Finite Group Databasebeta

← #26115255

Subgroups — #26115255

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251740C211yesmaximal, characteristic
2C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic
3C2×#15688859C214yesmaximal
4C47×C4⋊#95C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094003C2212yes—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×C4×C47⋊C8—21no—
9#15689029C412yescomplemented
10#15689021—21no—
11#15689021—21no—
12#15688993C411yescharacteristic, complemented
13#15688983—21no—
14#15688983—21no—
15#15688878—21no—
16#15688859—28no—
17C22×C4⋊C4×C47—21no—
18C22×C42×C47C2211yescharacteristic
19C2×#95×C47—21no—
20C2×#95×C47C2212yes—
21C22×Dic94C2×C412yes—
22C22×C47⋊C8—22no—
23#13094024—42no—
24#13094003—24no—
25#13094003—24no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)D414yes—
35C4×C47⋊C8—42no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C23×C4×C47—22no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C2×C4⋊C4×C47—24no—
43C2×C4⋊C4×C47—42no—
44C2×C4⋊C4×C47—42no—
45C2×C42×C47—42no—
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50#95×C47—24no—
51#95×C47—42no—
52C24×C47C2×D411yescharacteristic
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—24no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47SD1612yes—
71C22×C4×C47SD1612yes—
72C22×C4×C47Q1612yes—
73C22×C4×C47Q1612yes—
74C22×C4×C47C22×C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D412yes—
77C22×C4×C47#4112yes—
78C22×Dic47—22no—
79C22×Dic47—22no—
80C2×Dic94—24no—
81C2×Dic94—42no—
82C2×Dic94—42no—
83C4⋊C4×C47—48no—
84C42×C47—22no—
85C42×C47—22no—
86C42×C47—41no—
87C42×C47—41no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C4⋊Dic47—24no—
91C4⋊Dic47—42no—
92C4⋊Dic47—42no—
93C4⋊Dic47—42no—
94C4⋊Dic47—42no—
95Dic47⋊C4—44no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—42no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47C2×C22⋊C411yescharacteristic
104C23×C47#12411yescharacteristic
105C23×C47#12711yescharacteristic
106C23×C47C2×SD1611yescharacteristic
107C23×C47C2×SD1611yescharacteristic
108C23×C47C2×Q1611yescharacteristic
109C23×C47C2×Q1611yescharacteristic
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—24no—
121C2×C4×C47—24no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—44no—
127C2×C4×C47—44no—
128C2×C4×C47—44no—
129C2×C4×C47—82no—
130C2×C4×C47—82no—
131C2×C4×C47—82no—
132C2×C4×C47—82no—
133C2×C4×C47#9912yes—
134C2×C4×C47#10312yes—
135C2×C4×C47#10312yes—
136C2×C4×C47#10312yes—
137C2×C4×C47#10312yes—
138C2×C4×C47#10412yes—
139C2×C4×C47C4×D412yes—
140C2×C4×C47#12112yes—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147Dic94—84no—
148C47⋊C8—44no—
149#9679—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47#39011yescharacteristic
163C22×C47C2×#9911yescharacteristic
164C22×C47#41011yescharacteristic
165C22×C47C2×#10311yescharacteristic
166C22×C47C2×#10311yescharacteristic
167C22×C47C2×#10411yescharacteristic
168C22×C47#42111yescharacteristic
169C22×C47#48711yescharacteristic
170C22×C47#48711yescharacteristic
171C22×C47#49211yescharacteristic
172C22×C47#49411yescharacteristic
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—22no—
176C4×C47—22no—
177C4×C47—22no—
178C4×C47—84no—
179C4×C47#32814yescomplemented
180Dic47—82no—
181Dic47—82no—
182#2857—471no—
183C4×C22⋊C8—471no—
184C2×#328—474no—
185C4⋊#95D4711yescharacteristic
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47C2×#32811yescharacteristic
191C2×C47#150311yescharacteristic
192C2×C47#151811yescharacteristic
193C2×C47#152011yescharacteristic
194C2×C47#195611yescharacteristic
195C2×C47#196411yescharacteristic
196C2×C47#198011yescharacteristic
197C2×C4⋊Q8—471no—
198C2×C22⋊Q8—472no—
199C22×C4⋊C4—21no—
200C22×C42C2×D4711yescharacteristic
201C2×C22⋊C8—472no—
202C2×C4×C8—941no—
203#391—472no—
204#389—941no—
205#389—941no—
206#384—471no—
207#380—941no—
208#380—941no—
209C2×#95—21no—
210C2×#95C2×D4712yes—
211#336—941no—
212#328—948no—
213C47#967911yescharacteristic, complemented, Sylow
214C22×Q8—472no—
215C23×C4—22no—
216C23×C4C22⋊D4711yescharacteristic
217C23×C4C22⋊D4711yescharacteristic
218C23×C4C22×D4711yescharacteristic
219C22×C8—942no—
220C4⋊Q8—1882no—
221C22⋊Q8—944no—
222C22⋊Q8—944no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—472no—
227C2×C4⋊C4—941no—
228C2×C4⋊C4—941no—
229C2×C4⋊C4—941no—
230C2×C4⋊C4—941no—
231C2×C4⋊C4—942no—
232C2×C22⋊C4—942no—
233C2×C42—42no—
234C2×C42C4×D4711yescharacteristic
235C2×C42C4×D4711yescharacteristic
236C2×C42D18811yescharacteristic
237C2×C42D18811yescharacteristic
238C22⋊C8—474no—
239C22⋊C8—942no—
240C4×C8—1882no—
241#95—24no—
242#95—42no—
243#95—942no—
244#95—942no—
245C24C2×C22⋊D4711yescharacteristic
246C2×Q8—944no—
247C2×Q8—1882no—
248C2×Q8—1882no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—942no—
263C22×C4—942no—
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4(C2×D47)⋊C411yescharacteristic
266C22×C4(C2×D47)⋊C411yescharacteristic
267C22×C4(C2×D47)⋊C411yescharacteristic
268C22×C4#1062533512yes—
269C22×C4#1062533512yes—
270C22×C4#1062533712yes—
271C22×C4#1062533712yes—
272C22×C4C2×C4×D4711yescharacteristic
273C22×C4C2×D18811yescharacteristic
274C22×C4D4×D4712yes—
275C22×C4#1062535412yes—
276C2×C8—944no—
277C2×C8—1882no—
278C2×C8—1882no—
279C4⋊C4—48no—
280C4⋊C4—944no—
281C4⋊C4—1882no—
282C4⋊C4—1882no—
283C4⋊C4—1882no—
284C4⋊C4—1882no—
285C4⋊C4—1884no—
286C22⋊C4—1882no—
287C22⋊C4—1882no—
288C42—22no—
289C42—22no—
290C42—41no—
291C42—41no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23C2×(C2×D47)⋊C411yescharacteristic
297C23#1309401311yescharacteristic
298C23#1309402011yescharacteristic
299C23C2×#1062533511yescharacteristic
300C23C2×#1062533511yescharacteristic
301C23C2×#1062533711yescharacteristic
302C23C2×#1062533711yescharacteristic
303Q8—3764no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4#1309393812yes—
334C2×C4#1309394712yes—
335C2×C4#1309394712yes—
336C2×C4#1309394712yes—
337C2×C4#1309394712yes—
338C2×C4#1309395212yes—
339C2×C4#1309398612yes—
340C2×C4#1309399912yes—
341C8—1884no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22#1568902411yescharacteristic
355C22C2×#1309393811yescharacteristic
356C22#1568909611yescharacteristic
357C22C2×#1309394711yescharacteristic
358C22C2×#1309394711yescharacteristic
359C22C2×#1309395211yescharacteristic
360C22#1568914111yescharacteristic
361C22#1568947511yescharacteristic
362C22#1568947511yescharacteristic
363C22#1568949511yescharacteristic
364C22#1568950211yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—84no—
371C4—3762no—
372C4—3762no—
373C4#1568885914yescomplemented
374C2—21no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2C2×#1568885911yescharacteristic
379C2#2024188611yescharacteristic
380C2#2024195111yescharacteristic
381C2#2024195911yescharacteristic
382C2#2024418911yescharacteristic
383C2#2024422311yescharacteristic
384C2#2024430611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×C4⋊#95C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC4⋊#95D47
hypocenterC47#9679
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. C2×#13094024C4semidirect
9. #15689029C4semidirect
12. #15688993C4semidirect
179. C4×C47#328semidirect
213. C47#9679semidirect
373. C4#15688859semidirect

Maximal subgroups

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

subgroupindex
#202517402
C4×C22⋊(C47⋊C8)2
C2×#156888592
C47×C4⋊#952
#967947