Finite Group Databasebeta

← C2×#20290623 #28319188

Subgroups — C2×#20290623 #28319188

Order 13568.

1606 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811594C211yesmaximal, characteristic
2C2×C4×C22⋊#6157C211yesmaximal, characteristic
3#20290623C218yesmaximal, direct product factor
4C2×#16811493C212yesmaximal, complemented
5C2×#16811493C212yesmaximal, complemented
6C2×#117×D53C211yesmaximal, characteristic
7C22×#11719936C2212yes—
8C22×C22⋊#6157—21no—
9C22×C4×#6157C2212yes—
10#16811594—22no—
11#16811594C2214yes—
12C2×#13133274—21no—
13C2×C22⋊#88324—21no—
14C4×C22⋊#6157—22no—
15C4×C22⋊#6157C2214yes—
16C2×(C2×#6157)⋊C4—21no—
17#16811493C2218yescomplemented
18#16811493C2218yescomplemented
19C23×C4×D53C2211yescharacteristic
20C2×#13133176C411yescharacteristic, complemented
21C2×#13133175C2211yescharacteristic
22C2×#13133175C2211yescharacteristic
23C2×#13133174—21no—
24C2×#13133173—21no—
25#117×D53—22no—
26#117×D53—22no—
27#117×D53C414yescomplemented
28#117×D53C2214yes—
29C2×C4⋊C4×D53C411yescharacteristic, complemented
30C2×C4⋊C4×D53C411yescharacteristic, complemented
31C2×C22⋊C4×D53—21no—
32C2×C42×D53—21no—
33C2×#117×C53C411yescharacteristic, complemented
34C23×#6157—22no—
35C2×#11719936—28no—
36C2×#11719936—22no—
37C2×#11719936C2314yes—
38C22×#88324—22no—
39C2×C22⋊#6157—28no—
40C2×C22⋊#6157—24no—
41C2×C4×#6157—28no—
42C2×C4×#6157—22no—
43C2×C4×#6157C2314yes—
44#13133274—24no—
45C22⋊#88324—24no—
46(C2×#6157)⋊C4—24no—
47C24×D53D411yescharacteristic
48C23×Dic53D411yescharacteristic
49C22×C4×D53—28no—
50C22×C4×D53—21no—
51C22×C4×D53—21no—
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C412yes—
54C22×C4×D53D411yescharacteristic
55C22×C4×D53D412yes—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D412yes—
58C22×C4×D53C2312yes—
59C22×C4×D53C2311yescharacteristic
60#13133176—22no—
61#13133176C2×C414yescomplemented
62#13133175—21no—
63#13133175—21no—
64#13133175—21no—
65#13133175—21no—
66#13133175C2×C412yes—
67#13133175C2×C412yes—
68#13133175C2312yes—
69#13133175C2312yes—
70#13133174—24no—
71#13133174—24no—
72#13133173—24no—
73#13133173—24no—
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×C4⋊Dic53C2×C411yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53C2×C414yescomplemented
81C4⋊C4×D53C2×C414yescomplemented
82C2×C22⋊Dic53—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C22⋊C4×D53—24no—
86C22⋊C4×D53—24no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C42×D53—24no—
91C42×D53—24no—
92C23×C4×C53C2×C411yescharacteristic
93#117×C53—22no—
94#117×C53C2×C414yescomplemented
95C2×C4⋊C4×C53C2×C411yescharacteristic
96C2×C4⋊C4×C53C2×C411yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C42×C53—21no—
99C24×C53C22⋊C411yescharacteristic
100C23×D53—24no—
101C23×D53—24no—
102C23×D53—42no—
103C23×D53C22⋊C412yes—
104C23×D53C22⋊C411yescharacteristic
105C23×D53C2×D412yes—
106C23×D53C2×D411yescharacteristic
107C22×#6157—28no—
108C22×#6157—24no—
109C22×#6157—44no—
110C22×#6157—44no—
111C22×C4×C53—24no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53C22⋊C411yescharacteristic
115C22×C4×C53C22⋊C412yes—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C22×C411yescharacteristic
118C22×Dic53—24no—
119C22×Dic53—21no—
120C22×Dic53—21no—
121C22×Dic53C22⋊C411yescharacteristic
122C22×Dic53C22⋊C412yes—
123C22×Dic53C2×D412yes—
124C22×Dic53C2×D411yescharacteristic
125C2×C4×D53—24no—
126C2×C4×D53—24no—
127C2×C4×D53—24no—
128C2×C4×D53—28no—
129C2×C4×D53—28no—
130C2×C4×D53—28no—
131C2×C4×D53—28no—
132C2×C4×D53—24no—
133C2×C4×D53—24no—
134C2×C4×D53—44no—
135C2×C4×D53—44no—
136C2×C4×D53C22⋊C412yes—
137C2×C4×D53C22⋊C412yes—
138C2×C4×D53C22⋊C412yes—
139C2×C4×D53C22×C412yes—
140C2×C4×D53C22×C411yescharacteristic
141C2×C4×D53C2×D412yes—
142C2×C4×D53C2×D412yes—
143C2×C4×D53C2×D411yescharacteristic
144C2×C4×D53C2×D411yescharacteristic
145C2×C4×D53#4118yes—
146C2×C4×D53C2411yescharacteristic
147C22⋊#6157—216no—
148C2×#88324—24no—
149C2×#88324—44no—
150C4×#6157—28no—
151C4×#6157—44no—
152#11719936—28no—
153#11719936—44no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C22×C412yes—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—24no—
159C42×C53—24no—
160C22⋊Dic53—24no—
161#11719921—22no—
162#11719921—22no—
163#11719921—22no—
164#11719921—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53C22×C412yes—
168C4⋊Dic53C22×C412yes—
169#11719919—24no—
170#11719919—24no—
171C4×Dic53—24no—
172C4×Dic53—22no—
173C4×Dic53—22no—
174C4×Dic53—22no—
175C4×Dic53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53C2×C22⋊C412yes—
180C23×C53C2×C22⋊C411yescharacteristic
181C22×D53—22no—
182C22×D53—22no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—24no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—416no—
192C22×D53#10414yes—
193C22×D53C2×C22⋊C411yescharacteristic
194C22×D53C2×C22⋊C412yes—
195C22×D53#12314yes—
196C22×D53C22×D411yescharacteristic
197C2×#6157—28no—
198C2×#6157—416no—
199C2×#6157—44no—
200C2×#6157—44no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53C2×C22⋊C412yes—
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53#11714yes—
211C2×C4×C53C23×C411yescharacteristic
212C2×Dic53—22no—
213C2×Dic53—24no—
214C2×Dic53—24no—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53C2×C22⋊C412yes—
220C2×Dic53C2×C22⋊C411yescharacteristic
221C2×Dic53#12314yes—
222C2×Dic53C22×D411yescharacteristic
223C4×D53—24no—
224C4×D53—24no—
225C4×D53—24no—
226C4×D53—24no—
227C4×D53—24no—
228C4×D53—24no—
229C4×D53—48no—
230C4×D53—48no—
231C4×D53#11714yes—
232C4×D53#12314yes—
233C4×D53#12314yes—
234C4×D53C2×#4114yes—
235#88324—44no—
236C2×#1826—531nomaximal, Sylow
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—44no—
241C22×C53#38112yes—
242C22×C53C2×#10412yes—
243C22×C53#42112yes—
244C22×C53C22×C22⋊C411yescharacteristic
245C2×D53—28no—
246C2×D53—28no—
247C2×D53—21no—
248C2×D53—21no—
249C2×D53—22no—
250C2×D53—21no—
251C2×D53—21no—
252C2×D53—22no—
253C2×D53—48no—
254C2×D53—48no—
255C2×D53#38112yes—
256C2×D53C2×#10414yes—
257C2×D53C2×#12312yes—
258#6157—48no—
259C4×C53—22no—
260C4×C53—42no—
261C4×C53—42no—
262C4×C53#38112yes—
263C4×C53C2×#11712yes—
264Dic53—22no—
265Dic53—42no—
266Dic53—42no—
267Dic53#38112yes—
268Dic53C2×#12312yes—
269C22×#117—531no—
270C2×#407—531no—
271C2×C4×C22⋊C4—531no—
272#1826—538no—
273C2×#339—532no—
274C2×#339—532no—
275C2×C53—42no—
276C2×C53#182614yescomplemented
277C2×C53C2×#38111yescharacteristic
278C2×C53C22×#10411yescharacteristic
279C2×C53C2×#42111yescharacteristic
280D53—22no—
281D53—22no—
282D53—24no—
283C24×C4—531no—
284C22×M16—532no—
285C2×#117—531no—
286C2×#117—534no—
287C2×#117—534no—
288C2×#117—531no—
289C2×#117—531no—
290C2×#117—1061no—
291C2×#117—1061no—
292C2×#117—1062no—
293C2×#117—1062no—
294C2×#117#615711yescharacteristic, complemented
295C22×C4⋊C4—531no—
296C22×C4⋊C4—531no—
297C22×C22⋊C4—1061no—
298C22×C22⋊C4—1061no—
299C22×C42—532no—
300C22×C42—1061no—
301#407—534no—
302#407—1062no—
303C2×C22⋊C8—1061no—
304C2×C22⋊C8—1061no—
305C4×C22⋊C4—534no—
306C4×C22⋊C4—1062no—
307C2×#95—1061no—
308#339—538no—
309#339—538no—
310C53C2×#182611yescharacteristic, complemented, Sylow
311C25—531no—
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—532no—
316C23×C4—532no—
317C23×C4—531no—
318C23×C4—531no—
319C23×C4—532no—
320C23×C4—532no—
321C23×C4—1068no—
322C23×C4—1061no—
323C23×C4—1061no—
324C23×C4—1062no—
325C23×C4C2×#615711yescharacteristic
326C2×M16—534no—
327C2×M16—1062no—
328C2×M16—1068no—
329C22×C8—1062no—
330#117—22no—
331#117—534no—
332#117—532no—
333#117—532no—
334#117—532no—
335#117—532no—
336#117—1064no—
337#117—1064no—
338#117—1064no—
339#117—1064no—
340#117—1061no—
341#117—1061no—
342#117—1061no—
343#117—1061no—
344#117—1062no—
345#117C2×#615714yescomplemented
346C2×C4⋊C4—531no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—534no—
349C2×C4⋊C4—534no—
350C2×C4⋊C4—1062no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4C2×#615711yescharacteristic
355C2×C4⋊C4C2×#615711yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—1064no—
358C2×C22⋊C4—1064no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1064no—
362C2×C22⋊C4—1068no—
363C2×C22⋊C4—1061no—
364C2×C42—21no—
365C2×C42—534no—
366C2×C42—1064no—
367C2×C42—1064no—
368C2×C42—1061no—
369C2×C42—1061no—
370C2×C42—1061no—
371C2×C42—1068no—
372C2×C42—1062no—
373C22⋊C8—1064no—
374C22⋊C8—1064no—
375#95—1064no—
376C24—531no—
377C24—531no—
378C24—532no—
379C24—532no—
380C24—1064no—
381C24—1064no—
382C24—2122no—
383C24C22⋊#615711yescharacteristic
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—538no—
391C22×C4—531no—
392C22×C4—531no—
393C22×C4—532no—
394C22×C4—532no—
395C22×C4—531no—
396C22×C4—532no—
397C22×C4—532no—
398C22×C4—532no—
399C22×C4—532no—
400C22×C4—532no—
401C22×C4—532no—
402C22×C4—1064no—
403C22×C4—1068no—
404C22×C4—1068no—
405C22×C4—1068no—
406C22×C4—1068no—
407C22×C4—1064no—
408C22×C4—1064no—
409C22×C4—1064no—
410C22×C4—1064no—
411C22×C4—1064no—
412C22×C4—1064no—
413C22×C4—1068no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4—2124no—
417C22×C4—2124no—
418C22×C4—2124no—
419C22×C4—2124no—
420C22×C4C22⋊#615712yes—
421C22×C4C22⋊#615711yescharacteristic
422C22×C4C22×#615712yes—
423C22×C4C22×#615711yescharacteristic
424M16—1068no—
425M16—2124no—
426C2×C8—1064no—
427C2×C8—2124no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—532no—
431C4⋊C4—532no—
432C4⋊C4—1064no—
433C4⋊C4—1064no—
434C4⋊C4—1061no—
435C4⋊C4—1061no—
436C4⋊C4C22×#615712yes—
437C4⋊C4C22×#615712yes—
438C22⋊C4—24no—
439C22⋊C4—1062no—
440C22⋊C4—1062no—
441C22⋊C4—1062no—
442C22⋊C4—1062no—
443C22⋊C4—10616no—
444C22⋊C4—1064no—
445C42—24no—
446C42—1062no—
447C42—1062no—
448C42—1062no—
449C42—1062no—
450C42—1068no—
451C42—1064no—
452C42—2124no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—534no—
457C23—534no—
458C23—531no—
459C23—532no—
460C23—531no—
461C23—1064no—
462C23—1064no—
463C23—1064no—
464C23—1064no—
465C23—1062no—
466C23—1062no—
467C23—1062no—
468C23—1062no—
469C23—2122no—
470C23—2122no—
471C23—21216no—
472C23C2×C22⋊#615712yes—
473C23C2×C22⋊#615711yescharacteristic
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—534no—
482C2×C4—531no—
483C2×C4—531no—
484C2×C4—534no—
485C2×C4—534no—
486C2×C4—534no—
487C2×C4—534no—
488C2×C4—532no—
489C2×C4—1064no—
490C2×C4—1064no—
491C2×C4—1064no—
492C2×C4—1064no—
493C2×C4—1064no—
494C2×C4—1064no—
495C2×C4—1062no—
496C2×C4—1064no—
497C2×C4—1064no—
498C2×C4—1068no—
499C2×C4—1062no—
500C2×C4—1062no—
501C2×C4—2128no—
502C2×C4—2128no—
503C2×C4—2124no—
504C2×C4—2124no—
505C2×C4—21216no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4C2×C22⋊#615712yes—
509C2×C4C2×C22⋊#615711yescharacteristic
510C2×C4#1313329614yes—
511C2×C4C23×#615711yescharacteristic
512C8—2124no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—44no—
517C22—534no—
518C22—532no—
519C22—532no—
520C22—1061no—
521C22—1061no—
522C22—1061no—
523C22—1062no—
524C22—1061no—
525C22—1068no—
526C22—1068no—
527C22—1062no—
528C22—2128no—
529C22—2128no—
530C22#1681156612yes—
531C22D53⋊#10412yes—
532C22#1681162012yes—
533C22C22×C22⋊#615711yescharacteristic
534C4—22no—
535C4—42no—
536C4—42no—
537C4—532no—
538C4—532no—
539C4—1062no—
540C4—2128no—
541C4—2122no—
542C4—2122no—
543C4#1681156612yes—
544C4C2×#1313329612yes—
545C2—42no—
546C2—1062no—
547C2—1064no—
548C2—1062no—
549C2#2029062314yesdirect product factor
550C2C2×#1681156611yescharacteristic
551C2C2×D53⋊#10411yescharacteristic
552C2C2×#1681162011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC2×C4×D53C24
hypercenterC2×#117#6157
hypocenterC53C2×#1826
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #20290623C2direct (N × H)
4. C2×#16811493C2semidirect
5. C2×#16811493C2semidirect
17. #16811493C22semidirect
18. #16811493C22semidirect
20. C2×#13133176C4semidirect
27. #117×D53C4semidirect
29. C2×C4⋊C4×D53C4semidirect
30. C2×C4⋊C4×D53C4semidirect
33. C2×#117×C53C4semidirect
61. #13133176C2×C4semidirect
80. C4⋊C4×D53C2×C4semidirect
81. C4⋊C4×D53C2×C4semidirect
94. #117×C53C2×C4semidirect
276. C2×C53#1826semidirect
294. C2×#117#6157semidirect
310. C53C2×#1826semidirect
345. #117C2×#6157semidirect
549. C2#20290623direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115942
C2×C4×C22⋊#61572
#202906232
C2×#168114932
C2×#117×D532
C2×#182653