Finite Group Databasebeta

← C2×#20257098 #27124220

Subgroups — C2×#20257098 #27124220

Order 12032.

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#20257098C2132yesmaximal, direct product factor
2C2×#15689834C214yesmaximal, complemented
3C2×#15689826C214yesmaximal, complemented
4C2×#15689811C214yesmaximal, complemented
5C2×#15689776C214yesmaximal, complemented
6C2×#15689722C214yesmaximal, complemented
7C2×#15689704C211yesmaximal, characteristic, complemented
8C2×#15689636C214yesmaximal, complemented
9C2×#15689635C212yesmaximal, complemented
10C2×#15689603C212yesmaximal, complemented
11C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
12C2×C47×#544C211yesmaximal, characteristic, complemented
13C22×#10625354C2214yes—
14C22×#10625350C2212yes—
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2212yes—
17#15689834C22164yescomplemented
18#15689826C22164yescomplemented
19#15689811C22164yescomplemented
20#15689776C22164yescomplemented
21#15689722C22164yescomplemented
22#15689704C22116yescomplemented
23C2×#13094019C2218yescomplemented
24C2×#13094015C2218yescomplemented
25#15689636C22164yescomplemented
26#15689635C22132yescomplemented
27C2×#13094009C2218yes—
28C2×#13094009C2218yescomplemented
29C2×#13094008C2214yes—
30C2×#13094004C2212yes—
31C2×#13094004C2212yes—
32C2×#13094004C2214yescomplemented
33C2×#13094000C2214yes—
34C2×#13094000C2218yescomplemented
35C2×#13093999C2212yes—
36C2×#13093999C2212yes—
37C2×#13093999C2214yes—
38C2×#13093999C2214yescomplemented
39C2×#13093998C2214yes—
40C2×#13093994C2214yes—
41#15689603C22132yescomplemented
42C2×C4×Dic94C2214yescomplemented
43C2×C4×D188C2214yes—
44C2×C4×D188C2214yes—
45C2×C4×D188C2214yescomplemented
46C2×#13093986C2218yes—
47C2×(C22⋊D47)⋊C4C2218yes—
48C2×#13093982C2214yes—
49C2×#13093982C2214yescomplemented
50C22×C4⋊Dic47C2211yescharacteristic
51C22×Dic47⋊C4C2212yes—
52C2×C4⋊C4×D47C2214yes—
53C2×C4⋊C4×D47C2214yes—
54C2×C4⋊C4×D47C22116yescomplemented
55C2×C4⋊C4×D47C2214yescomplemented
56C47×#544C22116yescomplemented
57C2×C4⋊Q8×C47C2214yescomplemented
58C2×#125×C47C2214yescomplemented
59C2×C22⋊Q8×C47C2214yes—
60C2×#117×C47C2212yes—
61C22×C4⋊C4×C47C2211yescharacteristic
62C24×D47C2311yescharacteristic
63C2×#10625354—416no—
64C2×#10625354C23132yes—
65C2×#10625350—216no—
66C2×#10625350C23116yes—
67C22×Dic94C2312yes—
68C22×D188C2314yes—
69C22×D188C2314yes—
70C22×D188C2312yes—
71C22×D188C2314yes—
72C22×C22⋊D47C2314yes—
73C23×Dic47C2311yescharacteristic
74C23×Dic47C2312yes—
75C22×C4×D47—28no—
76C22×C4×D47—48no—
77C22×C4×D47C2314yes—
78C22×C4×D47C2312yes—
79C22×C4×D47C2312yes—
80C22×C4×D47C2318yes—
81C22×C4×D47C2314yes—
82C22×C4×D47C2314yes—
83C22×C4×D47C23116yes—
84C22×C4×D47C2318yes—
85#13094019C23164yescomplemented
86#13094015C23164yescomplemented
87#13094009C23164yes—
88#13094009C23164yescomplemented
89#13094008C23132yes—
90#13094004C23116yes—
91#13094004C23116yes—
92#13094004C23132yescomplemented
93#13094000C23132yes—
94#13094000C23164yescomplemented
95#13093999C23116yes—
96#13093999C23116yes—
97#13093999C23132yes—
98#13093999C23132yescomplemented
99#13093998C23132yes—
100#13093994C23132yes—
101C4×Dic94C23132yescomplemented
102C4×D188C23132yes—
103C4×D188C23132yes—
104C4×D188C23132yescomplemented
105#13093986C23164yes—
106(C22⋊D47)⋊C4C23164yes—
107#13093982C23132yes—
108#13093982C23132yescomplemented
109C2×C4⋊Dic47C2314yes—
110C2×C4⋊Dic47C2318yes—
111C2×C4⋊Dic47C2314yes—
112C2×C4⋊Dic47C2314yes—
113C2×C4⋊Dic47C2314yes—
114C2×Dic47⋊C4C2318yes—
115C2×Dic47⋊C4C23116yes—
116C2×Dic47⋊C4C2318yes—
117C4⋊C4×D47C23132yes—
118C4⋊C4×D47C23132yes—
119C4⋊C4×D47C23132yescomplemented
120C2×(C2×D47)⋊C4C2314yes—
121C2×(C2×D47)⋊C4C2318yes—
122C2×(C2×D47)⋊C4C2314yes—
123C2×(C2×D47)⋊C4C2318yes—
124C2×(C2×D47)⋊C4C2314yes—
125C2×(C2×D47)⋊C4C2314yes—
126C2×C4×Dic47C2318yes—
127C22×Q8×C47C2314yes—
128C23×C4×C47C2312yes—
129C23×C4×C47C2311yescharacteristic
130C4⋊Q8×C47C23132yescomplemented
131#125×C47C23132yescomplemented
132C22⋊Q8×C47C23132yes—
133#117×C47C23116yes—
134C2×C4⋊C4×C47C2314yes—
135C2×C4⋊C4×C47C2318yes—
136C2×C4⋊C4×C47C2312yes—
137C2×C4⋊C4×C47C2312yes—
138C2×C4⋊C4×C47C2314yes—
139C2×C4⋊C4×C47C2318yes—
140C2×C22⋊C4×C47C2314yes—
141C2×C42×C47C2314yes—
142C24×C47C2411yescharacteristic
143C23×D47—24no—
144C23×D47—24no—
145C23×D47—24no—
146C23×D47C2414yes—
147C23×D47C2414yes—
148C23×D47C2411yescharacteristic
149C23×D47C2411yescharacteristic
150C2×Q8×C47—48no—
151C2×Q8×C47C24116yes—
152C22×C4×C47—44no—
153C22×C4×C47#4118yescomplemented
154C22×C4×C47C2414yes—
155C22×C4×C47C2414yes—
156C22×C4×C47C2414yes—
157C22×C4×C47C2418yes—
158C22×C4×C47C2412yes—
159C22×C4×C47C2414yes—
160C2×C22⋊D47—216no—
161C2×C22⋊D47C24116yes—
162C22×Dic47—24no—
163C22×Dic47—44no—
164C22×Dic47C2414yes—
165C22×Dic47C2414yes—
166C22×Dic47C2412yes—
167C22×Dic47C2414yes—
168C22×Dic47C2418yes—
169#10625354—464no—
170#10625350—264no—
171C2×D188—216no—
172C2×D188—216no—
173C2×D188—28no—
174C2×D188—48no—
175C2×D188C24116yes—
176C2×D188C24116yes—
177C2×D188C2418yes—
178C2×D188C24116yes—
179C2×C4×D47—216no—
180C2×C4×D47—28no—
181C2×C4×D47—28no—
182C2×C4×D47—216no—
183C2×C4×D47—232no—
184C2×C4×D47—432no—
185C2×C4×D47—48no—
186C2×C4×D47—416no—
187C2×C4×D47C24116yes—
188C2×C4×D47C2418yes—
189C2×C4×D47C2418yes—
190C2×C4×D47C24116yes—
191C2×C4×D47C24116yes—
192C2×C4×D47C24132yes—
193C2×Dic94—28no—
194C2×Dic94C2418yes—
195C4⋊C4×C47C24116yes—
196C4⋊C4×C47C2418yes—
197C4⋊C4×C47C2418yes—
198C4⋊C4×C47C24116yes—
199C4⋊C4×C47C24132yes—
200C22⋊C4×C47C24116yes—
201C42×C47C24116yes—
202(C2×D47)⋊C4C24116yes—
203(C2×D47)⋊C4C24132yes—
204(C2×D47)⋊C4C24116yes—
205(C2×D47)⋊C4C24132yes—
206(C2×D47)⋊C4C24116yes—
207(C2×D47)⋊C4C24116yes—
208C4⋊Dic47C24116yes—
209C4⋊Dic47C24116yes—
210C4⋊Dic47C24116yes—
211C4⋊Dic47C24116yes—
212Dic47⋊C4C24132yes—
213Dic47⋊C4C24132yes—
214C4×Dic47C24132yes—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47C2×#4114yes—
218C23×C47C2512yes—
219C23×C47C2511yescharacteristic
220C22×D47—232no—
221C22×D47—28no—
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—28no—
227C22×D47—28no—
228C22×D47—28no—
229C22×D47—28no—
230C22×D47—28no—
231C22×D47H(5,𝔽2)14yes—
232C22×D472−1+414yes—
233C22×D47C2518yes—
234C22×D47C2512yes—
235C22×D47C2512yes—
236Q8×C47—416no—
237C2×C4×C47—28no—
238C2×C4×C47—28no—
239C2×C4×C47—28no—
240C2×C4×C47—48no—
241C2×C4×C47C2×#4118yes—
242C2×C4×C47C2×#41116yescomplemented
243C2×C4×C47C2518yes—
244C2×C4×C47C2518yes—
245C2×C4×C47C2518yes—
246C2×C4×C47C2514yes—
247C22⋊D47—232no—
248C2×Dic47—28no—
249C2×Dic47—28no—
250C2×Dic47—24no—
251C2×Dic47—28no—
252C2×Dic47—48no—
253C2×Dic47C2518yes—
254C2×Dic47C2518yes—
255C2×Dic47C2514yes—
256D188—232no—
257D188—232no—
258D188—216no—
259D188—416no—
260C4×D47—232no—
261C4×D47—216no—
262C4×D47—216no—
263C4×D47—232no—
264C4×D47—416no—
265C4×D47—432no—
266Dic94—216no—
267C2×#3492—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—28no—
271C22×C47C22×#4114yes—
272C22×C47C22×#4112yes—
273C22×C47C2×H(5,𝔽2)12yes—
274C22×C47C2×2−1+412yes—
275C22×C47C2611yescharacteristic
276C2×D47—232no—
277C2×D47—22no—
278C2×D47—22no—
279C2×D47—22no—
280C2×D47—22no—
281C2×D47—28no—
282C2×D47—28no—
283C2×D47—28no—
284C2×D47—28no—
285C2×D47—28no—
286C2×D47—416no—
287C2×D47C2×H(5,𝔽2)14yes—
288C2×D47C2×2−1+414yes—
289C4×C47—28no—
290C4×C47—28no—
291C4×C47—28no—
292C4×C47C22×#4118yes—
293Dic47—28no—
294Dic47—24no—
295Dic47—28no—
296#3492—4732no—
297C2×#554—474no—
298C2×#553—474no—
299C2×#552—474no—
300C2×#547—474no—
301C2×#544D4711yescharacteristic, complemented
302C2×#539—474no—
303C2×#537—471no—
304C2×#525—474no—
305C2×#525—472no—
306C2×#520—472no—
307C23×C4⋊C4—471no—
308C2×C47—24no—
309C2×C47#349214yescomplemented
310C2×C47C23×#4111yescharacteristic
311C2×C47C22×H(5,𝔽2)11yescharacteristic
312C2×C47C22×2−1+411yescharacteristic
313D47—24no—
314D47—24no—
315D47—48no—
316C22×#41—474no—
317C22×#41—472no—
318C24×C4—471no—
319C24×C4—472no—
320#554—4764no—
321#553—4764no—
322#552—4764no—
323#547—4764no—
324#544C2×D47116yescomplemented
325#539—4764no—
326#537—4716no—
327C2×C4⋊Q8C2×D4714yescomplemented
328C2×#126—478no—
329C2×#125—478no—
330C2×#125C2×D4714yescomplemented
331#525—4764no—
332#525—4732no—
333C2×#123—478no—
334C2×#123—478no—
335C2×#123—474no—
336C2×C22⋊Q8—474no—
337C2×C22⋊Q8—478no—
338C2×C22⋊Q8—472no—
339C2×C22⋊Q8—472no—
340C2×C22⋊Q8—474no—
341C2×C22⋊Q8C2×D4714yes—
342C2×#121—474no—
343C2×#121—472no—
344C2×#121—474no—
345C2×#121—472no—
346C2×#121—474no—
347C2×#120—474no—
348#520—4732no—
349C2×C4×Q8—474no—
350C2×C4×D4—478no—
351C2×C4×D4—478no—
352C2×C4×D4—474no—
353C2×C4×D4—474no—
354C2×C4×D4—474no—
355C2×#117—474no—
356C2×#117—474no—
357C2×#117C2×D4712yes—
358C22×C4⋊C4—471no—
359C22×C4⋊C4—472no—
360C22×C4⋊C4—474no—
361C22×C4⋊C4—474no—
362C22×C4⋊C4—4716no—
363C22×C4⋊C4—474no—
364C22×C4⋊C4C2×D4711yescharacteristic
365C47C2×#349211yescharacteristic, complemented, Sylow
366C25—471no—
367C2×#41—4732no—
368C2×#41—4716no—
369C2×#41—9416no—
370C2×#41—18816no—
371C22×Q8—472no—
372C22×Q8C22×D4714yes—
373C22×D4—474no—
374C22×D4—474no—
375C22×D4—472no—
376C22×D4—474no—
377C22×D4—474no—
378C23×C4—471no—
379C23×C4—472no—
380C23×C4—474no—
381C23×C4—472no—
382C23×C4—472no—
383C23×C4—478no—
384C23×C4—474no—
385C23×C4—474no—
386C23×C4—4716no—
387C23×C4—478no—
388C23×C4—948no—
389C23×C4—1888no—
390C23×C4C22×D4712yes—
391C23×C4C22×D4711yescharacteristic
392C4⋊Q8C22×D47132yescomplemented
393#126—4764no—
394#125—4764no—
395#125C22×D47132yescomplemented
396#123—4764no—
397#123—4764no—
398#123—4732no—
399C22⋊Q8—4732no—
400C22⋊Q8—4764no—
401C22⋊Q8—4716no—
402C22⋊Q8—4716no—
403C22⋊Q8—4732no—
404C22⋊Q8C22×D47132yes—
405#121—4732no—
406#121—4716no—
407#121—4732no—
408#121—4716no—
409#121—4732no—
410#120—4732no—
411C4×Q8—4732no—
412C4×D4—4764no—
413C4×D4—4764no—
414C4×D4—4732no—
415C4×D4—4732no—
416C4×D4—4732no—
417#117—4732no—
418#117—4732no—
419#117C22×D47116yes—
420C2×C4⋊C4—474no—
421C2×C4⋊C4—478no—
422C2×C4⋊C4—478no—
423C2×C4⋊C4—4716no—
424C2×C4⋊C4—478no—
425C2×C4⋊C4—474no—
426C2×C4⋊C4—474no—
427C2×C4⋊C4—474no—
428C2×C4⋊C4—4732no—
429C2×C4⋊C4—4732no—
430C2×C4⋊C4—4732no—
431C2×C4⋊C4C22×D4714yes—
432C2×C4⋊C4C22×D4718yes—
433C2×C4⋊C4C22×D4712yes—
434C2×C4⋊C4C22×D4712yes—
435C2×C4⋊C4C22×D4714yes—
436C2×C4⋊C4C22×D4718yes—
437C2×C22⋊C4—474no—
438C2×C22⋊C4—478no—
439C2×C22⋊C4—474no—
440C2×C22⋊C4—478no—
441C2×C22⋊C4—474no—
442C2×C22⋊C4—474no—
443C2×C22⋊C4C22×D4714yes—
444C2×C42—478no—
445C2×C42C22×D4714yes—
446C24—471no—
447C24—471no—
448C24—474no—
449C24—474no—
450C24—944no—
451C24—944no—
452C24—944no—
453C24C23×D4711yescharacteristic
454#41—9464no—
455#41—18864no—
456C2×Q8—48no—
457C2×Q8—478no—
458C2×Q8—948no—
459C2×Q8C23×D47116yes—
460C2×D4—4716no—
461C2×D4—4716no—
462C2×D4—478no—
463C2×D4—4716no—
464C2×D4—4716no—
465C2×D4—9416no—
466C2×D4—9416no—
467C2×D4—9416no—
468C2×D4—948no—
469C2×D4—1888no—
470C22×C4—44no—
471C22×C4—474no—
472C22×C4—474no—
473C22×C4—474no—
474C22×C4—472no—
475C22×C4—478no—
476C22×C4—4716no—
477C22×C4—478no—
478C22×C4—478no—
479C22×C4—4716no—
480C22×C4—4716no—
481C22×C4—4732no—
482C22×C4—9416no—
483C22×C4—944no—
484C22×C4—948no—
485C22×C4—948no—
486C22×C4—9416no—
487C22×C4—9432no—
488C22×C4—1888no—
489C22×C4—18816no—
490C22×C4—18832no—
491C22×C4—1884no—
492C22×C4#1062535418yescomplemented
493C22×C4C23×D4714yes—
494C22×C4C23×D4714yes—
495C22×C4C23×D4714yes—
496C22×C4C23×D4718yes—
497C22×C4C23×D4712yes—
498C22×C4C23×D4714yes—
499C4⋊C4—4732no—
500C4⋊C4—4732no—
501C4⋊C4—4716no—
502C4⋊C4—4716no—
503C4⋊C4—4716no—
504C4⋊C4—4716no—
505C4⋊C4C23×D47116yes—
506C4⋊C4C23×D4718yes—
507C4⋊C4C23×D4718yes—
508C4⋊C4C23×D47116yes—
509C4⋊C4C23×D47132yes—
510C22⋊C4—4716no—
511C22⋊C4—4732no—
512C22⋊C4—4716no—
513C22⋊C4—4732no—
514C22⋊C4—4716no—
515C22⋊C4—4716no—
516C22⋊C4C23×D47116yes—
517C42—4732no—
518C42C23×D47116yes—
519C23—22no—
520C23—22no—
521C23—474no—
522C23—474no—
523C23—472no—
524C23—472no—
525C23—478no—
526C23—948no—
527C23—948no—
528C23—9432no—
529C23—948no—
530C23—942no—
531C23—942no—
532C23—942no—
533C23—942no—
534C23—948no—
535C23—948no—
536C23—948no—
537C23C2×#1062535414yes—
538C23C24×D4712yes—
539C23C24×D4711yescharacteristic
540Q8—416no—
541Q8—9416no—
542D4—9432no—
543D4—9432no—
544D4—9432no—
545D4—9416no—
546D4—18816no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—48no—
551C2×C4—478no—
552C2×C4—478no—
553C2×C4—474no—
554C2×C4—9432no—
555C2×C4—948no—
556C2×C4—944no—
557C2×C4—948no—
558C2×C4—948no—
559C2×C4—9416no—
560C2×C4—9416no—
561C2×C4—9432no—
562C2×C4—18816no—
563C2×C4—18832no—
564C2×C4—1888no—
565C2×C4C2×#1062535418yes—
566C2×C4C2×#10625354116yescomplemented
567C2×C4C24×D4718yes—
568C2×C4C24×D4718yes—
569C2×C4C24×D4718yes—
570C2×C4C24×D4714yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—474no—
575C22—474no—
576C22—9432no—
577C22—948no—
578C22—948no—
579C22—942no—
580C22—942no—
581C22—942no—
582C22—942no—
583C22—948no—
584C22—948no—
585C22—948no—
586C22—18816no—
587C22C22×#1062535414yes—
588C22C22×#1062535412yes—
589C22H(5,𝔽2)×D4712yes—
590C222−1+4×D4712yes—
591C22C25×D4711yescharacteristic
592C4—28no—
593C4—28no—
594C4—28no—
595C4—944no—
596C4—948no—
597C4—948no—
598C4C22×#1062535418yes—
599C2—24no—
600C2—944no—
601C2—944no—
602C2—1888no—
603C2#2025709814yesdirect product factor
604C2C23×#1062535411yescharacteristic
605C2C2×H(5,𝔽2)×D4711yescharacteristic
606C2C2×2−1+4×D4711yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#544C2
socleC23×C47C25
hypercenterC2×#544D47
hypocenterC47C2×#3492
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. #20257098C2direct (N × H)
2. C2×#15689834C2semidirect
3. C2×#15689826C2semidirect
4. C2×#15689811C2semidirect
5. C2×#15689776C2semidirect
6. C2×#15689722C2semidirect
7. C2×#15689704C2semidirect
8. C2×#15689636C2semidirect
9. C2×#15689635C2semidirect
10. C2×#15689603C2semidirect
11. C22×C4⋊C4×D47C2semidirect
12. C2×C47×#544C2semidirect
17. #15689834C22semidirect
18. #15689826C22semidirect
19. #15689811C22semidirect
20. #15689776C22semidirect
21. #15689722C22semidirect
22. #15689704C22semidirect
23. C2×#13094019C22semidirect
24. C2×#13094015C22semidirect
25. #15689636C22semidirect
26. #15689635C22semidirect
28. C2×#13094009C22semidirect
32. C2×#13094004C22semidirect
34. C2×#13094000C22semidirect
38. C2×#13093999C22semidirect
41. #15689603C22semidirect
42. C2×C4×Dic94C22semidirect
45. C2×C4×D188C22semidirect
49. C2×#13093982C22semidirect
54. C2×C4⋊C4×D47C22semidirect
55. C2×C4⋊C4×D47C22semidirect
56. C47×#544C22semidirect
57. C2×C4⋊Q8×C47C22semidirect
58. C2×#125×C47C22semidirect
85. #13094019C23semidirect
86. #13094015C23semidirect
88. #13094009C23semidirect
92. #13094004C23semidirect
94. #13094000C23semidirect
98. #13093999C23semidirect
101. C4×Dic94C23semidirect
104. C4×D188C23semidirect
108. #13093982C23semidirect
119. C4⋊C4×D47C23semidirect
130. C4⋊Q8×C47C23semidirect
131. #125×C47C23semidirect
153. C22×C4×C47#41semidirect
242. C2×C4×C47C2×#41semidirect
301. C2×#544D47semidirect
309. C2×C47#3492semidirect
324. #544C2×D47semidirect
327. C2×C4⋊Q8C2×D47semidirect
330. C2×#125C2×D47semidirect
365. C47C2×#3492semidirect
392. C4⋊Q8C22×D47semidirect
395. #125C22×D47semidirect
492. C22×C4#10625354semidirect
566. C2×C4C2×#10625354semidirect
603. C2#20257098direct (N × H)

Maximal subgroups

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

subgroupindex
#202570982
C2×#156898342
C2×#156898262
C2×#156898112
C2×#156897762
C2×#156897222
C2×#156897042
C2×#156896362
C2×#156896352
C2×#156896032
C22×C4⋊C4×D472
C2×C47×#5442
C2×#349247