Finite Group Databasebeta

← C2×#20289999 #28314151

Subgroups — C2×#20289999 #28314151

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811643C211yesmaximal, characteristic, complemented
2C2×#16811639C211yesmaximal, characteristic, complemented
3#20289999C218yesmaximal, direct product factor
4C2×#16811468C212yesmaximal, complemented
5C2×#16811467C212yesmaximal
6C22×#13133212C211yesmaximal, characteristic
7#16811643—22no—
8#16811643C2214yescomplemented
9#16811639—22no—
10#16811639C2214yescomplemented
11C2×#13133284—21no—
12C2×C4⋊#88324—21no—
13C2×C22⋊#88324C2212yes—
14C2×C22⋊#88324C2212yes—
15C2×#13133272—21no—
16C2×C4×#88324—21no—
17#16811468C2218yescomplemented
18#16811467C2218yes—
19C2×#13133212—21no—
20C2×#13133212—21no—
21C2×#13133212—21no—
22C2×#13133212—21no—
23C2×#13133212—22no—
24C2×#13133212—22no—
25C2×#13133212C414yes—
26C2×#13133212C2214yes—
27C22×C4⋊Dic53C2211yescharacteristic
28C22×#11719919C411yescharacteristic
29C22×#11719921C411yescharacteristic
30C22×#11719921C411yescharacteristic
31C22×#11719921C2211yescharacteristic
32C22×C4×Dic53C2211yescharacteristic
33C22×C4⋊C4×C53C411yescharacteristic
34C22×#88324—22no—
35C22×#88324—22no—
36#13133284—24no—
37C4⋊#88324—24no—
38C22⋊#88324—22no—
39C22⋊#88324—22no—
40C22⋊#88324C2314yes—
41C22⋊#88324C2314yes—
42#13133272—24no—
43C4×#88324—24no—
44C24×D53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2311yescharacteristic
48#13133212—24no—
49#13133212—24no—
50#13133212—24no—
51#13133212—24no—
52#13133212—24no—
53#13133212—24no—
54#13133212—24no—
55#13133212—24no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53C2×C412yes—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—21no—
63C2×#11719919—21no—
64C2×#11719919—22no—
65C2×#11719919C2×C414yes—
66C2×#11719921—22no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×#11719921—21no—
70C2×#11719921—21no—
71C2×#11719921—22no—
72C2×#11719921—21no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921C2×C414yes—
77C2×#11719921C2×C414yes—
78C2×#11719921C2×C412yes—
79C2×#11719921C2312yes—
80C2×C4×Dic53C2×C412yes—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D412yes—
86C2×C4×Dic53D412yes—
87C2×C4×Dic53C2312yes—
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53C2×C414yes—
95C24×C53C22×C411yescharacteristic
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—24no—
99C23×D53—42no—
100C23×D53—42no—
101C23×D53C22×C412yes—
102C22×C4×C53—24no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—24no—
108C22×C4×C53—42no—
109C22×C4×C53C22⋊C412yes—
110C22×C4×C53C22⋊C411yescharacteristic
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C22×C412yes—
113C22×C4×C53C22×C412yes—
114C22×C4×C53C22×C412yes—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C22×Dic53—42no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53C22⋊C411yescharacteristic
121C22×Dic53C22⋊C411yescharacteristic
122C22×Dic53C22⋊C412yes—
123C22×Dic53C22×C412yes—
124C22×Dic53C22×C411yescharacteristic
125C22×Dic53C22×C412yes—
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2411yescharacteristic
131C2×#88324—24no—
132C2×#88324—24no—
133C2×#88324—44no—
134C2×#88324—44no—
135C4⋊C4×C53—24no—
136C4⋊C4×C53—24no—
137#11719921—24no—
138#11719921—24no—
139#11719921—24no—
140#11719921—24no—
141#11719921—22no—
142#11719921—22no—
143#11719921—22no—
144#11719921—22no—
145C4⋊Dic53—22no—
146C4⋊Dic53—22no—
147C4⋊Dic53—22no—
148C4⋊Dic53—22no—
149#11719919—24no—
150#11719919—24no—
151C4×Dic53C22⋊C412yes—
152C4×Dic53C22⋊C412yes—
153C4×Dic53C22⋊C412yes—
154C4×Dic53C22⋊C412yes—
155C4×Dic53C2×D412yes—
156C4×Dic53C2×D412yes—
157C4×Dic53C2×D412yes—
158C4×Dic53C2×D412yes—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53C2×C22⋊C411yescharacteristic
162C23×C53C2×C22⋊C411yescharacteristic
163C23×C53#13114yes—
164C23×C53C23×C411yescharacteristic
165C22×D53—22no—
166C22×D53—24no—
167C22×D53—42no—
168C22×D53—42no—
169C22×D53—42no—
170C22×D53—42no—
171C22×D53—42no—
172C22×D53—42no—
173C22×D53—42no—
174C22×D53—42no—
175C22×D53—88no—
176C2×C4×C53—24no—
177C2×C4×C53—24no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—22no—
182C2×C4×C53—24no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53C2×C22⋊C412yes—
189C2×C4×C53C2×C22⋊C412yes—
190C2×Dic53—22no—
191C2×Dic53—22no—
192C2×Dic53—22no—
193C2×Dic53—22no—
194C2×Dic53—22no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—42no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53—42no—
204C2×Dic53C2×C22⋊C412yes—
205C2×Dic53C2×C22⋊C412yes—
206C2×Dic53C2×C22⋊C411yescharacteristic
207C2×Dic53C2×C22⋊C411yescharacteristic
208C2×Dic53C22×D411yescharacteristic
209C2×Dic53C22×D411yescharacteristic
210#88324—44no—
211#88324—44no—
212C2×#1473—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—28no—
218C22×C53#40812yes—
219C22×C53#40812yes—
220C22×C53#41012yes—
221C22×C53#41312yes—
222C22×C53C22×C22⋊C411yescharacteristic
223C22×C53C2×#13112yes—
224C2×D53—42no—
225C2×D53—42no—
226C2×D53—42no—
227C2×D53—84no—
228C2×D53—84no—
229C2×D53—84no—
230C4×C53—22no—
231C4×C53—22no—
232C4×C53—42no—
233C4×C53—42no—
234Dic53—21no—
235Dic53—21no—
236Dic53—21no—
237Dic53—21no—
238Dic53—22no—
239Dic53—22no—
240Dic53—42no—
241C22×#126—531no—
242C2×#432—531no—
243C2×#431—531no—
244#1473—538no—
245C2×#324—532no—
246C2×#323—532no—
247C2×C53—22no—
248C2×C53—22no—
249C2×C53#147314yescomplemented
250C2×C53C2×#40811yescharacteristic
251C2×C53C2×#41011yescharacteristic
252C2×C53C2×#41311yescharacteristic
253D53—82no—
254C2×#126—534no—
255C2×#126—534no—
256C2×#126—1061no—
257C2×#126—1061no—
258C2×#126—1061no—
259C2×#126—1061no—
260C2×#126—1062no—
261C2×#126—1062no—
262C22×C4⋊C4—531no—
263C22×C4⋊C4—531no—
264C22×C4⋊C4#615711yescharacteristic
265C22×C22⋊C4—531no—
266C22×C22⋊C4—531no—
267C22×C22⋊C4—531no—
268C22×C42—531no—
269#432—534no—
270#432—1062no—
271#431—534no—
272#431—1062no—
273C2×C4⋊C8—1061no—
274C2×C4⋊C8—1061no—
275C2×C22⋊C8—532no—
276C2×C22⋊C8—532no—
277C2×#97—1061no—
278C2×C4×C8—1061no—
279#324—538no—
280#323—538no—
281C53C2×#147311yescharacteristic, complemented, Sylow
282C25—531no—
283C23×C4—531no—
284C23×C4—531no—
285C23×C4—531no—
286C23×C4C2×#615711yescharacteristic
287C23×C4C2×#615711yescharacteristic
288C23×C4C2×#615711yescharacteristic
289C22×C8—1062no—
290C22×C8—1062no—
291#126—1064no—
292#126—1064no—
293#126—1064no—
294#126—1064no—
295#126—1064no—
296#126—1064no—
297#126—1064no—
298#126—1064no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—532no—
303C2×C4⋊C4—534no—
304C2×C4⋊C4—532no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—1061no—
311C2×C4⋊C4—1061no—
312C2×C4⋊C4C2×#615714yes—
313C2×C22⋊C4—532no—
314C2×C22⋊C4—534no—
315C2×C22⋊C4—532no—
316C2×C22⋊C4—534no—
317C2×C22⋊C4—1062no—
318C2×C22⋊C4—1061no—
319C2×C22⋊C4—1061no—
320C2×C22⋊C4—1061no—
321C2×C22⋊C4—1061no—
322C2×C22⋊C4—1061no—
323C2×C22⋊C4—1061no—
324C2×C22⋊C4—1062no—
325C2×C22⋊C4—1061no—
326C2×C22⋊C4—1061no—
327C2×C42—531no—
328C2×C42—531no—
329C2×C42—531no—
330C2×C42—531no—
331C2×C42—532no—
332C2×C42—532no—
333C2×C42—532no—
334C2×C42—532no—
335C4⋊C8—1064no—
336C4⋊C8—1064no—
337C22⋊C8—534no—
338C22⋊C8—534no—
339C22⋊C8—1062no—
340C22⋊C8—1062no—
341#97—1064no—
342C4×C8—1064no—
343C24—532no—
344C24—1064no—
345C24—1061no—
346C24—1061no—
347C24—2122no—
348C24—2122no—
349C24C22×#615711yescharacteristic
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—42no—
357C22×C4—531no—
358C22×C4—531no—
359C22×C4—531no—
360C22×C4—531no—
361C22×C4—532no—
362C22×C4—531no—
363C22×C4—531no—
364C22×C4—531no—
365C22×C4—532no—
366C22×C4—531no—
367C22×C4—532no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—2122no—
371C22×C4—2122no—
372C22×C4—2122no—
373C22×C4C22⋊#615712yes—
374C22×C4C22⋊#615711yescharacteristic
375C22×C4C22⋊#615711yescharacteristic
376C22×C4C22×#615712yes—
377C22×C4C22×#615712yes—
378C22×C4C22×#615712yes—
379C2×C8—1064no—
380C2×C8—1064no—
381C2×C8—2124no—
382C2×C8—2124no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—1064no—
386C4⋊C4—1064no—
387C4⋊C4—1062no—
388C4⋊C4—1062no—
389C4⋊C4—1062no—
390C4⋊C4—1062no—
391C22⋊C4—1064no—
392C22⋊C4—1064no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C22⋊C4—1062no—
396C22⋊C4—1062no—
397C22⋊C4—1064no—
398C22⋊C4—1064no—
399C42—532no—
400C42—532no—
401C42—532no—
402C42—532no—
403C42—532no—
404C42—532no—
405C42—532no—
406C42—532no—
407C23—22no—
408C23—22no—
409C23—1064no—
410C23—1062no—
411C23—4248no—
412C23—2122no—
413C23—2122no—
414C23—2122no—
415C23—2122no—
416C23—2122no—
417C23—2122no—
418C23—2122no—
419C23—2122no—
420C23C2×C22⋊#615711yescharacteristic
421C23C2×C22⋊#615711yescharacteristic
422C23#1313330414yes—
423C23C23×#615711yescharacteristic
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—531no—
437C2×C4—531no—
438C2×C4—532no—
439C2×C4—532no—
440C2×C4—531no—
441C2×C4—531no—
442C2×C4—1062no—
443C2×C4—1062no—
444C2×C4—1062no—
445C2×C4—1062no—
446C2×C4—1062no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4—2122no—
454C2×C4—2122no—
455C2×C4—2122no—
456C2×C4C2×C22⋊#615712yes—
457C2×C4C2×C22⋊#615712yes—
458C8—2124no—
459C8—2124no—
460C22—28no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—4244no—
466C22—4244no—
467C22—4244no—
468C22—2122no—
469C22—2122no—
470C22—2122no—
471C22#1681159612yes—
472C22#1681159612yes—
473C22#1681160412yes—
474C22#1681160912yes—
475C22C22×C22⋊#615711yescharacteristic
476C22C2×#1313330412yes—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—1061no—
482C4—1061no—
483C4—1061no—
484C4—1061no—
485C4—1062no—
486C4—1062no—
487C4—2122no—
488C2—22no—
489C2—22no—
490C2—4242no—
491C2#2028999914yesdirect product factor
492C2C2×#1681159611yescharacteristic
493C2C2×#1681160411yescharacteristic
494C2C2×#1681160911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1473
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16811643C2semidirect
2. C2×#16811639C2semidirect
3. #20289999C2direct (N × H)
4. C2×#16811468C2semidirect
8. #16811643C22semidirect
10. #16811639C22semidirect
17. #16811468C22semidirect
249. C2×C53#1473semidirect
281. C53C2×#1473semidirect
491. C2#20289999direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168116432
C2×#168116392
#202899992
C2×#168114682
C2×#168114672
C22×#131332122
C2×#147353