Finite Group Databasebeta

← #26115658

Subgroups — #26115658

Order 12032.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253713C211yesmaximal, characteristic
2#20243518C211yesmaximal, characteristic, complemented
3#20241952C211yesmaximal, characteristic, complemented
4#20241949C211yesmaximal, characteristic
5#20241947C211yesmaximal, characteristic
6#20241672C211yesmaximal, characteristic, complemented
7C47×#1518C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689480—21no—
10#15689480—21no—
11#15689475C411yescharacteristic, complemented
12#15689475C411yescharacteristic, complemented
13#15689471C411yescharacteristic, complemented
14#15689471C411yescharacteristic, complemented
15#15689176C2211yescharacteristic
16#15689117—21no—
17#15689117—21no—
18C2×C4×C47⋊C8C2211yescharacteristic
19#15688989C2211yescharacteristic
20#15688885—21no—
21#15688885—21no—
22#15688860—22no—
23#15688858—22no—
24#15688856—22no—
25#15688845—21no—
26C8×C47⋊C8—21no—
27C47×#533C2211yescharacteristic
28C47×#431C2211yescharacteristic
29C47×#382C2211yescharacteristic
30C47×#328—22no—
31C2×#10625350—21no—
32C22×C47⋊C8D411yescharacteristic
33C22×C47⋊C8D411yescharacteristic
34#13094024C2×C411yescharacteristic
35#13094021C2×C411yescharacteristic
36#13094012—21no—
37#13093998—21no—
38#13093998—21no—
39C4×D188—21no—
40C4×D188—21no—
41C2×C4⋊Dic47—21no—
42C4⋊(C47⋊C8)—21no—
43#13093947—21no—
44#13093947—21no—
45#13093947—21no—
46#13093947—21no—
47#13093944—21no—
48#13093944—21no—
49#13093944—21no—
50#13093944—21no—
51C22⋊(C47⋊C8)D412yes—
52C4×C47⋊C8—21no—
53C4×C47⋊C8C2×C411yescharacteristic
54C4×C47⋊C8C2×C411yescharacteristic
55C4×C47⋊C8D411yescharacteristic
56C4×C47⋊C8D411yescharacteristic
57C47⋊#95—21no—
58C47⋊#95—21no—
59C4⋊Q8×C47C2×C411yescharacteristic
60C4⋊Q8×C47C2×C411yescharacteristic
61#125×C47—21no—
62C22⋊Q8×C47—21no—
63C22⋊Q8×C47—21no—
64C4×Q8×C47—21no—
65C4×Q8×C47—21no—
66#117×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47—21no—
70C22⋊C8×C47D412yes—
71C4×C8×C47—21no—
72#95×C47—21no—
73#95×C47—21no—
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C22×C4×C47—41no—
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C2×C22⋊D47—41no—
81C22×Dic47—41no—
82#10625350—41no—
83#10625350—41no—
84#10625350—41no—
85#10625350—41no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×C4×D47—41no—
90C2×Dic94—21no—
91C2×C8×C47—22no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C42×C47—41no—
102C42×C47C22×C411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47#4111yescharacteristic
105C42×C47#4111yescharacteristic
106(C2×D47)⋊C4—41no—
107(C2×D47)⋊C4—41no—
108C4⋊Dic47—21no—
109C4⋊Dic47—21no—
110C4⋊Dic47—41no—
111C4⋊Dic47—41no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—21no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8—22no—
117C2×C47⋊C8C22⋊C411yescharacteristic
118C2×C47⋊C8C22⋊C411yescharacteristic
119C2×C47⋊C8C22⋊C411yescharacteristic
120C2×C47⋊C8C22⋊C411yescharacteristic
121C23×C47#12711yescharacteristic
122C22×D47—41no—
123Q8×C47—81no—
124Q8×C47—81no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47C2×C22⋊C411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12411yescharacteristic
138C2×C4×C47#13511yescharacteristic
139C2×C4×C47#13511yescharacteristic
140C2×C4×C47#13511yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C22⋊D47—81no—
143C22⋊D47—81no—
144C2×Dic47—41no—
145C2×Dic47—81no—
146C2×Dic47—81no—
147C2×Dic47—81no—
148D188—41no—
149D188—41no—
150D188—81no—
151D188—81no—
152C4×D47—81no—
153C4×D47—81no—
154Dic94—41no—
155Dic94—41no—
156C8×C47—42no—
157C47⋊C8—21no—
158C47⋊C8—21no—
159C47⋊C8—21no—
160C47⋊C8—21no—
161C47⋊C8—42no—
162#9757—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47#39011yescharacteristic
166C22×C47#49511yescharacteristic
167C22×C47#49511yescharacteristic
168C2×D47—81no—
169C2×D47—81no—
170C2×D47—81no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47—161no—
178C4×C47#41611yescharacteristic
179C4×C47#41611yescharacteristic
180C4×C47#48811yescharacteristic
181C4×C47#48811yescharacteristic
182Dic47—81no—
183Dic47—161no—
184#3131—471no—
185#1835—471no—
186#1518—471no—
187#1518D4711yescharacteristic, complemented
188#1517—471no—
189#1517—471no—
190#1450—471no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47#195811yescharacteristic
194C2×C47#197611yescharacteristic
195C2×C47#197611yescharacteristic
196D47—161no—
197#533C2×D4711yescharacteristic
198#532—471no—
199#488—941no—
200#488—941no—
201#487—471no—
202#487—471no—
203#486—471no—
204#486—471no—
205#431—471no—
206#431C2×D4711yescharacteristic
207#416—941no—
208#416—941no—
209C2×C4×C8—471no—
210#382—471no—
211#382C2×D4711yescharacteristic
212#340—941no—
213#340—941no—
214#328—22no—
215#328—942no—
216#327—942no—
217#327—942no—
218#322—941no—
219C82—941no—
220C47#975711yescharacteristic, complemented, Sylow
221C2×#41—941no—
222C22×C8—471no—
223C22×C8—471no—
224C4⋊Q8—471no—
225C4⋊Q8C4×D4711yescharacteristic
226C4⋊Q8C4×D4711yescharacteristic
227#127—471no—
228#125—21no—
229#124—941no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—941no—
233#121—941no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—941no—
237C4×D4—941no—
238#117—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—941no—
241C2×C42C22×D4711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—941no—
244#103—941no—
245#103—941no—
246#103—941no—
247#103—941no—
248#102—941no—
249#102—941no—
250#102—941no—
251#102—941no—
252C22⋊C8—472no—
253C22⋊C8C22⋊D4712yes—
254C4×C8—21no—
255C4×C8—471no—
256C4×C8—471no—
257C4×C8—471no—
258C4×C8—471no—
259C4×C8—941no—
260#95—21no—
261#95—21no—
262#95—941no—
263#95—941no—
264#41—1881no—
265#41—1881no—
266#41—1881no—
267#41—1881no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—941no—
271C2×D4—941no—
272C2×D4—941no—
273C2×D4—941no—
274C2×D4—1881no—
275C22×C4—41no—
276C22×C4—1881no—
277C22×C4—1881no—
278C22×C4D4×D4711yescharacteristic
279C22×C4D4×D4711yescharacteristic
280C22×C4C2×C22⋊D4711yescharacteristic
281C2×C8—22no—
282C2×C8—471no—
283C2×C8—471no—
284C2×C8—471no—
285C2×C8—471no—
286C2×C8—941no—
287C2×C8—941no—
288C2×C8—941no—
289C2×C8—941no—
290C2×C8—942no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—941no—
300C4⋊C4—941no—
301C4⋊C4—1881no—
302C4⋊C4—1881no—
303C22⋊C4—41no—
304C22⋊C4—1881no—
305C22⋊C4—1881no—
306C42—41no—
307C42C2×C4×D4711yescharacteristic
308C42#1062535011yescharacteristic
309C42D4×D4711yescharacteristic
310C42#1062535211yescharacteristic
311C23—1881no—
312C23#1309402011yescharacteristic
313Q8—81no—
314Q8—81no—
315Q8—1881no—
316Q8—1881no—
317D4—3761no—
318D4—3761no—
319D4—3761no—
320D4—3761no—
321D4—1881no—
322D4—1881no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—3761no—
331C2×C4—3761no—
332C2×C4—3761no—
333C2×C4—3761no—
334C2×C4—3761no—
335C2×C4—1881no—
336C2×C4C22⋊C4×D4711yescharacteristic
337C2×C4(C22⋊D47)⋊C411yescharacteristic
338C2×C4C4×C22⋊D4711yescharacteristic
339C2×C4#1309399611yescharacteristic
340C2×C4#1309399711yescharacteristic
341C2×C4#1309401011yescharacteristic
342C2×C4#1309404511yescharacteristic
343C2×C4#1309404511yescharacteristic
344C2×C4#1309404711yescharacteristic
345C2×C4#1309404711yescharacteristic
346C8—42no—
347C8—941no—
348C8—941no—
349C8—941no—
350C8—941no—
351C8—1882no—
352C22—21no—
353C22—21no—
354C22—3761no—
355C22—3761no—
356C22—3761no—
357C22#1568902311yescharacteristic
358C22#1568950411yescharacteristic
359C22#1568950811yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—81no—
366C4—161no—
367C4—3761no—
368C4—7521no—
369C4#1568911811yescharacteristic
370C4#1568911811yescharacteristic
371C4#1568947911yescharacteristic
372C4#1568947911yescharacteristic
373C2—21no—
374C2—21no—
375C2—7521no—
376C2#2024419611yescharacteristic
377C2#2024428311yescharacteristic
378C2#2024428511yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1518C2
socleC22×C47#390
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1518D47
hypocenterC47#9757
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #20243518C2semidirect
3. #20241952C2semidirect
6. #20241672C2semidirect
7. C47×#1518C2semidirect
11. #15689475C4semidirect
12. #15689475C4semidirect
13. #15689471C4semidirect
14. #15689471C4semidirect
187. #1518D47semidirect
220. C47#9757semidirect

Maximal subgroups

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

subgroupindex
#202537132
#202435182
#202419522
#202419492
#202419472
#202416722
C47×#15182
#975747