Finite Group Databasebeta

← C2×#23774577 #35160345

Subgroups — C2×#23774577 #35160345

Order 20224.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099118C211yesmaximal, characteristic, complemented
2C2×#19099095C211yesmaximal, characteristic
3#23774577C2116yesmaximal, direct product factor
4C2×#19098964C212yesmaximal, complemented
5C2×#19098916C212yesmaximal, complemented
6C2×#19098906C212yesmaximal, complemented
7C2×#19098885C212yesmaximal
8C2×#19098859C211yesmaximal, characteristic, complemented
9C2×#19098845C211yesmaximal, characteristic, complemented
10C2×#19098853C211yesmaximal, characteristic, complemented
11C2×#19098615C211yesmaximal, characteristic, complemented
12C2×#474×C79C211yesmaximal, characteristic, complemented
13#19099118—24no—
14#19099118C2218yescomplemented
15C2×#14579161C2211yescharacteristic
16#19099095—24no—
17#19099095C2218yes—
18C2×#14579154—21no—
19C2×#14579152C2212yes—
20C2×#14579152C2211yescharacteristic
21C2×#14579148C2211yescharacteristic
22C2×#14579143—21no—
23C2×#14579141—21no—
24C2×#14579141C2211yescharacteristic
25C2×#14579137—21no—
26C2×#14579132C2211yescharacteristic
27C2×Q8×Dic79—21no—
28C2×C4×Dic158C2211yescharacteristic
29C2×#14579126—21no—
30C2×C4×C22⋊D79—21no—
31C2×#14579120—21no—
32C2×#14579119C2212yes—
33C2×C42×D79C2211yescharacteristic
34#19098964C22116yescomplemented
35#19098916C22116yescomplemented
36#19098906C22116yescomplemented
37#19098885C22116yes—
38#19098859C2218yescomplemented
39#19098853C2218yescomplemented
40#19098845C2218yescomplemented
41#19098615C2218yescomplemented
42C2×#14579095C2212yes—
43C2×#14579092C2212yes—
44C2×#14579092C2212yes—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46C2×#14579086C2212yes—
47C2×#14579084C2212yes—
48C2×#14579082C2212yes—
49C2×(C2×D79)⋊C8C2212yes—
50C2×C8×Dic79C2212yes—
51C2×C4⋊Q8×C79C2211yescharacteristic
52C2×C4×D4×C79C2211yescharacteristic
53#474×C79C2218yescomplemented
54C2×#107×C79C2212yes—
55C2×C4⋊C8×C79C2211yescharacteristic
56C2×#102×C79C2212yes—
57C22×Dic158—21no—
58C22×Dic158C2311yescharacteristic
59C22×C22⋊D79—21no—
60C22×C4×D79D411yescharacteristic
61C22×C4×D79D411yescharacteristic
62C22×C4×D79C2311yescharacteristic
63C22×C79⋊C8C2312yes—
64#14579161—22no—
65#14579161C2314yes—
66#14579154—28no—
67#14579152—22no—
68#14579152—24no—
69#14579152C2318yes—
70#14579152C2314yes—
71#14579148—22no—
72#14579148C2314yes—
73#14579143—28no—
74#14579141—28no—
75#14579141—22no—
76#14579141C2314yes—
77#14579137—28no—
78#14579132—22no—
79#14579132C2314yes—
80Q8×Dic79—28no—
81C4×Dic158—22no—
82C4×Dic158C2314yes—
83#14579126—28no—
84C4×C22⋊D79—28no—
85#14579120—28no—
86#14579119—24no—
87#14579119C2318yes—
88C2×C4⋊Dic79—21no—
89C2×C4⋊Dic79—21no—
90C2×C4⋊Dic79—21no—
91C2×C4⋊Dic79—22no—
92C2×C4⋊Dic79C2312yes—
93C2×C4⋊Dic79C2311yescharacteristic
94C2×Dic79⋊C4—21no—
95C2×Dic79⋊C4—21no—
96C2×Dic79⋊C4—22no—
97C2×C22⋊Dic79—21no—
98C2×C22⋊Dic79—21no—
99C2×C22⋊Dic79—21no—
100C2×(C2×D79)⋊C4—22no—
101C2×(C2×D79)⋊C4—21no—
102C2×C4×Dic79—21no—
103C2×C4×Dic79—22no—
104C2×C4×Dic79D411yescharacteristic
105C2×C4×Dic79D411yescharacteristic
106C2×C4×Dic79C2311yescharacteristic
107C42×D79D414yes—
108C42×D79C2314yes—
109#14579095C2318yes—
110#14579092C2318yes—
111#14579092C2318yes—
112C4⋊(C79⋊C8)C2314yes—
113#14579086C2318yes—
114#14579084C2318yes—
115#14579082C2318yes—
116(C2×D79)⋊C8C2318yes—
117C8×Dic79C2318yes—
118C22×Q8×C79—21no—
119C22×D4×C79C2311yescharacteristic
120C23×C4×C79—21no—
121C22×C8×C79C2312yes—
122C4⋊Q8×C79—22no—
123C4⋊Q8×C79C2314yes—
124C4×D4×C79—22no—
125C4×D4×C79C2314yes—
126C2×C4⋊C4×C79—21no—
127C2×C4⋊C4×C79C2311yescharacteristic
128C2×C4⋊C4×C79C2312yes—
129C2×C22⋊C4×C79—21no—
130C2×C42×C79C2311yescharacteristic
131#107×C79C2318yes—
132C4⋊C8×C79C2314yes—
133#102×C79C2318yes—
134C23×D79C2×D411yescharacteristic
135C2×Dic158—24no—
136C2×Dic158—21no—
137C2×Dic158—22no—
138C2×Dic158—22no—
139C2×Dic158—44no—
140C2×Dic158C2412yes—
141C2×C22⋊D79—24no—
142C2×C22⋊D79—44no—
143C22×Dic79—22no—
144C22×Dic79—21no—
145C22×Dic79—21no—
146C22×Dic79C2×D411yescharacteristic
147C22×Dic79C2×D411yescharacteristic
148C22×Dic79C2×D411yescharacteristic
149C2×C4×D79—24no—
150C2×C4×D79—22no—
151C2×C4×D79—22no—
152C2×C4×D79—42no—
153C2×C4×D79C2×D412yes—
154C2×C4×D79C2×D414yes—
155C2×C4×D79C2×D414yes—
156C2×C4×D79C2412yes—
157C2×C79⋊C8—24no—
158C2×C79⋊C8C2414yes—
159C4⋊Dic79—24no—
160C4⋊Dic79—24no—
161C4⋊Dic79—28no—
162C4⋊Dic79—24no—
163C4⋊Dic79—22no—
164C4⋊Dic79—21no—
165C4⋊Dic79C2414yes—
166C4⋊Dic79C2412yes—
167Dic79⋊C4—24no—
168Dic79⋊C4—28no—
169Dic79⋊C4—24no—
170C22⋊Dic79—24no—
171C22⋊Dic79—24no—
172C22⋊Dic79—24no—
173(C2×D79)⋊C4—28no—
174(C2×D79)⋊C4—24no—
175C4×Dic79—28no—
176C4×Dic79—24no—
177C4×Dic79C2×D412yes—
178C4×Dic79C2×D414yes—
179C4×Dic79C2×D414yes—
180C4×Dic79C2412yes—
181C24×C79—21no—
182C2×Q8×C79—24no—
183C2×Q8×C79—44no—
184C2×D4×C79—21no—
185C2×D4×C79—22no—
186C2×D4×C79—22no—
187C2×D4×C79C2412yes—
188C22×C4×C79—21no—
189C22×C4×C79—22no—
190C22×C4×C79—24no—
191C22×C4×C79—44no—
192C22×C4×C79C2×D411yescharacteristic
193C22×C4×C79C2×D411yescharacteristic
194C22×C4×C79C2411yescharacteristic
195C2×C8×C79—24no—
196C2×C8×C79C2414yes—
197C4⋊C4×C79—24no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—22no—
200C4⋊C4×C79C2414yes—
201C4⋊C4×C79C2412yes—
202C22⋊C4×C79—24no—
203C42×C79C2×D412yes—
204C42×C79C2412yes—
205C23×C79—22no—
206C23×C79—42no—
207C23×C79—42no—
208C23×C79—42no—
209C23×C79C22×D411yescharacteristic
210C22×D79—22no—
211C22×D79—22no—
212C22×D79—22no—
213C22×D79C22×D412yes—
214Q8×C79—48no—
215D4×C79—22no—
216D4×C79—22no—
217D4×C79—42no—
218C2×C4×C79—22no—
219C2×C4×C79—24no—
220C2×C4×C79—22no—
221C2×C4×C79—42no—
222C2×C4×C79—44no—
223C2×C4×C79—48no—
224C2×C4×C79#13514yes—
225C2×C4×C79C22×D412yes—
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C22×D412yes—
228C2×C4×C792−1+412yes—
229C2×C4×C792−1+412yes—
230C2×C4×C79C2511yescharacteristic
231C22⋊D79—48no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—22no—
235C2×Dic79—22no—
236C2×Dic79—24no—
237C2×Dic79—22no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—44no—
241C2×Dic79C22×D412yes—
242C2×Dic79C22×D412yes—
243C2×Dic79C22×D412yes—
244C4×D79—24no—
245C4×D79—28no—
246C4×D79—22no—
247C4×D79—22no—
248C4×D79—44no—
249Dic158—22no—
250Dic158—22no—
251Dic158—48no—
252Dic158—42no—
253C8×C79—24no—
254C79⋊C8—24no—
255C22×C79—42no—
256C22×C79—42no—
257C22×C79—42no—
258C22×C79—84no—
259C22×C79#53612yes—
260C22×C79C2×#13512yes—
261C22×C79#57512yes—
262C22×C79C23×D411yescharacteristic
263C2×D79—22no—
264C2×D79—22no—
265C2×D79—22no—
266C2×D79—44no—
267C4×C79—22no—
268C4×C79—42no—
269C4×C79—44no—
270C4×C79#53612yes—
271C4×C79C2×#13514yes—
272C4×C79C2×2−1+411yescharacteristic
273C4×C79C2×2−1+411yescharacteristic
274Dic79—22no—
275Dic79—22no—
276Dic79—22no—
277Dic79—42no—
278Dic79—42no—
279Dic79—44no—
280C2×#3233—791nomaximal, Sylow
281C2×C79—82no—
282C2×C79#323314yescomplemented
283C2×C79C2×#53611yescharacteristic
284C2×C79C22×#13511yescharacteristic
285C2×C79C2×#57511yescharacteristic
286D79—42no—
287C2×#533—791no—
288C2×#529—791no—
289#3233—7916no—
290C2×#499—792no—
291C2×#488—792no—
292C2×#485—792no—
293C2×#482—792no—
294C2×#477—791no—
295C2×#475—791no—
296C2×#476—791no—
297C2×#474D7911yescharacteristic, complemented
298C2×#431—791no—
299C79C2×#323311yescharacteristic, complemented, Sylow
300#533—798no—
301#533—1584no—
302C2×C4⋊Q8—791no—
303C2×C4⋊Q8C2×D7911yescharacteristic
304#529—798no—
305#529—1584no—
306C2×#126—1581no—
307C2×#125—792no—
308C2×#125—791no—
309C2×#124—791no—
310C2×#123—1581no—
311C2×C22⋊Q8—791no—
312C2×C22⋊Q8—1581no—
313C2×C22⋊Q8—1581no—
314C2×#121—791no—
315C2×C4×Q8—791no—
316C2×C4×Q8—1581no—
317C2×C4×Q8—1581no—
318C2×C4×D4—1581no—
319C2×C4×D4C2×D7911yescharacteristic
320C2×#117—792no—
321C2×#117—1581no—
322C22×C42—791no—
323#499—7916no—
324#488—7916no—
325#485—7916no—
326#482—7916no—
327#477—798no—
328#476—798no—
329#475—798no—
330#474C2×D7918yescomplemented
331#431—798no—
332C2×#107—792no—
333C2×#107C2×D7912yes—
334C2×#106—792no—
335C2×#106—792no—
336C2×C4⋊C8—791no—
337C2×C4⋊C8C2×D7911yescharacteristic
338C2×#103—792no—
339C2×#103—792no—
340C2×#102—792no—
341C2×#102C2×D7912yes—
342C2×C22⋊C8—792no—
343C2×C4×C8—792no—
344C22×Q8—21no—
345C22×Q8—791no—
346C22×Q8—1581no—
347C22×D4—1581no—
348C22×D4C22×D7911yescharacteristic
349C23×C4—21no—
350C23×C4—791no—
351C23×C4—791no—
352C23×C4—791no—
353C22×C8—792no—
354C22×C8C22×D7912yes—
355C4⋊Q8—22no—
356C4⋊Q8—794no—
357C4⋊Q8—1582no—
358C4⋊Q8C22×D7914yes—
359#126—1588no—
360#125—794no—
361#125—798no—
362#125—1582no—
363#125—1584no—
364#124—794no—
365#124—1582no—
366#123—1588no—
367C22⋊Q8—794no—
368C22⋊Q8—1582no—
369C22⋊Q8—1588no—
370C22⋊Q8—1588no—
371#121—794no—
372#121—1582no—
373C4×Q8—794no—
374C4×Q8—1588no—
375C4×Q8—1588no—
376C4×Q8—1582no—
377C4×D4—22no—
378C4×D4—1588no—
379C4×D4C22×D7914yes—
380#117—798no—
381#117—1588no—
382#117—1584no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—792no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4—1581no—
389C2×C4⋊C4—1582no—
390C2×C4⋊C4—1582no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4C22×D7912yes—
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1582no—
397C2×C22⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C42—791no—
402C2×C42—791no—
403C2×C42—791no—
404C2×C42—794no—
405C2×C42—794no—
406C2×C42—1582no—
407C2×C42—1581no—
408C2×C42C22×D7911yescharacteristic
409#107—798no—
410#107C22×D7918yes—
411#106—798no—
412#106—798no—
413C4⋊C8—794no—
414C4⋊C8C22×D7914yes—
415#103—798no—
416#103—798no—
417#102—798no—
418#102C22×D7918yes—
419C22⋊C8—798no—
420C4×C8—798no—
421C24—21no—
422C24—791no—
423C2×Q8—24no—
424C2×Q8—44no—
425C2×Q8—3164no—
426C2×Q8—792no—
427C2×Q8—1582no—
428C2×Q8—1582no—
429C2×Q8—1581no—
430C2×Q8—1584no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—3164no—
435C2×D4—1584no—
436C2×D4C23×D7912yes—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—3162no—
442C22×C4—794no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—792no—
447C22×C4—792no—
448C22×C4—794no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1582no—
452C22×C4—1582no—
453C22×C4—1582no—
454C22×C4—1584no—
455C22×C4D4×D7911yescharacteristic
456C22×C4D4×D7911yescharacteristic
457C22×C4C23×D7911yescharacteristic
458C2×C8—24no—
459C2×C8—794no—
460C2×C8—1584no—
461C2×C8C23×D7914yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—794no—
466C4⋊C4—792no—
467C4⋊C4—1584no—
468C4⋊C4—1588no—
469C4⋊C4—1584no—
470C4⋊C4—1584no—
471C4⋊C4—1588no—
472C4⋊C4—1584no—
473C4⋊C4—1582no—
474C4⋊C4—1581no—
475C4⋊C4—1584no—
476C4⋊C4C23×D7914yes—
477C4⋊C4C23×D7912yes—
478C22⋊C4—24no—
479C22⋊C4—1588no—
480C22⋊C4—1584no—
481C22⋊C4—1584no—
482C22⋊C4—1584no—
483C22⋊C4—1584no—
484C42—792no—
485C42—794no—
486C42—794no—
487C42—792no—
488C42—1588no—
489C42—1584no—
490C42D4×D7912yes—
491C42C23×D7912yes—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—792no—
497C23—1582no—
498C23—1582no—
499C23—1582no—
500C23C2×D4×D7911yescharacteristic
501Q8—48no—
502Q8—3168no—
503Q8—3162no—
504Q8—1582no—
505Q8—1582no—
506D4—22no—
507D4—22no—
508D4—42no—
509D4—3168no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—3164no—
517C2×C4—3162no—
518C2×C4—3162no—
519C2×C4—3164no—
520C2×C4—792no—
521C2×C4—792no—
522C2×C4—792no—
523C2×C4—1582no—
524C2×C4—1582no—
525C2×C4—1584no—
526C2×C4—1582no—
527C2×C4—1582no—
528C2×C4—1582no—
529C2×C4—1584no—
530C2×C4—1582no—
531C2×C4—1582no—
532C2×C4—1588no—
533C2×C4#1457918014yes—
534C2×C4C2×D4×D7912yes—
535C2×C4C2×D4×D7911yescharacteristic
536C2×C4C2×D4×D7912yes—
537C2×C4#1457920812yes—
538C2×C4#1457920912yes—
539C2×C4C24×D7911yescharacteristic
540C8—24no—
541C8—1584no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—84no—
546C22—3164no—
547C22—1582no—
548C22—1582no—
549C22—1582no—
550C22#1909913612yes—
551C22C2×#1457918012yes—
552C22#1909936212yes—
553C22C22×D4×D7911yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—3162no—
558C4—3164no—
559C4—3162no—
560C4—1582no—
561C4—1582no—
562C4—1582no—
563C4#1909913612yes—
564C4C2×#1457918014yes—
565C4C2×#1457920811yescharacteristic
566C4C2×#1457920911yescharacteristic
567C2—82no—
568C2—3162no—
569C2#2377457714yesdirect product factor
570C2C2×#1909913611yescharacteristic
571C2C22×#1457918011yescharacteristic
572C2C2×#1909936211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#474×C79C2
socleC23×C79C22×D4
prime radicalC2×#14579132C22
hypercenterC2×#474D79
hypocenterC79C2×#3233
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#19099118C2semidirect
3. #23774577C2direct (N × H)
4. C2×#19098964C2semidirect
5. C2×#19098916C2semidirect
6. C2×#19098906C2semidirect
8. C2×#19098859C2semidirect
9. C2×#19098845C2semidirect
10. C2×#19098853C2semidirect
11. C2×#19098615C2semidirect
12. C2×#474×C79C2semidirect
14. #19099118C22semidirect
34. #19098964C22semidirect
35. #19098916C22semidirect
36. #19098906C22semidirect
38. #19098859C22semidirect
39. #19098853C22semidirect
40. #19098845C22semidirect
41. #19098615C22semidirect
53. #474×C79C22semidirect
282. C2×C79#3233semidirect
297. C2×#474D79semidirect
299. C79C2×#3233semidirect
330. #474C2×D79semidirect
569. C2#23774577direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190991182
C2×#190990952
#237745772
C2×#190989642
C2×#190989162
C2×#190989062
C2×#190988852
C2×#190988592
C2×#190988452
C2×#190988532
C2×#190986152
C2×#474×C792
C2×#323379