Finite Group Databasebeta

← #26111664

Subgroups — #26111664

Order 12032.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689007C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3C2×#15688848C212yesmaximal
4#20241230C212yesmaximal
5C2×C47×#431C211yesmaximal, characteristic
6C22×Dic47⋊C4C411yescharacteristic
7C22×Dic47⋊C4C2211yescharacteristic
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689007—21no—
10#15689007—21no—
11#15689007—21no—
12#15689007—21no—
13C2×C47⋊#95C411yescharacteristic
14C2×C47⋊#95C411yescharacteristic
15C2×C47⋊#95C411yescharacteristic
16C2×C47⋊#95C2211yescharacteristic
17#15688878—21no—
18#15688878—21no—
19#15688848—24no—
20C22×C42×C47C2211yescharacteristic
21C47×#431—24no—
22C2×C4⋊C8×C47—21no—
23C2×C22⋊C8×C47C2212yes—
24C2×C4×C8×C47—21no—
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C23×Dic47C2×C411yescharacteristic
28C23×Dic47C2×C411yescharacteristic
29C22×C47⋊C8—22no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—22no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—22no—
36C2×Dic47⋊C4—22no—
37C2×Dic47⋊C4C814yescomplemented
38C22⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)—22no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C23×C4×C47C2×C411yescharacteristic
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47C2311yescharacteristic
51C22×C8×C47—22no—
52C2×C42×C47—22no—
53C2×C42×C47—22no—
54C2×C42×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47D411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C4⋊C8×C47—24no—
59C22⋊C8×C47—22no—
60C22⋊C8×C47D414yes—
61C4×C8×C47—24no—
62C24×C47C22×C411yescharacteristic
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C2×C812yes—
75C22×C4×C47M1612yes—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D412yes—
79C22×C4×C47#4112yes—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—22no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—22no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—22no—
89C22×Dic47—22no—
90C22×Dic47—41no—
91C22×Dic47—41no—
92C22×Dic47—41no—
93C22×Dic47—41no—
94C22×Dic47—41no—
95C22×Dic47—41no—
96C22×Dic47C2×C812yes—
97C22×Dic47C2×C812yes—
98C2×C8×C47—24no—
99C2×C8×C47—42no—
100C2×C8×C47—42no—
101C42×C47—21no—
102C42×C47—21no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—21no—
106C42×C47—21no—
107Dic47⋊C4—24no—
108Dic47⋊C4—24no—
109Dic47⋊C4—41no—
110Dic47⋊C4—42no—
111Dic47⋊C4—41no—
112C2×C47⋊C8—42no—
113C2×C47⋊C8—42no—
114C2×C47⋊C8—42no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47C4×D412yes—
121C23×C47C22×C811yescharacteristic
122C23×C47C2×M1611yescharacteristic
123C23×C47#13112yes—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—24no—
135C2×C4×C47—24no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—44no—
142C2×C4×C47C22⋊C812yes—
143C2×C4×C47C22⋊C812yes—
144C2×C4×C47#10112yes—
145C2×C4×C47#12011yescharacteristic
146C2×C4×C47#12112yes—
147C2×C4×C47#12311yescharacteristic
148C2×Dic47—22no—
149C2×Dic47—22no—
150C2×Dic47—24no—
151C2×Dic47—42no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—41no—
162C2×Dic47—41no—
163C2×Dic47—41no—
164C2×Dic47—82no—
165C2×Dic47—82no—
166C8×C47—44no—
167C47⋊C8—82no—
168#9082—471nomaximal, Sylow
169C22×C47—22no—
170C22×C47—22no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—22no—
176C22×C47—22no—
177C22×C47#38611yescharacteristic
178C22×C47C2×C22⋊C811yescharacteristic
179C22×C47#40811yescharacteristic
180C22×C47#41011yescharacteristic
181C22×C47#41011yescharacteristic
182C22×C47C2×#10111yescharacteristic
183C22×C47#41311yescharacteristic
184C22×C47C8×D412yes—
185C22×C47#43512yes—
186C4×C47—22no—
187C4×C47—22no—
188C4×C47—22no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—22no—
192C4×C47—41no—
193Dic47—42no—
194Dic47—42no—
195Dic47—81no—
196Dic47—81no—
197C2×#431D4711yescharacteristic
198C2×#387—471no—
199#1817—471no—
200C2×#324—472no—
201#1280—472no—
202C2×C47—22no—
203C2×C47—22no—
204C2×C47#145811yescharacteristic
205C2×C47#146111yescharacteristic
206C2×C47#147011yescharacteristic
207C2×C47#147311yescharacteristic
208C2×C47#186911yescharacteristic
209C2×C47#189911yescharacteristic
210C2×C47#190111yescharacteristic
211C22×C4⋊C4—471no—
212C22×C4⋊C4—471no—
213C22×C42C2×D4711yescharacteristic
214#431—24no—
215C2×C4⋊C8—21no—
216C2×C22⋊C8—472no—
217C2×C22⋊C8C2×D4712yes—
218C2×C4×C8—21no—
219#387—941no—
220#387—941no—
221#387—941no—
222#387—941no—
223C2×#95—471no—
224C2×#95—471no—
225C2×#95—471no—
226C2×#95—471no—
227#336—941no—
228#336—941no—
229#324—944no—
230C47#908211yescharacteristic, complemented, Sylow
231C23×C4—471no—
232C23×C4—471no—
233C23×C4—471no—
234C23×C4—471no—
235C23×C4C4×D4711yescharacteristic
236C23×C4C4×D4711yescharacteristic
237C23×C4C22×D4711yescharacteristic
238C22×C8—22no—
239C22×C8—942no—
240C2×C4⋊C4—474no—
241C2×C4⋊C4—942no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—942no—
246C2×C4⋊C4—942no—
247C2×C4⋊C4—941no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42D18811yescharacteristic
251C2×C42D18811yescharacteristic
252C2×C42C22⋊D4711yescharacteristic
253C2×C42C22⋊D4711yescharacteristic
254C4⋊C8—24no—
255C22⋊C8—22no—
256C22⋊C8—942no—
257C22⋊C8—942no—
258C22⋊C8C22⋊D4714yes—
259C4×C8—24no—
260#95—941no—
261#95—941no—
262#95—941no—
263#95—941no—
264#95—941no—
265#95—941no—
266#95—941no—
267#95—941no—
268C24C2×C4×D4711yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—472no—
277C22×C4—472no—
278C22×C4—941no—
279C22×C4—941no—
280C22×C4—942no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—942no—
284C22×C4—942no—
285C22×C4—942no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—1881no—
289C22×C4—1881no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4C8×D4712yes—
295C22×C4#1062532412yes—
296C22×C4(C2×D47)⋊C411yescharacteristic
297C22×C4(C2×D47)⋊C411yescharacteristic
298C22×C4(C2×D47)⋊C411yescharacteristic
299C22×C4(C2×D47)⋊C411yescharacteristic
300C22×C4C2×D18811yescharacteristic
301C22×C4#1062535012yes—
302C22×C4C2×C22⋊D4711yescharacteristic
303C22×C4C2×C22⋊D4712yes—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—1882no—
308C2×C8—1882no—
309C2×C8—1882no—
310C4⋊C4—944no—
311C4⋊C4—944no—
312C4⋊C4—1881no—
313C4⋊C4—1882no—
314C4⋊C4—1881no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23C2×(C2×D47)⋊C411yescharacteristic
326C23C4×C22⋊D4712yes—
327C23C2×C8×D4711yescharacteristic
328C23C2×#1062532411yescharacteristic
329C23#1309403012yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—942no—
349C2×C4—942no—
350C2×C4—944no—
351C2×C4—3762no—
352C2×C4—3762no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4—1881no—
356C2×C4—1881no—
357C2×C4—1881no—
358C2×C4—1881no—
359C2×C4—1882no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4—1881no—
363C2×C4—1881no—
364C2×C4—1882no—
365C2×C4—1882no—
366C2×C4(C2×D47)⋊C812yes—
367C2×C4(C2×D47)⋊C812yes—
368C2×C4#1309394212yes—
369C2×C4#1309399311yescharacteristic
370C2×C4#1309399812yes—
371C2×C4#1309400711yescharacteristic
372C8—44no—
373C8—3762no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22#1568900211yescharacteristic
383C22C2×(C2×D47)⋊C811yescharacteristic
384C22#1568908911yescharacteristic
385C22#1568909611yescharacteristic
386C22#1568909611yescharacteristic
387C22C2×#1309394211yescharacteristic
388C22#1568910511yescharacteristic
389C22C8×C22⋊D4712yes—
390C22#1568919412yes—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—41no—
398C4—3761no—
399C4—3761no—
400C4—1882no—
401C4—1882no—
402C2—22no—
403C2—22no—
404C2#2024170011yescharacteristic
405C2#2024171111yescharacteristic
406C2#2024174311yescharacteristic
407C2#2024175511yescharacteristic
408C2#2024368011yescharacteristic
409C2#2024384611yescharacteristic
410C2#2024385611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC2×C47×#431C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC23×C4×C47C23
hypercenterC2×#431D47
hypocenterC47#9082
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
37. C2×Dic47⋊C4C8semidirect
230. C47#9082semidirect

Maximal subgroups

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

subgroupindex
C2×#156890072
#202434562
C2×#156888482
#202412302
C2×C47×#4312
#908247