Finite Group Databasebeta

← C2×#23763709 #34289472

Subgroups — C2×#23763709 #34289472

Order 20224.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579137C211yesmaximal, characteristic
2C22×#14579119C211yesmaximal, characteristic
3C22×#14579085C211yesmaximal, characteristic, complemented
4C22×(C2×D79)⋊C8C211yesmaximal, characteristic
5#23763709C218yesmaximal, direct product factor
6C2×#19098326C211yesmaximal, characteristic, complemented
7C2×#19098325C211yesmaximal, characteristic, complemented
8C22×#103×C79C211yesmaximal, characteristic, complemented
9C23×C4×D79C2211yescharacteristic
10C23×C79⋊C8C2211yescharacteristic
11C2×#14579137—24no—
12C2×#14579137—22no—
13C2×#14579137C418yescomplemented
14C2×#14579137C2214yes—
15C2×#14579119—22no—
16C2×#14579119—24no—
17C2×#14579119—24no—
18C2×#14579119C2214yes—
19C22×C4⋊Dic79C2211yescharacteristic
20C22×C4⋊Dic79C2211yescharacteristic
21C22×Dic79⋊C4—21no—
22C22×(C2×D79)⋊C4—21no—
23C22×(C2×D79)⋊C4—21no—
24C22×C4×Dic79—21no—
25C2×#14579085—24no—
26C2×#14579085C2214yescomplemented
27C2×(C2×D79)⋊C8—24no—
28C2×(C2×D79)⋊C8C2214yes—
29#19098326C2214yescomplemented
30#19098325C2214yescomplemented
31C23×Q8×C79C2211yescharacteristic
32C23×C8×C79C2211yescharacteristic
33C22×C4⋊C4×C79C2211yescharacteristic
34C2×#103×C79—24no—
35C2×#103×C79C2214yescomplemented
36C24×D79D411yescharacteristic
37C23×Dic79—21no—
38C23×Dic79—21no—
39C23×Dic79D411yescharacteristic
40C22×C4×D79—22no—
41C22×C4×D79—22no—
42C22×C4×D79—22no—
43C22×C4×D79—22no—
44C22×C4×D79C2×C414yes—
45C22×C4×D79D412yes—
46C22×C4×D79D414yes—
47C22×C4×D79C2312yes—
48C22×C79⋊C8—22no—
49C22×C79⋊C8—22no—
50C22×C79⋊C8—22no—
51C22×C79⋊C8C2312yes—
52#14579137—216no—
53#14579137—28no—
54#14579137C2×C4116yescomplemented
55#14579119—216no—
56#14579119—28no—
57#14579119—28no—
58C2×C4⋊Dic79—21no—
59C2×C4⋊Dic79—22no—
60C2×C4⋊Dic79—21no—
61C2×C4⋊Dic79—22no—
62C2×C4⋊Dic79C2×C414yes—
63C2×C4⋊Dic79D414yes—
64C2×C4⋊Dic79C2312yes—
65C2×C4⋊Dic79C2312yes—
66C2×Dic79⋊C4—28no—
67C2×Dic79⋊C4—24no—
68C2×(C2×D79)⋊C4—28no—
69C2×(C2×D79)⋊C4—28no—
70C2×(C2×D79)⋊C4—24no—
71C2×(C2×D79)⋊C4—24no—
72C2×C4×Dic79—28no—
73C2×C4×Dic79—24no—
74#14579085—28no—
75(C2×D79)⋊C8—28no—
76C22×Q8×C79—22no—
77C22×Q8×C79—21no—
78C22×Q8×C79—22no—
79C22×Q8×C79—22no—
80C22×Q8×C79—22no—
81C22×Q8×C79—22no—
82C22×Q8×C79C2×C414yes—
83C22×Q8×C79C2312yes—
84C23×C4×C79—21no—
85C23×C4×C79—21no—
86C23×C4×C79C2311yescharacteristic
87C22×C8×C79—22no—
88C22×C8×C79—22no—
89C22×C8×C79—22no—
90C22×C8×C79C2312yes—
91C2×C4⋊C4×C79—21no—
92C2×C4⋊C4×C79—22no—
93C2×C4⋊C4×C79D414yes—
94C2×C4⋊C4×C79C2312yes—
95#103×C79—28no—
96C23×D79—22no—
97C23×D79—22no—
98C23×D79—22no—
99C23×D79—22no—
100C23×D79—22no—
101C23×D79—22no—
102C23×D79C22⋊C414yes—
103C23×D79C2×D412yes—
104C22×Dic79—24no—
105C22×Dic79—22no—
106C22×Dic79—24no—
107C22×Dic79—22no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—42no—
111C22×Dic79—42no—
112C22×Dic79—42no—
113C22×Dic79—42no—
114C22×Dic79C22⋊C414yes—
115C22×Dic79C2×D412yes—
116C2×C4×D79—28no—
117C2×C4×D79—28no—
118C2×C4×D79—24no—
119C2×C4×D79—28no—
120C2×C4×D79—28no—
121C2×C4×D79—24no—
122C2×C4×D79—22no—
123C2×C4×D79—22no—
124C2×C4×D79—22no—
125C2×C4×D79—22no—
126C2×C4×D79C22⋊C418yes—
127C2×C4×D79C22×C414yes—
128C2×C4×D79C2×D414yes—
129C2×C79⋊C8—22no—
130C2×C79⋊C8—22no—
131C2×C79⋊C8—22no—
132C2×C79⋊C8—44no—
133C4⋊Dic79—24no—
134C4⋊Dic79—24no—
135C4⋊Dic79—22no—
136C4⋊Dic79—22no—
137C4⋊Dic79C22×C414yes—
138C4⋊Dic79C2×D414yes—
139Dic79⋊C4—216no—
140(C2×D79)⋊C4—216no—
141(C2×D79)⋊C4—216no—
142C4×Dic79—216no—
143C24×C79C2×D411yescharacteristic
144C2×Q8×C79—24no—
145C2×Q8×C79—22no—
146C2×Q8×C79—28no—
147C2×Q8×C79—28no—
148C2×Q8×C79—22no—
149C2×Q8×C79—22no—
150C2×Q8×C79—22no—
151C2×Q8×C79—22no—
152C2×Q8×C79—42no—
153C2×Q8×C79—44no—
154C2×Q8×C79—44no—
155C2×Q8×C79—42no—
156C2×Q8×C79C22×C414yes—
157C22×C4×C79—24no—
158C22×C4×C79—24no—
159C22×C4×C79—22no—
160C22×C4×C79—22no—
161C22×C4×C79—42no—
162C22×C4×C79—42no—
163C22×C4×C79—42no—
164C22×C4×C79—42no—
165C22×C4×C79C22×C412yes—
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×D412yes—
168C22×C4×C79C2×D412yes—
169C22×C4×C79C2×D412yes—
170C22×C4×C79#4112yes—
171C22×C4×C79#4112yes—
172C22×C4×C79C2411yescharacteristic
173C2×C8×C79—22no—
174C2×C8×C79—22no—
175C2×C8×C79—22no—
176C2×C8×C79—44no—
177C4⋊C4×C79—24no—
178C4⋊C4×C79—22no—
179C4⋊C4×C79C2×D414yes—
180C23×C79C2×C22⋊C412yes—
181C23×C79#12012yes—
182C23×C79#12312yes—
183C23×C79#13512yes—
184C23×C79#13512yes—
185C23×C79#13712yes—
186C23×C79#13712yes—
187C23×C79C22×D411yescharacteristic
188C22×D79—22no—
189C22×D79—22no—
190C22×D79—22no—
191C22×D79—22no—
192C22×D79—22no—
193C22×D79—22no—
194C22×D79—28no—
195C22×D79—28no—
196C22×D79—28no—
197C22×D79—44no—
198C22×D79—44no—
199C22×D79—44no—
200C22×D79—44no—
201C22×D79C2×C22⋊C414yes—
202Q8×C79—24no—
203Q8×C79—24no—
204Q8×C79—42no—
205Q8×C79—42no—
206Q8×C79—48no—
207C2×C4×C79—24no—
208C2×C4×C79—24no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—48no—
214C2×C4×C79—48no—
215C2×C4×C79C2×C22⋊C412yes—
216C2×C4×C79C4×D414yes—
217C2×C4×C79C4×D414yes—
218C2×C4×C79#12012yes—
219C2×C4×C79#12114yes—
220C2×C4×C79#12114yes—
221C2×C4×C79#12312yes—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C22×D411yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—28no—
231C2×Dic79—24no—
232C2×Dic79—24no—
233C2×Dic79—42no—
234C2×Dic79—42no—
235C2×Dic79—42no—
236C2×Dic79—42no—
237C2×Dic79—48no—
238C2×Dic79—48no—
239C2×Dic79C2×C22⋊C414yes—
240C4×D79—24no—
241C4×D79—24no—
242C4×D79—24no—
243C4×D79—24no—
244C4×D79—216no—
245C8×C79—42no—
246C79⋊C8—42no—
247C22×C79#38614yes—
248C22×C79#41614yes—
249C22×C79#41914yes—
250C22×C79#45214yes—
251C22×C79#45214yes—
252C22×C79#48514yes—
253C22×C79#48514yes—
254C22×C79C22×C22⋊C411yescharacteristic
255C22×C79C2×#12011yescharacteristic
256C22×C79C2×#12311yescharacteristic
257C22×C79C2×#13511yescharacteristic
258C22×C79C2×#13511yescharacteristic
259C22×C79C2×#13711yescharacteristic
260C22×C79C2×#13711yescharacteristic
261C2×D79—24no—
262C2×D79—24no—
263C2×D79—24no—
264C2×D79—42no—
265C2×D79—42no—
266C2×D79—416no—
267C2×D79—42no—
268C2×D79—42no—
269C4×C79—44no—
270C4×C79—44no—
271C4×C79#38614yes—
272C4×C79C2×C4×D411yescharacteristic
273C4×C79C2×C4×D411yescharacteristic
274C4×C79C2×#12111yescharacteristic
275C4×C79C2×#12111yescharacteristic
276Dic79—24no—
277Dic79—44no—
278Dic79—44no—
279C2×#1873—791nomaximal, Sylow
280C2×C79#187318yescomplemented
281C2×C79C2×#38611yescharacteristic
282C2×C79C2×#41611yescharacteristic
283C2×C79C2×#41911yescharacteristic
284C2×C79C2×#45211yescharacteristic
285C2×C79C2×#45211yescharacteristic
286C2×C79C2×#48511yescharacteristic
287C2×C79C2×#48511yescharacteristic
288D79—44no—
289C22×C22⋊Q8—791no—
290C22×#117—791no—
291C22×#103—791no—
292C22×#103D7911yescharacteristic, complemented
293C22×C22⋊C8—791no—
294#1873—798no—
295C2×#340—791no—
296C2×#340—791no—
297C79C2×#187311yescharacteristic, complemented, Sylow
298C23×Q8C2×D7911yescharacteristic
299C24×C4—791no—
300C23×C8—791no—
301C23×C8C2×D7911yescharacteristic
302C2×C22⋊Q8—794no—
303C2×C22⋊Q8—798no—
304C2×C22⋊Q8—1582no—
305C2×C22⋊Q8—1584no—
306C2×#117—794no—
307C2×#117—1584no—
308C2×#117—1582no—
309C2×#117—1584no—
310C22×C4⋊C4—791no—
311C22×C4⋊C4—791no—
312C22×C4⋊C4—1581no—
313C22×C4⋊C4C2×D7911yescharacteristic
314C22×C22⋊C4—1581no—
315C22×C22⋊C4—1581no—
316C22×C42—1581no—
317C2×#103—24no—
318C2×#103—794no—
319C2×#103—1584no—
320C2×#103C2×D7914yescomplemented
321C2×C22⋊C8—794no—
322C2×C22⋊C8—1584no—
323#340—794no—
324#340—794no—
325C25—791no—
326C22×Q8—22no—
327C22×Q8—21no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8C4×D7914yes—
333C22×Q8C22×D7912yes—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—791no—
337C23×C4—794no—
338C23×C4—792no—
339C23×C4—794no—
340C23×C4—792no—
341C23×C4—1581no—
342C23×C4—1581no—
343C23×C4—1582no—
344C23×C4—1582no—
345C23×C4—1582no—
346C23×C4—1582no—
347C23×C4C22×D7911yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—792no—
352C22×C8—1582no—
353C22×C8—1582no—
354C22×C8—1582no—
355C22×C8C22×D7912yes—
356C22⋊Q8—7916no—
357C22⋊Q8—15816no—
358C22⋊Q8—1588no—
359#117—1588no—
360#117—15816no—
361#117—1588no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—794no—
365C2×C4⋊C4—794no—
366C2×C4⋊C4—792no—
367C2×C4⋊C4—792no—
368C2×C4⋊C4—1588no—
369C2×C4⋊C4—1584no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1582no—
373C2×C4⋊C4—1582no—
374C2×C4⋊C4C22⋊D7914yes—
375C2×C4⋊C4C22×D7912yes—
376C2×C22⋊C4—1584no—
377C2×C22⋊C4—1588no—
378C2×C22⋊C4—1584no—
379C2×C22⋊C4—1588no—
380C2×C42—1584no—
381C2×C42—1588no—
382#103—28no—
383#103—1588no—
384C22⋊C8—1588no—
385C24—794no—
386C24—792no—
387C24—1582no—
388C24—1582no—
389C24—1582no—
390C24—1582no—
391C24—1582no—
392C24—1582no—
393C24D4×D7911yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8C2×C4×D7914yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—3162no—
416C22×C4—3162no—
417C22×C4—3162no—
418C22×C4—3162no—
419C22×C4—794no—
420C22×C4—792no—
421C22×C4—798no—
422C22×C4—794no—
423C22×C4—794no—
424C22×C4—1584no—
425C22×C4—1584no—
426C22×C4—1582no—
427C22×C4—1582no—
428C22×C4—1582no—
429C22×C4—1582no—
430C22×C4—1582no—
431C22×C4—1582no—
432C22×C4—1588no—
433C22×C4—1588no—
434C22×C4—1584no—
435C22×C4—1588no—
436C22×C4—1588no—
437C22×C4—1584no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4C2×C4×D7912yes—
441C22×C4D4×D7911yescharacteristic
442C22×C4D4×D7912yes—
443C22×C4D4×D7912yes—
444C22×C4C2×C22⋊D7912yes—
445C22×C4#1192962312yes—
446C22×C4#1192962412yes—
447C22×C4C23×D7911yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—3164no—
453C2×C8—1582no—
454C2×C8—1582no—
455C2×C8—1582no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—794no—
459C4⋊C4—794no—
460C4⋊C4—15816no—
461C4⋊C4—1584no—
462C4⋊C4—1584no—
463C4⋊C4—1582no—
464C4⋊C4—1582no—
465C4⋊C4C2×C22⋊D7914yes—
466C22⋊C4—15816no—
467C22⋊C4—15816no—
468C42—15816no—
469C23—3164no—
470C23—3164no—
471C23—3164no—
472C23—3164no—
473C23—794no—
474C23—1588no—
475C23—1582no—
476C23—1582no—
477C23—1588no—
478C23—1588no—
479C23—1582no—
480C23—1582no—
481C23—1582no—
482C23—1582no—
483C23C22⋊C4×D7912yes—
484C23#1457912912yes—
485C23#1457914312yes—
486C23#1457918212yes—
487C23#1457918412yes—
488C23#1457919012yes—
489C23#1457919312yes—
490C23C2×D4×D7911yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—48no—
494Q8—42no—
495Q8—42no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—3168no—
505C2×C4—3168no—
506C2×C4—3162no—
507C2×C4—3162no—
508C2×C4—3162no—
509C2×C4—3162no—
510C2×C4—794no—
511C2×C4—1584no—
512C2×C4—1584no—
513C2×C4—1588no—
514C2×C4—1582no—
515C2×C4—1582no—
516C2×C4—15816no—
517C2×C4—1584no—
518C2×C4—1584no—
519C2×C4—1584no—
520C2×C4—1584no—
521C2×C4C22⋊C4×D7912yes—
522C2×C4(C22⋊D79)⋊C414yes—
523C2×C4C4×C22⋊D7914yes—
524C2×C4#1457912912yes—
525C2×C4#1457913214yes—
526C2×C4#1457913414yes—
527C2×C4#1457914312yes—
528C2×C4C22×C4×D7911yescharacteristic
529C2×C4C2×D4×D7911yescharacteristic
530C2×C4C2×D4×D7911yescharacteristic
531C2×C4C22×C22⋊D7911yescharacteristic
532C2×C4C2×#1192962311yescharacteristic
533C2×C4C2×#1192962411yescharacteristic
534C8—42no—
535C8—3162no—
536C22—3162no—
537C22—3162no—
538C22—3162no—
539C22—3162no—
540C22—31616no—
541C22—1584no—
542C22—1584no—
543C22—1584no—
544C22#1909843914yes—
545C22#1909855814yes—
546C22#1909857014yes—
547C22#1909872214yes—
548C22#1909872414yes—
549C22#1909890214yes—
550C22#1909890314yes—
551C22C2×C22⋊C4×D7911yescharacteristic
552C22C2×#1457912911yescharacteristic
553C22C2×#1457914311yescharacteristic
554C22C2×#1457918211yescharacteristic
555C22C2×#1457918411yescharacteristic
556C22C2×#1457919011yescharacteristic
557C22C2×#1457919311yescharacteristic
558C4—44no—
559C4—44no—
560C4—3164no—
561C4—3164no—
562C4—1584no—
563C4#1909843914yes—
564C4C2×(C22⋊D79)⋊C411yescharacteristic
565C4C2×C4×C22⋊D7911yescharacteristic
566C4C2×#1457913211yescharacteristic
567C4C2×#1457913411yescharacteristic
568C2—3164no—
569C2#2376370918yesdirect product factor
570C2C2×#1909843911yescharacteristic
571C2C2×#1909855811yescharacteristic
572C2C2×#1909857011yescharacteristic
573C2C2×#1909872211yescharacteristic
574C2C2×#1909872411yescharacteristic
575C2C2×#1909890211yescharacteristic
576C2C2×#1909890311yescharacteristic
subgroupG/H
centerC24D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC22×#103×C79C2
socleC24×C79C2×D4
prime radicalC24×D79D4
cosocleC22×C4×C79C24
hypercenterC22×#103D79
hypocenterC79C2×#1873
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14579085C2semidirect
5. #23763709C2direct (N × H)
6. C2×#19098326C2semidirect
7. C2×#19098325C2semidirect
8. C22×#103×C79C2semidirect
13. C2×#14579137C4semidirect
26. C2×#14579085C22semidirect
29. #19098326C22semidirect
30. #19098325C22semidirect
35. C2×#103×C79C22semidirect
54. #14579137C2×C4semidirect
280. C2×C79#1873semidirect
292. C22×#103D79semidirect
297. C79C2×#1873semidirect
320. C2×#103C2×D79semidirect
569. C2#23763709direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791372
C22×#145791192
C22×#145790852
C22×(C2×D79)⋊C82
#237637092
C2×#190983262
C2×#190983252
C22×#103×C792
C2×#187379