Finite Group Databasebeta

← #26113723

Subgroups — #26113723

Order 12032.

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

normal subgroups only

This group has 840 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
1#20248319C211yesmaximal, characteristic
2#20241903C212yesmaximal, complemented
3#20241715C211yesmaximal, characteristic, complemented
4#20241274C212yesmaximal, complemented
5C47×#2273C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8C2×#13093993—21no—
9#15689087C2212yes—
10#15689059C412yescomplemented
11#15689032C412yescomplemented
12#15689023—22no—
13C4×C22⋊Dic47C2211yescharacteristic
14#15688856—24no—
15#15688847—22no—
16C22×C22⋊C4×C47—21no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C4×C22⋊C4×C47—24no—
19C4×C22⋊C4×C47—21no—
20C4×C22⋊C4×C47C2212yes—
21C22×D188C2×C411yescharacteristic, complemented
22C22×C22⋊D47—21no—
23C22×C22⋊D47—22no—
24C2×#10625329—22no—
25#13094007—44no—
26#13093998—22no—
27#13093998—22no—
28#13093993—44no—
29C2×C4⋊Dic47C2×C411yescharacteristic, complemented
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—21no—
35C2×C4×Dic47—21no—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C47⋊#95C2×C412yescomplemented
39C25×C47D411yescharacteristic
40C23×C4×C47—22no—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47—28no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47C2×C412yes—
49C2×C22⋊C4×C47D412yes—
50C2×C42×C47—22no—
51C2×C42×C47—22no—
52#95×C47—22no—
53#95×C47D414yescomplemented
54C24×C47—21no—
55C24×C47—24no—
56C24×C47—21no—
57C24×C47—42no—
58C24×C47—81no—
59C24×C47C22⋊C412yes—
60C24×C47C2×D411yescharacteristic
61C23×D47—21no—
62C22×C4×C47—24no—
63C22×C4×C47—21no—
64C22×C4×C47—28no—
65C22×C4×C47—21no—
66C22×C4×C47—42no—
67C22×C4×C47—81no—
68C22×C4×C47—82no—
69C22×C4×C47—82no—
70C22×C4×C47C22⋊C412yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D412yes—
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47#4112yes—
75C2×C22⋊D47—44no—
76C2×C22⋊D47—42no—
77C2×C22⋊D47—42no—
78C2×C22⋊D47—44no—
79C2×C22⋊D47—41no—
80C2×C22⋊D47—41no—
81C2×C22⋊D47—82no—
82C22×Dic47—22no—
83C22×Dic47—21no—
84C2×D188—22no—
85C2×D188—81no—
86C22⋊C4×C47—28no—
87C22⋊C4×C47—28no—
88C22⋊C4×C47—24no—
89C22⋊C4×C47—24no—
90C42×C47—82no—
91C42×C47—82no—
92C22⋊Dic47—24no—
93C22⋊Dic47—44no—
94(C2×D47)⋊C4—44no—
95(C2×D47)⋊C4—42no—
96C4⋊Dic47—22no—
97Dic47⋊C4—44no—
98C4×Dic47—81no—
99#10625329—82no—
100C2×C47⋊C8—42no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—24no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—22no—
108C23×C47—41no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—42no—
113C23×C47—81no—
114C23×C47—81no—
115C23×C47—88no—
116C23×C47#10414yes—
117C23×C47C2×C22⋊C411yescharacteristic
118C23×C47#12011yescharacteristic
119C23×C47#12311yescharacteristic
120C22×D47—41no—
121C22×D47—42no—
122C22×D47—41no—
123C22×D47—41no—
124C22×D47—42no—
125C2×C4×C47—28no—
126C2×C4×C47—22no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—44no—
132C2×C4×C47—42no—
133C2×C4×C47—44no—
134C2×C4×C47—42no—
135C2×C4×C47—81no—
136C2×C4×C47—88no—
137C2×C4×C47C4×D412yes—
138C2×C4×C47#12112yes—
139C22⋊D47—88no—
140C22⋊D47—82no—
141C22⋊D47—82no—
142C2×Dic47—24no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148D188—82no—
149C47⋊C8—82no—
150#9409—471nomaximal, Sylow
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—24no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—44no—
160C22×C47—44no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—42no—
164C22×C47—42no—
165C22×C47—82no—
166C22×C47—84no—
167C22×C47#38611yescharacteristic
168C22×C47C2×#10412yes—
169C22×C47#45312yes—
170C2×D47—82no—
171C2×D47—82no—
172C2×D47—81no—
173C2×D47—81no—
174C2×D47—81no—
175C4×C47—22no—
176C4×C47—22no—
177C4×C47—48no—
178C4×C47—42no—
179Dic47—44no—
180Dic47—81no—
181#2539—471no—
182#2273D4711yescharacteristic, complemented
183#1507—472no—
184#1462—471no—
185#1297—472no—
186C2×C47—22no—
187C2×C47—22no—
188C2×C47—41no—
189C2×C47—44no—
190C2×C47#188711yescharacteristic
191C2×C47#189611yescharacteristic
192C2×C47#191311yescharacteristic
193D47—161no—
194C2×#123—941no—
195C2×#121—471no—
196C2×#120—941no—
197C22×C22⋊C4—21no—
198C22×C22⋊C4C2×D4711yescharacteristic
199#407—472no—
200#399—472no—
201#392—472no—
202#390—942no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—24no—
205C4×C22⋊C4—471no—
206C4×C22⋊C4C2×D4712yes—
207#327—944no—
208#323—942no—
209C47#940911yescharacteristic, complemented, Sylow
210C25C22⋊D4711yescharacteristic
211C22×D4—471no—
212C22×D4—942no—
213C22×D4—941no—
214C23×C4—22no—
215C23×C4C22⋊D4711yescharacteristic
216C23×C4C22×D4711yescharacteristic
217C2×M16—942no—
218#123—1884no—
219#121—942no—
220#121—942no—
221#120—1884no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—942no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—941no—
230C2×C22⋊C4—942no—
231C2×C22⋊C4—942no—
232C2×C22⋊C4—941no—
233C2×C22⋊C4C4×D4712yes—
234C2×C22⋊C4D18812yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—941no—
238C22⋊C8—942no—
239C22⋊C8—942no—
240#95—22no—
241#95—472no—
242#95C22⋊D4714yescomplemented
243C24—21no—
244C24—24no—
245C24—21no—
246C24—42no—
247C24—81no—
248C24—941no—
249C24(C2×D47)⋊C412yes—
250C24C2×C22⋊D4711yescharacteristic
251C2×D4—942no—
252C2×D4—3761no—
253C2×D4—3762no—
254C2×D4—1884no—
255C2×D4—1882no—
256C2×D4—1882no—
257C2×D4—1884no—
258C2×D4—1881no—
259C2×D4—1881no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—942no—
269C22×C4—941no—
270C22×C4(C2×D47)⋊C412yes—
271C22×C4C2×C4×D4711yescharacteristic
272C22×C4C2×D18811yescharacteristic
273C22×C4#1062535012yes—
274C22×C4C2×C22⋊D4712yes—
275M16—3762no—
276C2×C8—1882no—
277C4⋊C4—942no—
278C4⋊C4—1884no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—944no—
284C22⋊C4—1884no—
285C22⋊C4—1884no—
286C22⋊C4—1882no—
287C42—82no—
288C42—82no—
289C42—3761no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—88no—
305C23—1881no—
306C23—1882no—
307C23—1881no—
308C23—1881no—
309C23—1882no—
310C23#1309395214yes—
311C23C2×(C2×D47)⋊C411yescharacteristic
312C23#1309399311yescharacteristic
313C23#1309400711yescharacteristic
314D4—3762no—
315D4—3762no—
316D4—3762no—
317D4—3768no—
318C2×C4—28no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—88no—
330C2×C4—944no—
331C2×C4—1881no—
332C2×C4—1881no—
333C2×C4—1881no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4C4×C22⋊D4712yes—
337C2×C4#1309399812yes—
338C8—3762no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—84no—
355C22—3762no—
356C22—3762no—
357C22—3761no—
358C22—3761no—
359C22—3761no—
360C22#1568900211yescharacteristic
361C22C2×#1309395212yes—
362C22#1568929112yes—
363C4—22no—
364C4—22no—
365C4—48no—
366C4—42no—
367C4—3761no—
368C4—1884no—
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—7521no—
374C2#2024378611yescharacteristic
375C2#2024383111yescharacteristic
376C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2273C2
socleC22×C47#386
prime radical#20248319C2
cosocleC23×C4×C47C23
hypercenter#2273D47
hypocenterC47#9409
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
2. #20241903C2semidirect
3. #20241715C2semidirect
4. #20241274C2semidirect
5. C47×#2273C2semidirect
10. #15689059C4semidirect
11. #15689032C4semidirect
21. C22×D188C2×C4semidirect
29. C2×C4⋊Dic47C2×C4semidirect
38. C47⋊#95C2×C4semidirect
53. #95×C47D4semidirect
182. #2273D47semidirect
209. C47#9409semidirect
242. #95C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202483192
#202419032
#202417152
#202412742
C47×#22732
#940947