Finite Group Databasebeta

← #26112220

Subgroups — #26112220

Order 12032.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252860C211yesmaximal, characteristic
2#20241947C211yesmaximal, characteristic
3#20241882C211yesmaximal, characteristic
4#20241783C211yesmaximal, characteristic, complemented
5#20241291C211yesmaximal, characteristic, complemented
6#20241246C211yesmaximal, characteristic, complemented
7C47×#2351C211yesmaximal, characteristic, complemented
8C2×#13094055—21no—
9#15689673C2211yescharacteristic
10#15689345—22no—
11#15689333C412yescomplemented
12#15689323C412yescomplemented
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16#15689122—21no—
17#15688856—22no—
18#15688856—21no—
19#15688856—21no—
20Q8⋊(C47⋊C8)—22no—
21C2×C4×Q8×C47—21no—
22C2×C4×Q8×C47C2211yescharacteristic
23C47×#389—22no—
24C47×#389—21no—
25C47×#382—21no—
26C47×#382C2211yescharacteristic
27C4×C4⋊C4×C47—21no—
28C4×C4⋊C4×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094055—42no—
31#13094055—42no—
32C2×#10625337—22no—
33C2×#10625336—22no—
34C2×#10625329—21no—
35#13094024C2×C411yescharacteristic
36#13094021C2×C411yescharacteristic
37#13094012—21no—
38#13093998—21no—
39#13093998—21no—
40C4×D188—21no—
41C4×D188—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)C2×C412yes—
46#13093947—22no—
47#13093944—22no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8—21no—
52C22×Q8×C47D411yescharacteristic
53C4×Q8×C47—21no—
54C4×Q8×C47—21no—
55C4×Q8×C47—42no—
56C4×Q8×C47—42no—
57C4×Q8×C47C2×C412yes—
58C4×Q8×C47D412yes—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—22no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C42×C47—21no—
66C2×C42×C47—22no—
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69#95×C47—22no—
70#95×C47—21no—
71#95×C47D411yescharacteristic
72#95×C47D411yescharacteristic
73#95×C47D412yes—
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—21no—
77C2×Q8×C47C22⋊C412yes—
78C2×Q8×C47D812yes—
79C2×Q8×C47SD1612yes—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—22no—
84C22×C4×C47—21no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C2×C22⋊D47—41no—
89#10625350—41no—
90#10625350—41no—
91#10625350—41no—
92#10625350—41no—
93C2×D188—21no—
94C2×D188—21no—
95C2×D188—21no—
96C2×C4×D47—41no—
97C2×Dic94—21no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47—44no—
103C4⋊C4×C47—44no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47C22⋊C412yes—
107C42×C47—22no—
108C42×C47—22no—
109C42×C47—44no—
110C42×C47—41no—
111C42×C47—41no—
112C42×C47C22×C411yescharacteristic
113C42×C47C2×D411yescharacteristic
114C42×C47#4111yescharacteristic
115C42×C47#4111yescharacteristic
116#10625337—44no—
117#10625336—44no—
118(C2×D47)⋊C4—41no—
119(C2×D47)⋊C4—41no—
120C4⋊Dic47—21no—
121C4⋊Dic47—21no—
122#10625329—41no—
123#10625329—41no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C23×C47#12011yescharacteristic
128C22×D47—41no—
129Q8×C47—22no—
130Q8×C47—22no—
131Q8×C47—22no—
132Q8×C47#10214yes—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—42no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47C2×C22⋊C411yescharacteristic
150C2×C4×C47C4×D411yescharacteristic
151C2×C4×C47C4×D411yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12111yescharacteristic
154C2×C4×C47#12311yescharacteristic
155C2×C4×C47C2×D811yescharacteristic
156C2×C4×C47C2×SD1611yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47#13711yescharacteristic
159C22⋊D47—81no—
160C22⋊D47—81no—
161C2×Dic47—41no—
162D188—41no—
163D188—41no—
164D188—81no—
165D188—81no—
166C4×D47—81no—
167C4×D47—81no—
168Dic94—41no—
169Dic94—41no—
170C47⋊C8—42no—
171C47⋊C8—81no—
172C47⋊C8—81no—
173#9180—471nomaximal, Sylow
174C22×C47—21no—
175C22×C47—21no—
176C22×C47#38611yescharacteristic
177C22×C47#44711yescharacteristic
178C22×C47#44811yescharacteristic
179C2×D47—81no—
180C2×D47—81no—
181C2×D47—81no—
182C4×C47—24no—
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—42no—
188C4×C47—42no—
189C4×C47—82no—
190C4×C47—81no—
191C4×C47C2×#10211yescharacteristic
192C4×C47#41711yescharacteristic
193C4×C47#48011yescharacteristic
194C4×C47#48111yescharacteristic
195Dic47—81no—
196#3037—471no—
197#2351D4711yescharacteristic, complemented
198#1517—471no—
199#1502—471no—
200#1479—471no—
201#1303—471no—
202#1286—471no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47#187411yescharacteristic
206C2×C47#190511yescharacteristic
207C2×C47#190611yescharacteristic
208D47—161no—
209C2×#137—941no—
210#532—471no—
211C2×C4×Q8—21no—
212C2×C4×Q8C2×D4711yescharacteristic
213#464—942no—
214#462—472no—
215#460—472no—
216#431—471no—
217#431—471no—
218#423—471no—
219#417—941no—
220#389—21no—
221#389—22no—
222#382—21no—
223#382C2×D4711yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2×D4711yescharacteristic
226#327—941no—
227#327—941no—
228#327—942no—
229Q8⋊C8—942no—
230C47#918011yescharacteristic, complemented, Sylow
231C2×#41—941no—
232C22×Q8C22⋊D4711yescharacteristic
233#137—1882no—
234#137—1882no—
235C2×Q16—942no—
236C2×SD16—942no—
237C2×M16—941no—
238C4⋊Q8—471no—
239#127—471no—
240#124—941no—
241#121—941no—
242#121—941no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8C4×D4712yes—
248C4×Q8D18812yes—
249C4×D4—941no—
250C4×D4—941no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D4711yescharacteristic
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42C22×D4711yescharacteristic
261C4⋊C8—472no—
262C4⋊C8—941no—
263C4⋊C8—941no—
264C4⋊C8—941no—
265#103—942no—
266#102—942no—
267C22⋊C8—941no—
268C22⋊C8—941no—
269C4×C8—941no—
270C4×C8—941no—
271#95—22no—
272#95—21no—
273#95C22⋊D4711yescharacteristic
274#95C22⋊D4711yescharacteristic
275#95C22⋊D4712yes—
276#41—1881no—
277#41—1881no—
278#41—1881no—
279#41—1881no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—941no—
284C2×Q8(C2×D47)⋊C412yes—
285C2×Q8#1062533412yes—
286C2×Q8#1062533612yes—
287C2×D4—941no—
288C2×D4—941no—
289C2×D4—941no—
290C2×D4—1881no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—1881no—
297C22×C4C2×C22⋊D4711yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299C22×C4C2×C22⋊D4711yescharacteristic
300Q16—1884no—
301SD16—1884no—
302M16—1881no—
303M16—1881no—
304C2×C8—942no—
305C2×C8—1881no—
306C2×C8—1881no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—941no—
316C4⋊C4—941no—
317C4⋊C4(C2×D47)⋊C412yes—
318C22⋊C4—1881no—
319C22⋊C4—1881no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—41no—
324C42—41no—
325C42C2×C4×D4711yescharacteristic
326C42C2×D18811yescharacteristic
327C42#1062535011yescharacteristic
328C42#1062535011yescharacteristic
329C23—1881no—
330C23#1309399311yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—1881no—
335Q8—1881no—
336Q8#1309394414yes—
337D4—3761no—
338D4—3761no—
339D4—3761no—
340D4—3761no—
341D4—1881no—
342D4—1881no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—3761no—
360C2×C4—3761no—
361C2×C4—1881no—
362C2×C4C2×(C2×D47)⋊C411yescharacteristic
363C2×C4C4×C22⋊D4711yescharacteristic
364C2×C4C4×C22⋊D4711yescharacteristic
365C2×C4#1309399811yescharacteristic
366C2×C4#1309399811yescharacteristic
367C2×C4#1309400711yescharacteristic
368C2×C4C2×#1062533411yescharacteristic
369C2×C4C2×#1062533611yescharacteristic
370C2×C4#1309405011yescharacteristic
371C2×C4#1309405511yescharacteristic
372C8—3761no—
373C8—3761no—
374C8—1882no—
375C22—21no—
376C22—21no—
377C22—3761no—
378C22—3761no—
379C22—3761no—
380C22#1568900211yescharacteristic
381C22#1568925711yescharacteristic
382C22#1568926211yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—24no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—81no—
392C4—3761no—
393C4C2×#1309394411yescharacteristic
394C4#1568912211yescharacteristic
395C4#1568943611yescharacteristic
396C4#1568944111yescharacteristic
397C2—21no—
398C2—21no—
399C2—7521no—
400C2#2024370511yescharacteristic
401C2#2024388211yescharacteristic
402C2#2024388911yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2351C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2351D47
hypocenterC47#9180
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #20241783C2semidirect
5. #20241291C2semidirect
6. #20241246C2semidirect
7. C47×#2351C2semidirect
11. #15689333C4semidirect
12. #15689323C4semidirect
197. #2351D47semidirect
230. C47#9180semidirect

Maximal subgroups

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

subgroupindex
#202528602
#202419472
#202418822
#202417832
#202412912
#202412462
C47×#23512
#918047