Finite Group Databasebeta

← C2×#20289994 #28314145

Subgroups — C2×#20289994 #28314145

Order 13568.

1426 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811593C211yesmaximal, characteristic
2C22×C22⋊#88324C211yesmaximal, characteristic
3#20289994C218yesmaximal, direct product factor
4C2×#16811469C212yesmaximal, complemented
5C2×#16811468C212yesmaximal, complemented
6C22×#13133194C211yesmaximal, characteristic
7C22×#11719940—21no—
8C23×#88324—21no—
9#16811593—22no—
10#16811593C2214yes—
11C2×C22⋊#88324—22no—
12C2×C22⋊#88324—21no—
13C2×C22⋊#88324—21no—
14C2×C22⋊#88324C2212yes—
15C2×C22⋊#88324C2212yes—
16C2×C22⋊#88324C2214yes—
17#16811469C2218yescomplemented
18#16811468C2218yescomplemented
19C23×Dic106C411yescharacteristic
20C24×Dic53C2211yescharacteristic
21C2×#13133194—21no—
22C2×#13133194—21no—
23C2×#13133194—21no—
24C2×#13133194—22no—
25C2×#13133194—21no—
26C2×#13133194—22no—
27C2×#13133194C414yes—
28C2×#13133194C2214yes—
29C22×C4⋊Dic53C2211yescharacteristic
30C22×#11719919C411yescharacteristic
31C22×#11719919C2211yescharacteristic
32C22×C22⋊Dic53C411yescharacteristic
33C22×C22⋊C4×C53C411yescharacteristic
34C2×#11719940—24no—
35C2×#11719940—44no—
36C22×#88324—22no—
37C22×#88324—24no—
38C22×#88324—22no—
39C22×#88324—44no—
40C22⋊#88324—22no—
41C22⋊#88324—24no—
42C22⋊#88324—24no—
43C22⋊#88324—22no—
44C22⋊#88324C2314yes—
45C22⋊#88324C2314yes—
46C22×Dic106—21no—
47C22×Dic106—22no—
48C22×Dic106—21no—
49C22×Dic106—44no—
50C22×Dic106C2×C414yes—
51C23×Dic53—44no—
52C23×Dic53C2×C411yescharacteristic
53C23×Dic53C2×C411yescharacteristic
54C23×Dic53C2×C411yescharacteristic
55C23×Dic53C2×C412yes—
56C23×Dic53D411yescharacteristic
57C23×Dic53D411yescharacteristic
58C23×Dic53D412yes—
59C23×Dic53D411yescharacteristic
60C23×Dic53D411yescharacteristic
61C23×Dic53D412yes—
62C23×Dic53C2311yescharacteristic
63C23×Dic53C2312yes—
64#13133194—24no—
65#13133194—24no—
66#13133194—24no—
67#13133194—24no—
68#13133194—24no—
69#13133194—24no—
70#13133194—24no—
71#13133194—24no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2×C412yes—
77C2×C4⋊Dic53C2312yes—
78C2×#11719919—22no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919C2×C414yes—
86C2×#11719919C2×C412yes—
87C2×#11719919C2312yes—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—22no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53C2×C414yes—
92C25×C53C2×C411yescharacteristic
93C23×C4×C53C2×C411yescharacteristic
94C23×C4×C53C2×C411yescharacteristic
95C2×C22⋊C4×C53—21no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53C2×C414yes—
99C24×C53—24no—
100C24×C53—24no—
101C24×C53—42no—
102C24×C53C22⋊C411yescharacteristic
103C24×C53C22⋊C412yes—
104C24×C53C22⋊C411yescharacteristic
105C24×C53C22×C412yes—
106C24×C53C22×C411yescharacteristic
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—24no—
112C22×C4×C53—42no—
113C22×C4×C53C22×C412yes—
114C22×C4×C53C22×C412yes—
115C22×Dic53—24no—
116C22×Dic53—24no—
117C22×Dic53—24no—
118C22×Dic53—21no—
119C22×Dic53—21no—
120C22×Dic53—21no—
121C22×Dic53—21no—
122C22×Dic53—24no—
123C22×Dic53—44no—
124C22×Dic53—44no—
125C22×Dic53—44no—
126C22×Dic53—44no—
127C22×Dic53—42no—
128C22×Dic53—42no—
129C22×Dic53—42no—
130C22×Dic53—42no—
131C22×Dic53C22⋊C411yescharacteristic
132C22×Dic53C22⋊C412yes—
133C22×Dic53C22⋊C412yes—
134C22×Dic53C22⋊C412yes—
135C22×Dic53C22⋊C412yes—
136C22×Dic53C22⋊C412yes—
137C22×Dic53C22⋊C411yescharacteristic
138C22×Dic53C22×C412yes—
139C22×Dic53C22×C412yes—
140C22×Dic53C22×C412yes—
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D412yes—
145C22×Dic53C2×D412yes—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D412yes—
149C22×Dic53C2×D412yes—
150C22×Dic53C2411yescharacteristic
151C2×Dic106—24no—
152C2×Dic106—24no—
153C2×Dic106—44no—
154C2×Dic106—48no—
155C2×Dic106—44no—
156C2×Dic106—48no—
157#11719940—48no—
158C2×#88324—24no—
159C2×#88324—24no—
160C2×#88324—44no—
161C2×#88324—48no—
162C2×#88324—44no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—24no—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171#11719919—24no—
172#11719919—22no—
173#11719919—22no—
174#11719919—24no—
175#11719919—22no—
176#11719919—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—24no—
186C23×C53—22no—
187C23×C53—24no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—416no—
191C23×C53#10114yes—
192C23×C53C2×C22⋊C412yes—
193C23×C53C2×C22⋊C411yescharacteristic
194C23×C53C2×C22⋊C412yes—
195C23×C53C2×C22⋊C411yescharacteristic
196C23×C53#13114yes—
197C23×C53C23×C411yescharacteristic
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×Dic53—22no—
207C2×Dic53—22no—
208C2×Dic53—22no—
209C2×Dic53—24no—
210C2×Dic53—24no—
211C2×Dic53—24no—
212C2×Dic53—24no—
213C2×Dic53—24no—
214C2×Dic53—24no—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—24no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—41no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53#9914yes—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C411yescharacteristic
234C2×Dic53C2×C22⋊C411yescharacteristic
235C2×Dic53C2×C22⋊C412yes—
236C2×Dic53C22×D411yescharacteristic
237C2×Dic53C22×D411yescharacteristic
238Dic106—48no—
239Dic106—48no—
240#88324—44no—
241#88324—44no—
242C2×#1472—531nomaximal, Sylow
243C22×C53—22no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—22no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—28no—
250C22×C53—28no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—48no—
255C22×C53—48no—
256C22×C53—44no—
257C22×C53#40812yes—
258C22×C53#40812yes—
259C22×C53C2×#9912yes—
260C22×C53C2×#10114yes—
261C22×C53C2×#10112yes—
262C22×C53C22×C22⋊C411yescharacteristic
263C22×C53C2×#13112yes—
264C4×C53—42no—
265C4×C53—42no—
266Dic53—21no—
267Dic53—21no—
268Dic53—22no—
269Dic53—21no—
270Dic53—21no—
271Dic53—42no—
272Dic53—42no—
273Dic53C2×#9914yes—
274C22×C22⋊Q8—531no—
275C2×#407—531no—
276C22×C22⋊C8—531no—
277#1472—538no—
278C2×#324—532no—
279C2×#324—532no—
280C2×C53—22no—
281C2×C53—22no—
282C2×C53—24no—
283C2×C53—42no—
284C2×C53#147214yescomplemented
285C2×C53C2×#40811yescharacteristic
286C2×C53C22×#9911yescharacteristic
287C2×C53C22×#10111yescharacteristic
288C23×Q8—531no—
289C24×C4—531no—
290C22×M16—1061no—
291C23×C8—1061no—
292C2×C22⋊Q8—534no—
293C2×C22⋊Q8—534no—
294C2×C22⋊Q8—1061no—
295C2×C22⋊Q8—1061no—
296C2×C22⋊Q8—1062no—
297C2×C22⋊Q8—1061no—
298C2×C22⋊Q8—1061no—
299C2×C22⋊Q8—1062no—
300C22×C4⋊C4—531no—
301C22×C4⋊C4—531no—
302C22×C4⋊C4—531no—
303C22×C22⋊C4—531no—
304C22×C22⋊C4#615711yescharacteristic
305#407—534no—
306#407—1062no—
307C2×C22⋊C8—532no—
308C2×C22⋊C8—534no—
309C2×C22⋊C8—532no—
310C2×C22⋊C8—1062no—
311C2×C22⋊C8—1061no—
312C2×C22⋊C8—1061no—
313#324—538no—
314#324—538no—
315C53C2×#147211yescharacteristic, complemented, Sylow
316C25C2×#615711yescharacteristic
317C22×Q8—534no—
318C22×Q8—1062no—
319C22×Q8—1061no—
320C22×Q8—1061no—
321C22×Q8—2124no—
322C23×C4—531no—
323C23×C4—531no—
324C23×C4—531no—
325C23×C4—532no—
326C23×C4—531no—
327C23×C4—531no—
328C23×C4—532no—
329C23×C4—531no—
330C23×C4—532no—
331C23×C4—532no—
332C23×C4—531no—
333C23×C4—531no—
334C23×C4—2124no—
335C23×C4C2×#615711yescharacteristic
336C23×C4C2×#615711yescharacteristic
337C2×M16—1064no—
338C2×M16—2124no—
339C22×C8—1064no—
340C22×C8—1062no—
341C22×C8—1062no—
342C22×C8—2124no—
343C22⋊Q8—1064no—
344C22⋊Q8—1064no—
345C22⋊Q8—1064no—
346C22⋊Q8—1064no—
347C22⋊Q8—1064no—
348C22⋊Q8—1064no—
349C22⋊Q8—1064no—
350C22⋊Q8—1064no—
351C2×C4⋊C4—532no—
352C2×C4⋊C4—532no—
353C2×C4⋊C4—534no—
354C2×C4⋊C4—532no—
355C2×C4⋊C4—532no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4—1061no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—534no—
371C2×C22⋊C4—1062no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1061no—
374C2×C22⋊C4C2×#615714yes—
375C22⋊C8—534no—
376C22⋊C8—534no—
377C22⋊C8—1062no—
378C22⋊C8—1064no—
379C22⋊C8—1062no—
380C22⋊C8—1064no—
381C24—24no—
382C24—24no—
383C24—42no—
384C24C22⋊#615711yescharacteristic
385C24C22⋊#615712yes—
386C24C22⋊#615711yescharacteristic
387C24C22×#615712yes—
388C24C22×#615711yescharacteristic
389C2×Q8—1064no—
390C2×Q8—1064no—
391C2×Q8—2124no—
392C2×Q8—2128no—
393C2×Q8—2128no—
394C2×Q8—2124no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—42no—
401C22×C4—531no—
402C22×C4—532no—
403C22×C4—532no—
404C22×C4—532no—
405C22×C4—532no—
406C22×C4—532no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—531no—
410C22×C4—532no—
411C22×C4—532no—
412C22×C4—532no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—532no—
416C22×C4—532no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—1064no—
422C22×C4—1064no—
423C22×C4—1064no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1064no—
429C22×C4—2124no—
430C22×C4—2124no—
431C22×C4—2124no—
432C22×C4—2124no—
433C22×C4—2122no—
434C22×C4—2122no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4C22×#615712yes—
438C22×C4C22×#615712yes—
439M16—2128no—
440C2×C8—1064no—
441C2×C8—1064no—
442C2×C8—2128no—
443C2×C8—2124no—
444C2×C8—2124no—
445C4⋊C4—1064no—
446C4⋊C4—1064no—
447C4⋊C4—1062no—
448C4⋊C4—1062no—
449C4⋊C4—1062no—
450C4⋊C4—1062no—
451C4⋊C4—1062no—
452C4⋊C4—1062no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—1064no—
458C22⋊C4—1064no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—416no—
473C23#1313327814yes—
474C23C2×C22⋊#615712yes—
475C23C2×C22⋊#615711yescharacteristic
476C23C2×C22⋊#615712yes—
477C23C2×C22⋊#615711yescharacteristic
478C23#1313330414yes—
479C23C23×#615711yescharacteristic
480Q8—2128no—
481Q8—2128no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—534no—
491C2×C4—532no—
492C2×C4—531no—
493C2×C4—531no—
494C2×C4—532no—
495C2×C4—531no—
496C2×C4—531no—
497C2×C4—1064no—
498C2×C4—1064no—
499C2×C4—1064no—
500C2×C4—1064no—
501C2×C4—1064no—
502C2×C4—1064no—
503C2×C4—1062no—
504C2×C4—1062no—
505C2×C4—1062no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—1064no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4—2121no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C8—2124no—
523C8—2124no—
524C22—28no—
525C22—28no—
526C22—22no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—48no—
536C22—48no—
537C22—44no—
538C22#1681159612yes—
539C22#1681159612yes—
540C22D53⋊#9912yes—
541C22C2×#1313327814yes—
542C22C2×#1313327812yes—
543C22C22×C22⋊#615711yescharacteristic
544C22C2×#1313330412yes—
545C4—42no—
546C4—42no—
547C4—534no—
548C4—1061no—
549C4—1061no—
550C4—1061no—
551C4—1061no—
552C4—1062no—
553C4—2122no—
554C4—2122no—
555C2—22no—
556C2—24no—
557C2—22no—
558C2—42no—
559C2#2028999414yesdirect product factor
560C2C2×#1681159611yescharacteristic
561C2C2×D53⋊#9911yescharacteristic
562C2C22×#1313327811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C22⋊C4#6157
hypocenterC53C2×#1472
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289994C2direct (N × H)
4. C2×#16811469C2semidirect
5. C2×#16811468C2semidirect
17. #16811469C22semidirect
18. #16811468C22semidirect
284. C2×C53#1472semidirect
315. C53C2×#1472semidirect
559. C2#20289994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115932
C22×C22⋊#883242
#202899942
C2×#168114692
C2×#168114682
C22×#131331942
C2×#147253