4×Q8×C47—212no—
21C22×C4⋊C4×C47C2213yes—
22C22×C42×C47C2213yes—
23C24×D47C2311yescharacteristic
24C23×Dic47C2313yes—
25C23×Dic47C2311yescharacteristic
26C23×Dic47C2313yes—
27C2×Dic47⋊C4—26no—
28C2×Dic47⋊C4—46no—
29C2×(C2×D47)⋊C4—26no—
30C2×(C2×D47)⋊C4—26no—
31C2×(C2×D47)⋊C4—26no—
32C2×(C2×D47)⋊C4—26no—
33C2×(C2×D47)⋊C4—46no—
34C2×(C2×D47)⋊C4C2×C4112yes—
35C47⋊#95—26no—
36C47⋊#95—26no—
37C47⋊#95—26no—
38C47⋊#95—46no—
39C47⋊#95C2×C4112yes—
40C22×Q8×C47—26no—
41C22×Q8×C47—26no—
42C22×Q8×C47C2×C414yes—
43C23×C4×C47C2313yes—
44C23×C4×C47C2313yes—
45C23×C4×C47C2311yescharacteristic
46C4×Q8×C47—232no—
47C2×C4⋊C4×C47—212no—
48C2×C4⋊C4×C47—26no—
49C2×C42×C47—212no—
50C2×C42×C47D4112yes—
51C24×C47C2411yescharacteristic
52C23×D47—23no—
53C23×D47—23no—
54C23×D47—23no—
55C23×D47—21no—
56C23×D47—21no—
57C23×D47—23no—
58C23×D47C22×C412yes—
59C2×Q8×C47—28no—
60C2×Q8×C47—224no—
61C2×Q8×C47—224no—
62C22×C4×C47—26no—
63C22×C4×C47—26no—
64C22×C4×C47—21no—
65C22×C4×C47—23no—
66C22×C4×C47—26no—
67C22×C4×C47—23no—
68C22×C4×C47—26no—
69C22×C4×C47—23no—
70C22×C4×C47C22×C416yes—
71C22×C4×C47C2×D416yes—
72C22×C4×C47C2×D416yes—
73C22×C4×C47C2×D416yes—
74C22×C4×C47#4116yes—
75C22×Dic47—23no—
76C22×Dic47—23no—
77C22×Dic47—26no—
78C22×Dic47—26no—
79C22×Dic47—23no—
80C22×Dic47—23no—
81C22×Dic47—26no—
82C22×Dic47—26no—
83C22×Dic47—23no—
84C22×Dic47—23no—
85C22×Dic47—23no—
86C22×Dic47—21no—
87C22×Dic47C22×C416yes—
88C4⋊C4×C47—224no—
89C42×C47—224no—
90(C2×D47)⋊C4—224no—
91(C2×D47)⋊C4—412no—
92(C2×D47)⋊C4—44no—
93Dic47⋊C4—412no—
94C23×C47C23×C411yescharacteristic
95C23×C47C22×D413yes—
96C23×C47C2×#4113yes—
97C23×C47H(5,𝔽2)13yes—
98C23×C472−1+411yescharacteristic
99C23×C472−1+411yescharacteristic
100C23×C472−1+413yes—
101C22×D47—26no—
102C22×D47—22no—
103C22×D47—26no—
104C22×D47—41no—
105C22×D47—43no—
106C22×D47—43no—
107C22×D47—43no—
108C22×D47—43no—
109C22×D47—43no—
110C22×D47—43no—
111C22×D47—43no—
112C22×D47—46no—
113Q8×C47—232no—
114C2×C4×C47—212no—
115C2×C4×C47—26no—
116C2×C4×C47—26no—
117C2×C4×C47—212no—
118C2×C4×C47—26no—
119C2×C4×C47—26no—
120C2×C4×C47—26no—
121C2×C4×C47—26no—
122C2×C4×C47—26no—
123C2×C4×C47—46no—
124C2×C4×C47—46no—
125C2×C4×C47C4×D4112yes—
126C2×C4×C47#12014yes—
127C2×Dic47—26no—
128C2×Dic47—212no—
129C2×Dic47—43no—
130C2×Dic47—43no—
131C2×Dic47—46no—
132C2×Dic47—41no—
133C2×Dic47—43no—
134C2×Dic47—43no—
135C2×Dic47—43no—
136C2×Dic47—43no—
137C2×Dic47—43no—
138C2×Dic47—46no—
139C2×Dic47—46no—
140#14301—471nomaximal, Sylow
141C22×C47C2×C4×D413yes—
142C22×C47#51911yescharacteristic
143C22×C47#52013yes—
144C22×C47C2×#12011yescharacteristic
145C22×C47#52513yes—
146C22×C47#53513yes—
147C22×C47#53613yes—
148C22×C47#53613yes—
149C22×C47#53713yes—
150C22×C47#54113yes—
151C22×C47#54213yes—
152C22×C47#54216yes—
153C2×D47—46no—
154C2×D47—46no—
155C2×D47—42no—
156C2×D47—81no—
157C2×D47—81no—
158C2×D47—83no—
159C2×D47—83no—
160C4×C47—24no—
161C4×C47—212no—
162C4×C47—46no—
163Dic47—46no—
164Dic47—81no—
165Dic47—83no—
166C22×C4×Q8D4711yescharacteristic, complemented
167C2×#393—471no—
168C2×#388—473no—
169#1446—471no—
170#1444—473no—
171#1439—473no—
172#1434—473no—
173C2×C47C4×#12011yescharacteristic
174C2×C47#232313yes—
175C2×C47#232613yes—
176C2×C47#261411yescharacteristic
177C2×C47#261511yescharacteristic
178C2×C47#263213yes—
179C2×C47#263713yes—
180D47—82no—
181C23×Q8C2×D4711yescharacteristic
182C2×C4×Q8—212no—
183C22×C4⋊C4—473no—
184C22×C4⋊C4C2×D4713yes—
185C22×C22⋊C4—473no—
186C22×C22⋊C4—471no—
187C22×C22⋊C4—473no—
188C22×C42C2×D4713yes—
189#393—478no—
190#388—9412no—
191C2×#95—473no—
192C2×#95—473no—
193C2×#95—473no—
194C2×#95—473no—
195C2×#95—476no—
196C47#1430111yescharacteristic, complemented, Sylow
197C25—471no—
198C22×Q8—26no—
199C22×Q8—26no—
200C22×Q8C4×D4714yes—
201C23×C4—473no—
202C23×C4—471no—
203C23×C4—473no—
204C23×C4C22×D4713yes—
205C23×C4C22×D4713yes—
206C23×C4C22×D4711yescharacteristic
207C4×Q8—232no—
208C2×C4⋊C4—212no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—946no—
211C2×C4⋊C4—1886no—
212C2×C22⋊C4—4712no—
213C2×C22⋊C4—946no—
214C2×C22⋊C4—946no—
215C2×C22⋊C4—946no—
216C2×C22⋊C4—946no—
217C2×C22⋊C4—1886no—
218C2×C42—212no—
219C2×C42C22⋊D47112yes—
220#95—4712no—
221#95—946no—
222#95—946no—
223#95—946no—
224#95—1886no—
225C24—472no—
226C24—943no—
227C24—943no—
228C24—943no—
229C24—941no—
230C24—941no—
231C24—943no—
232C24C23×D4711yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—21no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—476no—
245C22×C4—943no—
246C22×C4—943no—
247C22×C4—946no—
248C22×C4—946no—
249C22×C4—943no—
250C22×C4—943no—
251C22×C4—946no—
252C22×C4—946no—
253C22×C4—943no—
254C22×C4—941no—
255C22×C4—943no—
256C22×C4—943no—
257C22×C4C2×C4×D4716yes—
258C22×C4#1062535016yes—
259C22×C4C2×C22⋊D4716yes—
260C22×C4C2×C22⋊D4716yes—
261C22×C4C2×C22⋊D4716yes—
262C4⋊C4—224no—
263C4⋊C4—18812no—
264C22⋊C4—9424no—
265C22⋊C4—18812no—
266C22⋊C4—1884no—
267C42—224no—
268C23—946no—
269C23—942no—
270C23—946no—
271C23—1883no—
272C23—1883no—
273C23—1886no—
274C23—1881no—
275C23—1883no—
276C23—1883no—
277C23—1883no—
278C23—1883no—
279C23—1883no—
280C23C22×C4×D4711yescharacteristic
281C23C22×C22⋊D4713yes—
282C23C2×#1062535013yes—
283C23#1309407013yes—
284C23#1309407111yescharacteristic
285C23#1309407213yes—
286C23#1309407211yescharacteristic
287Q8—232no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—212no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—946no—
300C2×C4—9412no—
301C2×C4—1886no—
302C2×C4—1886no—
303C2×C4—1881no—
304C2×C4—1883no—
305C2×C4—1883no—
306C2×C4—1883no—
307C2×C4—1883no—
308C2×C4—1883no—
309C2×C4—1886no—
310C2×C4—1883no—
311C2×C4—1883no—
312C2×C4C4×C22⋊D47112yes—
313C2×C4#1309399314yes—
314C22—3761no—
315C22—3761no—
316C22—3763no—
317C22—3763no—
318C22—1886no—
319C22—1886no—
320C22—1882no—
321C22C2×C4×C22⋊D4713yes—
322C22#1568959911yescharacteristic
323C22#1568960513yes—
324C22C2×#1309399311yescharacteristic
325C22#1568963413yes—
326C22#1568968713yes—
327C22#1568969713yes—
328C22#1568969713yes—
329C22#1568970113yes—
330C22#1568973113yes—
331C22#1568974513yes—
332C22#1568974516yes—
333C4—24no—
334C4—212no—
335C4—46no—
336C4—3761no—
337C4—3763no—
338C4—1886no—
339C2—3762no—
340C2C4×#1309399311yescharacteristic
341C2#2024606413yes—
342C2#2024609513yes—
343C2#2024910711yescharacteristic
344C2#2024911211yescharacteristic
345C2#2024924813yes—
346C2#2024929413yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×C4×Q8×C47C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC22×C4×Q8D47
hypocenterC47#14301
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20241659C2semidirect
4. #20241651C2semidirect
6. #20241615C2semidirect
7. C22×C4×Q8×C47C2semidirect
12. #15689034C4semidirect
166. C22×C4×Q8D47semidirect
196. C47#14301semidirect

Maximal subgroups

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

subgroupindex
C2×#156890342
C2×#156890142
#202416592
#202416512
#202416302
#202416152
C22×C4×Q8×C472
#1430147