Finite Group Databasebeta

← C22×#15688893 #26213730

Subgroups — C22×#15688893 #26213730

Order 12032.

2190 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C211yesmaximal, characteristic
2C23×#10625329C211yesmaximal, characteristic
3C2×#15688893C2112yesmaximal, direct product factor
4C22×#117×C47C211yesmaximal, characteristic
5C22×#10625329—21no—
6C22×#10625329—23no—
7C22×#10625329—23no—
8C22×#10625329—21no—
9C22×#10625329—21no—
10C22×#10625329—23no—
11C22×#10625329—23no—
12C22×#10625329C412yescomplemented
13C22×#10625329C416yescomplemented
14C22×#10625329C416yescomplemented
15C22×#10625329C2216yes—
16C22×#10625329C2216yes—
17C23×C47⋊C8—21no—
18C23×C47⋊C8C412yescomplemented
19#15688893C22116yesdirect product factor
20C24×C4×C47C2211yescharacteristic
21C2×#117×C47—23no—
22C2×#117×C47—23no—
23C2×#117×C47C416yes—
24C2×#117×C47C2216yes—
25C22×C4⋊C4×C47—21no—
26C22×C22⋊C4×C47C412yes—
27C22×C42×C47—21no—
28C2×#10625329—26no—
29C2×#10625329—26no—
30C2×#10625329—212no—
31C2×#10625329—26no—
32C2×#10625329—26no—
33C2×#10625329—212no—
34C2×#10625329—26no—
35C2×#10625329—26no—
36C2×#10625329—26no—
37C2×#10625329—26no—
38C2×#10625329—26no—
39C2×#10625329—26no—
40C2×#10625329—26no—
41C2×#10625329C2×C4112yescomplemented
42C2×#10625329C2×C4112yescomplemented
43C2×#10625329C2×C4112yescomplemented
44C2×#10625329C2314yes—
45C2×#10625329C2314yes—
46C22×C47⋊C8—26no—
47C22×C47⋊C8—22no—
48C22×C47⋊C8—26no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—26no—
51C22×C47⋊C8C2×C4112yescomplemented
52C25×C47C2×C411yescharacteristic
53C23×C4×C47—22no—
54C23×C4×C47C2×C413yes—
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47C2×C413yes—
58C23×C4×C47C2×C413yes—
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D413yes—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47D413yes—
64C23×C4×C47D413yes—
65C23×C4×C47Q811yescharacteristic
66C23×C4×C47Q813yes—
67C23×C4×C47C2313yes—
68#117×C47—212no—
69#117×C47—26no—
70#117×C47—26no—
71#117×C47C2×C4112yes—
72#117×C47C2314yes—
73C2×C4⋊C4×C47—26no—
74C2×C4⋊C4×C47—26no—
75C2×C22⋊C4×C47—26no—
76C2×C22⋊C4×C47C2×C4112yes—
77C2×C42×C47—26no—
78C2×C42×C47—26no—
79C24×C47—44no—
80C24×C47C4211yescharacteristic
81C24×C47C4213yes—
82C24×C47C22⋊C411yescharacteristic
83C24×C47C22⋊C413yes—
84C24×C47C4⋊C411yescharacteristic
85C24×C47C4⋊C413yes—
86C24×C47C22×C413yes—
87C22×C4×C47—212no—
88C22×C4×C47—48no—
89C22×C4×C47C4213yes—
90C22×C4×C47C4213yes—
91C22×C4×C47C4216yes—
92C22×C4×C47C22⋊C413yes—
93C22×C4×C47C22⋊C413yes—
94C22×C4×C47C22⋊C416yes—
95C22×C4×C47C22⋊C413yes—
96C22×C4×C47C22⋊C413yes—
97C22×C4×C47C22⋊C411yescharacteristic
98C22×C4×C47C22⋊C413yes—
99C22×C4×C47C22⋊C413yes—
100C22×C4×C47C22⋊C413yes—
101C22×C4×C47C22⋊C413yes—
102C22×C4×C47C22⋊C416yes—
103C22×C4×C47C22⋊C416yes—
104C22×C4×C47C22⋊C411yescharacteristic
105C22×C4×C47C4⋊C413yes—
106C22×C4×C47C4⋊C413yes—
107C22×C4×C47C4⋊C416yes—
108C22×C4×C47C4⋊C413yes—
109C22×C4×C47C4⋊C413yes—
110C22×C4×C47C4⋊C413yes—
111C22×C4×C47C4⋊C413yes—
112C22×C4×C47C4⋊C413yes—
113C22×C4×C47C4⋊C413yes—
114C22×C4×C47C4⋊C416yes—
115C22×C4×C47C4⋊C416yes—
116C22×C4×C47C4⋊C411yescharacteristic
117C22×C4×C47C4⋊C411yescharacteristic
118C22×C4×C47C22×C413yes—
119C22×C4×C47C22×C413yes—
120C22×C4×C47C22×C413yes—
121C22×C4×C47C22×C413yes—
122C22×C4×C47C22×C413yes—
123C22×C4×C47C2×D413yes—
124C22×C4×C47C2×D413yes—
125C22×C4×C47C2×D413yes—
126C22×C4×C47C2×D413yes—
127C22×C4×C47C2×D413yes—
128C22×C4×C47C2×D413yes—
129C22×C4×C47C2×Q813yes—
130C22×C4×C47C2×Q813yes—
131C22×C4×C47C2411yescharacteristic
132C4⋊C4×C47—212no—
133C4⋊C4×C47—24no—
134C22⋊C4×C47—212no—
135C22⋊C4×C47C22×C418yes—
136C42×C47—212no—
137C42×C47—24no—
138#10625329—212no—
139#10625329—212no—
140#10625329—212no—
141#10625329—212no—
142#10625329—24no—
143#10625329—24no—
144#10625329—24no—
145#10625329C22×C418yescomplemented
146C2×C47⋊C8—212no—
147C2×C47⋊C8—212no—
148C2×C47⋊C8—212no—
149C2×C47⋊C8—212no—
150C2×C47⋊C8—24no—
151C2×C47⋊C8C22×C418yescomplemented
152C23×C47—24no—
153C23×C47—24no—
154C23×C47—24no—
155C23×C47—46no—
156C23×C47—46no—
157C23×C47—46no—
158C23×C47—46no—
159C23×C47#9513yes—
160C23×C47#9513yes—
161C23×C47#9513yes—
162C23×C47#9516yes—
163C23×C47#9511yescharacteristic
164C23×C47C2×C4213yes—
165C23×C47C2×C4213yes—
166C23×C47C2×C22⋊C413yes—
167C23×C47C2×C22⋊C413yes—
168C23×C47C2×C4⋊C413yes—
169C23×C47C2×C4⋊C413yes—
170C23×C47C23×C411yescharacteristic
171C2×C4×C47—28no—
172C2×C4×C47—424no—
173C2×C4×C47#9516yes—
174C2×C4×C47#9516yes—
175C2×C4×C47#9516yes—
176C2×C4×C47#9516yes—
177C2×C4×C47#9516yes—
178C2×C4×C47#9516yes—
179C2×C4×C47#9513yes—
180C2×C4×C47#9513yes—
181C2×C4×C47#9513yes—
182C2×C4×C47#9513yes—
183C2×C4×C47C2×C4213yes—
184C2×C4×C47C2×C4213yes—
185C2×C4×C47C2×C22⋊C413yes—
186C2×C4×C47C2×C22⋊C413yes—
187C2×C4×C47C2×C22⋊C413yes—
188C2×C4×C47C2×C22⋊C413yes—
189C2×C4×C47C2×C22⋊C413yes—
190C2×C4×C47C2×C22⋊C413yes—
191C2×C4×C47C2×C22⋊C413yes—
192C2×C4×C47C2×C22⋊C413yes—
193C2×C4×C47C2×C22⋊C413yes—
194C2×C4×C47C2×C22⋊C413yes—
195C2×C4×C47C2×C4⋊C413yes—
196C2×C4×C47C2×C4⋊C413yes—
197C2×C4×C47C2×C4⋊C413yes—
198C2×C4×C47C2×C4⋊C413yes—
199C2×C4×C47C2×C4⋊C413yes—
200C2×C4×C47C2×C4⋊C413yes—
201C2×C4×C47C2×C4⋊C413yes—
202C2×C4×C47C2×C4⋊C413yes—
203C2×C4×C47C2×C4⋊C413yes—
204C2×C4×C47C2×C4⋊C413yes—
205C2×C4×C47C23×C411yescharacteristic
206C2×C4×C47C23×C411yescharacteristic
207C2×C4×C47C22×D411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C22×Q811yescharacteristic
211C47⋊C8—28no—
212C47⋊C8—28no—
213C22×#344—471nomaximal, Sylow
214C22×C47—26no—
215C22×C47—26no—
216C22×C47—26no—
217C22×C47—26no—
218C22×C47—26no—
219C22×C47—26no—
220C22×C47—46no—
221C22×C47—43no—
222C22×C47—43no—
223C22×C47—43no—
224C22×C47—41no—
225C22×C47#34414yescomplemented
226C22×C47C2×#9513yes—
227C22×C47C2×#9513yes—
228C22×C47C2×#9513yes—
229C22×C47C2×#9513yes—
230C22×C47C22×C4211yescharacteristic
231C22×C47C22×C22⋊C411yescharacteristic
232C22×C47C22×C4⋊C411yescharacteristic
233C4×C47—48no—
234C4×C47C2×#9513yes—
235C4×C47C2×#9513yes—
236C4×C47C2×#9513yes—
237C4×C47C2×#9513yes—
238C4×C47C22×C22⋊C411yescharacteristic
239C4×C47C22×C22⋊C411yescharacteristic
240C4×C47C22×C4⋊C411yescharacteristic
241C4×C47C22×C4⋊C411yescharacteristic
242C23×M16—471no—
243C23×M16—471no—
244C22×#117D4711yescharacteristic
245C2×#344—4712no—
246C2×C47—21no—
247C2×C47—21no—
248C2×C47—23no—
249C2×C47—23no—
250C2×C47—23no—
251C2×C47—21no—
252C2×C47C2×#34416yescomplemented
253C2×C47C22×#9511yescharacteristic
254C24×C4C2×D4711yescharacteristic
255C22×M16—476no—
256C22×M16—476no—
257C22×M16—472no—
258C22×M16—476no—
259C22×M16—476no—
260C22×M16—943no—
261C22×M16—941no—
262C22×M16—943no—
263C22×M16—943no—
264C22×M16—941no—
265C22×M16—943no—
266C22×M16—941no—
267C23×C8—472no—
268C23×C8—941no—
269C2×#117—23no—
270C2×#117—23no—
271C2×#117Dic4716yes—
272C2×#117C2×D4716yes—
273C22×C4⋊C4—21no—
274C22×C22⋊C4Dic4712yes—
275C22×C42—21no—
276#344—4716no—
277C47C22×#34411yescharacteristic, complemented, Sylow
278C25C2×Dic4711yescharacteristic
279C23×C4—22no—
280C23×C4Dic9411yescharacteristic
281C23×C4Dic9413yes—
282C23×C4C4×D4711yescharacteristic
283C23×C4C4×D4713yes—
284C23×C4C4×D4713yes—
285C23×C4C4×D4711yescharacteristic
286C23×C4D18813yes—
287C23×C4D18811yescharacteristic
288C23×C4C2×Dic4713yes—
289C23×C4C22⋊D4711yescharacteristic
290C23×C4C22⋊D4713yes—
291C23×C4C22⋊D4713yes—
292C23×C4C22⋊D4711yescharacteristic
293C23×C4C22×D4713yes—
294C2×M16—4712no—
295C2×M16—474no—
296C2×M16—4712no—
297C2×M16—4712no—
298C2×M16—474no—
299C2×M16—9412no—
300C2×M16—946no—
301C2×M16—946no—
302C2×M16—946no—
303C2×M16—946no—
304C2×M16—9412no—
305C2×M16—946no—
306C2×M16—946no—
307C2×M16—946no—
308C2×M16—946no—
309C2×M16—946no—
310C2×M16—946no—
311C2×M16—946no—
312C22×C8—4712no—
313C22×C8—946no—
314C22×C8—942no—
315C22×C8—946no—
316C22×C8—942no—
317C22×C8—946no—
318#117—212no—
319#117—26no—
320#117—26no—
321#117C2×Dic47112yes—
322#117C22×D4714yes—
323C2×C4⋊C4—26no—
324C2×C4⋊C4—26no—
325C2×C22⋊C4—26no—
326C2×C22⋊C4C2×Dic47112yes—
327C2×C42—26no—
328C2×C42—26no—
329C24—44no—
330C24C4×Dic4711yescharacteristic
331C24C4×Dic4713yes—
332C24C4⋊Dic4711yescharacteristic
333C24C4⋊Dic4713yes—
334C24C22⋊Dic4711yescharacteristic
335C24C22⋊Dic4713yes—
336C24C22×Dic4713yes—
337C22×C4—212no—
338C22×C4—48no—
339C22×C4C4×Dic4713yes—
340C22×C4C4×Dic4716yes—
341C22×C4C4×Dic4713yes—
342C22×C4Dic47⋊C413yes—
343C22×C4Dic47⋊C413yes—
344C22×C4Dic47⋊C413yes—
345C22×C4Dic47⋊C413yes—
346C22×C4Dic47⋊C416yes—
347C22×C4Dic47⋊C416yes—
348C22×C4Dic47⋊C411yescharacteristic
349C22×C4Dic47⋊C411yescharacteristic
350C22×C4Dic47⋊C413yes—
351C22×C4Dic47⋊C413yes—
352C22×C4C4⋊Dic4713yes—
353C22×C4C4⋊Dic4716yes—
354C22×C4C4⋊Dic4713yes—
355C22×C4(C2×D47)⋊C411yescharacteristic
356C22×C4(C2×D47)⋊C413yes—
357C22×C4(C2×D47)⋊C413yes—
358C22×C4(C2×D47)⋊C413yes—
359C22×C4(C2×D47)⋊C413yes—
360C22×C4(C2×D47)⋊C416yes—
361C22×C4(C2×D47)⋊C416yes—
362C22×C4(C2×D47)⋊C413yes—
363C22×C4(C2×D47)⋊C413yes—
364C22×C4(C2×D47)⋊C411yescharacteristic
365C22×C4C22⋊Dic4713yes—
366C22×C4C22⋊Dic4713yes—
367C22×C4C22⋊Dic4716yes—
368C22×C4C2×Dic9413yes—
369C22×C4C2×Dic9413yes—
370C22×C4C2×C4×D4713yes—
371C22×C4C2×C4×D4713yes—
372C22×C4C2×C4×D4713yes—
373C22×C4C2×C4×D4713yes—
374C22×C4C2×D18813yes—
375C22×C4C2×D18813yes—
376C22×C4C22×Dic4713yes—
377C22×C4C2×C22⋊D4713yes—
378C22×C4C2×C22⋊D4713yes—
379C22×C4C2×C22⋊D4713yes—
380C22×C4C2×C22⋊D4713yes—
381C22×C4C23×D4711yescharacteristic
382M16—478no—
383M16—9412no—
384M16—9412no—
385M16—9412no—
386M16—9412no—
387M16—944no—
388M16—944no—
389M16—944no—
390C2×C8—478no—
391C2×C8—9412no—
392C2×C8—9412no—
393C2×C8—9412no—
394C2×C8—9412no—
395C2×C8—944no—
396C4⋊C4—212no—
397C4⋊C4—24no—
398C22⋊C4—212no—
399C22⋊C4C22×Dic4718yes—
400C42—212no—
401C42—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—46no—
406C23—46no—
407C23—46no—
408C23—46no—
409C23C47⋊#9513yes—
410C23C47⋊#9513yes—
411C23C47⋊#9513yes—
412C23C47⋊#9516yes—
413C23C47⋊#9511yescharacteristic
414C23C2×C4×Dic4713yes—
415C23C2×C4×Dic4713yes—
416C23C2×C22⋊Dic4713yes—
417C23C2×C22⋊Dic4713yes—
418C23C2×C4⋊Dic4713yes—
419C23C2×C4⋊Dic4713yes—
420C23C23×Dic4711yescharacteristic
421C2×C4—28no—
422C2×C4—424no—
423C2×C4C47⋊#9516yes—
424C2×C4C47⋊#9516yes—
425C2×C4C47⋊#9516yes—
426C2×C4C47⋊#9516yes—
427C2×C4C47⋊#9516yes—
428C2×C4C47⋊#9516yes—
429C2×C4C47⋊#9513yes—
430C2×C4C47⋊#9513yes—
431C2×C4C47⋊#9513yes—
432C2×C4C47⋊#9513yes—
433C2×C4C2×C4×Dic4713yes—
434C2×C4C2×C4×Dic4713yes—
435C2×C4C2×(C2×D47)⋊C413yes—
436C2×C4C2×(C2×D47)⋊C413yes—
437C2×C4C2×(C2×D47)⋊C413yes—
438C2×C4C2×(C2×D47)⋊C413yes—
439C2×C4C2×(C2×D47)⋊C413yes—
440C2×C4C2×(C2×D47)⋊C413yes—
441C2×C4C2×(C2×D47)⋊C413yes—
442C2×C4C2×(C2×D47)⋊C413yes—
443C2×C4C2×C22⋊Dic4713yes—
444C2×C4C2×C22⋊Dic4713yes—
445C2×C4C2×Dic47⋊C413yes—
446C2×C4C2×Dic47⋊C413yes—
447C2×C4C2×Dic47⋊C413yes—
448C2×C4C2×Dic47⋊C413yes—
449C2×C4C2×Dic47⋊C413yes—
450C2×C4C2×Dic47⋊C413yes—
451C2×C4C2×Dic47⋊C413yes—
452C2×C4C2×Dic47⋊C413yes—
453C2×C4C2×C4⋊Dic4713yes—
454C2×C4C2×C4⋊Dic4713yes—
455C2×C4C22×C4×D4711yescharacteristic
456C2×C4C22×C4×D4711yescharacteristic
457C2×C4C22×C22⋊D4711yescharacteristic
458C2×C4C22×C22⋊D4711yescharacteristic
459C2×C4C22×D18811yescharacteristic
460C2×C4C22×Dic9411yescharacteristic
461C8—948no—
462C8—948no—
463C22—26no—
464C22—26no—
465C22—26no—
466C22—26no—
467C22—26no—
468C22—26no—
469C22—46no—
470C22—43no—
471C22—43no—
472C22—43no—
473C22—41no—
474C22#1568889314yesdirect product factor
475C22C2×C47⋊#9513yes—
476C22C2×C47⋊#9513yes—
477C22C2×C47⋊#9513yes—
478C22C2×C47⋊#9513yes—
479C22C22×C4×Dic4711yescharacteristic
480C22C22×C22⋊Dic4711yescharacteristic
481C22C22×C4⋊Dic4711yescharacteristic
482C4—48no—
483C4C2×C47⋊#9513yes—
484C4C2×C47⋊#9513yes—
485C4C2×C47⋊#9513yes—
486C4C2×C47⋊#9513yes—
487C4C22×(C2×D47)⋊C411yescharacteristic
488C4C22×(C2×D47)⋊C411yescharacteristic
489C4C22×Dic47⋊C411yescharacteristic
490C4C22×Dic47⋊C411yescharacteristic
491C2—21no—
492C2—23no—
493C2—23no—
494C2—23no—
495C2—21no—
496C2—21no—
497C2C2×#1568889316yesdirect product factor
498C2C22×C47⋊#9511yescharacteristic
subgroupG/H
centerC22×C4(C2×D47)⋊C4
commutatorC22×C47C22×C42
FrattiniC22×C4C23×D47
FittingC22×#117×C47C2
socleC23×C47#95
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenterC22×#117D47
hypocenterC47C22×#344
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#15688893C2direct (N × H)
12. C22×#10625329C4semidirect
13. C22×#10625329C4semidirect
14. C22×#10625329C4semidirect
18. C23×C47⋊C8C4semidirect
19. #15688893C22direct (N × H)
41. C2×#10625329C2×C4semidirect
42. C2×#10625329C2×C4semidirect
43. C2×#10625329C2×C4semidirect
51. C22×C47⋊C8C2×C4semidirect
145. #10625329C22×C4semidirect
151. C2×C47⋊C8C22×C4semidirect
225. C22×C47#344semidirect
252. C2×C47C2×#344semidirect
277. C47C22×#344semidirect
474. C22#15688893direct (N × H)
497. C2C2×#15688893direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253292
C2×#156888932
C22×#117×C472
C22×#34447