Finite Group Databasebeta

← C2×#23763623 #34289120

Subgroups — C2×#23763623 #34289120

Order 20224.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099043C211yesmaximal, characteristic
2C22×#14579087C211yesmaximal, characteristic
3C2×#19098562C211yesmaximal, characteristic
4C2×#19098432C211yesmaximal, characteristic, complemented
5#23763623C218yesmaximal, direct product factor
6C2×#19098331C211yesmaximal, characteristic, complemented
7C2×#19098322C211yesmaximal, characteristic, complemented
8C2×#428×C79C211yesmaximal, characteristic, complemented
9C22×#11929624C2211yescharacteristic
10C22×#11929608C2211yescharacteristic
11#19099043—22no—
12#19099043—22no—
13#19099043—22no—
14#19099043C2214yes—
15C2×C4×Dic158—21no—
16C2×C4×D316—21no—
17C2×(C22⋊D79)⋊C4—22no—
18C2×#14579119—21no—
19C22×C4⋊Dic79C2211yescharacteristic
20C22×Dic79⋊C4—22no—
21C2×C4⋊C4×D79—21no—
22C22×C4×Dic79C2211yescharacteristic
23C2×#14579087C418yescomplemented
24C2×#14579087C2214yes—
25#19098562—22no—
26#19098562C2214yes—
27C2×#14579084—21no—
28C2×#14579081—21no—
29#19098432—22no—
30#19098432C2214yescomplemented
31#19098331C2214yescomplemented
32#19098322C2214yescomplemented
33C22×M16×C79C2211yescharacteristic
34C2×#117×C79C2211yescharacteristic
35C22×C4⋊C4×C79C2211yescharacteristic
36#428×C79—22no—
37#428×C79C2214yescomplemented
38C2×#107×C79—21no—
39C2×#106×C79—21no—
40C2×#11929624—21no—
41C2×#11929624—21no—
42C2×#11929624—22no—
43C2×#11929624—22no—
44C2×#11929624C2×C414yes—
45C2×#11929624D412yes—
46C2×#11929624Q814yes—
47C2×#11929624C2312yes—
48C22×Dic158D411yescharacteristic
49C22×D316D411yescharacteristic
50C22×C22⋊D79—21no—
51C23×Dic79—21no—
52C23×Dic79D412yes—
53C22×C4×D79—21no—
54C22×C4×D79—22no—
55C2×#11929608—21no—
56C2×#11929608—22no—
57C2×#11929608—22no—
58C2×#11929608C2312yes—
59C22×C79⋊C8—21no—
60C4×Dic158—24no—
61C4×Dic158—24no—
62C4×D316—24no—
63C4×D316—24no—
64(C22⋊D79)⋊C4—28no—
65(C22⋊D79)⋊C4—28no—
66#14579119—24no—
67#14579119—24no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79D412yes—
73C2×C4⋊Dic79C2312yes—
74C2×Dic79⋊C4—22no—
75C2×Dic79⋊C4—28no—
76C2×Dic79⋊C4—48no—
77C4⋊C4×D79—24no—
78C4⋊C4×D79—24no—
79C2×(C2×D79)⋊C4—22no—
80C2×C4×Dic79—22no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—22no—
83C2×C4×Dic79C2×C414yescomplemented
84C2×C4×Dic79C2312yes—
85#14579087C2×C4116yescomplemented
86#14579084—24no—
87#14579081—24no—
88C23×C4×C79—21no—
89C23×C4×C79C2311yescharacteristic
90C2×M16×C79—21no—
91C2×M16×C79—22no—
92C2×M16×C79C2×C414yescomplemented
93C2×M16×C79C2312yes—
94C22×C8×C79—21no—
95#117×C79—21no—
96#117×C79—21no—
97#117×C79D412yes—
98#117×C79C2312yes—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79D411yescharacteristic
105C2×C4⋊C4×C79D411yescharacteristic
106C2×C4⋊C4×C79C2312yes—
107C2×C22⋊C4×C79—21no—
108C2×C42×C79—21no—
109#107×C79—24no—
110#106×C79—24no—
111C23×D79—21no—
112#11929624—24no—
113#11929624—24no—
114#11929624—24no—
115#11929624—24no—
116#11929624—24no—
117#11929624—22no—
118#11929624—22no—
119#11929624C4⋊C418yescomplemented
120#11929624C22×C414yes—
121#11929624C2×Q814yes—
122C2×Dic158—21no—
123C2×Dic158—22no—
124C2×Dic158C4⋊C414yescomplemented
125C2×Dic158C2×D412yes—
126C2×D316—21no—
127C2×D316—22no—
128C2×D316C4⋊C414yescomplemented
129C2×D316C2×D412yes—
130C2×C22⋊D79—24no—
131C2×C22⋊D79—22no—
132C2×C22⋊D79—24no—
133C2×C22⋊D79—22no—
134C22×Dic79—24no—
135C22×Dic79—24no—
136C22×Dic79—24no—
137C22×Dic79—21no—
138C22×Dic79—22no—
139C22×Dic79—22no—
140C22×Dic79—22no—
141C22×Dic79—44no—
142C22×Dic79C2×D414yes—
143C2×C4×D79—28no—
144C2×C4×D79—24no—
145C2×C4×D79—22no—
146C2×C4×D79—24no—
147C2×C4×D79—22no—
148C2×C4×D79—84no—
149#11929608—22no—
150#11929608—22no—
151#11929608—42no—
152C2×C79⋊C8—22no—
153C2×C79⋊C8—42no—
154C4⋊Dic79—22no—
155C4⋊Dic79—22no—
156C4⋊Dic79—22no—
157C4⋊Dic79—22no—
158Dic79⋊C4—28no—
159Dic79⋊C4—416no—
160(C2×D79)⋊C4—28no—
161C4×Dic79—28no—
162C4×Dic79—24no—
163C4×Dic79—22no—
164C4×Dic79C22×C414yescomplemented
165C24×C79C2×D411yescharacteristic
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—22no—
169C22×C4×C79—44no—
170C22×C4×C79—41no—
171C22×C4×C79—41no—
172C22×C4×C79C22×C412yes—
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×Q812yes—
179C22×C4×C79#4112yes—
180C22×C4×C79#4112yes—
181C22×C4×C79C2411yescharacteristic
182M16×C79—24no—
183M16×C79—22no—
184M16×C79C22×C414yescomplemented
185C2×C8×C79—24no—
186C2×C8×C79—22no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—22no—
190C4⋊C4×C79—22no—
191C4⋊C4×C79—41no—
192C4⋊C4×C79—41no—
193C4⋊C4×C79C2×D412yes—
194C4⋊C4×C79C2×D412yes—
195C22⋊C4×C79—22no—
196C22⋊C4×C79—22no—
197C42×C79—22no—
198C42×C79—22no—
199C23×C79—22no—
200C23×C79—22no—
201C23×C79C4×D412yes—
202C23×C79#12011yescharacteristic
203C23×C79#12011yescharacteristic
204C23×C79C22⋊Q812yes—
205C23×C79C22×D411yescharacteristic
206C22×D79—22no—
207C22×D79—24no—
208C22×D79—42no—
209C22×D79—42no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—42no—
213C2×C4×C79—42no—
214C2×C4×C79—44no—
215C2×C4×C79—42no—
216C2×C4×C79—82no—
217C2×C4×C79—82no—
218C2×C4×C79C2×C4⋊C412yes—
219C2×C4×C79C2×C4⋊C412yes—
220C2×C4×C79C4×D412yes—
221C2×C4×C79C4×D412yes—
222C2×C4×C79C4×D412yes—
223C2×C4×C79#12011yescharacteristic
224C2×C4×C79#12011yescharacteristic
225C2×C4×C79C22⋊Q812yes—
226C2×C4×C79C22⋊Q812yes—
227C2×C4×C79C22⋊Q812yes—
228C2×C4×C79#12312yes—
229C2×C4×C79#12312yes—
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C22×D411yescharacteristic
232C2×C4×C79C22×D411yescharacteristic
233C2×C4×C79C22×Q811yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C22⋊D79—24no—
237C22⋊D79—24no—
238C22⋊D79—28no—
239C2×Dic79—24no—
240C2×Dic79—216no—
241C2×Dic79—24no—
242C2×Dic79—24no—
243C2×Dic79—22no—
244C2×Dic79—24no—
245C2×Dic79—24no—
246C2×Dic79—44no—
247C2×Dic79—44no—
248C2×Dic79—82no—
249D316—22no—
250D316—24no—
251D316C2×C4⋊C414yescomplemented
252C4×D79—24no—
253C4×D79—24no—
254C4×D79—28no—
255C4×D79—88no—
256Dic158—22no—
257Dic158—24no—
258Dic158C2×C4⋊C414yescomplemented
259C8×C79—24no—
260C79⋊C8—42no—
261C22×C79—24no—
262C22×C79—24no—
263C22×C79—21no—
264C22×C79—21no—
265C22×C79—21no—
266C22×C79—21no—
267C22×C79#38512yes—
268C22×C79#38512yes—
269C22×C79#45312yes—
270C22×C79#45612yes—
271C22×C79C2×C4×D411yescharacteristic
272C22×C79C2×#12011yescharacteristic
273C22×C79C2×C22⋊Q811yescharacteristic
274C2×D79—24no—
275C2×D79—48no—
276C2×D79—42no—
277C2×D79—42no—
278C4×C79—42no—
279C4×C79—42no—
280C4×C79—82no—
281C4×C79#38512yes—
282C4×C79#38512yes—
283C4×C79C22×C4⋊C411yescharacteristic
284C4×C79C2×C4×D411yescharacteristic
285C4×C79C2×C22⋊Q811yescharacteristic
286C4×C79C2×#12311yescharacteristic
287Dic79—28no—
288Dic79—24no—
289Dic79—44no—
290C2×#1859—791nomaximal, Sylow
291C2×C79—21no—
292C2×C79—21no—
293C2×C79—22no—
294C2×C79#185914yescomplemented
295C2×C79C2×#38511yescharacteristic
296C2×C79C2×#45311yescharacteristic
297C2×C79C2×#45611yescharacteristic
298D79—44no—
299C2×#520—791no—
300C2×#428D7911yescharacteristic, complemented
301C22×#104—791no—
302C2×#417—791no—
303C2×#384—791no—
304#1859—798no—
305C2×#343—791no—
306C2×#339—791no—
307C79C2×#185911yescharacteristic, complemented, Sylow
308C22×#41—791no—
309C22×M16—791no—
310C22×M16C2×D7911yescharacteristic
311#520—794no—
312#520—1582no—
313#520—1582no—
314#520—1582no—
315C2×C4×Q8—1581no—
316C2×C4×D4—1581no—
317C2×C4×D4—1582no—
318C2×#117—1581no—
319C2×#117C2×D7911yescharacteristic
320C22×C4⋊C4—791no—
321C22×C4⋊C4—1582no—
322C22×C4⋊C4—1581no—
323C22×C4⋊C4C2×D7911yescharacteristic
324C22×C42—791no—
325#428—22no—
326#428C2×D7914yescomplemented
327C2×#107—21no—
328C2×#106—21no—
329C2×#104—794no—
330C2×#104—798no—
331#417—794no—
332#417—1582no—
333C2×#103—1581no—
334C2×#102—1581no—
335#384—794no—
336#384—1582no—
337#343—794no—
338#339—794no—
339C2×#41—794no—
340C2×#41—792no—
341C2×#41—794no—
342C2×#41—792no—
343C2×#41—1581no—
344C2×#41—1581no—
345C2×#41—1582no—
346C2×#41—1582no—
347C22×Q8—791no—
348C22×D4—791no—
349C22×D4—1581no—
350C23×C4—21no—
351C23×C4—792no—
352C23×C4—1582no—
353C23×C4—1581no—
354C23×C4—1581no—
355C23×C4C22×D7911yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—792no—
359C2×M16—1582no—
360C2×M16—1582no—
361C2×M16—1581no—
362C2×M16C4×D7914yescomplemented
363C2×M16C22×D7912yes—
364C22×C8—21no—
365C22×C8—1581no—
366C4×Q8—1584no—
367C4×Q8—1584no—
368C4×D4—1584no—
369C4×D4—1584no—
370C4×D4—1588no—
371C4×D4—1588no—
372#117—21no—
373#117—21no—
374#117—1584no—
375#117—1584no—
376#117C22⋊D7912yes—
377#117C22×D7912yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—3168no—
384C2×C4⋊C4—792no—
385C2×C4⋊C4—792no—
386C2×C4⋊C4—1584no—
387C2×C4⋊C4—1588no—
388C2×C4⋊C4—1582no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1584no—
394C2×C4⋊C4C22⋊D7911yescharacteristic
395C2×C4⋊C4C22⋊D7911yescharacteristic
396C2×C4⋊C4C22×D7912yes—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1582no—
399C2×C42—21no—
400C2×C42—794no—
401C2×C42—792no—
402C2×C42—1582no—
403C2×C42—1581no—
404C2×C42—1582no—
405#107—24no—
406#106—24no—
407#104—7916no—
408#103—1584no—
409#102—1584no—
410C24—1581no—
411C24D4×D7911yescharacteristic
412#41—794no—
413#41—798no—
414#41—794no—
415#41—1582no—
416#41—1582no—
417#41—1584no—
418#41—1584no—
419#41—1584no—
420#41—1584no—
421#41—1584no—
422C2×Q8—794no—
423C2×Q8—792no—
424C2×Q8—1581no—
425C2×Q8—1582no—
426C2×D4—794no—
427C2×D4—792no—
428C2×D4—1584no—
429C2×D4—1582no—
430C2×D4—1584no—
431C2×D4—1582no—
432C2×D4—1581no—
433C2×D4—1582no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—3164no—
441C22×C4—794no—
442C22×C4—6324no—
443C22×C4—1588no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1584no—
447C22×C4—1584no—
448C22×C4—1584no—
449C22×C4—1581no—
450C22×C4—1582no—
451C22×C4—1584no—
452C22×C4—1582no—
453C22×C4—1584no—
454C22×C4—1582no—
455C22×C4C2×C4×D7912yes—
456C22×C4D4×D7911yescharacteristic
457C22×C4D4×D7911yescharacteristic
458C22×C4D4×D7911yescharacteristic
459C22×C4C2×C22⋊D7911yescharacteristic
460C22×C4C2×C22⋊D7911yescharacteristic
461C22×C4Q8×D7912yes—
462C22×C4#1192962412yes—
463C22×C4#1192962512yes—
464C22×C4C23×D7911yescharacteristic
465M16—22no—
466M16—24no—
467M16—3162no—
468M16—1582no—
469M16—1582no—
470M16C2×C4×D7914yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—3162no—
474C2×C8—1582no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—31616no—
482C4⋊C4—1588no—
483C4⋊C4—1582no—
484C4⋊C4—1582no—
485C4⋊C4—1582no—
486C4⋊C4—1582no—
487C4⋊C4C2×C22⋊D7912yes—
488C4⋊C4C2×C22⋊D7912yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—1588no—
492C42—22no—
493C42—22no—
494C42—794no—
495C42—1588no—
496C42—1582no—
497C42—1584no—
498C23—22no—
499C23—22no—
500C23—3162no—
501C23—3162no—
502C23—1584no—
503C23—1582no—
504C23#1457912312yes—
505C23#1457912911yescharacteristic
506C23#1457912911yescharacteristic
507C23#1457913912yes—
508C23C2×D4×D7911yescharacteristic
509Q8—794no—
510Q8—1584no—
511Q8—1582no—
512D4—794no—
513D4—1588no—
514D4—1584no—
515D4—1584no—
516D4—1584no—
517D4—1582no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—3164no—
527C2×C4—3164no—
528C2×C4—6328no—
529C2×C4—6322no—
530C2×C4—1584no—
531C2×C4—1584no—
532C2×C4—1584no—
533C2×C4—15816no—
534C2×C4—1584no—
535C2×C4—1584no—
536C2×C4—1582no—
537C2×C4—1588no—
538C2×C4—1584no—
539C2×C4—1584no—
540C2×C4C4⋊C4×D7912yes—
541C2×C4C4⋊C4×D7912yes—
542C2×C4C4×C22⋊D7912yes—
543C2×C4C4×C22⋊D7912yes—
544C2×C4#1457912312yes—
545C2×C4#1457912911yescharacteristic
546C2×C4#1457912911yescharacteristic
547C2×C4#1457913712yes—
548C2×C4#1457913712yes—
549C2×C4#1457913912yes—
550C2×C4#1457914612yes—
551C2×C4#1457914612yes—
552C2×C4C22×C4×D7911yescharacteristic
553C2×C4C2×D4×D7911yescharacteristic
554C2×C4C22×C22⋊D7911yescharacteristic
555C2×C4C2×Q8×D7911yescharacteristic
556C2×C4C2×#1192962411yescharacteristic
557C2×C4C2×#1192962511yescharacteristic
558C8—24no—
559C8—3162no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—3162no—
567C22—3162no—
568C22—3168no—
569C22—1584no—
570C22#1909843412yes—
571C22#1909843412yes—
572C22#1909872912yes—
573C22#1909874412yes—
574C22C2×#1457912311yescharacteristic
575C22C2×#1457912911yescharacteristic
576C22C2×#1457913911yescharacteristic
577C4—42no—
578C4—42no—
579C4—82no—
580C4—3164no—
581C4—1588no—
582C4—1584no—
583C4#1909843412yes—
584C4#1909843412yes—
585C4C2×C4⋊C4×D7911yescharacteristic
586C4C2×C4×C22⋊D7911yescharacteristic
587C4C2×#1457913711yescharacteristic
588C4C2×#1457914611yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—3164no—
593C2#2376362314yesdirect product factor
594C2C2×#1909843411yescharacteristic
595C2C2×#1909872911yescharacteristic
596C2C2×#1909874411yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#428×C79C2
socleC23×C79#120
prime radicalC22×#11929624C22
cosocleC22×C4×C79C24
hypercenterC2×#428D79
hypocenterC79C2×#1859
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×#19098432C2semidirect
5. #23763623C2direct (N × H)
6. C2×#19098331C2semidirect
7. C2×#19098322C2semidirect
8. C2×#428×C79C2semidirect
23. C2×#14579087C4semidirect
30. #19098432C22semidirect
31. #19098331C22semidirect
32. #19098322C22semidirect
37. #428×C79C22semidirect
83. C2×C4×Dic79C2×C4semidirect
85. #14579087C2×C4semidirect
92. C2×M16×C79C2×C4semidirect
119. #11929624C4⋊C4semidirect
124. C2×Dic158C4⋊C4semidirect
128. C2×D316C4⋊C4semidirect
164. C4×Dic79C22×C4semidirect
184. M16×C79C22×C4semidirect
251. D316C2×C4⋊C4semidirect
258. Dic158C2×C4⋊C4semidirect
294. C2×C79#1859semidirect
300. C2×#428D79semidirect
307. C79C2×#1859semidirect
326. #428C2×D79semidirect
362. C2×M16C4×D79semidirect
470. M16C2×C4×D79semidirect
593. C2#23763623direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190990432
C22×#145790872
C2×#190985622
C2×#190984322
#237636232
C2×#190983312
C2×#190983222
C2×#428×C792
C2×#185979