Finite Group Databasebeta

← #26106791

Subgroups — #26106791

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688974C211yesmaximal, characteristic
2#20243152C212yesmaximal
3C2×#15688848C212yesmaximal
4#20241231C211yesmaximal, characteristic
5C47×#1280C211yesmaximal, characteristic
6C22×Dic47⋊C4C411yescharacteristic
7C22×Dic47⋊C4C2211yescharacteristic
8C22×C4×Dic47C411yescharacteristic
9C22×C4×Dic47C411yescharacteristic
10C22×C4×Dic47C2211yescharacteristic
11C2×C22⋊(C47⋊C8)C2212yes—
12C2×#13093934—22no—
13#15688974—22no—
14#15688974—22no—
15C2×C47⋊#95C411yescharacteristic
16#15688877—22no—
17#15688848C418yescomplemented
18C2×C22⋊C8×C47C2212yes—
19C2×#95×C47C2211yescharacteristic
20C23×Dic47C2×C411yescharacteristic
21C23×Dic47C2×C411yescharacteristic
22C23×Dic47C2×C411yescharacteristic
23C23×Dic47C2×C411yescharacteristic
24C22×C47⋊C8—22no—
25C2×Dic47⋊C4—22no—
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4C2×C414yes—
32C2×C4×Dic47—22no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—21no—
35C2×C4×Dic47—22no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—42no—
39C2×C4×Dic47—42no—
40C2×C4×Dic47D412yes—
41C2×C4×Dic47D412yes—
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)C2×C414yes—
44#13093934—44no—
45C47⋊#95—22no—
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47C2311yescharacteristic
49C22×C8×C47—22no—
50C22⋊C8×C47—22no—
51C22⋊C8×C47C2×C414yes—
52#95×C47—22no—
53C24×C47C22×C411yescharacteristic
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47M1612yes—
65C22×C4×C47M1612yes—
66C22×C4×C47C22×C412yes—
67C22×C4×C47C2×D412yes—
68C22×C4×C47#4111yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C22×Dic47—21no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—22no—
77C22×Dic47—22no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—41no—
83C22×Dic47—41no—
84C22×Dic47—41no—
85C22×Dic47—41no—
86C22×Dic47C4212yes—
87C22×Dic47C4212yes—
88C22×Dic47C22⋊C412yes—
89C22×Dic47C22⋊C411yescharacteristic
90C22×Dic47C22⋊C411yescharacteristic
91C22×Dic47M1612yes—
92C22×Dic47M1612yes—
93C2×C8×C47—24no—
94C2×C8×C47—42no—
95C2×C8×C47—42no—
96Dic47⋊C4—28no—
97Dic47⋊C4—24no—
98Dic47⋊C4—24no—
99C4×Dic47—24no—
100C4×Dic47—24no—
101C4×Dic47—24no—
102C4×Dic47—42no—
103C4×Dic47—42no—
104C4×Dic47—42no—
105C2×C47⋊C8—24no—
106C2×C47⋊C8—42no—
107C2×C47⋊C8—42no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47C2×C4211yescharacteristic
113C23×C47C2×C22⋊C411yescharacteristic
114C23×C47#11711yescharacteristic
115C23×C47C2×M1611yescharacteristic
116C23×C47C2×M1611yescharacteristic
117C23×C47C2×M1611yescharacteristic
118C23×C47C2×M1611yescharacteristic
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
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×C47C4×D412yes—
135C2×C4×C47C4×D412yes—
136C2×Dic47—24no—
137C2×Dic47—24no—
138C2×Dic47—22no—
139C2×Dic47—22no—
140C2×Dic47—21no—
141C2×Dic47—21no—
142C2×Dic47—22no—
143C2×Dic47—22no—
144C2×Dic47—21no—
145C2×Dic47—21no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—22no—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47#9714yescomplemented
159C2×Dic47#9912yes—
160C2×Dic47#10112yes—
161C8×C47—44no—
162C47⋊C8—44no—
163#8153—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47C4×C22⋊C411yescharacteristic
177C22×C47C2×#9711yescharacteristic
178C22×C47C4×M1611yescharacteristic
179C22×C47#40711yescharacteristic
180C22×C47#40711yescharacteristic
181C22×C47C2×#9911yescharacteristic
182C22×C47#41011yescharacteristic
183C22×C47C2×#10111yescharacteristic
184C22×C47#41311yescharacteristic
185C22×C47#43211yescharacteristic
186C22×C47#43211yescharacteristic
187C4×C47—22no—
188C4×C47—22no—
189C4×C47—42no—
190C4×C47—42no—
191Dic47—22no—
192Dic47—22no—
193Dic47—24no—
194Dic47—42no—
195Dic47—42no—
196C2×C4×C4⋊C4—471no—
197#1749—472no—
198C2×#324—472no—
199#1280—471no—
200#1280D4711yescharacteristic
201C2×C47—21no—
202C2×C47—21no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47#146311yescharacteristic
206C2×C47#146411yescharacteristic
207C2×C47#146511yescharacteristic
208C2×C47#146711yescharacteristic
209C2×C47#174911yescharacteristic
210C2×C47C4×#9911yescharacteristic
211C2×C47C4×#10111yescharacteristic
212C22×C4⋊C4—471no—
213C22×C4⋊C4—471no—
214C22×C42—471no—
215C22×C42—471no—
216C22×C42—471no—
217C2×C22⋊C8—472no—
218C2×C22⋊C8C2×D4712yes—
219C2×#97—942no—
220C4×C4⋊C4—942no—
221C4×C4⋊C4—942no—
222C2×#95—471no—
223C2×#95C2×D4711yescharacteristic
224#336—942no—
225#324—478no—
226C47#815311yescharacteristic, complemented, Sylow
227C23×C4—471no—
228C23×C4—471no—
229C23×C4—471no—
230C23×C4—471no—
231C23×C4C4×D4711yescharacteristic
232C23×C4C4×D4711yescharacteristic
233C23×C4C22×D4711yescharacteristic
234C22×C8—22no—
235C22×C8—942no—
236C2×C4⋊C4—474no—
237C2×C4⋊C4—942no—
238C2×C4⋊C4—942no—
239C2×C4⋊C4—942no—
240C2×C4⋊C4—942no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—941no—
243C2×C42—472no—
244C2×C42—472no—
245C2×C42—941no—
246C2×C42—941no—
247C2×C42—942no—
248C2×C42—942no—
249C2×C42—941no—
250C2×C42—941no—
251C2×C42—1882no—
252C2×C42—1882no—
253C22⋊C8—22no—
254C22⋊C8—474no—
255C22⋊C8—942no—
256C22⋊C8C4×D4714yes—
257#97—1884no—
258#95—22no—
259#95—942no—
260C24C2×C4×D4711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—472no—
272C22×C4—471no—
273C22×C4—471no—
274C22×C4—472no—
275C22×C4—472no—
276C22×C4—472no—
277C22×C4—472no—
278C22×C4—942no—
279C22×C4—942no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4#1062532412yes—
295C22×C4#1062532412yes—
296C22×C4C2×C4×D4712yes—
297C22×C4D4×D4712yes—
298C22×C4#1062535211yescharacteristic
299C22×C4#1062535411yescharacteristic
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—944no—
304C2×C8—1882no—
305C2×C8—1882no—
306C4⋊C4—948no—
307C4⋊C4—944no—
308C4⋊C4—944no—
309C42—944no—
310C42—944no—
311C42—944no—
312C42—1882no—
313C42—1882no—
314C42—1882no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C42×D4711yescharacteristic
320C23C22⋊C4×D4711yescharacteristic
321C23#1309398211yescharacteristic
322C23M16×D4711yescharacteristic
323C23M16×D4711yescharacteristic
324C23C2×#1062532411yescharacteristic
325C23C2×#1062532411yescharacteristic
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—474no—
342C2×C4—472no—
343C2×C4—472no—
344C2×C4—944no—
345C2×C4—944no—
346C2×C4—942no—
347C2×C4—942no—
348C2×C4—941no—
349C2×C4—941no—
350C2×C4—942no—
351C2×C4—942no—
352C2×C4—941no—
353C2×C4—941no—
354C2×C4—942no—
355C2×C4—942no—
356C2×C4—942no—
357C2×C4—942no—
358C2×C4—942no—
359C2×C4—942no—
360C2×C4—1882no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4—1881no—
364C2×C4—1881no—
365C2×C4—1882no—
366C2×C4(C22⋊D47)⋊C412yes—
367C2×C4#1309398612yes—
368C8—44no—
369C8—1884no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#1568897011yescharacteristic
383C22#97×D4711yescharacteristic
384C22C4×#1062532411yescharacteristic
385C22#1568908511yescharacteristic
386C22#1568908511yescharacteristic
387C22#99×D4711yescharacteristic
388C22#1568909511yescharacteristic
389C22#101×D4711yescharacteristic
390C22#1568910411yescharacteristic
391C22#1568917911yescharacteristic
392C22#1568917911yescharacteristic
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—942no—
398C4—942no—
399C4—944no—
400C4—1882no—
401C4—1882no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2#2024171611yescharacteristic
407C2#2024171911yescharacteristic
408C2#2024172311yescharacteristic
409C2#2024173011yescharacteristic
410C2#2024314911yescharacteristic
411C2#2024317511yescharacteristic
412C2#2024318311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1280C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC23×C4×C47C23
hypercenter#1280D47
hypocenterC47#8153
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
17. #15688848C4semidirect
158. C2×Dic47#97semidirect
226. C47#8153semidirect

Maximal subgroups

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

subgroupindex
C2×#156889742
#202431522
C2×#156888482
#202412312
C47×#12802
#815347