Finite Group Databasebeta

← #26109652

Subgroups — #26109652

Order 12032.

940 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688983C211yesmaximal, characteristic
2#20241611C211yesmaximal, characteristic
3#20241268C214yesmaximal, complemented
4C47×#1779C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C22×C4⋊Dic47C411yescharacteristic, complemented
7C22×C22⋊Dic47—21no—
8C22×C22⋊Dic47—21no—
9#15688983—21no—
10#15688983—21no—
11#15688983C412yescomplemented
12#15688983C2212yes—
13C2×C47⋊#95—21no—
14C2×C47⋊#95—21no—
15C2×C47⋊#95C2212yes—
16C24×C4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C2×C22⋊C8×C47—21no—
19C2×C22⋊C8×C47C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C23×Dic47—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—42no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47C2×C412yes—
28C2×C4⋊Dic47C2×C412yes—
29C2×C22⋊Dic47—24no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—42no—
33C2×C22⋊Dic47—42no—
34C2×C22⋊Dic47—42no—
35C2×C22⋊Dic47—42no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—24no—
39C47⋊#95—42no—
40C25×C47D411yescharacteristic
41C23×C4×C47—24no—
42C23×C4×C47—42no—
43C23×C4×C47—42no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C22×C8×C47—22no—
51C22⋊C8×C47—24no—
52C22⋊C8×C47—42no—
53C22⋊C8×C47—42no—
54C24×C47—22no—
55C24×C47—22no—
56C24×C47—22no—
57C24×C47—22no—
58C24×C47—41no—
59C24×C47—41no—
60C24×C47C22⋊C411yescharacteristic
61C24×C47C22⋊C411yescharacteristic
62C24×C47D812yes—
63C24×C47SD1612yes—
64C24×C47C2×D411yescharacteristic
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47—416no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47#4114yes—
88C22×Dic47—24no—
89C22×Dic47—22no—
90C22×Dic47—22no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—42no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C22×Dic47—42no—
98C22×Dic47—42no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22×Dic47—42no—
103C2×C8×C47—42no—
104C2×C8×C47—42no—
105C2×C8×C47—42no—
106C22⋊Dic47—48no—
107C22⋊Dic47—44no—
108C22⋊Dic47—42no—
109C22⋊Dic47—42no—
110C4⋊Dic47—42no—
111C4⋊Dic47—42no—
112C4⋊Dic47—42no—
113C4⋊Dic47—42no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—21no—
121C23×C47—21no—
122C23×C47—24no—
123C23×C47—24no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—21no—
128C23×C47—21no—
129C23×C47—21no—
130C23×C47—24no—
131C23×C47—24no—
132C23×C47—42no—
133C23×C47—42no—
134C23×C47—44no—
135C23×C47—44no—
136C23×C47—44no—
137C23×C47—44no—
138C23×C47#9912yes—
139C23×C47#10212yes—
140C23×C47#10212yes—
141C23×C47#10412yes—
142C23×C47C2×C22⋊C411yescharacteristic
143C23×C47#12312yes—
144C23×C47C2×D811yescharacteristic
145C23×C47C2×SD1611yescharacteristic
146C23×C47#13712yes—
147C2×C4×C47—28no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—44no—
158C2×C4×C47—44no—
159C2×C4×C47—44no—
160C2×C4×C47—44no—
161C2×C4×C47—44no—
162C2×C4×C47#9912yes—
163C2×C4×C47#11712yes—
164C2×C4×C47#12312yes—
165C2×Dic47—44no—
166C2×Dic47—44no—
167C2×Dic47—44no—
168C2×Dic47—42no—
169C2×Dic47—42no—
170C2×Dic47—42no—
171C2×Dic47—42no—
172C2×Dic47—42no—
173C2×Dic47—42no—
174C2×Dic47—82no—
175C2×Dic47—82no—
176C2×Dic47—82no—
177C2×Dic47—84no—
178C2×Dic47—84no—
179C2×Dic47—82no—
180C8×C47—82no—
181#8712—471nomaximal, Sylow
182C22×C47—22no—
183C22×C47—22no—
184C22×C47—22no—
185C22×C47—22no—
186C22×C47—22no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—22no—
190C22×C47—22no—
191C22×C47—22no—
192C22×C47—22no—
193C22×C47—22no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—22no—
197C22×C47—22no—
198C22×C47—22no—
199C22×C47—48no—
200C22×C47—48no—
201C22×C47—41no—
202C22×C47—41no—
203C22×C47—41no—
204C22×C47—41no—
205C22×C47#32714yes—
206C22×C47#38111yescharacteristic
207C22×C47C2×#9911yescharacteristic
208C22×C47C2×#9911yescharacteristic
209C22×C47C2×#10211yescharacteristic
210C22×C47#41911yescharacteristic
211C22×C47C2×#10411yescharacteristic
212C22×C47#42111yescharacteristic
213C22×C47#48012yes—
214C22×C47#48312yes—
215C4×C47—22no—
216C4×C47—22no—
217C4×C47—22no—
218C4×C47—41no—
219Dic47—84no—
220Dic47—82no—
221Dic47—82no—
222C2×#380—471no—
223#1779D4711yescharacteristic
224C22⋊#95—471no—
225#1294—474no—
226C2×C47—22no—
227C2×C47—24no—
228C2×C47—22no—
229C2×C47—42no—
230C2×C47C2×#32711yescharacteristic
231C2×C47#150811yescharacteristic
232C2×C47#151111yescharacteristic
233C2×C47#151911yescharacteristic
234C2×C47#181911yescharacteristic
235C2×C47#182311yescharacteristic
236C2×C47#182611yescharacteristic
237C24×C4C2×D4711yescharacteristic
238C22×C4⋊C4—471no—
239C22×C4⋊C4—471no—
240C22×C22⋊C4—941no—
241C22×C22⋊C4—941no—
242C2×C22⋊C8—21no—
243C2×C22⋊C8—21no—
244C2×C22⋊C8C2×D4712yes—
245#380—472no—
246#380—472no—
247#380—941no—
248#380—941no—
249C2×#95—472no—
250C2×#95—941no—
251C2×#95—941no—
252C47#871211yescharacteristic, complemented, Sylow
253C25D18811yescharacteristic
254C23×C4—24no—
255C23×C4—42no—
256C23×C4—42no—
257C23×C4—941no—
258C23×C4—941no—
259C23×C4—942no—
260C23×C4C4×D4711yescharacteristic
261C23×C4C4×D4711yescharacteristic
262C23×C4D18811yescharacteristic
263C23×C4C22⋊D4711yescharacteristic
264C23×C4C22⋊D4711yescharacteristic
265C23×C4C22×D4711yescharacteristic
266C22×C8—22no—
267C2×C4⋊C4—472no—
268C2×C4⋊C4—472no—
269C2×C4⋊C4—941no—
270C2×C4⋊C4—941no—
271C2×C4⋊C4—1882no—
272C2×C4⋊C4—1882no—
273C2×C22⋊C4—944no—
274C2×C22⋊C4—942no—
275C2×C22⋊C4—942no—
276C2×C22⋊C4—1882no—
277C2×C22⋊C4—1882no—
278C2×C22⋊C4—1882no—
279C2×C22⋊C4—1882no—
280C22⋊C8—24no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283#95—942no—
284#95—942no—
285#95—944no—
286#95—1882no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24#1062532512yes—
294C24D37612yes—
295C24(C2×D47)⋊C411yescharacteristic
296C24(C2×D47)⋊C411yescharacteristic
297C24C2×D18811yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—416no—
316C22×C4—944no—
317C22×C4—942no—
318C22×C4—942no—
319C22×C4—1882no—
320C22×C4—1882no—
321C22×C4—1882no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4—1882no—
325C22×C4—1882no—
326C22×C4—1882no—
327C22×C4—1882no—
328C22×C4—1882no—
329C22×C4—1882no—
330C22×C4—1882no—
331C22×C4(C2×D47)⋊C411yescharacteristic
332C22×C4(C2×D47)⋊C411yescharacteristic
333C22×C4C2×C4×D4711yescharacteristic
334C22×C4#1062535214yes—
335C22×C4C2×C22⋊D4711yescharacteristic
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—1882no—
340C4⋊C4—1882no—
341C4⋊C4—1882no—
342C4⋊C4—1882no—
343C22⋊C4—1888no—
344C22⋊C4—1884no—
345C22⋊C4—1882no—
346C22⋊C4—1882no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—24no—
359C23—24no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—24no—
364C23—24no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23#1309393812yes—
372C23#1309394612yes—
373C23#1309394612yes—
374C23#1309395012yes—
375C23C2×(C2×D47)⋊C411yescharacteristic
376C23#1309400812yes—
377C23C2×D37611yescharacteristic
378C23C2×#1062532511yescharacteristic
379C23#1309405712yes—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—3762no—
396C2×C4—3762no—
397C2×C4—3762no—
398C2×C4—3764no—
399C2×C4—3764no—
400C2×C4—3762no—
401C2×C4—1884no—
402C2×C4—1884no—
403C2×C4—1884no—
404C2×C4—1882no—
405C2×C4—1882no—
406C2×C4—1882no—
407C2×C4—1882no—
408C2×C4—1882no—
409C2×C4—1882no—
410C2×C4#1309393812yes—
411C2×C4#1309398012yes—
412C2×C4#1309400512yes—
413C8—82no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—48no—
432C22—48no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22#1568885814yes—
438C22#1568898411yescharacteristic
439C22C2×#1309393811yescharacteristic
440C22C2×#1309393811yescharacteristic
441C22C2×#1309394611yescharacteristic
442C22#1568913311yescharacteristic
443C22C2×#1309395011yescharacteristic
444C22#1568913911yescharacteristic
445C22#1568943712yes—
446C22#1568945412yes—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—3764no—
452C4—3762no—
453C4—3762no—
454C2—22no—
455C2—24no—
456C2—22no—
457C2—42no—
458C2C2×#1568885811yescharacteristic
459C2#2024190911yescharacteristic
460C2#2024192211yescharacteristic
461C2#2024195711yescharacteristic
462C2#2024346011yescharacteristic
463C2#2024347511yescharacteristic
464C2#2024348511yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1779C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#1779D47
hypocenterC47#8712
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #20241268C2semidirect
5. C22×C4⋊Dic47C4semidirect
6. C22×C4⋊Dic47C4semidirect
11. #15688983C4semidirect
252. C47#8712semidirect

Maximal subgroups

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

subgroupindex
C2×#156889832
#202416112
#202412682
C47×#17792
#871247