Finite Group Databasebeta

← C4×#16810518 #27305283

Subgroups — C4×#16810518 #27305283

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810518C212yesmaximal
2C2×C4×#11719919C211yesmaximal, characteristic
3C2×C4×#11719919C211yesmaximal, characteristic
4C2×C4×#11719919C211yesmaximal, characteristic
5C2×C42×Dic53C211yesmaximal, characteristic
6#20272950C212yesmaximal
7#20272947C212yesmaximal
8#20272943C212yesmaximal
9C4×#13133124C211yesmaximal, characteristic
10C4×#13133124C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic
12C22×#11719919C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×#11719919C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810518C4116yesdirect product factor
23C4×#11719919—24no—
24C4×#11719919—24no—
25C4×#11719919C418yescomplemented
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C42×Dic53—24no—
35C22×C4⋊C4×C53C2212yes—
36C22×C42×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C4×C4⋊C4×C53—24no—
40C2×#95×C53C2212yes—
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46C23×Dic53C2311yescharacteristic
47C2×#11719919—24no—
48C2×#11719919—24no—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—24no—
52C2×#11719919C2×C418yes—
53C2×#11719919C2×C418yes—
54C2×#11719919C2×C414yes—
55C2×#11719919C2×C418yes—
56C2×#11719919C2×C414yes—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—22no—
59C2×C4×Dic53—22no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—22no—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53C2×C418yes—
70C2×C4×Dic53C2×C414yes—
71C2×C4×Dic53C2×C414yes—
72C2×C4×Dic53D414yes—
73C2×C4×Dic53Q814yes—
74C2×C4×Dic53Q814yes—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124C2×C418yes—
80#13133124C2×C418yes—
81#13133124C2×C414yes—
82#13133124C2×C414yes—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53—24no—
89C2×C4⋊C4×C53—24no—
90C2×C4⋊C4×C53C2×C418yes—
91C2×C42×C53—22no—
92C2×C42×C53—22no—
93C2×C42×C53—22no—
94C2×C42×C53—22no—
95C2×C42×C53—22no—
96C2×C42×C53—22no—
97C2×C42×C53—22no—
98C2×C42×C53C2×C414yes—
99C2×C42×C53D414yes—
100#95×C53—24no—
101C24×C53C2411yescharacteristic
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C414yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C22×C414yes—
123C22×C4×C53C22×C414yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D414yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×Q812yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53#4114yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53C22×C412yes—
153C22×Dic53C22×C414yes—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C414yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C414yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C22×C414yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×Q812yes—
163C22×Dic53C2×Q814yes—
164C22×Dic53C2×Q812yes—
165C22×Dic53#4114yes—
166C22×Dic53#4114yes—
167C4⋊C4×C53—216no—
168C42×C53—28no—
169C42×C53—28no—
170C42×C53—44no—
171#11719919—216no—
172#11719919—28no—
173#11719919—216no—
174#11719919C42116yescomplemented
175C4×Dic53—216no—
176C4×Dic53—28no—
177C4×Dic53—28no—
178C4×Dic53—216no—
179C4×Dic53—44no—
180C4×Dic53C4⋊C4116yescomplemented
181C23×C53C23×C411yescharacteristic
182C23×C53C23×C412yes—
183C23×C53C22×D411yescharacteristic
184C23×C53C22×D411yescharacteristic
185C23×C53C22×Q811yescharacteristic
186C23×C53C22×Q811yescharacteristic
187C23×C53C2×#4112yes—
188C23×C53C2×#4112yes—
189C23×C53C2×#4111yescharacteristic
190C23×C53C2×#4111yescharacteristic
191C23×C53C2×#4112yes—
192C2×C4×C53—24no—
193C2×C4×C53—22no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—24no—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53C2×C4214yes—
215C2×C4×C53C2×C4⋊C414yes—
216C2×C4×C53C4×D418yes—
217C2×C4×C53C4×D414yes—
218C2×C4×C53C4×Q814yes—
219C2×C4×C53#12114yes—
220C2×C4×C53C22⋊Q814yes—
221C2×C4×C53#12514yes—
222C2×C4×C53C4⋊Q814yes—
223C2×Dic53—24no—
224C2×Dic53—22no—
225C2×Dic53—24no—
226C2×Dic53—22no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—22no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53C2×C4214yes—
250C2×Dic53C2×C4214yes—
251C2×Dic53C2×C4⋊C414yes—
252C2×Dic53C2×C4⋊C414yes—
253C2×Dic53#11714yes—
254C2×Dic53C4×D418yes—
255C2×Dic53C4×Q818yes—
256C2×Dic53C4×Q818yes—
257C4×#389—531nomaximal, Sylow
258C22×C53C22×C4211yescharacteristic
259C22×C53C22×C4⋊C411yescharacteristic
260C22×C53C2×#11711yescharacteristic
261C22×C53C2×C4×D412yes—
262C22×C53C2×C4×D412yes—
263C22×C53C2×C4×D411yescharacteristic
264C22×C53C2×C4×Q812yes—
265C22×C53C2×C4×Q812yes—
266C22×C53C2×C4×Q811yescharacteristic
267C22×C53C4×#4112yes—
268C22×C53C4×#4111yescharacteristic
269C22×C53C4×#4112yes—
270C22×C53C4×#4111yescharacteristic
271C22×C53C2×#12111yescharacteristic
272C22×C53C2×C22⋊Q811yescharacteristic
273C22×C53#52512yes—
274C22×C53C2×#12511yescharacteristic
275C22×C53#52912yes—
276C22×C53#52912yes—
277C22×C53C2×C4⋊Q811yescharacteristic
278C22×C53#53212yes—
279C22×C53#53312yes—
280C22×C53#53312yes—
281C4×C53—24no—
282C4×C53—28no—
283C4×C53—42no—
284C4×C53—44no—
285C4×C53#38918yescomplemented
286Dic53—24no—
287Dic53—24no—
288Dic53—28no—
289Dic53—42no—
290Dic53—44no—
291Dic53C4×C4⋊C418yescomplemented
292C2×#389—532no—
293C2×C4×C4⋊C4—531no—
294C2×C4×C4⋊C4—531no—
295C2×C4×C4⋊C4—531no—
296C2×C4×C4⋊C4D5311yescharacteristic
297C2×C43—531no—
298#1436—532no—
299#1436—532no—
300C4⋊#95—532no—
301C4×#95—531no—
302C4×#95—531no—
303C2×C53C2×C4×C4⋊C411yescharacteristic
304C2×C53C42×D411yescharacteristic
305C2×C53C42×Q811yescharacteristic
306C2×C53C2×#38911yescharacteristic
307C2×C53#229311yescharacteristic
308C2×C53#229511yescharacteristic
309C2×C53#229611yescharacteristic
310C2×C53C4×#12112yes—
311C2×C53C4×C22⋊Q812yes—
312C2×C53C4×#12512yes—
313C2×C53C4×C4⋊Q812yes—
314C22×C4⋊C4—531no—
315C22×C4⋊C4—531no—
316C22×C4⋊C4—532no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4C2×D5312yes—
319C22×C42—531no—
320C22×C42—532no—
321C22×C42—531no—
322C22×C42—531no—
323C22×C42—532no—
324C22×C42—531no—
325C22×C42C2×D5311yescharacteristic
326C22×C42C2×D5311yescharacteristic
327C22×C42C2×D5311yescharacteristic
328#389—5316no—
329C4×C4⋊C4—24no—
330C4×C4⋊C4—538no—
331C4×C4⋊C4—1064no—
332C4×C4⋊C4—1064no—
333C2×#95—532no—
334C2×#95—532no—
335C2×#95—531no—
336C2×#95—531no—
337C2×#95—532no—
338C2×#95—532no—
339C2×#95—532no—
340C2×#95—532no—
341C2×#95C2×D5312yes—
342C43—1064no—
343C53C4×#38911yescharacteristic, complemented, Sylow
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—532no—
348C23×C4—532no—
349C23×C4—531no—
350C23×C4C22×D5312yes—
351C23×C4C22×D5311yescharacteristic
352C23×C4C22×D5311yescharacteristic
353C23×C4C22×D5312yes—
354C23×C4C22×D5311yescharacteristic
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—534no—
358C2×C4⋊C4—538no—
359C2×C4⋊C4—534no—
360C2×C4⋊C4—538no—
361C2×C4⋊C4—538no—
362C2×C4⋊C4—1064no—
363C2×C4⋊C4—1064no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4C4×D5318yes—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—534no—
376C2×C42—534no—
377C2×C42—534no—
378C2×C42—534no—
379C2×C42—538no—
380C2×C42—534no—
381C2×C42—1064no—
382C2×C42—1064no—
383C2×C42—1064no—
384C2×C42—1064no—
385C2×C42—1064no—
386C2×C42—1062no—
387C2×C42—1062no—
388C2×C42—1062no—
389C2×C42—1062no—
390C2×C42—1062no—
391C2×C42—1062no—
392C2×C42—1062no—
393C2×C42C4×D5314yes—
394C2×C42C22⋊D5314yes—
395#95—24no—
396#95—534no—
397#95—534no—
398#95—538no—
399#95—538no—
400#95—1064no—
401#95—1064no—
402#95—1064no—
403#95—1064no—
404C24C23×D5311yescharacteristic
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—532no—
423C22×C4—534no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—534no—
427C22×C4—532no—
428C22×C4—534no—
429C22×C4—532no—
430C22×C4—534no—
431C22×C4—532no—
432C22×C4—532no—
433C22×C4—534no—
434C22×C4—534no—
435C22×C4—532no—
436C22×C4—534no—
437C22×C4—1062no—
438C22×C4—1061no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1061no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1061no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1062no—
456C22×C4C2×C4×D5312yes—
457C22×C4C2×C4×D5314yes—
458C22×C4C2×C4×D5312yes—
459C22×C4C2×C4×D5314yes—
460C22×C4C2×C4×D5314yes—
461C22×C4C2×C4×D5312yes—
462C22×C4#1171994514yes—
463C22×C4#1171994512yes—
464C22×C4D4×D5312yes—
465C22×C4#1171994712yes—
466C22×C4Q8×D5312yes—
467C22×C4Q8×D5312yes—
468C22×C4C2×C22⋊D5314yes—
469C22×C4C2×C22⋊D5312yes—
470C4⋊C4—216no—
471C4⋊C4—5316no—
472C4⋊C4—1068no—
473C4⋊C4—10616no—
474C4⋊C4—10616no—
475C42—28no—
476C42—28no—
477C42—44no—
478C42—5316no—
479C42—10616no—
480C42—10616no—
481C42—1068no—
482C42—1068no—
483C42—2124no—
484C23C22×C4×D5311yescharacteristic
485C23C22×C4×D5312yes—
486C23C2×D4×D5311yescharacteristic
487C23C22×C22⋊D5311yescharacteristic
488C23C2×Q8×D5311yescharacteristic
489C23C2×Q8×D5311yescharacteristic
490C23#41×D5312yes—
491C23#41×D5312yes—
492C23C2×#1171994711yescharacteristic
493C23C2×#1171994511yescharacteristic
494C23C2×#1171994512yes—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—534no—
518C2×C4—534no—
519C2×C4—534no—
520C2×C4—538no—
521C2×C4—538no—
522C2×C4—534no—
523C2×C4—534no—
524C2×C4—538no—
525C2×C4—1064no—
526C2×C4—1062no—
527C2×C4—1064no—
528C2×C4—1062no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1064no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1064no—
538C2×C4—1064no—
539C2×C4—1062no—
540C2×C4—1064no—
541C2×C4—1062no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1062no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4C42×D5314yes—
552C2×C4C4⋊C4×D5314yes—
553C2×C4C4×C22⋊D5318yes—
554C2×C4C4×C22⋊D5314yes—
555C2×C4#1313318214yes—
556C2×C4#1313318914yes—
557C2×C4#1313319514yes—
558C2×C4#1313320714yes—
559C2×C4#1313321514yes—
560C22C2×C42×D5311yescharacteristic
561C22C2×C4⋊C4×D5311yescharacteristic
562C22#117×D5311yescharacteristic
563C22C4×D4×D5312yes—
564C22C2×C4×C22⋊D5312yes—
565C22C2×C4×C22⋊D5311yescharacteristic
566C22C4×Q8×D5312yes—
567C22C4×Q8×D5312yes—
568C22C2×#1313318211yescharacteristic
569C22C4×#1171994712yes—
570C22#1681109111yescharacteristic
571C22C4×#1171994512yes—
572C22C4×#1171994511yescharacteristic
573C22C2×#1313318911yescharacteristic
574C22C2×#1313319511yescharacteristic
575C22#1681113212yes—
576C22C2×#1313320711yescharacteristic
577C22#1681115112yes—
578C22#1681115312yes—
579C22C2×#1313321511yescharacteristic
580C22#1681116912yes—
581C22#1681117712yes—
582C22#1681117712yes—
583C4—24no—
584C4—28no—
585C4—42no—
586C4—44no—
587C4—538no—
588C4—1068no—
589C4—1064no—
590C4—1064no—
591C4—2122no—
592C4—2124no—
593C4#1681051818yesdirect product factor
594C2C4×C4⋊C4×D5311yescharacteristic
595C2C42×C22⋊D5311yescharacteristic
596C2C4×#1313318211yescharacteristic
597C2C2×#1681051811yescharacteristic
598C2#2027717411yescharacteristic
599C2#2027719111yescharacteristic
600C2#2027719511yescharacteristic
601C2C4×#1313318912yes—
602C2C4×#1313319512yes—
603C2C4×#1313320712yes—
604C2C4×#1313321512yes—
subgroupG/H
centerC23×C4C22×D53
commutatorC22×C53C22×C42
FrattiniC24C23×D53
FittingC2×C4×C4⋊C4×C53C2
socle, prime radical, cosocleC24×C53C24
hypercenterC2×C4×C4⋊C4D53
hypocenterC53C4×#389
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
22. #16810518C4direct (N × H)
25. C4×#11719919C4semidirect
174. #11719919C42semidirect
180. C4×Dic53C4⋊C4semidirect
285. C4×C53#389semidirect
291. Dic53C4×C4⋊C4semidirect
343. C53C4×#389semidirect
593. C4#16810518direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168105182
C2×C4×#117199192
C2×C42×Dic532
#202729502
#202729472
#202729432
C4×#131331242
C2×C4×C4⋊C4×C532
C4×#38953