Finite Group Databasebeta

← #26112351

Subgroups — #26112351

Order 12032.

690 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 115 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#20249122C211yesmaximal, characteristic
2#20241955C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic, complemented
4#20241255C212yesmaximal, complemented
5C47×#2312C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7C2×#13093993—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689048C412yescomplemented
10#15689039—22no—
11#15689032C412yescomplemented
12#15689002—21no—
13#15688983C2211yescharacteristic
14#15688856—24no—
15#15688847—22no—
16C22×C22⋊C4×C47C2211yescharacteristic
17C47×#388—24no—
18C47×#380C2212yes—
19C47×#379—21no—
20C4×C22⋊C4×C47—21no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C22×C22⋊D47—22no—
24C22×C47⋊C8—22no—
25#13093998—22no—
26#13093998—22no—
27#13093993—44no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—21no—
32C2×(C2×D47)⋊C4—21no—
33C2×(C2×D47)⋊C4—22no—
34C22⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C47⋊#95—22no—
37C47⋊#95C2×C412yes—
38C25×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C4⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C22⋊C4×C47D412yes—
49C2×C42×C47—22no—
50#95×C47—22no—
51#95×C47D414yescomplemented
52C24×C47—21no—
53C24×C47—21no—
54C24×C47—22no—
55C24×C47—22no—
56C24×C47—81no—
57C24×C47—81no—
58C24×C47C22⋊C412yes—
59C24×C47C2×D411yescharacteristic
60C23×D47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—24no—
64C22×C4×C47—81no—
65C22×C4×C47—81no—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4112yes—
71C2×C22⋊D47—44no—
72C2×C22⋊D47—41no—
73C2×C22⋊D47—41no—
74C2×C22⋊D47—42no—
75C2×C22⋊D47—42no—
76C2×C22⋊D47—82no—
77C2×C22⋊D47—82no—
78C22×Dic47—21no—
79C22×Dic47—22no—
80C2×D188—22no—
81C2×D188—81no—
82C4⋊C4×C47—82no—
83C22⋊C4×C47—24no—
84C22⋊C4×C47—24no—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—82no—
88C22⋊C4×C47—82no—
89C42×C47—82no—
90C22⋊Dic47—44no—
91C22⋊Dic47—42no—
92(C2×D47)⋊C4—42no—
93(C2×D47)⋊C4—82no—
94C4⋊Dic47—22no—
95C4⋊Dic47—81no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—82no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—81no—
110C23×C47—81no—
111C23×C47—84no—
112C23×C47—84no—
113C23×C47C2×C22⋊C411yescharacteristic
114C23×C47#12011yescharacteristic
115C23×C47#12311yescharacteristic
116C23×C47#13512yes—
117C23×C47AGL(1,ℤ/8)12yes—
118C22×D47—42no—
119C22×D47—42no—
120C22×D47—41no—
121C22×D47—41no—
122C22×D47—41no—
123C2×C4×C47—22no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—44no—
131C2×C4×C47—44no—
132C2×C4×C47—44no—
133C2×C4×C47—81no—
134C2×C4×C47C4×D412yes—
135C2×C4×C47#12112yes—
136C22⋊D47—88no—
137C22⋊D47—82no—
138C22⋊D47—82no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—42no—
144C2×Dic47—41no—
145D188—82no—
146C47⋊C8—82no—
147#9205—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47—44no—
157C22×C47—42no—
158C22×C47—42no—
159C22×C47—42no—
160C22×C47—82no—
161C22×C47—84no—
162C22×C47—84no—
163C22×C47#38611yescharacteristic
164C22×C47#41611yescharacteristic
165C22×C47#41811yescharacteristic
166C22×C47#44912yes—
167C22×C47#48212yes—
168C2×D47—82no—
169C2×D47—82no—
170C2×D47—81no—
171C2×D47—81no—
172C2×D47—81no—
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—42no—
176C4×C47—84no—
177Dic47—82no—
178Dic47—81no—
179#2617—471no—
180#2312D4711yescharacteristic, complemented
181#1519—472no—
182C2×#323—471no—
183#1289—472no—
184C2×C47—21no—
185C2×C47—22no—
186C2×C47—21no—
187C2×C47—22no—
188C2×C47—82no—
189C2×C47#187511yescharacteristic
190C2×C47#191111yescharacteristic
191C2×C47#191311yescharacteristic
192D47—161no—
193C2×#121—471no—
194C2×#120—941no—
195C22×C22⋊C4C2×D4711yescharacteristic
196C2×C22⋊C8—472no—
197#397—472no—
198#394—942no—
199#392—472no—
200#388—24no—
201#386—941no—
202#380—471no—
203#380C2×D4712yes—
204#379—21no—
205C4×C22⋊C4—21no—
206#327—944no—
207#323—942no—
208C47#920511yescharacteristic, complemented, Sylow
209C25C22⋊D4711yescharacteristic
210C22×D4—471no—
211C22×D4—942no—
212C22×D4—941no—
213C23×C4C22⋊D4711yescharacteristic
214C23×C4C22×D4711yescharacteristic
215C22×C8—942no—
216#121—942no—
217#121—942no—
218#120—1884no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—942no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—941no—
230C2×C22⋊C4—941no—
231C2×C22⋊C4C4×D4712yes—
232C2×C22⋊C4D18812yes—
233C2×C42—22no—
234C22⋊C8—942no—
235C22⋊C8—942no—
236#95—22no—
237#95—472no—
238#95—942no—
239#95C22⋊D4714yescomplemented
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—81no—
245C24—81no—
246C24—941no—
247C24(C2×D47)⋊C412yes—
248C24C2×C22⋊D4711yescharacteristic
249C2×D4—942no—
250C2×D4—3761no—
251C2×D4—3762no—
252C2×D4—3762no—
253C2×D4—1884no—
254C2×D4—1881no—
255C2×D4—1881no—
256C2×D4—1882no—
257C2×D4—1882no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—942no—
264C22×C4—941no—
265C22×C4(C2×D47)⋊C412yes—
266C22×C4C2×C4×D4711yescharacteristic
267C22×C4C2×D18811yescharacteristic
268C22×C4#1062535012yes—
269C22×C4C2×C22⋊D4712yes—
270C2×C8—3762no—
271C2×C8—1882no—
272C4⋊C4—82no—
273C4⋊C4—942no—
274C4⋊C4—3761no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C22⋊C4—3762no—
282C22⋊C4—1884no—
283C22⋊C4—1882no—
284C22⋊C4—1882no—
285C42—82no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—81no—
298C23—81no—
299C23—84no—
300C23—84no—
301C23—1882no—
302C23—1882no—
303C23—1881no—
304C23—1881no—
305C23—1881no—
306C23C2×(C2×D47)⋊C411yescharacteristic
307C23#1309399311yescharacteristic
308C23#1309400711yescharacteristic
309C23#1309404412yes—
310C23#1309404912yes—
311D4—3762no—
312D4—3768no—
313D4—3762no—
314D4—3762no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—81no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1882no—
331C2×C4—1881no—
332C2×C4C4×C22⋊D4712yes—
333C2×C4#1309399812yes—
334C8—3762no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—84no—
349C22—84no—
350C22—3762no—
351C22—3762no—
352C22—3761no—
353C22—3761no—
354C22—3761no—
355C22#1568900211yescharacteristic
356C22#1568911711yescharacteristic
357C22#1568912711yescharacteristic
358C22#1568926812yes—
359C22#1568944712yes—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—84no—
364C4—3762no—
365C4—3761no—
366C2—22no—
367C2—21no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—7521no—
372C2#2024371011yescharacteristic
373C2#2024392011yescharacteristic
374C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2312C2
socleC22×C47#386
prime radical#20249122C2
cosocleC23×C4×C47C23
hypercenter#2312D47
hypocenterC47#9205
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20241955C2semidirect
3. C2×#15688847C2semidirect
4. #20241255C2semidirect
5. C47×#2312C2semidirect
9. #15689048C4semidirect
11. #15689032C4semidirect
51. #95×C47D4semidirect
180. #2312D47semidirect
208. C47#9205semidirect
239. #95C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202491222
#202419552
C2×#156888472
#202412552
C47×#23122
#920547