Finite Group Databasebeta

← #26111329

Subgroups — #26111329

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689054C211yesmaximal, characteristic
2#20245990C211yesmaximal, characteristic
3#20243474C211yesmaximal, characteristic
4C2×#15688848C211yesmaximal, characteristic
5#20241268C211yesmaximal, characteristic
6#20241266C211yesmaximal, characteristic
7C47×#1823C211yesmaximal, characteristic
8C22×Dic47⋊C4—21no—
9C22×Dic47⋊C4C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11#15689054—22no—
12#15689054C414yescomplemented
13#15689010—22no—
14#15689004—22no—
15#15688984—21no—
16#15688983C2211yescharacteristic
17C22⋊C4×Dic47—21no—
18C2×C47⋊#95—21no—
19C2×C47⋊#95C2211yescharacteristic
20#15688885—22no—
21#15688848—22no—
22C22×C4⋊C4×C47C2211yescharacteristic
23C2×C22⋊C8×C47C2211yescharacteristic
24C47×#380C2211yescharacteristic
25C47×#340—22no—
26C23×Dic47—21no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30C2×C4⋊Dic47D412yes—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—24no—
36C2×Dic47⋊C4—42no—
37C2×Dic47⋊C4—42no—
38C2×Dic47⋊C4C2×C412yes—
39C2×Dic47⋊C4D412yes—
40C2×C22⋊Dic47—21no—
41C2×C4×Dic47—22no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C47⋊#95—22no—
45C47⋊#95—24no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—24no—
49C47⋊#95C2×C412yes—
50C23×C4×C47—21no—
51C23×C4×C47C2311yescharacteristic
52C22×C8×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47C2×C412yes—
59C2×C4⋊C4×C47D412yes—
60C2×C22⋊C4×C47—21no—
61C22⋊C8×C47—21no—
62C22⋊C8×C47—21no—
63#95×C47—21no—
64C24×C47C2×D411yescharacteristic
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—42no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47C22×C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C22×C4×C47#4111yescharacteristic
80C22×Dic47—24no—
81C22×Dic47—22no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—22no—
91C22×Dic47—42no—
92C22×Dic47—41no—
93C22×Dic47—41no—
94C22×Dic47—42no—
95C22×Dic47C22⋊C412yes—
96C22×Dic47C22⋊C412yes—
97C2×C8×C47—42no—
98C2×C8×C47—41no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C22⋊C4×C47—41no—
105C22⋊C4×C47—41no—
106C22⋊Dic47—41no—
107C22⋊Dic47—41no—
108C4⋊Dic47—42no—
109Dic47⋊C4—22no—
110Dic47⋊C4—22no—
111Dic47⋊C4—22no—
112Dic47⋊C4—22no—
113Dic47⋊C4—48no—
114C4×Dic47—82no—
115C2×C47⋊C8—42no—
116C2×C47⋊C8—41no—
117C23×C47—22no—
118C23×C47—22no—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47#12011yescharacteristic
121C23×C47#12311yescharacteristic
122C23×C47#13511yescharacteristic
123C23×C47#13511yescharacteristic
124C23×C47AGL(1,ℤ/8)11yescharacteristic
125C23×C47AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47—81no—
141C2×C4×C47—81no—
142C2×C4×C47C4×D411yescharacteristic
143C2×C4×C47C4×D411yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×Dic47—24no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—22no—
150C2×Dic47—22no—
151C2×Dic47—44no—
152C2×Dic47—44no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—42no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—81no—
165C8×C47—81no—
166C47⋊C8—81no—
167#9026—471nomaximal, Sylow
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—22no—
171C22×C47—22no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47#38611yescharacteristic
177C22×C47#41611yescharacteristic
178C22×C47#41811yescharacteristic
179C22×C47#44911yescharacteristic
180C22×C47#44911yescharacteristic
181C22×C47#45612yes—
182C22×C47#45812yes—
183C22×C47#48211yescharacteristic
184C22×C47#48211yescharacteristic
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—82no—
190C4×C47—82no—
191Dic47—44no—
192Dic47—42no—
193Dic47—42no—
194Dic47—82no—
195C2×#398—471no—
196#2313—471no—
197#1823—471no—
198#1823D4711yescharacteristic
199C2×#324—471no—
200#1294—471no—
201#1294—471no—
202C2×C47—21no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47#160912yes—
207C2×C47#160912yes—
208C2×C47#187511yescharacteristic
209C2×C47#188611yescharacteristic
210C2×C47#189011yescharacteristic
211C22×C4⋊C4—471no—
212C22×C4⋊C4—941no—
213C22×C4⋊C4C2×D4711yescharacteristic
214C2×C22⋊C8—471no—
215C2×C22⋊C8C2×D4711yescharacteristic
216#398—474no—
217#398—942no—
218#387—942no—
219#387—942no—
220#381—941no—
221#380—471no—
222#380C2×D4711yescharacteristic
223C4×C22⋊C4—941no—
224C2×#95—471no—
225C2×#95—941no—
226#340—22no—
227#340—942no—
228#324—942no—
229C47#902611yescharacteristic, complemented, Sylow
230C23×C4—21no—
231C23×C4—471no—
232C23×C4—471no—
233C23×C4—941no—
234C23×C4C22×D4711yescharacteristic
235C22×C8—21no—
236C22×C8—941no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—472no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4—472no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4—941no—
249C2×C4⋊C4—944no—
250C2×C4⋊C4—1882no—
251C2×C4⋊C4—1882no—
252C2×C4⋊C4C4×D4712yes—
253C2×C4⋊C4C22⋊D4712yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—941no—
256C2×C42—942no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—941no—
260C22⋊C8—941no—
261#95—21no—
262#95—472no—
263#95—944no—
264#95—942no—
265#95—944no—
266#95—941no—
267#95—941no—
268C24D4×D4711yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—472no—
279C22×C4—472no—
280C22×C4—941no—
281C22×C4—944no—
282C22×C4—942no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—942no—
291C22×C4—1882no—
292C22×C4—1882no—
293C22×C4—1881no—
294C22×C4—1881no—
295C22×C4C2×C4×D4711yescharacteristic
296C22×C4#1062535011yescharacteristic
297C22×C4D4×D4711yescharacteristic
298C22×C4D4×D4711yescharacteristic
299C22×C4#1062535211yescharacteristic
300C22×C4C2×C22⋊D4711yescharacteristic
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—1882no—
304C2×C8—1881no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—942no—
311C4⋊C4—942no—
312C4⋊C4—942no—
313C4⋊C4—942no—
314C4⋊C4—1882no—
315C4⋊C4—1888no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—1881no—
319C22⋊C4—1881no—
320C42—3762no—
321C23—22no—
322C23—22no—
323C23C22⋊C4×D4711yescharacteristic
324C23#1309399211yescharacteristic
325C23#1309400611yescharacteristic
326C23#1309404311yescharacteristic
327C23#1309404511yescharacteristic
328C23#1309404811yescharacteristic
329C23#1309405111yescharacteristic
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—944no—
347C2×C4—942no—
348C2×C4—942no—
349C2×C4—942no—
350C2×C4—942no—
351C2×C4—3761no—
352C2×C4—1882no—
353C2×C4—1882no—
354C2×C4—1882no—
355C2×C4—1882no—
356C2×C4—1884no—
357C2×C4—1884no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1882no—
361C2×C4—1882no—
362C2×C4—1881no—
363C2×C4—1881no—
364C2×C4—1882no—
365C2×C4(C22⋊D47)⋊C411yescharacteristic
366C2×C4C4×C22⋊D4711yescharacteristic
367C2×C4#1309399511yescharacteristic
368C2×C4#1309399711yescharacteristic
369C8—81no—
370C8—3761no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22#1568899911yescharacteristic
380C22#1568911611yescharacteristic
381C22#1568912611yescharacteristic
382C22#1568926511yescharacteristic
383C22#1568926711yescharacteristic
384C22#1568930412yes—
385C22#1568931212yes—
386C22#1568944511yescharacteristic
387C22#1568944611yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—3762no—
395C4—1884no—
396C4—1882no—
397C4—1882no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#2024238212yes—
403C2#2024238412yes—
404C2#2024370711yescharacteristic
405C2#2024377811yescharacteristic
406C2#2024379811yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1823C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1823D47
hypocenterC47#9026
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
12. #15689054C4semidirect
229. C47#9026semidirect

Maximal subgroups

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

subgroupindex
C2×#156890542
#202459902
#202434742
C2×#156888482
#202412682
#202412662
C47×#18232
#902647