Finite Group Databasebeta

← #26112736

Subgroups — #26112736

Order 12032.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689439C211yesmaximal, characteristic
2#20246267C211yesmaximal, characteristic, complemented
3#20244225C211yesmaximal, characteristic
4#20243479C211yesmaximal, characteristic, complemented
5#20241255C211yesmaximal, characteristic, complemented
6#20241234C211yesmaximal, characteristic, complemented
7C47×#1519C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9#15689439—21no—
10#15689439—21no—
11#15689439—22no—
12C2×#13093955C412yescomplemented
13C2×#13093945C412yescomplemented
14C2×#13093939—21no—
15C2×#13093939—21no—
16C2×C22⋊(C47⋊C8)C2211yescharacteristic
17#15689019—21no—
18#15689008—22no—
19#15688983C2211yescharacteristic
20#15688981—21no—
21C4⋊C4×Dic47—21no—
22C22⋊C4×Dic47—21no—
23C4×C22⋊Dic47C2211yescharacteristic
24#15688886—22no—
25C2×#121×C47C2211yescharacteristic
26C2×C22⋊C8×C47C2211yescharacteristic
27C47×#380C2211yescharacteristic
28C47×#327—22no—
29C23×Dic47—21no—
30C22×C47⋊C8C2×C412yes—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2×C412yes—
36C2×C4⋊Dic47D411yescharacteristic
37C2×C4⋊Dic47D411yescharacteristic
38C2×C4⋊Dic47D412yes—
39C2×Dic47⋊C4—22no—
40C2×C22⋊Dic47—21no—
41C2×C22⋊Dic47D411yescharacteristic
42C2×C22⋊Dic47D411yescharacteristic
43C2×C4×Dic47—21no—
44C2×C4×Dic47—22no—
45#13093955—22no—
46#13093955—22no—
47#13093945—22no—
48#13093945—22no—
49#13093939—42no—
50#13093939—42no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)—21no—
53C47⋊#95—22no—
54C47⋊#95—22no—
55C47⋊#95—21no—
56C47⋊#95—21no—
57C22×D4×C47—21no—
58C22×D4×C47C2×C411yescharacteristic
59C23×C4×C47C2311yescharacteristic
60C22×C8×C47—21no—
61#121×C47—21no—
62#121×C47—21no—
63#121×C47—21no—
64#121×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47C2×C411yescharacteristic
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C22⋊C8×C47—21no—
70C22⋊C8×C47—21no—
71#95×C47D412yes—
72C24×C47—21no—
73C24×C47C2×D411yescharacteristic
74C2×D4×C47—21no—
75C2×D4×C47—21no—
76C2×D4×C47—42no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—41no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C2×D4×C47—41no—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47—42no—
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47#4111yescharacteristic
91C22×C4×C47#4111yescharacteristic
92C22×Dic47—21no—
93C22×Dic47—21no—
94C22×Dic47—21no—
95C22×Dic47—22no—
96C22×Dic47—22no—
97C22×Dic47—21no—
98C22×Dic47—41no—
99C22×Dic47—82no—
100C2×C8×C47—42no—
101C2×C8×C47—41no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—42no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊Dic47—21no—
110C22⋊Dic47—21no—
111C22⋊Dic47—21no—
112C22⋊Dic47—41no—
113C22⋊Dic47—41no—
114C22⋊Dic47SD1612yes—
115C4⋊Dic47—22no—
116C4⋊Dic47—22no—
117C4⋊Dic47—21no—
118C4⋊Dic47—21no—
119C4⋊Dic47—42no—
120C4⋊Dic47C22⋊C412yes—
121C4⋊Dic47C22⋊C412yes—
122Dic47⋊C4—82no—
123C4×Dic47—42no—
124C4×Dic47—82no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—42no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—81no—
133C23×C47—81no—
134C23×C47C4×D411yescharacteristic
135C23×C47#12011yescharacteristic
136C23×C47#12111yescharacteristic
137C23×C47C2×SD1611yescharacteristic
138C23×C47#13511yescharacteristic
139C23×C47AGL(1,ℤ/8)11yescharacteristic
140C23×C47#13711yescharacteristic
141D4×C47—42no—
142D4×C47—81no—
143D4×C47—81no—
144D4×C47—81no—
145D4×C47—81no—
146D4×C47—81no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47C2×C22⋊C411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12311yescharacteristic
163C2×Dic47—22no—
164C2×Dic47—22no—
165C2×Dic47—22no—
166C2×Dic47—41no—
167C2×Dic47—41no—
168C2×Dic47—42no—
169C2×Dic47—42no—
170C2×Dic47—41no—
171C2×Dic47—41no—
172C2×Dic47—41no—
173C2×Dic47—82no—
174C2×Dic47—82no—
175C2×Dic47—81no—
176C2×Dic47—81no—
177C2×Dic47—81no—
178C8×C47—81no—
179C47⋊C8—42no—
180#9261—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—41no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47—41no—
191C22×C47—81no—
192C22×C47—81no—
193C22×C47—81no—
194C22×C47—161no—
195C22×C47—161no—
196C22×C47#38611yescharacteristic
197C22×C47C4×SD1611yescharacteristic
198C22×C47#44011yescharacteristic
199C22×C47#44811yescharacteristic
200C22×C47#44911yescharacteristic
201C22×C47#46011yescharacteristic
202C22×C47#46411yescharacteristic
203C4×C47—21no—
204C4×C47—21no—
205C4×C47—21no—
206C4×C47—82no—
207C4×C47—81no—
208C4×C47#41012yes—
209Dic47—42no—
210Dic47—42no—
211Dic47—42no—
212Dic47—81no—
213Dic47—161no—
214C2×#481—471no—
215#2345—471no—
216#1964—471no—
217#1824—471no—
218#1519D4711yescharacteristic
219#1289—471no—
220#1282—471no—
221C2×C47—21no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—21no—
225C2×C47—21no—
226C2×C47—21no—
227C2×C47—161no—
228C2×C47#188011yescharacteristic
229C2×C47#189911yescharacteristic
230C2×C47#190411yescharacteristic
231C2×#121C2×D4711yescharacteristic
232C22×C4⋊C4—471no—
233#481—942no—
234#481—941no—
235#481—941no—
236C2×#106—472no—
237C2×#102—472no—
238C2×#99—941no—
239C2×#99—941no—
240C2×C22⋊C8—471no—
241C2×C22⋊C8C2×D4711yescharacteristic
242#389—941no—
243#387—942no—
244#380—471no—
245#380—941no—
246#380C2×D4711yescharacteristic
247C4×C4⋊C4—941no—
248C4×C22⋊C4—471no—
249C4×C22⋊C4—941no—
250#340—942no—
251#327—22no—
252C47#926111yescharacteristic, complemented, Sylow
253C22×D4—21no—
254C22×D4C4×D4711yescharacteristic
255C23×C4—941no—
256C23×C4C22×D4711yescharacteristic
257C22×C8—21no—
258C22×C8—472no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—472no—
265C2×C4⋊C4—472no—
266C2×C4⋊C4—471no—
267C2×C4⋊C4—471no—
268C2×C4⋊C4—941no—
269C2×C4⋊C4—941no—
270C2×C4⋊C4—941no—
271C2×C4⋊C4—942no—
272C2×C4⋊C4—941no—
273C2×C4⋊C4C4×D4711yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—471no—
277C2×C22⋊C4—471no—
278C2×C22⋊C4—941no—
279C2×C42—942no—
280C2×C42—941no—
281#106—942no—
282#106—942no—
283#102—942no—
284#102—942no—
285#99—1882no—
286#99—1882no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—941no—
290C22⋊C8—941no—
291#95—942no—
292#95—942no—
293#95—941no—
294#95—941no—
295#95C22⋊D4712yes—
296C24—21no—
297C24D4×D4711yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—942no—
311C22×C4—942no—
312C22×C4—941no—
313C22×C4—941no—
314C22×C4—941no—
315C22×C4—941no—
316C22×C4—3762no—
317C22×C4—1881no—
318C22×C4C2×C4×D4711yescharacteristic
319C22×C4#1062535011yescharacteristic
320C22×C4D4×D4711yescharacteristic
321C22×C4D4×D4711yescharacteristic
322C22×C4#1062535211yescharacteristic
323C22×C4C2×C22⋊D4711yescharacteristic
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—942no—
327C2×C8—1882no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—472no—
332C4⋊C4—472no—
333C4⋊C4—942no—
334C4⋊C4—942no—
335C4⋊C4—941no—
336C4⋊C4—941no—
337C4⋊C4—3762no—
338C4⋊C4—1882no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—472no—
344C22⋊C4—941no—
345C22⋊C4—941no—
346C22⋊C4—941no—
347C22⋊C4—1881no—
348C22⋊C4—1881no—
349C42—3762no—
350C42—1882no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23—81no—
358C23(C22⋊D47)⋊C411yescharacteristic
359C23#1309399211yescharacteristic
360C23#1309399711yescharacteristic
361C23SD16×D4711yescharacteristic
362C23#1309404311yescharacteristic
363C23#1309404811yescharacteristic
364C23#1309405311yescharacteristic
365D4—42no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—942no—
384C2×C4—942no—
385C2×C4—942no—
386C2×C4—3762no—
387C2×C4—3762no—
388C2×C4—3761no—
389C2×C4—3761no—
390C2×C4—3761no—
391C2×C4—1881no—
392C2×C4—1881no—
393C2×C4—1881no—
394C2×C4—1881no—
395C2×C4—1882no—
396C2×C4—1881no—
397C2×C4—1882no—
398C2×C4C22⋊C4×D4711yescharacteristic
399C2×C4C4×C22⋊D4711yescharacteristic
400C2×C4#1309399511yescharacteristic
401C2×C4#1309400611yescharacteristic
402C8—81no—
403C8—1882no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—161no—
418C22—161no—
419C22#1568899911yescharacteristic
420C22#1568921411yescharacteristic
421C22#1568922011yescharacteristic
422C22#1568926011yescharacteristic
423C22#1568926511yescharacteristic
424C22#1568932211yescharacteristic
425C22#1568934411yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—81no—
431C4—3761no—
432C4—1882no—
433C4—1882no—
434C4—1882no—
435C4—7521no—
436C4#1568909512yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—161no—
444C2#2024374011yescharacteristic
445C2#2024384311yescharacteristic
446C2#2024387211yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1519C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#1519D47
hypocenterC47#9261
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #20246267C2semidirect
4. #20243479C2semidirect
5. #20241255C2semidirect
6. #20241234C2semidirect
12. C2×#13093955C4semidirect
13. C2×#13093945C4semidirect
252. C47#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#156894392
#202462672
#202442252
#202434792
#202412552
#202412342
C47×#15192
#926147