Finite Group Databasebeta

← #26113175

Subgroups — #26113175

Order 12032.

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

normal subgroups only

This group has 612 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#20245749C211yesmaximal, characteristic
2#20244333C211yesmaximal, characteristic, complemented
3#20241900C211yesmaximal, characteristic, complemented
4#20241831C211yesmaximal, characteristic
5#20241280C211yesmaximal, characteristic, complemented
6#20241245C211yesmaximal, characteristic, complemented
7C47×#3110C211yesmaximal, characteristic
8C2×D4×Dic47—21no—
9#15689180C2211yescharacteristic
10C4×#10625329C2211yescharacteristic
11C4⋊C4×Dic47—21no—
12C4×C4⋊Dic47C2211yescharacteristic
13C22⋊C4×Dic47—22no—
14C4×C22⋊Dic47—21no—
15#15688969C2211yescharacteristic
16C2×C47⋊#95—21no—
17#15688882—22no—
18#15688860—21no—
19#15688860—21no—
20D4⋊(C47⋊C8)—22no—
21C47×#533C2211yescharacteristic
22C2×C4×D4×C47C2211yescharacteristic
23C47×#484—21no—
24C47×#474C412yescomplemented
25C47×#461C412yescomplemented
26C47×#450—21no—
27C47×#440—22no—
28C47×#432C2211yescharacteristic
29C23×Dic47—22no—
30C2×#10625329—21no—
31D4×Dic47—28no—
32C2×C4⋊Dic47—21no—
33C2×C22⋊Dic47—22no—
34C2×C4×Dic47—21no—
35C2×C4×Dic47—22no—
36C2×C4×Dic47D412yes—
37C4⋊(C47⋊C8)—21no—
38C22⋊(C47⋊C8)—21no—
39#13093935—21no—
40#13093935—21no—
41C4×C47⋊C8—21no—
42C47⋊#95—22no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95D411yescharacteristic
46C47⋊#95D411yescharacteristic
47C22×D4×C47D411yescharacteristic
48C23×C4×C47—21no—
49C2×SD16×C47—22no—
50C4⋊Q8×C47C2×C411yescharacteristic, complemented
51C4⋊Q8×C47C2×C411yescharacteristic, complemented
52#125×C47—21no—
53C22⋊Q8×C47—21no—
54C22⋊Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×D4×C47—21no—
58C4×D4×C47—21no—
59C4×D4×C47C2×C412yes—
60C4×D4×C47D412yes—
61#117×C47—21no—
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C22⋊C4×C47—21no—
64C2×C42×C47C2311yescharacteristic
65#107×C47—22no—
66C4⋊C8×C47C2×C412yescomplemented
67#103×C47—22no—
68#102×C47—22no—
69C22⋊C8×C47—21no—
70#97×C47—21no—
71C24×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×D4×C47—24no—
75C2×D4×C47—21no—
76C2×D4×C47C22⋊C412yes—
77C22×C4×C47—21no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—82no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×Dic47—21no—
85C22×Dic47—22no—
86C22×Dic47—21no—
87C22×Dic47—28no—
88C22×Dic47—82no—
89C22×Dic47—82no—
90SD16×C47—44no—
91C2×C8×C47—22no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47C22⋊C412yes—
100C22⋊C4×C47—22no—
101C22⋊C4×C47—22no—
102C22⋊C4×C47—41no—
103C42×C47—41no—
104C42×C47C22×C411yescharacteristic
105C42×C47C2×D411yescharacteristic
106C42×C47#4111yescharacteristic
107C42×C47#4111yescharacteristic
108C22⋊Dic47—28no—
109C4⋊Dic47—24no—
110C4×Dic47—24no—
111C4×Dic47—42no—
112C4×Dic47—82no—
113#10625329—22no—
114#10625329—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C23×C47—24no—
118C23×C47—22no—
119C23×C47—42no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47#12011yescharacteristic
123Q8×C47—42no—
124Q8×C47—81no—
125D4×C47—24no—
126D4×C47—22no—
127D4×C47#10414yes—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—82no—
137C2×C4×C47—82no—
138C2×C4×C47C2×C22⋊C411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12311yescharacteristic
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—24no—
147C2×Dic47—41no—
148C2×Dic47—42no—
149C2×Dic47—41no—
150C2×Dic47—42no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—88no—
154C8×C47—42no—
155C47⋊C8—42no—
156C47⋊C8—81no—
157#9323—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—24no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—42no—
164C22×C47—42no—
165C22×C47#38611yescharacteristic
166C22×C47#45511yescharacteristic
167C22×C47#45611yescharacteristic
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—42no—
174C4×C47—81no—
175C4×C47—81no—
176C4×C47C2×#10412yes—
177Dic47—42no—
178Dic47—44no—
179Dic47—42no—
180#3110D4711yescharacteristic
181#2275—471no—
182#1986—471no—
183#1506—471no—
184#1490—471no—
185#1300—471no—
186#1285—471no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—44no—
190C2×C47#188211yescharacteristic
191C2×C47#189611yescharacteristic
192C2×C47#190411yescharacteristic
193#533C2×D4711yescharacteristic
194C2×C4×D4—941no—
195C2×C4×D4C2×D4711yescharacteristic
196#484—21no—
197#474Dic4712yescomplemented
198#461Dic4712yescomplemented
199#450—21no—
200#440—22no—
201#432—471no—
202#432C2×D4711yescharacteristic
203C4×M16—471no—
204C4×C4⋊C4—471no—
205C4×C4⋊C4—941no—
206C4×C22⋊C4—941no—
207C4×C22⋊C4—942no—
208#376—471no—
209C2×#95—941no—
210#339—942no—
211#328—941no—
212#328—941no—
213D4⋊C8—942no—
214C47#932311yescharacteristic, complemented, Sylow
215C22×D4C22⋊D4711yescharacteristic
216C23×C4—21no—
217C23×C4—942no—
218C2×SD16—22no—
219C2×M16—941no—
220C4⋊Q8C2×Dic4711yescharacteristic, complemented
221C4⋊Q8C2×Dic4711yescharacteristic, complemented
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—948no—
230C4×D4C2×Dic4712yes—
231C4×D4C22⋊D4712yes—
232#117—21no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4C22⋊D4711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—942no—
237C2×C42—472no—
238C2×C42—941no—
239C2×C42—942no—
240C2×C42C22×D4711yescharacteristic
241#107—22no—
242C4⋊C8—941no—
243C4⋊C8C2×Dic4712yescomplemented
244#103—22no—
245#102—22no—
246C22⋊C8—21no—
247C22⋊C8—941no—
248#97—21no—
249#97—941no—
250#97—941no—
251C4×C8—941no—
252#95—471no—
253#95—471no—
254#95—942no—
255#95—941no—
256#95—941no—
257C24—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—24no—
261C2×D4—21no—
262C2×D4C22⋊Dic4712yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—82no—
267C22×C4—948no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4—942no—
271C22×C4—3762no—
272C22×C4—3762no—
273C22×C4D4×D4711yescharacteristic
274C22×C4D4×D4711yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276SD16—44no—
277M16—942no—
278M16—1881no—
279C2×C8—22no—
280C2×C8—1881no—
281C2×C8—1881no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—944no—
290C4⋊C4C22⋊Dic4712yes—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—948no—
295C42—41no—
296C42—944no—
297C42—3762no—
298C42—1882no—
299C42#1062535211yescharacteristic
300C42#1062535211yescharacteristic
301C42C22×Dic4711yescharacteristic
302C42C2×C22⋊D4711yescharacteristic
303C23—24no—
304C23—22no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23#1309399211yescharacteristic
309Q8—42no—
310Q8—81no—
311D4—24no—
312D4—22no—
313D4#1309395114yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—942no—
325C2×C4—942no—
326C2×C4—944no—
327C2×C4—3768no—
328C2×C4—1881no—
329C2×C4—1882no—
330C2×C4—1881no—
331C2×C4—1882no—
332C2×C4—1881no—
333C2×C4—1881no—
334C2×C4C2×C22⋊Dic4711yescharacteristic
335C2×C4D4×Dic4711yescharacteristic
336C2×C4D4×Dic4711yescharacteristic
337C2×C4#1309399611yescharacteristic
338C2×C4#1309399611yescharacteristic
339C2×C4#1309400511yescharacteristic
340C8—42no—
341C8—3761no—
342C8—1882no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22#1568900311yescharacteristic
351C22#1568929911yescharacteristic
352C22#1568930411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—1882no—
362C4—1884no—
363C4—1882no—
364C4C2×#1309395112yes—
365C2—21no—
366C2—21no—
367C2—44no—
368C2#2024375911yescharacteristic
369C2#2024383211yescharacteristic
370C2#2024387711yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3110C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#3110D47
hypocenterC47#9323
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #20244333C2semidirect
3. #20241900C2semidirect
5. #20241280C2semidirect
6. #20241245C2semidirect
24. C47×#474C4semidirect
25. C47×#461C4semidirect
50. C4⋊Q8×C47C2×C4semidirect
51. C4⋊Q8×C47C2×C4semidirect
66. C4⋊C8×C47C2×C4semidirect
197. #474Dic47semidirect
198. #461Dic47semidirect
214. C47#9323semidirect
220. C4⋊Q8C2×Dic47semidirect
221. C4⋊Q8C2×Dic47semidirect
243. C4⋊C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202457492
#202443332
#202419002
#202418312
#202412802
#202412452
C47×#31102
#932347