Finite Group Databasebeta

← C2×#13128546 #16770861

Subgroups — C2×#13128546 #16770861

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#13128546C2132yesmaximal, direct product factor
2C2×#11719435C214yesmaximal, complemented
3C2×#11719400C214yesmaximal, complemented
4C2×#11719399C214yesmaximal, complemented
5C2×#11719431C214yesmaximal, complemented
6C2×#11719350C214yesmaximal, complemented
7C2×#11719215C211yesmaximal, characteristic, complemented
8C2×#11719332C214yesmaximal, complemented
9C2×#11719229C212yesmaximal, complemented
10C2×#11719228C212yesmaximal, complemented
11C22×C4⋊C4×D13C211yesmaximal, characteristic, complemented
12C2×C13×#544C211yesmaximal, characteristic, complemented
13C22×#6131C2214yes—
14C22×#6127C2212yes—
15C23×C4×D13C2211yescharacteristic
16C23×C4×D13C2212yes—
17C13×#544C22116yescomplemented
18C2×C13×C4⋊Q8C2214yescomplemented
19C2×C13×#125C2214yescomplemented
20C2×C13×C22⋊Q8C2214yes—
21C2×C13×#117C2212yes—
22C22×C13×C4⋊C4C2211yescharacteristic
23C2×#88200C2214yes—
24C2×#88200C2218yescomplemented
25C2×#88184C2214yes—
26C22×C4⋊Dic13C2211yescharacteristic
27C22×#6101C2212yes—
28#11719435C22164yescomplemented
29#11719431C22164yescomplemented
30#11719400C22164yescomplemented
31#11719399C22164yescomplemented
32#11719350C22164yescomplemented
33#11719332C22164yescomplemented
34#11719229C22132yescomplemented
35#11719228C22132yescomplemented
36C2×#88152C2218yescomplemented
37#11719215C22116yescomplemented
38C2×#88151C2212yes—
39C2×#88151C2212yes—
40C2×#88151C2214yescomplemented
41C2×#88149C2212yes—
42C2×#88149C2212yes—
43C2×#88149C2214yes—
44C2×#88149C2214yescomplemented
45C2×#88148C2218yes—
46C2×#88148C2218yescomplemented
47C2×#88147C2218yes—
48C2×#88146C2214yes—
49C2×#88146C2214yescomplemented
50C2×C4⋊C4×D13C2214yes—
51C2×C4⋊C4×D13C2214yes—
52C2×C4⋊C4×D13C22116yescomplemented
53C2×C4⋊C4×D13C2214yescomplemented
54C2×#88144C2218yescomplemented
55C2×#88140C2214yes—
56C2×#88136C2214yes—
57C2×#88135C2218yes—
58C2×C4×D52C2214yes—
59C2×C4×D52C2214yes—
60C2×C4×D52C2214yescomplemented
61C2×C4×Dic26C2214yescomplemented
62C24×D13C2311yescharacteristic
63C22×Q8×C13C2314yes—
64C23×C4×C13C2312yes—
65C23×C4×C13C2311yescharacteristic
66C22×C22⋊D13C2314yes—
67C23×Dic13C2311yescharacteristic
68C23×Dic13C2312yes—
69C2×#6131—416no—
70C2×#6131C23132yes—
71C2×#6127—216no—
72C2×#6127C23116yes—
73C22×D52C2314yes—
74C22×D52C2314yes—
75C22×D52C2312yes—
76C22×D52C2314yes—
77C22×C4×D13—28no—
78C22×C4×D13—48no—
79C22×C4×D13C2312yes—
80C22×C4×D13C2312yes—
81C22×C4×D13C2314yes—
82C22×C4×D13C2314yes—
83C22×C4×D13C2318yes—
84C22×C4×D13C2318yes—
85C22×C4×D13C2314yes—
86C22×C4×D13C23116yes—
87C22×Dic26C2312yes—
88C13×C4⋊Q8C23132yescomplemented
89C13×#125C23132yescomplemented
90C13×C22⋊Q8C23132yes—
91C13×#117C23116yes—
92C2×C13×C4⋊C4C2318yes—
93C2×C13×C4⋊C4C2314yes—
94C2×C13×C4⋊C4C2312yes—
95C2×C13×C4⋊C4C2318yes—
96C2×C13×C4⋊C4C2314yes—
97C2×C13×C4⋊C4C2312yes—
98C2×C13×C22⋊C4C2314yes—
99C2×C42×C13C2314yes—
100#88200C23132yes—
101#88200C23164yescomplemented
102#88184C23132yes—
103C2×#6103C2314yes—
104C2×#6103C2314yes—
105C2×#6103C2314yes—
106C2×#6103C2314yes—
107C2×#6103C2318yes—
108C2×#6103C2318yes—
109C2×C4⋊Dic13C2314yes—
110C2×C4⋊Dic13C2314yes—
111C2×C4⋊Dic13C2314yes—
112C2×C4⋊Dic13C2314yes—
113C2×C4⋊Dic13C2318yes—
114C2×#6101C2318yes—
115C2×#6101C2318yes—
116C2×#6101C23116yes—
117C2×C4×Dic13C2318yes—
118#88152C23164yescomplemented
119#88151C23116yes—
120#88151C23116yes—
121#88151C23132yescomplemented
122#88149C23116yes—
123#88149C23116yes—
124#88149C23132yes—
125#88149C23132yescomplemented
126#88148C23164yes—
127#88148C23164yescomplemented
128#88147C23164yes—
129#88146C23132yes—
130#88146C23132yescomplemented
131C4⋊C4×D13C23132yes—
132C4⋊C4×D13C23132yes—
133C4⋊C4×D13C23132yescomplemented
134#88144C23164yescomplemented
135#88140C23132yes—
136#88136C23132yes—
137#88135C23164yes—
138C4×D52C23132yes—
139C4×D52C23132yes—
140C4×D52C23132yescomplemented
141C4×Dic26C23132yescomplemented
142C2×#3492—131nomaximal, Sylow
143C24×C13C2411yescharacteristic
144C23×D13—24no—
145C23×D13—24no—
146C23×D13—24no—
147C23×D13C2414yes—
148C23×D13C2414yes—
149C23×D13C2411yescharacteristic
150C23×D13C2411yescharacteristic
151C2×Q8×C13—48no—
152C2×Q8×C13C24116yes—
153C22×C4×C13—44no—
154C22×C4×C13#4118yescomplemented
155C22×C4×C13C2414yes—
156C22×C4×C13C2414yes—
157C22×C4×C13C2414yes—
158C22×C4×C13C2418yes—
159C22×C4×C13C2412yes—
160C22×C4×C13C2414yes—
161C2×C22⋊D13—216no—
162C2×C22⋊D13C24116yes—
163C22×Dic13—24no—
164C22×Dic13—44no—
165C22×Dic13C2414yes—
166C22×Dic13C2414yes—
167C22×Dic13C2414yes—
168C22×Dic13C2412yes—
169C22×Dic13C2418yes—
170#6131—464no—
171#6127—264no—
172C2×D52—216no—
173C2×D52—216no—
174C2×D52—28no—
175C2×D52—48no—
176C2×D52C24116yes—
177C2×D52C24116yes—
178C2×D52C2418yes—
179C2×D52C24116yes—
180C2×C4×D13—216no—
181C2×C4×D13—28no—
182C2×C4×D13—28no—
183C2×C4×D13—216no—
184C2×C4×D13—232no—
185C2×C4×D13—432no—
186C2×C4×D13—48no—
187C2×C4×D13—416no—
188C2×C4×D13C2418yes—
189C2×C4×D13C2418yes—
190C2×C4×D13C24116yes—
191C2×C4×D13C24116yes—
192C2×C4×D13C24132yes—
193C2×C4×D13C24116yes—
194C2×Dic26—28no—
195C2×Dic26C2418yes—
196C13×C4⋊C4C24132yes—
197C13×C4⋊C4C24116yes—
198C13×C4⋊C4C2418yes—
199C13×C4⋊C4C24116yes—
200C13×C4⋊C4C2418yes—
201C13×C22⋊C4C24116yes—
202C42×C13C24116yes—
203#6103C24116yes—
204#6103C24116yes—
205#6103C24116yes—
206#6103C24116yes—
207#6103C24132yes—
208#6103C24132yes—
209C4⋊Dic13C24116yes—
210C4⋊Dic13C24116yes—
211C4⋊Dic13C24116yes—
212C4⋊Dic13C24116yes—
213#6101C24132yes—
214#6101C24132yes—
215C4×Dic13C24132yes—
216#3492—1332no—
217C2×#554—134no—
218C2×#553—134no—
219C2×#552—134no—
220C2×#547—134no—
221C2×#544D1311yescharacteristic, complemented
222C2×#539—134no—
223C2×#537—131no—
224C2×#525—134no—
225C2×#525—132no—
226C2×#520—132no—
227C23×C4⋊C4—131no—
228C23×C13—22no—
229C23×C13—22no—
230C23×C13C2×#4114yes—
231C23×C13C2512yes—
232C23×C13C2511yescharacteristic
233C22×D13—232no—
234C22×D13—28no—
235C22×D13—22no—
236C22×D13—22no—
237C22×D13—22no—
238C22×D13—22no—
239C22×D13—28no—
240C22×D13—28no—
241C22×D13—28no—
242C22×D13—28no—
243C22×D13—28no—
244C22×D13H(5,𝔽2)14yes—
245C22×D132−1+414yes—
246C22×D13C2518yes—
247C22×D13C2512yes—
248C22×D13C2512yes—
249Q8×C13—416no—
250C2×C4×C13—28no—
251C2×C4×C13—28no—
252C2×C4×C13—28no—
253C2×C4×C13—48no—
254C2×C4×C13C2×#4118yes—
255C2×C4×C13C2×#41116yescomplemented
256C2×C4×C13C2518yes—
257C2×C4×C13C2518yes—
258C2×C4×C13C2518yes—
259C2×C4×C13C2514yes—
260C22⋊D13—232no—
261C2×Dic13—28no—
262C2×Dic13—28no—
263C2×Dic13—24no—
264C2×Dic13—28no—
265C2×Dic13—48no—
266C2×Dic13C2518yes—
267C2×Dic13C2518yes—
268C2×Dic13C2514yes—
269D52—232no—
270D52—232no—
271D52—216no—
272D52—416no—
273C4×D13—232no—
274C4×D13—216no—
275C4×D13—216no—
276C4×D13—232no—
277C4×D13—416no—
278C4×D13—432no—
279Dic26—216no—
280C22×#41—134no—
281C22×#41—132no—
282C24×C4—131no—
283C24×C4—132no—
284#554—1364no—
285#553—1364no—
286#552—1364no—
287#547—1364no—
288#544C2×D13116yescomplemented
289#539—1364no—
290#537—1316no—
291C2×C4⋊Q8C2×D1314yescomplemented
292C2×#126—138no—
293C2×#125—138no—
294C2×#125C2×D1314yescomplemented
295#525—1364no—
296#525—1332no—
297C2×#123—138no—
298C2×#123—138no—
299C2×#123—134no—
300C2×C22⋊Q8—134no—
301C2×C22⋊Q8—138no—
302C2×C22⋊Q8—132no—
303C2×C22⋊Q8—132no—
304C2×C22⋊Q8—134no—
305C2×C22⋊Q8C2×D1314yes—
306C2×#121—134no—
307C2×#121—132no—
308C2×#121—134no—
309C2×#121—132no—
310C2×#121—134no—
311C2×#120—134no—
312#520—1332no—
313C2×C4×Q8—134no—
314C2×C4×D4—138no—
315C2×C4×D4—138no—
316C2×C4×D4—134no—
317C2×C4×D4—134no—
318C2×C4×D4—134no—
319C2×#117—134no—
320C2×#117—134no—
321C2×#117C2×D1312yes—
322C22×C4⋊C4—132no—
323C22×C4⋊C4—131no—
324C22×C4⋊C4—134no—
325C22×C4⋊C4—134no—
326C22×C4⋊C4—1316no—
327C22×C4⋊C4—134no—
328C22×C4⋊C4C2×D1311yescharacteristic
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—28no—
332C22×C13C22×#4114yes—
333C22×C13C22×#4112yes—
334C22×C13C2×H(5,𝔽2)12yes—
335C22×C13C2×2−1+412yes—
336C22×C13C2611yescharacteristic
337C2×D13—232no—
338C2×D13—22no—
339C2×D13—22no—
340C2×D13—22no—
341C2×D13—22no—
342C2×D13—28no—
343C2×D13—28no—
344C2×D13—28no—
345C2×D13—28no—
346C2×D13—28no—
347C2×D13—416no—
348C2×D13C2×H(5,𝔽2)14yes—
349C2×D13C2×2−1+414yes—
350C4×C13—28no—
351C4×C13—28no—
352C4×C13—28no—
353C4×C13C22×#4118yes—
354Dic13—28no—
355Dic13—24no—
356Dic13—28no—
357C25—131no—
358C2×#41—1332no—
359C2×#41—1316no—
360C2×#41—2616no—
361C2×#41—5216no—
362C22×Q8—132no—
363C22×Q8C22×D1314yes—
364C22×D4—134no—
365C22×D4—134no—
366C22×D4—132no—
367C22×D4—134no—
368C22×D4—134no—
369C23×C4—131no—
370C23×C4—132no—
371C23×C4—134no—
372C23×C4—132no—
373C23×C4—132no—
374C23×C4—138no—
375C23×C4—134no—
376C23×C4—134no—
377C23×C4—1316no—
378C23×C4—138no—
379C23×C4—268no—
380C23×C4—528no—
381C23×C4C22×D1312yes—
382C23×C4C22×D1311yescharacteristic
383C4⋊Q8C22×D13132yescomplemented
384#126—1364no—
385#125—1364no—
386#125C22×D13132yescomplemented
387#123—1364no—
388#123—1364no—
389#123—1332no—
390C22⋊Q8—1332no—
391C22⋊Q8—1364no—
392C22⋊Q8—1316no—
393C22⋊Q8—1316no—
394C22⋊Q8—1332no—
395C22⋊Q8C22×D13132yes—
396#121—1332no—
397#121—1316no—
398#121—1332no—
399#121—1316no—
400#121—1332no—
401#120—1332no—
402C4×Q8—1332no—
403C4×D4—1364no—
404C4×D4—1364no—
405C4×D4—1332no—
406C4×D4—1332no—
407C4×D4—1332no—
408#117—1332no—
409#117—1332no—
410#117C22×D13116yes—
411C2×C4⋊C4—134no—
412C2×C4⋊C4—138no—
413C2×C4⋊C4—138no—
414C2×C4⋊C4—1316no—
415C2×C4⋊C4—138no—
416C2×C4⋊C4—134no—
417C2×C4⋊C4—134no—
418C2×C4⋊C4—134no—
419C2×C4⋊C4—1332no—
420C2×C4⋊C4—1332no—
421C2×C4⋊C4—1332no—
422C2×C4⋊C4C22×D1314yes—
423C2×C4⋊C4C22×D1318yes—
424C2×C4⋊C4C22×D1312yes—
425C2×C4⋊C4C22×D1312yes—
426C2×C4⋊C4C22×D1314yes—
427C2×C4⋊C4C22×D1318yes—
428C2×C22⋊C4—134no—
429C2×C22⋊C4—138no—
430C2×C22⋊C4—134no—
431C2×C22⋊C4—138no—
432C2×C22⋊C4—134no—
433C2×C22⋊C4—134no—
434C2×C22⋊C4C22×D1314yes—
435C2×C42—138no—
436C2×C42C22×D1314yes—
437C2×C13—24no—
438C2×C13#349214yescomplemented
439C2×C13C23×#4111yescharacteristic
440C2×C13C22×H(5,𝔽2)11yescharacteristic
441C2×C13C22×2−1+411yescharacteristic
442D13—24no—
443D13—24no—
444D13—48no—
445C24—131no—
446C24—131no—
447C24—134no—
448C24—134no—
449C24—264no—
450C24—264no—
451C24—264no—
452C24C23×D1311yescharacteristic
453#41—2664no—
454#41—5264no—
455C2×Q8—48no—
456C2×Q8—138no—
457C2×Q8—268no—
458C2×Q8C23×D13116yes—
459C2×D4—1316no—
460C2×D4—1316no—
461C2×D4—138no—
462C2×D4—1316no—
463C2×D4—1316no—
464C2×D4—2616no—
465C2×D4—2616no—
466C2×D4—2616no—
467C2×D4—268no—
468C2×D4—528no—
469C22×C4—44no—
470C22×C4—134no—
471C22×C4—134no—
472C22×C4—134no—
473C22×C4—132no—
474C22×C4—138no—
475C22×C4—1316no—
476C22×C4—138no—
477C22×C4—138no—
478C22×C4—1316no—
479C22×C4—1316no—
480C22×C4—1332no—
481C22×C4—2616no—
482C22×C4—264no—
483C22×C4—268no—
484C22×C4—268no—
485C22×C4—2616no—
486C22×C4—2632no—
487C22×C4—528no—
488C22×C4—5216no—
489C22×C4—5232no—
490C22×C4—524no—
491C22×C4#613118yescomplemented
492C22×C4C23×D1314yes—
493C22×C4C23×D1314yes—
494C22×C4C23×D1314yes—
495C22×C4C23×D1318yes—
496C22×C4C23×D1312yes—
497C22×C4C23×D1314yes—
498C4⋊C4—1316no—
499C4⋊C4—1332no—
500C4⋊C4—1332no—
501C4⋊C4—1316no—
502C4⋊C4—1316no—
503C4⋊C4—1316no—
504C4⋊C4C23×D13116yes—
505C4⋊C4C23×D1318yes—
506C4⋊C4C23×D1318yes—
507C4⋊C4C23×D13116yes—
508C4⋊C4C23×D13132yes—
509C22⋊C4—1316no—
510C22⋊C4—1332no—
511C22⋊C4—1316no—
512C22⋊C4—1332no—
513C22⋊C4—1316no—
514C22⋊C4—1316no—
515C22⋊C4C23×D13116yes—
516C42—1332no—
517C42C23×D13116yes—
518C13C2×#349211yescharacteristic, complemented, Sylow
519C23—22no—
520C23—22no—
521C23—134no—
522C23—134no—
523C23—132no—
524C23—132no—
525C23—138no—
526C23—268no—
527C23—268no—
528C23—2632no—
529C23—268no—
530C23—262no—
531C23—262no—
532C23—262no—
533C23—262no—
534C23—268no—
535C23—268no—
536C23—268no—
537C23C2×#613114yes—
538C23C24×D1312yes—
539C23C24×D1311yescharacteristic
540Q8—416no—
541Q8—2616no—
542D4—2632no—
543D4—2632no—
544D4—2632no—
545D4—2616no—
546D4—5216no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—48no—
551C2×C4—138no—
552C2×C4—138no—
553C2×C4—134no—
554C2×C4—2632no—
555C2×C4—268no—
556C2×C4—264no—
557C2×C4—268no—
558C2×C4—268no—
559C2×C4—2616no—
560C2×C4—2616no—
561C2×C4—2632no—
562C2×C4—5216no—
563C2×C4—5232no—
564C2×C4—528no—
565C2×C4C2×#613118yes—
566C2×C4C2×#6131116yescomplemented
567C2×C4C24×D1318yes—
568C2×C4C24×D1318yes—
569C2×C4C24×D1318yes—
570C2×C4C24×D1314yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—134no—
575C22—134no—
576C22—2632no—
577C22—268no—
578C22—268no—
579C22—262no—
580C22—262no—
581C22—262no—
582C22—262no—
583C22—268no—
584C22—268no—
585C22—268no—
586C22—5216no—
587C22C22×#613114yes—
588C22C22×#613112yes—
589C22D13×H(5,𝔽2)12yes—
590C22D13×2−1+412yes—
591C22C25×D1311yescharacteristic
592C4—28no—
593C4—28no—
594C4—28no—
595C4—264no—
596C4—268no—
597C4—268no—
598C4C22×#613118yes—
599C2—24no—
600C2—264no—
601C2—264no—
602C2—528no—
603C2#1312854614yesdirect product factor
604C2C23×#613111yescharacteristic
605C2C2×D13×H(5,𝔽2)11yescharacteristic
606C2C2×D13×2−1+411yescharacteristic
subgroupG/H
centerC23C24×D13
commutator, cosocleC22×C13C26
FrattiniC22C25×D13
FittingC2×C13×#544C2
socleC23×C13C25
hypercenterC2×#544D13
hypocenterC13C2×#3492
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. #13128546C2direct (N × H)
2. C2×#11719435C2semidirect
3. C2×#11719400C2semidirect
4. C2×#11719399C2semidirect
5. C2×#11719431C2semidirect
6. C2×#11719350C2semidirect
7. C2×#11719215C2semidirect
8. C2×#11719332C2semidirect
9. C2×#11719229C2semidirect
10. C2×#11719228C2semidirect
11. C22×C4⋊C4×D13C2semidirect
12. C2×C13×#544C2semidirect
17. C13×#544C22semidirect
18. C2×C13×C4⋊Q8C22semidirect
19. C2×C13×#125C22semidirect
24. C2×#88200C22semidirect
28. #11719435C22semidirect
29. #11719431C22semidirect
30. #11719400C22semidirect
31. #11719399C22semidirect
32. #11719350C22semidirect
33. #11719332C22semidirect
34. #11719229C22semidirect
35. #11719228C22semidirect
36. C2×#88152C22semidirect
37. #11719215C22semidirect
40. C2×#88151C22semidirect
44. C2×#88149C22semidirect
46. C2×#88148C22semidirect
49. C2×#88146C22semidirect
52. C2×C4⋊C4×D13C22semidirect
53. C2×C4⋊C4×D13C22semidirect
54. C2×#88144C22semidirect
60. C2×C4×D52C22semidirect
61. C2×C4×Dic26C22semidirect
88. C13×C4⋊Q8C23semidirect
89. C13×#125C23semidirect
101. #88200C23semidirect
118. #88152C23semidirect
121. #88151C23semidirect
125. #88149C23semidirect
127. #88148C23semidirect
130. #88146C23semidirect
133. C4⋊C4×D13C23semidirect
134. #88144C23semidirect
140. C4×D52C23semidirect
141. C4×Dic26C23semidirect
154. C22×C4×C13#41semidirect
221. C2×#544D13semidirect
255. C2×C4×C13C2×#41semidirect
288. #544C2×D13semidirect
291. C2×C4⋊Q8C2×D13semidirect
294. C2×#125C2×D13semidirect
383. C4⋊Q8C22×D13semidirect
386. #125C22×D13semidirect
438. C2×C13#3492semidirect
491. C22×C4#6131semidirect
518. C13C2×#3492semidirect
566. C2×C4C2×#6131semidirect
603. C2#13128546direct (N × H)

Maximal subgroups

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

subgroupindex
#131285462
C2×#117194352
C2×#117194002
C2×#117193992
C2×#117194312
C2×#117193502
C2×#117192152
C2×#117193322
C2×#117192292
C2×#117192282
C22×C4⋊C4×D132
C2×C13×#5442
C2×#349213