Finite Group Databasebeta

← #26113170

Subgroups — #26113170

Order 12032.

612 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 97 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#20253483C211yesmaximal, characteristic
2#20245750C211yesmaximal, characteristic
3#20244333C211yesmaximal, characteristic, complemented
4#20241830C211yesmaximal, characteristic
5#20241281C211yesmaximal, characteristic, complemented
6#20241243C211yesmaximal, characteristic, complemented
7C47×#1506C211yesmaximal, characteristic, complemented
8C2×C4×D188—21no—
9C2×C4×D188C2211yescharacteristic
10#15689456—21no—
11#15689400C412yescomplemented
12#15689325C412yescomplemented
13#15689273—21no—
14#15689221—22no—
15#15689180C2211yescharacteristic
16C4×#10625329C2211yescharacteristic
17C4×C4⋊Dic47—21no—
18C4×C4⋊Dic47C2211yescharacteristic
19#15688970—21no—
20#15688970—22no—
21#95×D47—21no—
22#15688882—22no—
23#15688850—22no—
24C47×#533C2211yescharacteristic
25C47×#432C2211yescharacteristic
26C47×#376C2211yescharacteristic
27C47×#328—21no—
28C47×#328—21no—
29C22×D188D411yescharacteristic
30C22×C4×D47—22no—
31C22×C4×D47—21no—
32C2×#10625336—22no—
33C2×#10625329—21no—
34C4×D188—28no—
35C4×D188—21no—
36C4×D188—21no—
37C4×D188C2×C412yes—
38C4×D188D412yes—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47D411yescharacteristic
41C2×(C2×D47)⋊C4—22no—
42C2×(C2×D47)⋊C4—21no—
43C2×C4×Dic47—22no—
44C2×C4×Dic47D412yes—
45#13093957—22no—
46C4⋊(C47⋊C8)C2×C412yescomplemented
47#13093948—22no—
48#13093944—22no—
49C22⋊(C47⋊C8)—21no—
50#13093935—21no—
51#13093935—21no—
52C4×C47⋊C8—21no—
53C47⋊#95—22no—
54C47⋊#95—21no—
55C4⋊Q8×C47C2×C411yescharacteristic, complemented
56C4⋊Q8×C47C2×C411yescharacteristic, complemented
57#125×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47—21no—
62#117×C47—21no—
63C2×C42×C47—21no—
64C2×C42×C47C2311yescharacteristic
65C4⋊C8×C47—21no—
66C22⋊C8×C47—21no—
67#97×C47—21no—
68#95×C47—21no—
69#95×C47D411yescharacteristic
70#95×C47D411yescharacteristic
71C23×D47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—21no—
80C22×Dic47—22no—
81C2×D188—21no—
82C2×D188—24no—
83C2×D188C22⋊C412yes—
84C2×C4×D47—28no—
85C2×C4×D47—22no—
86C2×C4×D47—22no—
87C2×C4×D47—82no—
88C2×C4×D47—82no—
89C2×C4×D47—82no—
90C2×C8×C47—41no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C42×C47—24no—
99C42×C47—41no—
100C42×C47C22×C411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47#4111yescharacteristic
103C42×C47#4111yescharacteristic
104#10625336—44no—
105(C2×D47)⋊C4—28no—
106(C2×D47)⋊C4—22no—
107(C2×D47)⋊C4—22no—
108C4⋊Dic47—24no—
109C4⋊Dic47—21no—
110C4⋊Dic47C22⋊C412yes—
111C4×Dic47—42no—
112C4×Dic47—82no—
113#10625329—22no—
114#10625329—41no—
115C2×C47⋊C8—22no—
116C2×C47⋊C8—41no—
117C23×C47#12011yescharacteristic
118C22×D47—24no—
119C22×D47—22no—
120C22×D47—42no—
121C22×D47—41no—
122C22×D47—41no—
123Q8×C47—42no—
124Q8×C47—81no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47C2×C22⋊C411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12311yescharacteristic
143C2×Dic47—24no—
144C2×Dic47—22no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148D188—24no—
149D188—22no—
150D188#10414yes—
151C4×D47—82no—
152C4×D47—82no—
153C4×D47—88no—
154C8×C47—81no—
155C47⋊C8—42no—
156C47⋊C8—42no—
157#9323—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#38611yescharacteristic
161C22×C47#45511yescharacteristic
162C22×C47#45611yescharacteristic
163C2×D47—24no—
164C2×D47—44no—
165C2×D47—44no—
166C2×D47—42no—
167C2×D47—42no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—42no—
174C4×C47—42no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47C2×#10412yes—
178Dic47—44no—
179Dic47—42no—
180#3110—471no—
181#2275—471no—
182#1986—471no—
183#1506D4711yescharacteristic, complemented
184#1490—471no—
185#1300—471no—
186#1285—471no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47#188211yescharacteristic
190C2×C47#189611yescharacteristic
191C2×C47#190411yescharacteristic
192D47—44no—
193#533C2×D4711yescharacteristic
194C2×C4×D4—471no—
195C2×C4×D4—941no—
196#484—941no—
197#474—472no—
198#461—472no—
199#450—941no—
200#440—942no—
201#432—471no—
202#432C2×D4711yescharacteristic
203C4×M16—471no—
204C4×C4⋊C4—471no—
205C4×C4⋊C4—941no—
206C4×C22⋊C4—942no—
207C4×C22⋊C4—941no—
208#376C2×D4711yescharacteristic
209C2×#95—941no—
210#339—942no—
211#328—21no—
212#328—21no—
213D4⋊C8—942no—
214C47#932311yescharacteristic, complemented, Sylow
215C22×D4—471no—
216C23×C4—942no—
217C23×C4—941no—
218C2×SD16—942no—
219C2×M16—941no—
220C4⋊Q8C4×D4711yescharacteristic, complemented
221C4⋊Q8C4×D4711yescharacteristic, complemented
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—472no—
228C4×D4—472no—
229C4×D4—948no—
230C4×D4—941no—
231C4×D4—941no—
232#117—21no—
233C2×C4⋊C4—471no—
234C2×C4⋊C4—941no—
235C2×C22⋊C4—942no—
236C2×C22⋊C4—941no—
237C2×C42—21no—
238C2×C42—472no—
239C2×C42—942no—
240C2×C42C22×D4711yescharacteristic
241#107—942no—
242C4⋊C8—21no—
243C4⋊C8—472no—
244#103—942no—
245#102—942no—
246C22⋊C8—21no—
247C22⋊C8—941no—
248#97—21no—
249#97—941no—
250#97—941no—
251C4×C8—941no—
252#95—21no—
253#95—942no—
254#95—941no—
255#95C22⋊D4711yescharacteristic
256#95C22⋊D4711yescharacteristic
257C24—941no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—472no—
261C2×D4—944no—
262C2×D4—941no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—948no—
266C22×C4—942no—
267C22×C4—941no—
268C22×C4—942no—
269C22×C4—942no—
270C22×C4—3762no—
271C22×C4—3762no—
272C22×C4—3762no—
273C22×C4D4×D4711yescharacteristic
274C22×C4D4×D4711yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276SD16—1884no—
277M16—942no—
278M16—1881no—
279C2×C8—41no—
280C2×C8—942no—
281C2×C8—1881no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—472no—
289C4⋊C4—944no—
290C4⋊C4—941no—
291C22⋊C4—41no—
292C22⋊C4—948no—
293C22⋊C4—942no—
294C22⋊C4—942no—
295C42—24no—
296C42—41no—
297C42—3762no—
298C42—1882no—
299C42C2×C4×D4711yescharacteristic
300C42#1062535011yescharacteristic
301C42D4×D4711yescharacteristic
302C42#1062535211yescharacteristic
303C23—944no—
304C23—942no—
305C23—1882no—
306C23—1881no—
307C23—1881no—
308C23#1309399211yescharacteristic
309Q8—42no—
310Q8—81no—
311D4—474no—
312D4—944no—
313D4—942no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—944no—
327C2×C4—942no—
328C2×C4—3762no—
329C2×C4—3762no—
330C2×C4—3768no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4C22⋊C4×D4711yescharacteristic
335C2×C4(C22⋊D47)⋊C411yescharacteristic
336C2×C4C4×C22⋊D4711yescharacteristic
337C2×C4#1309399511yescharacteristic
338C2×C4#1309399711yescharacteristic
339C2×C4#1309400611yescharacteristic
340C8—81no—
341C8—1882no—
342C8—1882no—
343C22—21no—
344C22—21no—
345C22—944no—
346C22—1884no—
347C22—1884no—
348C22—1882no—
349C22—1882no—
350C22#1568899911yescharacteristic
351C22#1568929911yescharacteristic
352C22#1568930411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—1884no—
363C4—1882no—
364C4#104×D4712yes—
365C2—21no—
366C2—21no—
367C2—1884no—
368C2#2024375411yescharacteristic
369C2#2024382811yescharacteristic
370C2#2024387811yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1506C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1506D47
hypocenterC47#9323
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
3. #20244333C2semidirect
5. #20241281C2semidirect
6. #20241243C2semidirect
7. C47×#1506C2semidirect
11. #15689400C4semidirect
12. #15689325C4semidirect
46. C4⋊(C47⋊C8)C2×C4semidirect
55. C4⋊Q8×C47C2×C4semidirect
56. C4⋊Q8×C47C2×C4semidirect
183. #1506D47semidirect
214. C47#9323semidirect
220. C4⋊Q8C4×D47semidirect
221. C4⋊Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202534832
#202457502
#202443332
#202418302
#202412812
#202412432
C47×#15062
#932347