Finite Group Databasebeta

← #26116563

Subgroups — #26116563

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689451C211yesmaximal, characteristic
2C2×#15689053C211yesmaximal, characteristic
3#20243472C211yesmaximal, characteristic
4C2×#15688860C211yesmaximal, characteristic
5C2×#15688849C211yesmaximal, characteristic
6#20241266C211yesmaximal, characteristic
7C47×#2627C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9C22×Dic47⋊C4—21no—
10#15689451—22no—
11#15689451—22no—
12C2×#13093955—21no—
13C2×#13093948—21no—
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15C2×C22⋊(C47⋊C8)C2211yescharacteristic
16#15689053—22no—
17#15689053—22no—
18C2×C47⋊#95—21no—
19C2×C47⋊#95C2211yescharacteristic
20#15688885—22no—
21#15688860—22no—
22#15688849—22no—
23C2×C22⋊Q8×C47C2211yescharacteristic
24C22×C4⋊C4×C47C2211yescharacteristic
25C47×#400—21no—
26C47×#398—22no—
27C47×#395—21no—
28C47×#385—21no—
29C47×#381—21no—
30C47×#380C2211yescharacteristic
31C23×Dic47—21no—
32C23×Dic47—21no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47D412yes—
41C2×Dic47⋊C4—24no—
42C2×Dic47⋊C4—42no—
43C2×Dic47⋊C4—42no—
44#13093955—24no—
45#13093948—24no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)D412yes—
50C47⋊#95—24no—
51C47⋊#95—21no—
52C47⋊#95D412yes—
53C22×Q8×C47D411yescharacteristic
54C23×C4×C47D411yescharacteristic
55C23×C4×C47D411yescharacteristic
56C23×C4×C47C2311yescharacteristic
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C4⋊C4×C47D412yes—
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47—21no—
73#95×C47—22no—
74#95×C47—22no—
75#95×C47—22no—
76#95×C47—21no—
77C24×C47C2×D411yescharacteristic
78C2×Q8×C47—21no—
79C2×Q8×C47—21no—
80C2×Q8×C47—42no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—42no—
94C22×C4×C47—42no—
95C22×C4×C47SD1612yes—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47#4111yescharacteristic
102C22×Dic47—24no—
103C22×Dic47—22no—
104C22×Dic47—22no—
105C22×Dic47—42no—
106C22×Dic47—41no—
107C22×Dic47—41no—
108C22×Dic47—41no—
109C22×Dic47—41no—
110C22×Dic47—41no—
111C22×Dic47—41no—
112C22×Dic47—41no—
113C22×Dic47—41no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47—21no—
116C4⋊C4×C47—42no—
117C4⋊C4×C47—41no—
118C4⋊C4×C47—41no—
119C4⋊C4×C47—41no—
120C4⋊C4×C47—41no—
121C4⋊C4×C47—41no—
122C4⋊C4×C47—41no—
123C4⋊C4×C47—82no—
124C22⋊C4×C47—41no—
125C22⋊C4×C47—41no—
126C22⋊C4×C47—41no—
127C22⋊C4×C47—41no—
128C4⋊Dic47—22no—
129C4⋊Dic47—22no—
130C4⋊Dic47—41no—
131C4⋊Dic47—41no—
132Dic47⋊C4—48no—
133C2×C47⋊C8—22no—
134C2×C47⋊C8—42no—
135C2×C47⋊C8—42no—
136C2×C47⋊C8—41no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47#12011yescharacteristic
140C23×C47#12011yescharacteristic
141C23×C47#12111yescharacteristic
142C23×C47C2×SD1611yescharacteristic
143C23×C47#13511yescharacteristic
144C23×C47AGL(1,ℤ/8)11yescharacteristic
145C23×C47#13711yescharacteristic
146Q8×C47—42no—
147Q8×C47—81no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—42no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47—42no—
168C2×C4×C47—42no—
169C2×C4×C47—42no—
170C2×C4×C47—42no—
171C2×C4×C47—81no—
172C2×C4×C47#12011yescharacteristic
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47#12111yescharacteristic
175C2×C4×C47#12711yescharacteristic
176C2×Dic47—24no—
177C2×Dic47—44no—
178C2×Dic47—44no—
179C2×Dic47—42no—
180C2×Dic47—42no—
181C2×Dic47—42no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47—81no—
185C2×Dic47—81no—
186C2×Dic47—81no—
187C2×Dic47—81no—
188C47⋊C8—42no—
189C47⋊C8—81no—
190#9927—471nomaximal, Sylow
191C22×C47—22no—
192C22×C47—22no—
193C22×C47—22no—
194C22×C47—22no—
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—21no—
199C22×C47#39211yescharacteristic
200C22×C47#44811yescharacteristic
201C22×C47#44911yescharacteristic
202C22×C47#45011yescharacteristic
203C22×C47#45211yescharacteristic
204C22×C47#45612yes—
205C22×C47#45812yes—
206C22×C47#46511yescharacteristic
207C22×C47#46811yescharacteristic
208C4×C47—21no—
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—42no—
213C4×C47—82no—
214C4×C47—81no—
215C4×C47—81no—
216C4×C47—81no—
217Dic47—44no—
218Dic47—82no—
219C2×#483—471no—
220#2627D4711yescharacteristic
221C2×#398—471no—
222#1823—471no—
223C2×#328—471no—
224C2×#324—471no—
225#1294—471no—
226C2×C47—21no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47—21no—
230C2×C47#160912yes—
231C2×C47#161112yes—
232C2×C47#200011yescharacteristic
233C2×C47#200511yescharacteristic
234C2×C47#202211yescharacteristic
235C2×C22⋊Q8C2×D4711yescharacteristic
236C22×C4⋊C4—471no—
237C22×C4⋊C4—941no—
238C22×C4⋊C4C2×D4711yescharacteristic
239#483—942no—
240#483—942no—
241C2×#106—941no—
242C2×#103—941no—
243C2×C22⋊C8—471no—
244C2×C22⋊C8—471no—
245#400—21no—
246#398—22no—
247#398—942no—
248#398—942no—
249#395—21no—
250#385—21no—
251#381—21no—
252#380C2×D4711yescharacteristic
253C2×#95—471no—
254C2×#95—941no—
255#340—942no—
256#328—942no—
257#324—942no—
258C47#992711yescharacteristic, complemented, Sylow
259C22×Q8C22⋊D4711yescharacteristic
260C23×C4—941no—
261C23×C4—941no—
262C23×C4C22⋊D4711yescharacteristic
263C23×C4C22⋊D4711yescharacteristic
264C23×C4C22×D4711yescharacteristic
265C22×C8—941no—
266C22×C8—941no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—472no—
280C2×C4⋊C4—941no—
281C2×C4⋊C4—941no—
282C2×C4⋊C4—941no—
283C2×C4⋊C4—941no—
284C2×C4⋊C4—944no—
285C2×C4⋊C4—941no—
286C2×C4⋊C4—1882no—
287C2×C4⋊C4—1882no—
288C2×C4⋊C4C22⋊D4711yescharacteristic
289C2×C4⋊C4C22⋊D4712yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292#106—944no—
293#103—944no—
294C22⋊C8—472no—
295C22⋊C8—941no—
296C22⋊C8—941no—
297C22⋊C8—941no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95—472no—
303#95—944no—
304#95—941no—
305C24C2×C22⋊D4711yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—944no—
324C22×C4—942no—
325C22×C4—942no—
326C22×C4—1881no—
327C22×C4—1881no—
328C22×C4—1881no—
329C22×C4—1881no—
330C22×C4—1881no—
331C22×C4—1881no—
332C22×C4—1882no—
333C22×C4—1881no—
334C22×C4—1881no—
335C22×C4#1062533612yes—
336C22×C4D4×D4711yescharacteristic
337C22×C4D4×D4711yescharacteristic
338C22×C4D4×D4711yescharacteristic
339C22×C4#1062535211yescharacteristic
340C22×C4C2×C22⋊D4711yescharacteristic
341C22×C4C2×C22⋊D4711yescharacteristic
342C2×C8—942no—
343C2×C8—1882no—
344C2×C8—1881no—
345C2×C8—1882no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—82no—
356C4⋊C4—942no—
357C4⋊C4—942no—
358C4⋊C4—1888no—
359C4⋊C4—1881no—
360C4⋊C4—1881no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C23—22no—
366C23—22no—
367C23#1309399211yescharacteristic
368C23#1309399311yescharacteristic
369C23#1309399611yescharacteristic
370C23C2×#1062533611yescharacteristic
371C23#1309404411yescharacteristic
372C23#1309404911yescharacteristic
373C23#1309405511yescharacteristic
374Q8—42no—
375Q8—81no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—944no—
401C2×C4—3761no—
402C2×C4—3761no—
403C2×C4—3761no—
404C2×C4—3761no—
405C2×C4—1882no—
406C2×C4—1882no—
407C2×C4—1882no—
408C2×C4—1882no—
409C2×C4—1884no—
410C2×C4—1884no—
411C2×C4—1882no—
412C2×C4#1309399211yescharacteristic
413C2×C4#1309399511yescharacteristic
414C2×C4#1309399611yescharacteristic
415C2×C4#1309402011yescharacteristic
416C8—3761no—
417C8—1882no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22#1568903111yescharacteristic
427C22#1568926211yescharacteristic
428C22#1568926811yescharacteristic
429C22#1568927311yescharacteristic
430C22#1568928311yescharacteristic
431C22#1568930612yes—
432C22#1568931312yes—
433C22#1568935411yescharacteristic
434C22#1568937111yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—82no—
444C4—3762no—
445C4—1884no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#2024238512yes—
451C2#2024239612yes—
452C2#2024438911yescharacteristic
453C2#2024442211yescharacteristic
454C2#2024451711yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2627C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#2627D47
hypocenterC47#9927
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
258. C47#9927semidirect

Maximal subgroups

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

subgroupindex
C2×#156894512
C2×#156890532
#202434722
C2×#156888602
C2×#156888492
#202412662
C47×#26272
#992747