Finite Group Databasebeta

← C2×#23763706 #34289462

Subgroups — C2×#23763706 #34289462

Order 20224.

2270 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579141C211yesmaximal, characteristic
2C22×C4⋊C4×D79C211yesmaximal, characteristic
3C22×#14579086C211yesmaximal, characteristic, complemented
4C22×#14579084C211yesmaximal, characteristic, complemented
5C22×(C2×D79)⋊C8C211yesmaximal, characteristic
6#23763706C218yesmaximal, direct product factor
7C2×#19098325C211yesmaximal, characteristic, complemented
8C2×#340×C79C211yesmaximal, characteristic, complemented
9C23×Dic158C2211yescharacteristic
10C23×C4×D79—21no—
11C23×C4×D79C2211yescharacteristic
12C23×C79⋊C8C2211yescharacteristic
13C2×#14579141—24no—
14C2×#14579141—22no—
15C2×#14579141C418yescomplemented
16C2×#14579141C2214yes—
17C22×C4⋊Dic79—21no—
18C22×C4⋊Dic79C2211yescharacteristic
19C22×Dic79⋊C4—21no—
20C2×C4⋊C4×D79—22no—
21C2×C4⋊C4×D79—24no—
22C2×C4⋊C4×D79—24no—
23C2×C4⋊C4×D79C2214yes—
24C22×(C2×D79)⋊C4—21no—
25C2×#14579086—24no—
26C2×#14579086C2214yescomplemented
27C2×#14579084—24no—
28C2×#14579084C2214yescomplemented
29C2×(C2×D79)⋊C8—24no—
30C2×(C2×D79)⋊C8C2214yes—
31#19098325C2214yescomplemented
32C23×C8×C79C2211yescharacteristic
33C22×C4⋊C4×C79C2211yescharacteristic
34C22×C4⋊C4×C79C2211yescharacteristic
35#340×C79C2214yescomplemented
36C24×D79D411yescharacteristic
37C22×Dic158—22no—
38C22×Dic158—21no—
39C22×Dic158—22no—
40C22×Dic158—22no—
41C22×Dic158—22no—
42C22×Dic158—22no—
43C22×Dic158C2×C414yescomplemented
44C22×Dic158C2312yes—
45C23×Dic79—21no—
46C23×Dic79—21no—
47C23×Dic79D411yescharacteristic
48C22×C4×D79—22no—
49C22×C4×D79—22no—
50C22×C4×D79—22no—
51C22×C4×D79—22no—
52C22×C4×D79—28no—
53C22×C4×D79—24no—
54C22×C4×D79—44no—
55C22×C4×D79—44no—
56C22×C4×D79C2×C414yes—
57C22×C4×D79D412yes—
58C22×C4×D79D414yes—
59C22×C4×D79C2312yes—
60C22×C79⋊C8—22no—
61C22×C79⋊C8—22no—
62C22×C79⋊C8—22no—
63C22×C79⋊C8C2312yes—
64#14579141—216no—
65#14579141—28no—
66#14579141C2×C4116yescomplemented
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—22no—
69C2×C4⋊Dic79—28no—
70C2×C4⋊Dic79—24no—
71C2×C4⋊Dic79D414yes—
72C2×C4⋊Dic79C2312yes—
73C2×Dic79⋊C4—28no—
74C2×Dic79⋊C4—24no—
75C4⋊C4×D79—216no—
76C4⋊C4×D79—28no—
77C4⋊C4×D79—28no—
78C2×(C2×D79)⋊C4—28no—
79C2×(C2×D79)⋊C4—24no—
80#14579086—28no—
81#14579084—28no—
82(C2×D79)⋊C8—28no—
83C23×C4×C79—21no—
84C23×C4×C79—21no—
85C23×C4×C79C2311yescharacteristic
86C22×C8×C79—22no—
87C22×C8×C79—22no—
88C22×C8×C79—22no—
89C22×C8×C79C2312yes—
90C2×C4⋊C4×C79—21no—
91C2×C4⋊C4×C79—22no—
92C2×C4⋊C4×C79—21no—
93C2×C4⋊C4×C79—22no—
94C2×C4⋊C4×C79C2×C414yescomplemented
95C2×C4⋊C4×C79D414yes—
96C2×C4⋊C4×C79C2312yes—
97C2×C4⋊C4×C79C2312yes—
98C23×D79—22no—
99C23×D79—22no—
100C23×D79—22no—
101C23×D79—22no—
102C23×D79C22⋊C414yes—
103C23×D79SD1614yes—
104C23×D79Q1614yes—
105C23×D79C2×D412yes—
106C2×Dic158—24no—
107C2×Dic158—28no—
108C2×Dic158—28no—
109C2×Dic158—22no—
110C2×Dic158—22no—
111C2×Dic158—22no—
112C2×Dic158—22no—
113C2×Dic158—22no—
114C2×Dic158—42no—
115C2×Dic158—44no—
116C2×Dic158—44no—
117C2×Dic158—42no—
118C2×Dic158C22×C414yescomplemented
119C22×Dic79—24no—
120C22×Dic79—22no—
121C22×Dic79—24no—
122C22×Dic79—22no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—42no—
126C22×Dic79—42no—
127C22×Dic79—42no—
128C22×Dic79—42no—
129C22×Dic79C22⋊C414yes—
130C22×Dic79C2×D412yes—
131C2×C4×D79—216no—
132C2×C4×D79—28no—
133C2×C4×D79—28no—
134C2×C4×D79—24no—
135C2×C4×D79—28no—
136C2×C4×D79—28no—
137C2×C4×D79—24no—
138C2×C4×D79—22no—
139C2×C4×D79—22no—
140C2×C4×D79—22no—
141C2×C4×D79—22no—
142C2×C4×D79—48no—
143C2×C4×D79—432no—
144C2×C4×D79—48no—
145C2×C4×D79C22⋊C418yes—
146C2×C4×D79C22×C414yes—
147C2×C4×D79C2×D414yes—
148C2×C79⋊C8—22no—
149C2×C79⋊C8—22no—
150C2×C79⋊C8—22no—
151C2×C79⋊C8—44no—
152C4⋊Dic79—216no—
153C4⋊Dic79—24no—
154C4⋊Dic79—22no—
155C4⋊Dic79C2×D414yes—
156Dic79⋊C4—216no—
157(C2×D79)⋊C4—216no—
158C24×C79C2×D411yescharacteristic
159C22×C4×C79—24no—
160C22×C4×C79—24no—
161C22×C4×C79—22no—
162C22×C4×C79—22no—
163C22×C4×C79—42no—
164C22×C4×C79—42no—
165C22×C4×C79—42no—
166C22×C4×C79—42no—
167C22×C4×C79C22×C412yes—
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D412yes—
170C22×C4×C79C2×D412yes—
171C22×C4×C79C2×D412yes—
172C22×C4×C79#4112yes—
173C22×C4×C79#4112yes—
174C22×C4×C79C2411yescharacteristic
175C2×C8×C79—22no—
176C2×C8×C79—22no—
177C2×C8×C79—22no—
178C2×C8×C79—44no—
179C4⋊C4×C79—24no—
180C4⋊C4×C79—24no—
181C4⋊C4×C79—22no—
182C4⋊C4×C79—22no—
183C4⋊C4×C79C22×C414yescomplemented
184C4⋊C4×C79C2×D414yes—
185C23×C79C2×C22⋊C412yes—
186C23×C79#12012yes—
187C23×C79#12312yes—
188C23×C79C2×SD1612yes—
189C23×C79C2×Q1612yes—
190C23×C79AGL(1,ℤ/8)12yes—
191C23×C79#13712yes—
192C23×C79C22×D411yescharacteristic
193C22×D79—22no—
194C22×D79—22no—
195C22×D79—22no—
196C22×D79—22no—
197C22×D79—28no—
198C22×D79—28no—
199C22×D79—28no—
200C22×D79—28no—
201C22×D79—28no—
202C22×D79—28no—
203C22×D79#103116yescomplemented
204C22×D79C2×C22⋊C414yes—
205C22×D79C2×SD1614yes—
206C22×D79C2×Q1614yes—
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—
245C4×D79—432no—
246Dic158—24no—
247Dic158—24no—
248Dic158—42no—
249Dic158—42no—
250Dic158—48no—
251C8×C79—42no—
252C79⋊C8—42no—
253C22×C79#38614yes—
254C22×C79C2×#10314yes—
255C22×C79#41714yes—
256C22×C79#45014yes—
257C22×C79#45114yes—
258C22×C79#48314yes—
259C22×C79#48414yes—
260C22×C79C22×C22⋊C411yescharacteristic
261C22×C79C2×#12011yescharacteristic
262C22×C79C2×#12311yescharacteristic
263C22×C79C22×SD1611yescharacteristic
264C22×C79C22×Q1611yescharacteristic
265C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C79C2×#13711yescharacteristic
267C2×D79—24no—
268C2×D79—24no—
269C2×D79—24no—
270C2×D79—232no—
271C2×D79—24no—
272C2×D79—24no—
273C2×D79—24no—
274C2×D79C2×#10318yescomplemented
275C4×C79—44no—
276C4×C79—44no—
277C4×C79#38614yes—
278C4×C79C2×C4×D411yescharacteristic
279C4×C79C2×C4×D411yescharacteristic
280C4×C79C2×#12111yescharacteristic
281C4×C79C2×#12111yescharacteristic
282Dic79—24no—
283Dic79—44no—
284Dic79—44no—
285C2×#1872—791nomaximal, Sylow
286C2×C79#187218yescomplemented
287C2×C79C2×#38611yescharacteristic
288C2×C79C22×#10311yescharacteristic
289C2×C79C2×#41711yescharacteristic
290C2×C79C2×#45011yescharacteristic
291C2×C79C2×#45111yescharacteristic
292C2×C79C2×#48311yescharacteristic
293C2×C79C2×#48411yescharacteristic
294D79—28no—
295C22×C22⋊Q8—791no—
296C23×C4⋊C4—791no—
297C22×#103—791no—
298C22×#103—791no—
299C22×C22⋊C8—791no—
300#1872—798no—
301C2×#340—791no—
302C2×#340D7911yescharacteristic, complemented
303C79C2×#187211yescharacteristic, complemented, Sylow
304C23×Q8—791no—
305C24×C4—791no—
306C24×C4—1581no—
307C23×C8—791no—
308C23×C8C2×D7911yescharacteristic
309C2×C22⋊Q8—794no—
310C2×C22⋊Q8—798no—
311C2×C22⋊Q8—1582no—
312C2×C22⋊Q8—1584no—
313C22×C4⋊C4—791no—
314C22×C4⋊C4—794no—
315C22×C4⋊C4—1584no—
316C22×C4⋊C4—1582no—
317C22×C4⋊C4—1581no—
318C22×C4⋊C4—1581no—
319C22×C4⋊C4—1584no—
320C22×C4⋊C4C2×D7911yescharacteristic
321C22×C4⋊C4C2×D7911yescharacteristic
322C22×C22⋊C4—1581no—
323C2×#103—794no—
324C2×#103—794no—
325C2×#103—1584no—
326C2×#103—1584no—
327C2×C22⋊C8—794no—
328C2×C22⋊C8—1584no—
329#340—794no—
330#340C2×D7914yescomplemented
331C25—791no—
332C22×Q8—794no—
333C22×Q8—792no—
334C22×Q8—1582no—
335C22×Q8—1582no—
336C22×Q8—1582no—
337C22×Q8—1582no—
338C22×Q8—1582no—
339C22×Q8—1581no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—3164no—
343C23×C4—3164no—
344C23×C4—791no—
345C23×C4—794no—
346C23×C4—792no—
347C23×C4—794no—
348C23×C4—792no—
349C23×C4—1584no—
350C23×C4—1588no—
351C23×C4—1581no—
352C23×C4—1581no—
353C23×C4—1582no—
354C23×C4—1582no—
355C23×C4—1582no—
356C23×C4—1582no—
357C23×C4C22×D7911yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—792no—
362C22×C8—1582no—
363C22×C8—1582no—
364C22×C8—1582no—
365C22×C8C22×D7912yes—
366C22⋊Q8—7916no—
367C22⋊Q8—15816no—
368C22⋊Q8—1588no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—794no—
374C2×C4⋊C4—792no—
375C2×C4⋊C4—1588no—
376C2×C4⋊C4—15816no—
377C2×C4⋊C4—1584no—
378C2×C4⋊C4—1588no—
379C2×C4⋊C4—1584no—
380C2×C4⋊C4—1588no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4—1582no—
383C2×C4⋊C4—1588no—
384C2×C4⋊C4C4×D7914yescomplemented
385C2×C4⋊C4D31614yes—
386C2×C4⋊C4C22×D7912yes—
387C2×C4⋊C4C22×D7912yes—
388C2×C22⋊C4—1588no—
389C2×C22⋊C4—1584no—
390#103—1588no—
391#103—1588no—
392C22⋊C8—1588no—
393C24—794no—
394C24—794no—
395C24—794no—
396C24—792no—
397C24—1582no—
398C24—1582no—
399C24—1582no—
400C24—1582no—
401C24D4×D7911yescharacteristic
402C2×Q8—3162no—
403C2×Q8—3164no—
404C2×Q8—3164no—
405C2×Q8—3162no—
406C2×Q8—794no—
407C2×Q8—1582no—
408C2×Q8—1582no—
409C2×Q8—1588no—
410C2×Q8—1588no—
411C2×Q8—1582no—
412C2×Q8—1582no—
413C2×Q8—1584no—
414C2×Q8—1582no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—3168no—
424C22×C4—31632no—
425C22×C4—3168no—
426C22×C4—3162no—
427C22×C4—3162no—
428C22×C4—3162no—
429C22×C4—3162no—
430C22×C4—794no—
431C22×C4—792no—
432C22×C4—798no—
433C22×C4—794no—
434C22×C4—794no—
435C22×C4—15816no—
436C22×C4—1584no—
437C22×C4—1584no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1582no—
442C22×C4—1582no—
443C22×C4—1582no—
444C22×C4—1588no—
445C22×C4—1588no—
446C22×C4—1584no—
447C22×C4—1588no—
448C22×C4—1588no—
449C22×C4—1584no—
450C22×C4—1582no—
451C22×C4—1582no—
452C22×C4C2×C4×D7912yes—
453C22×C4D4×D7911yescharacteristic
454C22×C4D4×D7912yes—
455C22×C4D4×D7912yes—
456C22×C4C2×D31612yes—
457C22×C4#1192962412yes—
458C22×C4#1192962512yes—
459C22×C4C23×D7911yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—3164no—
465C2×C8—1582no—
466C2×C8—1582no—
467C2×C8—1582no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—794no—
473C4⋊C4—15816no—
474C4⋊C4—15816no—
475C4⋊C4—1584no—
476C4⋊C4—1582no—
477C4⋊C4C2×C4×D7914yescomplemented
478C4⋊C4C2×D31614yes—
479C22⋊C4—15816no—
480C23—794no—
481C23—7916no—
482C23—794no—
483C23—794no—
484C23—1588no—
485C23—1588no—
486C23—1588no—
487C23—1588no—
488C23—1588no—
489C23—1582no—
490C23—1582no—
491C23—1588no—
492C23—1582no—
493C23—1582no—
494C23C22⋊C4×D7912yes—
495C23#1457913112yes—
496C23#1457914612yes—
497C23SD16×D7912yes—
498C23Q16×D7912yes—
499C23#1457918512yes—
500C23#1457919012yes—
501C23C2×D4×D7911yescharacteristic
502Q8—3168no—
503Q8—3162no—
504Q8—3162no—
505Q8—1584no—
506Q8—1584no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—31632no—
516C2×C4—3168no—
517C2×C4—3168no—
518C2×C4—3162no—
519C2×C4—3162no—
520C2×C4—3162no—
521C2×C4—3162no—
522C2×C4—794no—
523C2×C4—1584no—
524C2×C4—1584no—
525C2×C4—1588no—
526C2×C4—1582no—
527C2×C4—1582no—
528C2×C4—15816no—
529C2×C4—1584no—
530C2×C4—1584no—
531C2×C4—1584no—
532C2×C4—1584no—
533C2×C4C22⋊C4×D7912yes—
534C2×C4#1457912314yes—
535C2×C4C4×D31614yes—
536C2×C4#1457913112yes—
537C2×C4#1457913414yes—
538C2×C4#1457913614yes—
539C2×C4#1457914612yes—
540C2×C4C22×C4×D7911yescharacteristic
541C2×C4C2×D4×D7911yescharacteristic
542C2×C4C2×D4×D7911yescharacteristic
543C2×C4C22×D31611yescharacteristic
544C2×C4C2×#1192962411yescharacteristic
545C2×C4C2×#1192962511yescharacteristic
546C8—42no—
547C8—3162no—
548C22—798no—
549C22—1584no—
550C22—1584no—
551C22—1584no—
552C22—1584no—
553C22—1584no—
554C22—1584no—
555C22—15832no—
556C22#1909844114yes—
557C22#103×D7914yes—
558C22#1909856114yes—
559C22#1909871514yes—
560C22#1909872014yes—
561C22#1909889514yes—
562C22#1909890014yes—
563C22C2×C22⋊C4×D7911yescharacteristic
564C22C2×#1457913111yescharacteristic
565C22C2×#1457914611yescharacteristic
566C22C2×SD16×D7911yescharacteristic
567C22C2×Q16×D7911yescharacteristic
568C22C2×#1457918511yescharacteristic
569C22C2×#1457919011yescharacteristic
570C4—44no—
571C4—44no—
572C4—3164no—
573C4—3164no—
574C4—1584no—
575C4#1909844114yes—
576C4C2×#1457912311yescharacteristic
577C4C2×C4×D31611yescharacteristic
578C4C2×#1457913411yescharacteristic
579C4C2×#1457913611yescharacteristic
580C2—1588no—
581C2#2376370618yesdirect product factor
582C2C2×#1909844111yescharacteristic
583C2C2×#103×D7911yescharacteristic
584C2C2×#1909856111yescharacteristic
585C2C2×#1909871511yescharacteristic
586C2C2×#1909872011yescharacteristic
587C2C2×#1909889511yescharacteristic
588C2C2×#1909890011yescharacteristic
subgroupG/H
centerC24D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#340×C79C2
socleC24×C79C2×D4
prime radicalC24×D79D4
cosocleC22×C4×C79C24
hypercenterC2×#340D79
hypocenterC79C2×#1872
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#14579086C2semidirect
4. C22×#14579084C2semidirect
6. #23763706C2direct (N × H)
7. C2×#19098325C2semidirect
8. C2×#340×C79C2semidirect
15. C2×#14579141C4semidirect
26. C2×#14579086C22semidirect
28. C2×#14579084C22semidirect
31. #19098325C22semidirect
35. #340×C79C22semidirect
43. C22×Dic158C2×C4semidirect
66. #14579141C2×C4semidirect
94. C2×C4⋊C4×C79C2×C4semidirect
118. C2×Dic158C22×C4semidirect
183. C4⋊C4×C79C22×C4semidirect
203. C22×D79#103semidirect
274. C2×D79C2×#103semidirect
286. C2×C79#1872semidirect
302. C2×#340D79semidirect
303. C79C2×#1872semidirect
330. #340C2×D79semidirect
384. C2×C4⋊C4C4×D79semidirect
477. C4⋊C4C2×C4×D79semidirect
581. C2#23763706direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791412
C22×C4⋊C4×D792
C22×#145790862
C22×#145790842
C22×(C2×D79)⋊C82
#237637062
C2×#190983252
C2×#340×C792
C2×#187279