Finite Group Databasebeta

← #26111311

Subgroups — #26111311

Order 12032.

572 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689063C211yesmaximal, characteristic
2#20245990C211yesmaximal, characteristic
3#20243479C211yesmaximal, characteristic
4#20243476C211yesmaximal, characteristic
5C2×#15688848C211yesmaximal, characteristic
6#20241267C211yesmaximal, characteristic
7C47×#1294C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9C22×Dic47⋊C4C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11#15689063—22no—
12#15689063C414yescomplemented
13#15689010—22no—
14#15689004—22no—
15#15688984—21no—
16#15688983C2211yescharacteristic
17C22⋊C4×Dic47—21no—
18C2×C47⋊#95—21no—
19C2×C47⋊#95—21no—
20#15688886—22no—
21#15688885—22no—
22#15688848—22no—
23C2×C22⋊C8×C47C2211yescharacteristic
24C47×#380C2211yescharacteristic
25C2×#95×C47C2211yescharacteristic
26C23×Dic47—21no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2×C412yes—
36C2×C4⋊Dic47D412yes—
37C2×Dic47⋊C4—21no—
38C2×Dic47⋊C4—21no—
39C2×Dic47⋊C4—21no—
40C2×Dic47⋊C4—21no—
41C2×Dic47⋊C4C2×C412yes—
42C2×Dic47⋊C4D412yes—
43C2×C22⋊Dic47—21no—
44C2×C4×Dic47—22no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47C47⋊#95—22no—
48C47⋊#95—24no—
49C47⋊#95—21no—
50C47⋊#95—24no—
51C47⋊#95—24no—
52C23×C4×C47—21no—
53C23×C4×C47C2311yescharacteristic
54C22×C8×C47—21no—
55C2×C4⋊C4×C47D412yes—
56C2×C22⋊C4×C47—21no—
57C22⋊C8×C47—21no—
58C22⋊C8×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47C2×C412yes—
62C24×C47C2×D411yescharacteristic
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47#4111yescharacteristic
75C22×C4×C47#4111yescharacteristic
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47—42no—
87C22×Dic47—44no—
88C22×Dic47—41no—
89C22×Dic47—41no—
90C22×Dic47C22⋊C412yes—
91C22×Dic47C22⋊C412yes—
92C2×C8×C47—42no—
93C2×C8×C47—41no—
94C4⋊C4×C47—42no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C22⋊Dic47—41no—
98C22⋊Dic47—41no—
99C4⋊Dic47—22no—
100C4⋊Dic47—22no—
101C4⋊Dic47—41no—
102C4⋊Dic47—41no—
103C4⋊Dic47—42no—
104Dic47⋊C4—22no—
105Dic47⋊C4—22no—
106Dic47⋊C4—22no—
107Dic47⋊C4—22no—
108C4×Dic47—82no—
109C2×C47⋊C8—42no—
110C2×C47⋊C8—41no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47C2×C22⋊C411yescharacteristic
114C23×C47#12011yescharacteristic
115C23×C47#12311yescharacteristic
116C23×C47#13511yescharacteristic
117C23×C47#13511yescharacteristic
118C23×C47#13711yescharacteristic
119C23×C47#13711yescharacteristic
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—82no—
133C2×C4×C47—82no—
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×Dic47—22no—
139C2×Dic47—22no—
140C2×Dic47—22no—
141C2×Dic47—22no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—42no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—42no—
148C2×Dic47—44no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—81no—
154C2×Dic47—82no—
155C2×Dic47—82no—
156C8×C47—81no—
157C47⋊C8—81no—
158#9022—471nomaximal, Sylow
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—22no—
162C22×C47—22no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47#38611yescharacteristic
168C22×C47#41611yescharacteristic
169C22×C47#41911yescharacteristic
170C22×C47#45211yescharacteristic
171C22×C47#45211yescharacteristic
172C22×C47#45412yes—
173C22×C47#45812yes—
174C22×C47#48511yescharacteristic
175C22×C47#48511yescharacteristic
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—82no—
181C4×C47—82no—
182Dic47—42no—
183Dic47—42no—
184Dic47—82no—
185Dic47—82no—
186C2×#400—471no—
187#2313—471no—
188#1824—471no—
189#1824—471no—
190C2×#324—471no—
191#1294—471no—
192#1294D4711yescharacteristic
193C2×C47—21no—
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47#161512yes—
198C2×C47#161512yes—
199C2×C47#187311yescharacteristic
200C2×C47#188811yescharacteristic
201C2×C47#189011yescharacteristic
202C22×C4⋊C4—471no—
203C22×C4⋊C4—471no—
204C2×C22⋊C8—471no—
205C2×C22⋊C8C2×D4711yescharacteristic
206#400—474no—
207#400—942no—
208#387—942no—
209#387—942no—
210#381—941no—
211#380—471no—
212#380C2×D4711yescharacteristic
213C4×C22⋊C4—941no—
214C2×#95—941no—
215C2×#95—941no—
216C2×#95C2×D4711yescharacteristic
217#340—942no—
218#340—942no—
219#324—942no—
220C47#902211yescharacteristic, complemented, Sylow
221C23×C4—21no—
222C23×C4—471no—
223C23×C4—471no—
224C23×C4—941no—
225C23×C4C22×D4711yescharacteristic
226C22×C8—21no—
227C22×C8—941no—
228C2×C4⋊C4—472no—
229C2×C4⋊C4—472no—
230C2×C4⋊C4—472no—
231C2×C4⋊C4—472no—
232C2×C4⋊C4—941no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4—941no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4D18812yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—941no—
244C2×C42—942no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—941no—
248C22⋊C8—941no—
249#95—21no—
250#95—21no—
251#95—944no—
252#95—944no—
253#95—942no—
254#95—944no—
255#95—941no—
256#95C4×D4712yes—
257C24D4×D4711yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—472no—
266C22×C4—472no—
267C22×C4—941no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—941no—
274C22×C4—941no—
275C22×C4—942no—
276C22×C4—942no—
277C22×C4—1884no—
278C22×C4—1881no—
279C22×C4—1881no—
280C22×C4—1882no—
281C22×C4C2×C4×D4711yescharacteristic
282C22×C4C2×D18811yescharacteristic
283C22×C4#1062535011yescharacteristic
284C22×C4D4×D4711yescharacteristic
285C22×C4D4×D4711yescharacteristic
286C22×C4#1062535411yescharacteristic
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—1882no—
290C2×C8—1881no—
291C4⋊C4—42no—
292C4⋊C4—942no—
293C4⋊C4—942no—
294C4⋊C4—942no—
295C4⋊C4—942no—
296C4⋊C4—942no—
297C4⋊C4—942no—
298C4⋊C4—1882no—
299C4⋊C4—1881no—
300C4⋊C4—1881no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—1881no—
304C22⋊C4—1881no—
305C42—3762no—
306C23—22no—
307C23—22no—
308C23C22⋊C4×D4711yescharacteristic
309C23#1309399411yescharacteristic
310C23#1309400911yescharacteristic
311C23#1309404311yescharacteristic
312C23#1309404511yescharacteristic
313C23#1309405311yescharacteristic
314C23#1309405611yescharacteristic
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—942no—
330C2×C4—942no—
331C2×C4—942no—
332C2×C4—942no—
333C2×C4—3762no—
334C2×C4—3762no—
335C2×C4—3761no—
336C2×C4—1884no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1882no—
344C2×C4—1881no—
345C2×C4—1881no—
346C2×C4—1882no—
347C2×C4#1309398611yescharacteristic
348C2×C4C4×D18811yescharacteristic
349C2×C4#1309399711yescharacteristic
350C2×C4#1309399911yescharacteristic
351C8—81no—
352C8—3761no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22#1568900111yescharacteristic
362C22#1568911611yescharacteristic
363C22#1568913011yescharacteristic
364C22#1568928611yescharacteristic
365C22#1568928811yescharacteristic
366C22#1568929712yes—
367C22#1568931412yes—
368C22#1568946611yescharacteristic
369C22#1568946711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—82no—
376C4—3762no—
377C4—3762no—
378C4—1882no—
379C4—1882no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#2024242012yes—
385C2#2024242212yes—
386C2#2024369911yescharacteristic
387C2#2024379011yescharacteristic
388C2#2024380011yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1294C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1294D47
hypocenterC47#9022
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
12. #15689063C4semidirect
220. C47#9022semidirect

Maximal subgroups

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

subgroupindex
C2×#156890632
#202459902
#202434792
#202434762
C2×#156888482
#202412672
C47×#12942
#902247