Finite Group Databasebeta

← #26190404

Subgroups — #26190404

Order 12032.

1058 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 167 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×#15689749C211yesmaximal, characteristic
2#20254209C212yesmaximal, complemented
3#20242378C214yesmaximal, complemented
4#20242344C214yesmaximal, complemented
5#20241907C212yesmaximal, complemented
6#20241715C211yesmaximal, characteristic
7C47×#2561C211yesmaximal, characteristic
8#15689749—24no—
9#15689749—22no—
10#15689749C2214yes—
11C2×#13094003—21no—
12C2×#13094003C2212yes—
13C2×#13094001—22no—
14C2×#13094001—22no—
15C22×C22⋊Dic47—21no—
16#15689453C2214yescomplemented
17#15689451—22no—
18#15689379—24no—
19#15689279C2214yescomplemented
20#15689278—22no—
21#15689087C2212yes—
22#15688859C2218yescomplemented
23#15688847C2214yes—
24C2×C22⋊Q8×C47C2211yescharacteristic
25C22×C22⋊C4×C47C2211yescharacteristic
26C47×#397C2212yes—
27C47×#393C2212yes—
28C47×#388—22no—
29C47×#381—21no—
30C47×#379—21no—
31C4×C22⋊C4×C47C2211yescharacteristic
32C22×Dic94—21no—
33C22×Dic94D412yes—
34C23×Dic47—21no—
35C23×Dic47—21no—
36C2×#10625337—24no—
37C2×#10625329—22no—
38#13094003—24no—
39#13094003—22no—
40#13094003—22no—
41#13094003—24no—
42#13094003D414yes—
43#13094003C2314yes—
44#13094001—28no—
45#13094001—28no—
46#13094001—28no—
47#13094001—28no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47D412yes—
50C2×Dic47⋊C4—22no—
51C2×Dic47⋊C4—22no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—22no—
54C2×C22⋊Dic47—24no—
55C2×C22⋊Dic47—22no—
56#13093955—24no—
57#13093948—24no—
58#13093947—24no—
59C22⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)C2314yes—
61C25×C47D411yescharacteristic
62C22×Q8×C47C2311yescharacteristic
63C23×C4×C47D411yescharacteristic
64C23×C4×C47C2311yescharacteristic
65C22⋊Q8×C47—22no—
66C22⋊Q8×C47—22no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47C2311yescharacteristic
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—22no—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47D412yes—
76C2×C22⋊C4×C47C2312yes—
77C2×C42×C47—21no—
78#95×C47—22no—
79#95×C47—22no—
80#95×C47C2312yes—
81C24×C47—21no—
82C24×C47—21no—
83C24×C47—24no—
84C24×C47—42no—
85C24×C47—81no—
86C24×C47C2×D411yescharacteristic
87C24×C47C2×D412yes—
88C2×Q8×C47—22no—
89C2×Q8×C47—81no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—42no—
96C22×C4×C47—81no—
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D412yes—
100C22×C4×C47C2×D412yes—
101C22×C4×C47C2411yescharacteristic
102C22×Dic47—22no—
103C22×Dic47—22no—
104C22×Dic47—24no—
105C22×Dic47—24no—
106C22×Dic47—82no—
107C22×Dic47—82no—
108C2×Dic94—24no—
109C2×Dic94—22no—
110C2×Dic94—82no—
111C2×Dic94—82no—
112C2×Dic94C2×D414yes—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—42no—
115C22⋊C4×C47—24no—
116C22⋊C4×C47—24no—
117C22⋊C4×C47—42no—
118C22⋊C4×C47—41no—
119C22⋊C4×C47—41no—
120C22⋊C4×C47—44no—
121C22⋊C4×C47—82no—
122C42×C47—81no—
123C22⋊Dic47—28no—
124C22⋊Dic47—24no—
125C22⋊Dic47—24no—
126#10625337—84no—
127C4⋊Dic47—24no—
128C4⋊Dic47—22no—
129C4⋊Dic47C2×D414yes—
130Dic47⋊C4—28no—
131Dic47⋊C4—24no—
132Dic47⋊C4—24no—
133#10625329—82no—
134C2×C47⋊C8—24no—
135C23×C47—24no—
136C23×C47—24no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—22no—
140C23×C47—22no—
141C23×C47—41no—
142C23×C47—42no—
143C23×C47—42no—
144C23×C47—42no—
145C23×C47—42no—
146C23×C47—81no—
147C23×C47—81no—
148C23×C47—88no—
149C23×C47#12011yescharacteristic
150C23×C47#12011yescharacteristic
151C23×C47#13714yes—
152C23×C47C22×D411yescharacteristic
153Q8×C47—44no—
154C2×C4×C47—22no—
155C2×C4×C47—42no—
156C2×C4×C47—41no—
157C2×C4×C47—42no—
158C2×C4×C47—41no—
159C2×C4×C47—42no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—42no—
167C2×C4×C47—42no—
168C2×C4×C47—81no—
169C2×C4×C47#12012yes—
170C2×C4×C47C22×D412yes—
171C2×Dic47—24no—
172C2×Dic47—24no—
173C2×Dic47—44no—
174C2×Dic47—44no—
175C2×Dic47—84no—
176C2×Dic47—84no—
177Dic94—44no—
178Dic94—82no—
179Dic94—84no—
180C47⋊C8—44no—
181#18319—471nomaximal, Sylow
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—24no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—44no—
191C22×C47—44no—
192C22×C47—48no—
193C22×C47—42no—
194C22×C47—42no—
195C22×C47—82no—
196C22×C47—84no—
197C22×C47#45512yes—
198C22×C47C2×#12011yescharacteristic
199C22×C47C2×#13712yes—
200C4×C47—22no—
201C4×C47—22no—
202C4×C47—42no—
203C4×C47—82no—
204C4×C47—81no—
205C4×C47—81no—
206Dic47—44no—
207Dic47—44no—
208C2×#543—471no—
209#3195—472no—
210#2561D4711yescharacteristic
211#1608—474no—
212#1602—474no—
213#1508—472no—
214#1462—471no—
215C2×C47—22no—
216C2×C47—22no—
217C2×C47—41no—
218C2×C47—44no—
219C2×C47#300411yescharacteristic
220C2×C47C2×#45511yescharacteristic
221C2×C47#302611yescharacteristic
222#543—474no—
223#543—944no—
224#543—942no—
225C2×C22⋊Q8—472no—
226C2×C22⋊Q8—942no—
227C2×C22⋊Q8—942no—
228C2×C22⋊Q8—941no—
229C2×C22⋊Q8C2×D4711yescharacteristic
230C22×C22⋊C4—941no—
231C22×C22⋊C4C2×D4711yescharacteristic
232#483—474no—
233#483—942no—
234#470—944no—
235#451—474no—
236#451—942no—
237#407—472no—
238#397C2×D4712yes—
239#393C2×D4712yes—
240#388—22no—
241#381—21no—
242#379—21no—
243C4×C22⋊C4C2×D4711yescharacteristic
244#328—478no—
245#323—474no—
246C47#1831911yescharacteristic, complemented, Sylow
247C25C22⋊D4711yescharacteristic
248C22×Q8—472no—
249C22×Q8—941no—
250C22×Q8C22×D4711yescharacteristic
251C23×C4—941no—
252C23×C4—941no—
253C23×C4C22⋊D4711yescharacteristic
254C23×C4C22×D4711yescharacteristic
255C2×Q16—944no—
256C2×M16—942no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—474no—
260C22⋊Q8—474no—
261C22⋊Q8—948no—
262C22⋊Q8—948no—
263C22⋊Q8—948no—
264C22⋊Q8—948no—
265C22⋊Q8—944no—
266C22⋊Q8—942no—
267C22⋊Q8—942no—
268C22⋊Q8—944no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—472no—
271C2×C4⋊C4—942no—
272C2×C4⋊C4—941no—
273C2×C4⋊C4—942no—
274C2×C4⋊C4C22×D4711yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—942no—
282C2×C22⋊C4—944no—
283C2×C22⋊C4—942no—
284C2×C22⋊C4—944no—
285C2×C22⋊C4C22⋊D4712yes—
286C2×C22⋊C4C22×D4712yes—
287C2×C42—21no—
288#106—944no—
289#103—944no—
290#103—944no—
291C22⋊C8—474no—
292C22⋊C8—942no—
293#95—22no—
294#95—22no—
295#95C22×D4712yes—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—42no—
300C24—81no—
301C24C2×C22⋊D4711yescharacteristic
302C24C2×C22⋊D4712yes—
303C2×Q8—22no—
304C2×Q8—81no—
305C2×Q8—474no—
306C2×Q8—944no—
307C2×Q8—942no—
308C2×Q8—3762no—
309C2×Q8—3762no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—81no—
317C22×C4—944no—
318C22×C4—944no—
319C22×C4—942no—
320C22×C4—942no—
321C22×C4—3762no—
322C22×C4—3762no—
323C22×C4D4×D4712yes—
324C22×C4D4×D4712yes—
325C22×C4C2×C22⋊D4712yes—
326C22×C4C2×C22⋊D4711yescharacteristic
327C22×C4C23×D4711yescharacteristic
328Q16—3764no—
329M16—3762no—
330C2×C8—944no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—474no—
334C4⋊C4—948no—
335C4⋊C4—944no—
336C4⋊C4—944no—
337C4⋊C4—942no—
338C4⋊C4—944no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—948no—
347C22⋊C4—944no—
348C22⋊C4—944no—
349C42—81no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—88no—
364C23#1309399211yescharacteristic
365C23#1309399211yescharacteristic
366C23#1309405414yes—
367C23C22×C22⋊D4711yescharacteristic
368Q8—44no—
369Q8—3762no—
370Q8—3764no—
371Q8—1884no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—944no—
388C2×C4—944no—
389C2×C4—3764no—
390C2×C4—3764no—
391C2×C4—1884no—
392C2×C4—1884no—
393C2×C4#1309399212yes—
394C2×C4C2×D4×D4712yes—
395C8—1884no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22#1568930012yes—
412C22C2×#1309399211yescharacteristic
413C22C2×#1309405412yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4—1884no—
421C4—1884no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#2025262111yescharacteristic
427C2C2×#1568930011yescharacteristic
428C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2561C2
socleC22×C47C2×#120
prime radicalC25×C47D4
hypercenter#2561D47
hypocenterC47#18319
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #20254209C2semidirect
3. #20242378C2semidirect
4. #20242344C2semidirect
5. #20241907C2semidirect
16. #15689453C22semidirect
19. #15689279C22semidirect
22. #15688859C22semidirect
246. C47#18319semidirect

Maximal subgroups

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

subgroupindex
C2×#156897492
#202542092
#202423782
#202423442
#202419072
#202417152
C47×#25612
#1831947