Finite Group Databasebeta

← #26188694

Subgroups — #26188694

Order 12032.

1054 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 183 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#20253332C212yesmaximal, complemented
2#20247662C211yesmaximal, characteristic
3#20242413C214yesmaximal, complemented
4#20242379C214yesmaximal, complemented
5#20241921C212yesmaximal, complemented
6#20241715C211yesmaximal, characteristic
7C2×C47×#543C211yesmaximal, characteristic
8C2×#13093983C2211yescharacteristic
9#15689461—24no—
10#15689456C2214yescomplemented
11#15689451C2214yescomplemented
12#15689163—22no—
13#15689132—22no—
14#15689087C2212yes—
15#15689059C2212yes—
16#15689048C2212yes—
17#15689011—22no—
18#15688996—21no—
19#15688986C2211yescharacteristic
20#15688979—21no—
21#15688860C2218yescomplemented
22#15688847C2214yes—
23C47×#543—24no—
24C47×#543—22no—
25C47×#543C2214yes—
26C2×C22⋊Q8×C47—22no—
27C2×C22⋊Q8×C47—21no—
28C2×C22⋊Q8×C47—22no—
29C2×C22⋊Q8×C47C2212yes—
30C22×C22⋊C4×C47—21no—
31C22×C22⋊C4×C47C2211yescharacteristic
32C2×#10625329—22no—
33#13093983—22no—
34#13093983—22no—
35C2×C4⋊Dic47C2311yescharacteristic
36C2×C4⋊Dic47C2311yescharacteristic
37C2×Dic47⋊C4—22no—
38C2×C22⋊Dic47—21no—
39C2×C22⋊Dic47—21no—
40C2×C22⋊Dic47—22no—
41C2×C22⋊Dic47—22no—
42C2×C4×Dic47—21no—
43#13093957—24no—
44#13093955—24no—
45#13093948—24no—
46#13093948—24no—
47C22⋊(C47⋊C8)—22no—
48C22⋊(C47⋊C8)C2314yes—
49C47⋊#95—21no—
50C47⋊#95—22no—
51C47⋊#95C2312yes—
52C25×C47D411yescharacteristic
53C22×Q8×C47—21no—
54C22×Q8×C47D412yes—
55C23×C4×C47—21no—
56C23×C4×C47—21no—
57C23×C4×C47D411yescharacteristic
58C23×C4×C47C2311yescharacteristic
59C22⋊Q8×C47—28no—
60C22⋊Q8×C47—28no—
61C22⋊Q8×C47—24no—
62C22⋊Q8×C47—28no—
63C22⋊Q8×C47—24no—
64C22⋊Q8×C47—22no—
65C22⋊Q8×C47—22no—
66C22⋊Q8×C47—28no—
67C22⋊Q8×C47D414yes—
68C22⋊Q8×C47C2314yes—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—22no—
71C2×C4⋊C4×C47—22no—
72C2×C4⋊C4×C47D412yes—
73C2×C22⋊C4×C47—22no—
74C2×C22⋊C4×C47—24no—
75C2×C22⋊C4×C47—22no—
76C2×C22⋊C4×C47—24no—
77C2×C22⋊C4×C47—22no—
78C2×C22⋊C4×C47—22no—
79C2×C22⋊C4×C47D412yes—
80C2×C22⋊C4×C47C2312yes—
81C24×C47—21no—
82C24×C47—21no—
83C24×C47—24no—
84C24×C47—42no—
85C24×C47—81no—
86C24×C47C2×D411yescharacteristic
87C24×C47C2×D412yes—
88C2×Q8×C47—24no—
89C2×Q8×C47—22no—
90C2×Q8×C47—44no—
91C2×Q8×C47—82no—
92C2×Q8×C47C2×D414yes—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—24no—
97C22×C4×C47—24no—
98C22×C4×C47—21no—
99C22×C4×C47—42no—
100C22×C4×C47—81no—
101C22×C4×C47—82no—
102C22×C4×C47—82no—
103C22×C4×C47C2×D412yes—
104C22×C4×C47C2×D412yes—
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2411yescharacteristic
108C22×Dic47—22no—
109C22×Dic47—21no—
110C22×Dic47—21no—
111C4⋊C4×C47—24no—
112C4⋊C4×C47—28no—
113C4⋊C4×C47—24no—
114C4⋊C4×C47—22no—
115C4⋊C4×C47—24no—
116C4⋊C4×C47C2×D414yes—
117C22⋊C4×C47—28no—
118C22⋊C4×C47—24no—
119C22⋊C4×C47—24no—
120C22⋊C4×C47—24no—
121C22⋊C4×C47—24no—
122C22⋊Dic47—42no—
123C22⋊Dic47—41no—
124C22⋊Dic47—41no—
125C22⋊Dic47—44no—
126C22⋊Dic47—82no—
127C4⋊Dic47—22no—
128C4⋊Dic47—22no—
129Dic47⋊C4—82no—
130C4×Dic47—42no—
131#10625329—44no—
132C2×C47⋊C8—24no—
133C23×C47—24no—
134C23×C47—24no—
135C23×C47—22no—
136C23×C47—22no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—41no—
140C23×C47—42no—
141C23×C47—42no—
142C23×C47—42no—
143C23×C47—42no—
144C23×C47—81no—
145C23×C47—81no—
146C23×C47—88no—
147C23×C47#12011yescharacteristic
148C23×C47#12011yescharacteristic
149C23×C47#13714yes—
150C23×C47C22×D411yescharacteristic
151Q8×C47—44no—
152Q8×C47—82no—
153Q8×C47—84no—
154C2×C4×C47—24no—
155C2×C4×C47—24no—
156C2×C4×C47—22no—
157C2×C4×C47—24no—
158C2×C4×C47—42no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—42no—
162C2×C4×C47—44no—
163C2×C4×C47—44no—
164C2×C4×C47—42no—
165C2×C4×C47—81no—
166C2×C4×C47—84no—
167C2×C4×C47—84no—
168C2×C4×C47#12012yes—
169C2×C4×C47C22×D412yes—
170C2×Dic47—42no—
171C2×Dic47—42no—
172C2×Dic47—42no—
173C2×Dic47—41no—
174C2×Dic47—41no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C47⋊C8—44no—
178#18138—471nomaximal, Sylow
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—24no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—44no—
188C22×C47—44no—
189C22×C47—48no—
190C22×C47—42no—
191C22×C47—42no—
192C22×C47—82no—
193C22×C47—84no—
194C22×C47#45812yes—
195C22×C47C2×#12011yescharacteristic
196C22×C47C2×#13712yes—
197C4×C47—22no—
198C4×C47—22no—
199C4×C47—42no—
200C4×C47—44no—
201C4×C47—44no—
202Dic47—82no—
203Dic47—81no—
204Dic47—81no—
205C2×#543D4711yescharacteristic
206#3095—472no—
207#2482—471no—
208#1614—474no—
209#1608—474no—
210#1511—472no—
211#1462—471no—
212C2×C47—22no—
213C2×C47—22no—
214C2×C47—41no—
215C2×C47—44no—
216C2×C47#300411yescharacteristic
217C2×C47#301911yescharacteristic
218C2×C47C2×#45811yescharacteristic
219#543—24no—
220#543—22no—
221#543C2×D4714yes—
222C2×C22⋊Q8—22no—
223C2×C22⋊Q8—21no—
224C2×C22⋊Q8—22no—
225C2×C22⋊Q8C2×D4712yes—
226C2×#117—471no—
227C22×C22⋊C4—21no—
228C22×C22⋊C4C2×D4711yescharacteristic
229#485—944no—
230#484—474no—
231#483—474no—
232#428—942no—
233#419—942no—
234#407—472no—
235#399—472no—
236#397—472no—
237#388—942no—
238#385—941no—
239#381—471no—
240#379—941no—
241#328—478no—
242#323—474no—
243C47#1813811yescharacteristic, complemented, Sylow
244C25C22⋊D4711yescharacteristic
245C22×Q8—21no—
246C22×Q8C22⋊D4712yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C22⋊D4711yescharacteristic
250C23×C4C22×D4711yescharacteristic
251C2×M16—942no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—28no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—28no—
260C22⋊Q8C22⋊D4714yes—
261C22⋊Q8C22×D4714yes—
262#117—942no—
263#117—942no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—471no—
268C2×C4⋊C4—471no—
269C2×C4⋊C4—942no—
270C2×C4⋊C4C22⋊D4712yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—942no—
278C2×C22⋊C4—941no—
279C2×C22⋊C4—941no—
280C2×C22⋊C4—942no—
281C2×C22⋊C4C22⋊D4712yes—
282C2×C22⋊C4C22×D4712yes—
283C2×C42—941no—
284#107—944no—
285#106—944no—
286#103—944no—
287#103—944no—
288C22⋊C8—474no—
289C22⋊C8—942no—
290#95—472no—
291#95—941no—
292#95—942no—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—81no—
298C24C2×C22⋊D4711yescharacteristic
299C24C2×C22⋊D4712yes—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊D4714yes—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—942no—
316C22×C4—941no—
317C22×C4—941no—
318C22×C4C2×C22⋊D4712yes—
319C22×C4C2×C22⋊D4712yes—
320C22×C4C2×C22⋊D4711yescharacteristic
321C22×C4C2×C22⋊D4712yes—
322C22×C4C23×D4711yescharacteristic
323M16—1884no—
324C2×C8—944no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—942no—
331C4⋊C4—942no—
332C4⋊C4—3762no—
333C4⋊C4C2×C22⋊D4714yes—
334C22⋊C4—28no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—3762no—
340C22⋊C4—1882no—
341C22⋊C4—1881no—
342C22⋊C4—1881no—
343C22⋊C4—1884no—
344C42—1882no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—88no—
359C23#1309399311yescharacteristic
360C23#1309399311yescharacteristic
361C23#1309405514yes—
362C23C22×C22⋊D4711yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4—1882no—
383C2×C4—1881no—
384C2×C4—1881no—
385C2×C4—1882no—
386C2×C4—1882no—
387C2×C4#1309399312yes—
388C2×C4C22×C22⋊D4712yes—
389C8—1884no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1568931312yes—
406C22C2×#1309399311yescharacteristic
407C22C2×#1309405512yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—3762no—
414C4—3761no—
415C4—3761no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#2025262211yescharacteristic
421C2#2025274511yescharacteristic
422C2C2×#1568931311yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#543C2
socleC22×C47C2×#120
prime radicalC25×C47D4
hypercenterC2×#543D47
hypocenterC47#18138
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #20253332C2semidirect
3. #20242413C2semidirect
4. #20242379C2semidirect
5. #20241921C2semidirect
10. #15689456C22semidirect
11. #15689451C22semidirect
21. #15688860C22semidirect
243. C47#18138semidirect

Maximal subgroups

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

subgroupindex
#202533322
#202476622
#202424132
#202423792
#202419212
#202417152
C2×C47×#5432
#1813847