Finite Group Databasebeta

← C2×#13128543 #16770859

Subgroups — C2×#13128543 #16770859

Order 3328.

7910 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128543C2132yesmaximal, direct product factor
2C2×#11719340C214yesmaximal, complemented
3C2×#11719319C214yesmaximal, complemented
4C2×#11719344C214yesmaximal, complemented
5C2×#11719316C214yesmaximal, complemented
6C2×#11719205C211yesmaximal, characteristic, complemented
7C2×#11719307C214yesmaximal, complemented
8C2×#11719311C214yesmaximal, complemented
9C2×#11719533C212yesmaximal, complemented
10C2×#11719532C212yesmaximal, complemented
11C22×C4⋊C4×D13C211yesmaximal, characteristic, complemented
12C2×C13×#537C211yesmaximal, characteristic, complemented
13C22×C13×#41C2212yes—
14C22×#6129C2214yes—
15C23×C4×D13C2211yescharacteristic
16C23×C4×D13C2212yes—
17C13×#537C22116yescomplemented
18C2×C13×C22⋊Q8C2214yescomplemented
19C2×C13×#121C2214yes—
20C2×C13×#121C2214yescomplemented
21C22×C13×C4⋊C4C2211yescharacteristic
22#11719533C22132yescomplemented
23#11719532C22132yescomplemented
24C2×#88200C2212yes—
25C2×#88200C2212yes—
26C2×Q8×Dic13C2214yescomplemented
27C2×#88192C2212yes—
28C2×#88192C2212yes—
29C2×#88192C2214yes—
30C2×#88191C2214yes—
31C2×#88189C2214yes—
32C2×#88189C2218yescomplemented
33C2×D4×Dic13C2218yes—
34C2×D4×Dic13C2214yescomplemented
35C2×C4×C22⋊D13C2214yes—
36C2×C4×C22⋊D13C2214yes—
37C2×#88180C2212yes—
38C2×#88180C2214yescomplemented
39C22×C4⋊Dic13C2211yescharacteristic
40C22×C4⋊Dic13C2214yescomplemented
41C2×#88178C2214yes—
42C22×#6101C2212yes—
43#11719344C22164yescomplemented
44#11719340C22164yescomplemented
45#11719319C22164yescomplemented
46#11719316C22164yescomplemented
47#11719311C22164yescomplemented
48#11719307C22164yescomplemented
49C2×#88151C2214yes—
50C2×#88146C2214yes—
51C2×C4⋊C4×D13C2214yes—
52C2×C4⋊C4×D13C2214yes—
53C2×C4⋊C4×D13C22116yescomplemented
54#11719205C22116yescomplemented
55C2×#88144C2214yescomplemented
56C2×#88143C2218yescomplemented
57C2×#88142C2214yescomplemented
58C2×#88137C2218yes—
59C2×#88135C2218yes—
60C2×#88133C2218yescomplemented
61C2×#88132C2218yescomplemented
62C24×D13C2311yescharacteristic
63C2×C13×#41—216no—
64C2×C13×#41C23116yes—
65C22×Q8×C13C2312yes—
66C22×D4×C13C2314yes—
67C22×D4×C13C2312yes—
68C22×D4×C13C2314yes—
69C23×C4×C13C2314yes—
70C23×C4×C13C2312yes—
71C23×C4×C13C2311yescharacteristic
72C22×C22⋊D13C2314yes—
73C22×C22⋊D13C2314yes—
74C23×Dic13C2311yescharacteristic
75C23×Dic13C2318yes—
76C23×Dic13C2312yes—
77C2×#6129—416no—
78C2×#6129C23132yes—
79C22×C4×D13—28no—
80C22×C4×D13—48no—
81C22×C4×D13C2314yes—
82C22×C4×D13C2312yes—
83C22×C4×D13C2312yes—
84C22×C4×D13C2318yes—
85C22×C4×D13C2314yes—
86C22×C4×D13C23116yes—
87C22×Dic26C2314yes—
88C13×C22⋊Q8C23132yescomplemented
89C13×#121C23132yes—
90C13×#121C23132yescomplemented
91C2×C13×C4⋊C4C2314yes—
92C2×C13×C4⋊C4C2314yes—
93C2×C13×C4⋊C4C2318yes—
94C2×C13×C22⋊C4C2318yes—
95C2×C22⋊Dic13C2314yes—
96C2×C22⋊Dic13C2314yes—
97C2×C22⋊Dic13C2318yes—
98C2×C22⋊Dic13C2314yes—
99#88200C23116yes—
100#88200C23116yes—
101Q8×Dic13C23132yescomplemented
102#88192C23116yes—
103#88192C23116yes—
104#88192C23132yes—
105#88191C23132yes—
106#88189C23132yes—
107#88189C23164yescomplemented
108D4×Dic13C23164yes—
109D4×Dic13C23132yescomplemented
110C4×C22⋊D13C23132yes—
111C4×C22⋊D13C23132yes—
112C2×#6103C2314yes—
113C2×#6103C2314yes—
114#88180C23116yes—
115#88180C23132yescomplemented
116C2×C4⋊Dic13C2318yes—
117C2×C4⋊Dic13C2314yes—
118C2×C4⋊Dic13C2312yes—
119C2×C4⋊Dic13C2312yes—
120C2×C4⋊Dic13C2314yes—
121C2×C4⋊Dic13C2318yes—
122C2×C4⋊Dic13C23132yescomplemented
123#88178C23132yes—
124C2×#6101C2314yes—
125C2×#6101C2318yes—
126C2×#6101C2314yes—
127C2×#6101C2318yes—
128C2×#6101C23116yes—
129C2×C4×Dic13C2318yes—
130C2×C4×Dic13C2314yes—
131#88151C23132yes—
132#88146C23132yes—
133C4⋊C4×D13C23132yes—
134C4⋊C4×D13C23132yes—
135#88144C23132yescomplemented
136#88143C23164yescomplemented
137#88142C23132yescomplemented
138#88137C23164yes—
139#88135C23164yes—
140#88133C23164yescomplemented
141#88132C23164yescomplemented
142C2×#3492—131nomaximal, Sylow
143C24×C13C2414yes—
144C24×C13C2411yescharacteristic
145C23×D13—24no—
146C23×D13—24no—
147C23×D13—24no—
148C23×D13C2414yes—
149C23×D13C2411yescharacteristic
150C23×D13C2411yescharacteristic
151C13×#41—264no—
152C2×Q8×C13—28no—
153C2×Q8×C13C2418yes—
154C2×D4×C13—216no—
155C2×D4×C13—28no—
156C2×D4×C13—48no—
157C2×D4×C13C24116yes—
158C2×D4×C13C2418yes—
159C2×D4×C13C24116yes—
160C22×C4×C13—216no—
161C22×C4×C13—44no—
162C22×C4×C13#4118yescomplemented
163C22×C4×C13C2414yes—
164C22×C4×C13C2414yes—
165C22×C4×C13C24116yes—
166C22×C4×C13C2418yes—
167C22×C4×C13C2412yes—
168C22×C4×C13C2414yes—
169C2×C22⋊D13—216no—
170C2×C22⋊D13—216no—
171C2×C22⋊D13C24116yes—
172C2×C22⋊D13C24116yes—
173C22×Dic13—24no—
174C22×Dic13—44no—
175C22×Dic13—416no—
176C22×Dic13C2414yes—
177C22×Dic13C2414yes—
178C22×Dic13C2414yes—
179C22×Dic13C2412yes—
180C22×Dic13C2414yes—
181C22×Dic13C24132yes—
182C22×Dic13C2418yes—
183#6129—464no—
184C2×C4×D13—216no—
185C2×C4×D13—28no—
186C2×C4×D13—28no—
187C2×C4×D13—232no—
188C2×C4×D13—432no—
189C2×C4×D13—48no—
190C2×C4×D13C24116yes—
191C2×C4×D13C2418yes—
192C2×C4×D13C2418yes—
193C2×C4×D13C24116yes—
194C2×Dic26—48no—
195C2×Dic26C24116yes—
196C13×C4⋊C4C24116yes—
197C13×C4⋊C4C24116yes—
198C13×C22⋊C4C24132yes—
199C22⋊Dic13C24116yes—
200C22⋊Dic13C24116yes—
201C22⋊Dic13C24132yes—
202C22⋊Dic13C24116yes—
203#6103C24116yes—
204#6103C24116yes—
205C4⋊Dic13C24132yes—
206C4⋊Dic13C24116yes—
207C4⋊Dic13C2418yes—
208C4⋊Dic13C2418yes—
209C4⋊Dic13C24116yes—
210#6101C24116yes—
211#6101C24132yes—
212#6101C24116yes—
213#6101C24132yes—
214C4×Dic13C24132yes—
215C4×Dic13C24116yes—
216#3492—1332no—
217C2×#554—134no—
218C2×#553—134no—
219C2×#552—134no—
220C2×#547—134no—
221C2×#544—131no—
222C2×#539—134no—
223C2×#537D1311yescharacteristic, complemented
224C2×#525—132no—
225C2×#525—134no—
226C2×#520—132no—
227C23×C4⋊C4—131no—
228C23×C13—22no—
229C23×C13—22no—
230C23×C13—28no—
231C23×C13—28no—
232C23×C13—28no—
233C23×C13C2×#4114yes—
234C23×C13C2518yes—
235C23×C13C2512yes—
236C23×C13C2511yescharacteristic
237C22×D13—232no—
238C22×D13—28no—
239C22×D13—22no—
240C22×D13—22no—
241C22×D13—22no—
242C22×D13—22no—
243C22×D13—28no—
244C22×D13—28no—
245C22×D13H(5,𝔽2)14yes—
246C22×D132−1+414yes—
247C22×D13C2512yes—
248C22×D13C2512yes—
249Q8×C13—216no—
250D4×C13—232no—
251D4×C13—216no—
252D4×C13—416no—
253C2×C4×C13—28no—
254C2×C4×C13—28no—
255C2×C4×C13—232no—
256C2×C4×C13—48no—
257C2×C4×C13C2×#4118yes—
258C2×C4×C13C2×#41116yescomplemented
259C2×C4×C13C2518yes—
260C2×C4×C13C2518yes—
261C2×C4×C13C2514yes—
262C22⋊D13—232no—
263C22⋊D13—232no—
264C2×Dic13—28no—
265C2×Dic13—24no—
266C2×Dic13—28no—
267C2×Dic13—28no—
268C2×Dic13—28no—
269C2×Dic13—48no—
270C2×Dic13—432no—
271C2×Dic13C2518yes—
272C2×Dic13C2518yes—
273C2×Dic13C2518yes—
274C2×Dic13C2514yes—
275C4×D13—232no—
276C4×D13—216no—
277C4×D13—216no—
278C4×D13—416no—
279Dic26—416no—
280C22×#41—134no—
281C22×#41C2×D1312yes—
282C24×C4—131no—
283C24×C4—132no—
284#554—1364no—
285#553—1364no—
286#552—1364no—
287#547—1364no—
288#544—1316no—
289#539—1364no—
290#537C2×D13116yescomplemented
291C2×C4⋊Q8—134no—
292C2×#126—138no—
293C2×#125—138no—
294C2×#125—134no—
295#525—1332no—
296#525—1364no—
297C2×#123—134no—
298C2×#123—138no—
299C2×#123—138no—
300C2×C22⋊Q8—132no—
301C2×C22⋊Q8—132no—
302C2×C22⋊Q8—134no—
303C2×C22⋊Q8—134no—
304C2×C22⋊Q8—138no—
305C2×C22⋊Q8C2×D1314yescomplemented
306C2×#121—132no—
307C2×#121—134no—
308C2×#121—132no—
309C2×#121C2×D1314yes—
310C2×#121C2×D1314yescomplemented
311C2×#120—134no—
312#520—1332no—
313C2×C4×Q8—134no—
314C2×C4×D4—138no—
315C2×C4×D4—134no—
316C2×C4×D4—134no—
317C2×C4×D4—134no—
318C2×C4×D4—138no—
319C2×#117—132no—
320C2×#117—134no—
321C2×#117—134no—
322C22×C4⋊C4—132no—
323C22×C4⋊C4—131no—
324C22×C4⋊C4—134no—
325C22×C4⋊C4—134no—
326C22×C4⋊C4—134no—
327C22×C4⋊C4—1316no—
328C22×C4⋊C4C2×D1311yescharacteristic
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—28no—
332C22×C13—28no—
333C22×C13—28no—
334C22×C13—28no—
335C22×C13—416no—
336C22×C13C22×#4114yes—
337C22×C13C22×#4112yes—
338C22×C13C2×H(5,𝔽2)12yes—
339C22×C13C2×2−1+412yes—
340C22×C13C2611yescharacteristic
341C2×D13—232no—
342C2×D13—22no—
343C2×D13—22no—
344C2×D13—22no—
345C2×D13—22no—
346C2×D13—28no—
347C2×D13—28no—
348C2×D13C2×H(5,𝔽2)14yes—
349C2×D13C2×2−1+414yes—
350C4×C13—28no—
351C4×C13—28no—
352C4×C13C22×#4118yes—
353Dic13—24no—
354Dic13—28no—
355Dic13—28no—
356Dic13—28no—
357C25—131no—
358C2×#41—216no—
359C2×#41—1332no—
360C2×#41—5216no—
361C2×#41C22×D13116yes—
362C22×Q8—134no—
363C22×Q8C22×D1312yes—
364C22×D4—134no—
365C22×D4—134no—
366C22×D4C22×D1314yes—
367C22×D4C22×D1312yes—
368C22×D4C22×D1314yes—
369C23×C4—131no—
370C23×C4—132no—
371C23×C4—138no—
372C23×C4—134no—
373C23×C4—132no—
374C23×C4—132no—
375C23×C4—138no—
376C23×C4—134no—
377C23×C4—1316no—
378C23×C4—268no—
379C23×C4—528no—
380C23×C4C22×D1314yes—
381C23×C4C22×D1312yes—
382C23×C4C22×D1311yescharacteristic
383C4⋊Q8—1332no—
384#126—1364no—
385#125—1364no—
386#125—1332no—
387#123—1332no—
388#123—1364no—
389#123—1364no—
390C22⋊Q8—1316no—
391C22⋊Q8—1316no—
392C22⋊Q8—1332no—
393C22⋊Q8—1332no—
394C22⋊Q8—1364no—
395C22⋊Q8C22×D13132yescomplemented
396#121—1316no—
397#121—1332no—
398#121—1316no—
399#121C22×D13132yes—
400#121C22×D13132yescomplemented
401#120—1332no—
402C4×Q8—1332no—
403C4×D4—1364no—
404C4×D4—1332no—
405C4×D4—1332no—
406C4×D4—1332no—
407C4×D4—1364no—
408#117—1316no—
409#117—1332no—
410#117—1332no—
411C2×C4⋊C4—134no—
412C2×C4⋊C4—138no—
413C2×C4⋊C4—138no—
414C2×C4⋊C4—1316no—
415C2×C4⋊C4—132no—
416C2×C4⋊C4—138no—
417C2×C4⋊C4—1332no—
418C2×C4⋊C4—132no—
419C2×C4⋊C4—134no—
420C2×C4⋊C4—134no—
421C2×C4⋊C4—138no—
422C2×C4⋊C4—134no—
423C2×C4⋊C4—1332no—
424C2×C4⋊C4—1332no—
425C2×C4⋊C4C22×D1314yes—
426C2×C4⋊C4C22×D1318yes—
427C2×C4⋊C4C22×D1314yes—
428C2×C22⋊C4—134no—
429C2×C22⋊C4—138no—
430C2×C22⋊C4—134no—
431C2×C22⋊C4—134no—
432C2×C22⋊C4—134no—
433C2×C22⋊C4—134no—
434C2×C22⋊C4C22×D1318yes—
435C2×C42—138no—
436C2×C42—134no—
437C2×C13—24no—
438C2×C13—48no—
439C2×C13#349214yescomplemented
440C2×C13C23×#4111yescharacteristic
441C2×C13C22×H(5,𝔽2)11yescharacteristic
442C2×C13C22×2−1+411yescharacteristic
443D13—24no—
444D13—24no—
445C24—131no—
446C24—131no—
447C24—134no—
448C24—264no—
449C24—264no—
450C24—264no—
451C24C23×D1314yes—
452C24C23×D1311yescharacteristic
453#41—264no—
454#41—5264no—
455C2×Q8—28no—
456C2×Q8—1316no—
457C2×Q8—528no—
458C2×Q8C23×D1318yes—
459C2×D4—216no—
460C2×D4—28no—
461C2×D4—48no—
462C2×D4—1316no—
463C2×D4—1316no—
464C2×D4—2616no—
465C2×D4—2616no—
466C2×D4C23×D13116yes—
467C2×D4C23×D1318yes—
468C2×D4C23×D13116yes—
469C22×C4—216no—
470C22×C4—44no—
471C22×C4—134no—
472C22×C4—134no—
473C22×C4—134no—
474C22×C4—134no—
475C22×C4—132no—
476C22×C4—138no—
477C22×C4—1332no—
478C22×C4—1316no—
479C22×C4—138no—
480C22×C4—138no—
481C22×C4—1316no—
482C22×C4—2616no—
483C22×C4—264no—
484C22×C4—268no—
485C22×C4—268no—
486C22×C4—2632no—
487C22×C4—528no—
488C22×C4—5232no—
489C22×C4—524no—
490C22×C4—5216no—
491C22×C4#612918yescomplemented
492C22×C4C23×D13116yes—
493C22×C4C23×D1314yes—
494C22×C4C23×D1314yes—
495C22×C4C23×D1318yes—
496C22×C4C23×D1312yes—
497C22×C4C23×D1314yes—
498C4⋊C4—1316no—
499C4⋊C4—1332no—
500C4⋊C4—138no—
501C4⋊C4—1332no—
502C4⋊C4—138no—
503C4⋊C4—1316no—
504C4⋊C4—1316no—
505C4⋊C4—1332no—
506C4⋊C4—1316no—
507C4⋊C4C23×D13116yes—
508C4⋊C4C23×D13116yes—
509C22⋊C4—1316no—
510C22⋊C4—1332no—
511C22⋊C4—1316no—
512C22⋊C4—1316no—
513C22⋊C4—1316no—
514C22⋊C4—1316no—
515C22⋊C4C23×D13132yes—
516C42—1332no—
517C42—1316no—
518C13C2×#349211yescharacteristic, complemented, Sylow
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—134no—
525C23—134no—
526C23—132no—
527C23—132no—
528C23—268no—
529C23—268no—
530C23—2632no—
531C23—268no—
532C23—262no—
533C23—262no—
534C23—262no—
535C23—262no—
536C23C2×#612914yes—
537C23C24×D1318yes—
538C23C24×D1312yes—
539C23C24×D1311yescharacteristic
540Q8—216no—
541Q8—5216no—
542D4—232no—
543D4—216no—
544D4—416no—
545D4—2632no—
546D4—2632no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—48no—
551C2×C4—138no—
552C2×C4—138no—
553C2×C4—138no—
554C2×C4—134no—
555C2×C4—2632no—
556C2×C4—268no—
557C2×C4—264no—
558C2×C4—268no—
559C2×C4—268no—
560C2×C4—268no—
561C2×C4—2616no—
562C2×C4—2616no—
563C2×C4—5216no—
564C2×C4—528no—
565C2×C4—5232no—
566C2×C4C2×#612918yes—
567C2×C4C2×#6129116yescomplemented
568C2×C4C24×D1318yes—
569C2×C4C24×D1318yes—
570C2×C4C24×D1314yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—416no—
578C22—134no—
579C22—134no—
580C22—2632no—
581C22—268no—
582C22—268no—
583C22—262no—
584C22—262no—
585C22—262no—
586C22—262no—
587C22C22×#612914yes—
588C22C22×#612912yes—
589C22D13×H(5,𝔽2)12yes—
590C22D13×2−1+412yes—
591C22C25×D1311yescharacteristic
592C4—28no—
593C4—28no—
594C4—264no—
595C4—268no—
596C4—268no—
597C4—268no—
598C4C22×#612918yes—
599C2—24no—
600C2—48no—
601C2—264no—
602C2—264no—
603C2#1312854314yesdirect product factor
604C2C23×#612911yescharacteristic
605C2C2×D13×H(5,𝔽2)11yescharacteristic
606C2C2×D13×2−1+411yescharacteristic
subgroupG/H
centerC23C24×D13
commutator, cosocleC22×C13C26
FrattiniC22C25×D13
FittingC2×C13×#537C2
socleC23×C13C25
hypercenterC2×#537D13
hypocenterC13C2×#3492
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #13128543C2direct (N × H)
2. C2×#11719340C2semidirect
3. C2×#11719319C2semidirect
4. C2×#11719344C2semidirect
5. C2×#11719316C2semidirect
6. C2×#11719205C2semidirect
7. C2×#11719307C2semidirect
8. C2×#11719311C2semidirect
9. C2×#11719533C2semidirect
10. C2×#11719532C2semidirect
11. C22×C4⋊C4×D13C2semidirect
12. C2×C13×#537C2semidirect
17. C13×#537C22semidirect
18. C2×C13×C22⋊Q8C22semidirect
20. C2×C13×#121C22semidirect
22. #11719533C22semidirect
23. #11719532C22semidirect
26. C2×Q8×Dic13C22semidirect
32. C2×#88189C22semidirect
34. C2×D4×Dic13C22semidirect
38. C2×#88180C22semidirect
40. C22×C4⋊Dic13C22semidirect
43. #11719344C22semidirect
44. #11719340C22semidirect
45. #11719319C22semidirect
46. #11719316C22semidirect
47. #11719311C22semidirect
48. #11719307C22semidirect
53. C2×C4⋊C4×D13C22semidirect
54. #11719205C22semidirect
55. C2×#88144C22semidirect
56. C2×#88143C22semidirect
57. C2×#88142C22semidirect
60. C2×#88133C22semidirect
61. C2×#88132C22semidirect
88. C13×C22⋊Q8C23semidirect
90. C13×#121C23semidirect
101. Q8×Dic13C23semidirect
107. #88189C23semidirect
109. D4×Dic13C23semidirect
115. #88180C23semidirect
122. C2×C4⋊Dic13C23semidirect
135. #88144C23semidirect
136. #88143C23semidirect
137. #88142C23semidirect
140. #88133C23semidirect
141. #88132C23semidirect
162. C22×C4×C13#41semidirect
223. C2×#537D13semidirect
258. C2×C4×C13C2×#41semidirect
290. #537C2×D13semidirect
305. C2×C22⋊Q8C2×D13semidirect
310. C2×#121C2×D13semidirect
395. C22⋊Q8C22×D13semidirect
400. #121C22×D13semidirect
439. C2×C13#3492semidirect
491. C22×C4#6129semidirect
518. C13C2×#3492semidirect
567. C2×C4C2×#6129semidirect
603. C2#13128543direct (N × H)

Maximal subgroups

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

subgroupindex
#131285432
C2×#117193402
C2×#117193192
C2×#117193442
C2×#117193162
C2×#117192052
C2×#117193072
C2×#117193112
C2×#117195332
C2×#117195322
C22×C4⋊C4×D132
C2×C13×#5372
C2×#349213