Finite Group Databasebeta

← #26190269

Subgroups — #26190269

Order 12032.

912 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256793C211yesmaximal, characteristic
2#20254336C212yesmaximal, complemented
3#20242361C214yesmaximal, complemented
4#20242344C214yesmaximal
5#20241951C212yesmaximal, complemented
6#20241702C211yesmaximal, characteristic
7C47×#2628C211yesmaximal, characteristic
8#15689754—21no—
9#15689754C2212yes—
10#15689751—22no—
11#15689751—22no—
12#15689749—21no—
13#15689749C2212yes—
14#15689680—21no—
15#15689680C2212yes—
16#15689676—24no—
17C2×#13093983—21no—
18#15689502—22no—
19#15689473—22no—
20#15689330—24no—
21#15689279C2218yes—
22#15689176C2212yes—
23#15688859C2218yes—
24#15688849C2212yes—
25#15688847C2212yes—
26C2×C4⋊Q8×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C47×#400—21no—
29C47×#399—21no—
30C47×#395C2212yes—
31C47×#393C2212yes—
32C47×#389—21no—
33C47×#382C2211yescharacteristic
34C47×#380—21no—
35C2×#10625337—28no—
36#13094024—21no—
37#13094024—21no—
38#13094024D412yes—
39#13094024D412yes—
40#13094023—24no—
41#13094023—24no—
42#13094016—22no—
43#13094016—22no—
44#13094015—24no—
45#13094015—24no—
46#13094003—22no—
47#13094003—22no—
48#13094003—22no—
49#13094003C2314yes—
50#13094001—24no—
51#13094001—24no—
52#13094001—24no—
53#13094001—24no—
54C4×Dic94—22no—
55C4×Dic94—22no—
56C4×Dic94—22no—
57C4×Dic94D414yes—
58#13093983—22no—
59#13093983—22no—
60#13093983—22no—
61#13093983—22no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C22⋊Dic47—22no—
65C2×C4×Dic47—22no—
66C4⋊(C47⋊C8)—22no—
67#13093948—28no—
68C22⋊(C47⋊C8)C2314yes—
69C4×C47⋊C8—22no—
70C22×Q8×C47C2312yes—
71C23×C4×C47D411yescharacteristic
72C4⋊Q8×C47—22no—
73C4⋊Q8×C47—22no—
74#117×C47—22no—
75#117×C47—22no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47D411yescharacteristic
81C2×C4⋊C4×C47C2311yescharacteristic
82C2×C22⋊C4×C47—22no—
83C2×C22⋊C4×C47D411yescharacteristic
84C2×C22⋊C4×C47C2311yescharacteristic
85C2×C42×C47D411yescharacteristic
86C2×C42×C47C2311yescharacteristic
87#95×C47—22no—
88#95×C47—21no—
89#95×C47—22no—
90#95×C47C2312yes—
91C24×C47C2×D411yescharacteristic
92C2×Q8×C47—24no—
93C2×Q8×C47—82no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47Q1614yescomplemented
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2411yescharacteristic
106C22×Dic47—22no—
107C22×Dic47—22no—
108C2×Dic94—22no—
109C2×Dic94—22no—
110C2×Dic94—22no—
111C2×Dic94C2×D414yes—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—42no—
115C4⋊C4×C47—41no—
116C4⋊C4×C47—41no—
117C4⋊C4×C47—42no—
118C4⋊C4×C47—82no—
119C22⋊C4×C47—22no—
120C22⋊C4×C47—22no—
121C22⋊C4×C47—82no—
122C42×C47—21no—
123C42×C47—21no—
124C42×C47C2×D412yes—
125C42×C47C2×D412yes—
126C22⋊Dic47—24no—
127C22⋊Dic47—24no—
128#10625337—88no—
129C4⋊Dic47—22no—
130C4⋊Dic47—22no—
131C4⋊Dic47—22no—
132C4⋊Dic47—22no—
133C4⋊Dic47—22no—
134C4⋊Dic47—22no—
135C4⋊Dic47—22no—
136C4⋊Dic47C2×D414yes—
137Dic47⋊C4—24no—
138Dic47⋊C4—24no—
139Dic47⋊C4—24no—
140Dic47⋊C4—24no—
141C4×Dic47—24no—
142C4×Dic47—24no—
143C2×C47⋊C8—24no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47C2×Q1612yes—
149C23×C47C22×D411yescharacteristic
150Q8×C47—48no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—24no—
154C2×C4×C47—24no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47#12012yes—
168C2×C4×C47#12011yescharacteristic
169C2×C4×C47#12011yescharacteristic
170C2×C4×C47C2×Q1612yes—
171C2×C4×C47C2×Q1612yes—
172C2×C4×C47#13712yes—
173C2×C4×C47C22×D412yes—
174C2×Dic47—24no—
175C2×Dic47—24no—
176C2×Dic47—82no—
177C2×Dic47—82no—
178Dic94—44no—
179Dic94—82no—
180Dic94—82no—
181Dic94—82no—
182C47⋊C8—44no—
183#18303—471nomaximal, Sylow
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—22no—
187C22×C47—42no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47—81no—
191C22×C47C2×#12011yescharacteristic
192C22×C47C22×Q1611yescharacteristic
193C22×C47C2×#13711yescharacteristic
194C4×C47—22no—
195C4×C47—22no—
196C4×C47—22no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—82no—
200C4×C47—82no—
201C4×C47#45112yes—
202C4×C47#45112yes—
203Dic47—44no—
204Dic47—44no—
205#3458—471no—
206#3209—472no—
207#2628D4711yescharacteristic
208#1605—474no—
209#1602—474no—
210#1518—472no—
211#1458—471no—
212C2×C47—21no—
213C2×C47—21no—
214C2×C47—42no—
215C2×C47C2×#45111yescharacteristic
216C2×C47#302211yescharacteristic
217C2×C47#302711yescharacteristic
218#544—472no—
219#544—942no—
220#544—942no—
221#544—941no—
222#543—472no—
223#543—941no—
224#533—472no—
225#533—944no—
226#533—941no—
227C2×C4⋊Q8C2×D4711yescharacteristic
228C2×#117—941no—
229C2×#117C2×D4711yescharacteristic
230#494—942no—
231#487—942no—
232#462—944no—
233#451—478no—
234#431—472no—
235#400—21no—
236#399—21no—
237#395C2×D4712yes—
238#393C2×D4712yes—
239#389—21no—
240#382C2×D4711yescharacteristic
241#380—21no—
242#328—478no—
243#324—472no—
244#323—472no—
245C47#1830311yescharacteristic, complemented, Sylow
246C22×Q8C22×D4712yes—
247C23×C4C22⋊D4711yescharacteristic
248C2×Q16—948no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—472no—
252C4⋊Q8—472no—
253C4⋊Q8—941no—
254C4⋊Q8—941no—
255C4⋊Q8—944no—
256C4⋊Q8—944no—
257#125—942no—
258#125—944no—
259#125—944no—
260#125—942no—
261C22⋊Q8—474no—
262C22⋊Q8—944no—
263C22⋊Q8—944no—
264C22⋊Q8—944no—
265C22⋊Q8—944no—
266C22⋊Q8—942no—
267C22⋊Q8—942no—
268C22⋊Q8—942no—
269C4×Q8—474no—
270C4×Q8—942no—
271C4×Q8—942no—
272C4×Q8—942no—
273#117—22no—
274#117—22no—
275#117—942no—
276#117—942no—
277#117—942no—
278#117—942no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—941no—
284C2×C4⋊C4—941no—
285C2×C4⋊C4C22⋊D4711yescharacteristic
286C2×C4⋊C4C22×D4711yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—942no—
289C2×C22⋊C4C22⋊D4711yescharacteristic
290C2×C22⋊C4C22×D4711yescharacteristic
291C2×C42—942no—
292C2×C42C22⋊D4711yescharacteristic
293C2×C42C22×D4711yescharacteristic
294C4⋊C8—942no—
295#103—948no—
296C22⋊C8—474no—
297C4×C8—942no—
298#95—22no—
299#95—21no—
300#95—22no—
301#95C22×D4712yes—
302C24C2×C22⋊D4711yescharacteristic
303C2×Q8—24no—
304C2×Q8—82no—
305C2×Q8—474no—
306C2×Q8—942no—
307C2×Q8—942no—
308C2×Q8—942no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—942no—
315C22×C4—942no—
316C22×C4#1062533714yescomplemented
317C22×C4C2×C22⋊D4711yescharacteristic
318C22×C4C2×C22⋊D4711yescharacteristic
319C22×C4C2×C22⋊D4711yescharacteristic
320C22×C4C2×C22⋊D4711yescharacteristic
321C22×C4C2×C22⋊D4711yescharacteristic
322C22×C4C23×D4711yescharacteristic
323Q16—3768no—
324C2×C8—944no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—474no—
333C4⋊C4—944no—
334C4⋊C4—944no—
335C4⋊C4—942no—
336C4⋊C4—942no—
337C4⋊C4—942no—
338C4⋊C4—942no—
339C4⋊C4—944no—
340C4⋊C4—942no—
341C4⋊C4—942no—
342C4⋊C4—942no—
343C4⋊C4—944no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—82no—
347C22⋊C4—944no—
348C22⋊C4—944no—
349C42—21no—
350C42—21no—
351C42—944no—
352C42—944no—
353C42D4×D4712yes—
354C42D4×D4712yes—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23C2×#1062533712yes—
360C23C22×C22⋊D4711yescharacteristic
361Q8—48no—
362Q8—3762no—
363Q8—3762no—
364Q8—3762no—
365Q8—1884no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—944no—
383C2×C4—944no—
384C2×C4—3762no—
385C2×C4—3762no—
386C2×C4#1309399212yes—
387C2×C4#1309399211yescharacteristic
388C2×C4#1309399211yescharacteristic
389C2×C4C2×#1062533712yes—
390C2×C4C2×#1062533712yes—
391C2×C4#1309405412yes—
392C2×C4C2×D4×D4712yes—
393C8—1884no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C2×#1309399211yescharacteristic
402C22C22×#1062533711yescharacteristic
403C22C2×#1309405411yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—82no—
410C4—82no—
411C4—1884no—
412C4—1884no—
413C4#1568927812yes—
414C4#1568927812yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2C2×#1568927811yescharacteristic
419C2#2025276611yescharacteristic
420C2#2025278911yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2628C2
socleC22×C47C2×#120
prime radicalC24×C47C2×D4
hypercenter#2628D47
hypocenterC47#18303
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #20254336C2semidirect
3. #20242361C2semidirect
5. #20241951C2semidirect
99. C22×C4×C47Q16semidirect
245. C47#18303semidirect
316. C22×C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202567932
#202543362
#202423612
#202423442
#202419512
#202417022
C47×#26282
#1830347