Finite Group Databasebeta

← C2×#23763594 #34288981

Subgroups — C2×#23763594 #34288981

Order 20224.

1966 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D79C211yesmaximal, characteristic
2C22×#14579093C211yesmaximal, characteristic, complemented
3C22×#14579092C211yesmaximal, characteristic, complemented
4C22×(C2×D79)⋊C8C211yesmaximal, characteristic
5C2×#19098420C211yesmaximal, characteristic
6#23763594C218yesmaximal, direct product factor
7C2×#19098325C211yesmaximal, characteristic, complemented
8C2×#340×C79C211yesmaximal, characteristic, complemented
9C23×C4×D79—21no—
10C23×C4×D79C2211yescharacteristic
11C23×C79⋊C8C2211yescharacteristic
12C22×C4⋊Dic79C2211yescharacteristic
13C22×C4⋊Dic79C2211yescharacteristic
14C22×Dic79⋊C4—21no—
15C2×C4⋊C4×D79—22no—
16C2×C4⋊C4×D79—24no—
17C2×C4⋊C4×D79—24no—
18C2×C4⋊C4×D79C2214yes—
19C22×(C2×D79)⋊C4—21no—
20C2×#14579093—24no—
21C2×#14579093C2214yescomplemented
22C2×#14579092—24no—
23C2×#14579092C2214yescomplemented
24C2×(C2×D79)⋊C8C418yescomplemented
25C2×(C2×D79)⋊C8C2214yes—
26#19098420—22no—
27#19098420C2214yes—
28C2×C79⋊#95—21no—
29#19098325C2214yescomplemented
30C23×C8×C79C2211yescharacteristic
31C22×C4⋊C4×C79C2211yescharacteristic
32C22×C4⋊C4×C79C2211yescharacteristic
33#340×C79C2214yescomplemented
34C24×D79D411yescharacteristic
35C23×Dic79—21no—
36C23×Dic79—21no—
37C23×Dic79D411yescharacteristic
38C22×C4×D79—22no—
39C22×C4×D79—22no—
40C22×C4×D79—22no—
41C22×C4×D79—22no—
42C22×C4×D79—28no—
43C22×C4×D79—24no—
44C22×C4×D79—44no—
45C22×C4×D79—44no—
46C22×C4×D79C2×C414yes—
47C22×C4×D79D412yes—
48C22×C4×D79Q814yes—
49C22×C4×D79C2312yes—
50C22×C79⋊C8—22no—
51C22×C79⋊C8—22no—
52C22×C79⋊C8C2×C414yescomplemented
53C22×C79⋊C8C2312yes—
54C2×C4⋊Dic79—21no—
55C2×C4⋊Dic79—22no—
56C2×C4⋊Dic79—22no—
57C2×C4⋊Dic79—22no—
58C2×C4⋊Dic79—21no—
59C2×C4⋊Dic79D414yes—
60C2×C4⋊Dic79C2312yes—
61C2×C4⋊Dic79C2312yes—
62C2×Dic79⋊C4—28no—
63C2×Dic79⋊C4—24no—
64C4⋊C4×D79—216no—
65C4⋊C4×D79—28no—
66C4⋊C4×D79—28no—
67C2×(C2×D79)⋊C4—24no—
68C2×(C2×D79)⋊C4—44no—
69#14579093—28no—
70#14579092—28no—
71(C2×D79)⋊C8C2×C4116yescomplemented
72C79⋊#95—24no—
73C23×C4×C79—21no—
74C23×C4×C79—21no—
75C23×C4×C79C2311yescharacteristic
76C22×C8×C79—22no—
77C22×C8×C79—22no—
78C22×C8×C79C2×C414yescomplemented
79C22×C8×C79C2312yes—
80C2×C4⋊C4×C79—21no—
81C2×C4⋊C4×C79—22no—
82C2×C4⋊C4×C79—22no—
83C2×C4⋊C4×C79—22no—
84C2×C4⋊C4×C79—21no—
85C2×C4⋊C4×C79D414yes—
86C2×C4⋊C4×C79C2312yes—
87C2×C4⋊C4×C79C2312yes—
88C23×D79—22no—
89C23×D79—22no—
90C23×D79—22no—
91C23×D79—22no—
92C23×D79C4⋊C414yes—
93C23×D79SD1614yes—
94C23×D79SD1614yes—
95C23×D79C2×D412yes—
96C22×Dic79—22no—
97C22×Dic79—24no—
98C22×Dic79—22no—
99C22×Dic79—22no—
100C22×Dic79—22no—
101C22×Dic79—42no—
102C22×Dic79—42no—
103C22×Dic79—42no—
104C22×Dic79—42no—
105C22×Dic79—42no—
106C22×Dic79C4⋊C414yes—
107C22×Dic79C2×D412yes—
108C2×C4×D79—216no—
109C2×C4×D79—28no—
110C2×C4×D79—28no—
111C2×C4×D79—24no—
112C2×C4×D79—28no—
113C2×C4×D79—28no—
114C2×C4×D79—24no—
115C2×C4×D79—22no—
116C2×C4×D79—22no—
117C2×C4×D79—22no—
118C2×C4×D79—22no—
119C2×C4×D79—48no—
120C2×C4×D79—432no—
121C2×C4×D79—48no—
122C2×C4×D79C4⋊C418yes—
123C2×C4×D79C22×C414yes—
124C2×C4×D79C2×Q814yes—
125C2×C79⋊C8—28no—
126C2×C79⋊C8—22no—
127C2×C79⋊C8—22no—
128C2×C79⋊C8C22×C414yescomplemented
129C4⋊Dic79—24no—
130C4⋊Dic79—22no—
131C4⋊Dic79—22no—
132C4⋊Dic79—22no—
133C4⋊Dic79—42no—
134C4⋊Dic79C2×D414yes—
135Dic79⋊C4—216no—
136(C2×D79)⋊C4—48no—
137C24×C79C2×D411yescharacteristic
138C22×C4×C79—24no—
139C22×C4×C79—22no—
140C22×C4×C79—22no—
141C22×C4×C79—42no—
142C22×C4×C79—42no—
143C22×C4×C79—42no—
144C22×C4×C79—42no—
145C22×C4×C79—42no—
146C22×C4×C79C22×C412yes—
147C22×C4×C79C2×D411yescharacteristic
148C22×C4×C79C2×D412yes—
149C22×C4×C79C2×D412yes—
150C22×C4×C79C2×Q812yes—
151C22×C4×C79#4112yes—
152C22×C4×C79#4112yes—
153C22×C4×C79C2411yescharacteristic
154C2×C8×C79—28no—
155C2×C8×C79—22no—
156C2×C8×C79—22no—
157C2×C8×C79C22×C414yescomplemented
158C4⋊C4×C79—24no—
159C4⋊C4×C79—22no—
160C4⋊C4×C79—22no—
161C4⋊C4×C79—22no—
162C4⋊C4×C79—42no—
163C4⋊C4×C79C2×D414yes—
164C23×C79C2×C4⋊C412yes—
165C23×C79#12012yes—
166C23×C79#12312yes—
167C23×C79C2×SD1612yes—
168C23×C79C2×SD1612yes—
169C23×C79AGL(1,ℤ/8)12yes—
170C23×C79#13712yes—
171C23×C79C22×D411yescharacteristic
172C22×D79—22no—
173C22×D79—22no—
174C22×D79—22no—
175C22×D79—22no—
176C22×D79—28no—
177C22×D79—28no—
178C22×D79—28no—
179C22×D79—28no—
180C22×D79—28no—
181C22×D79—28no—
182C22×D79#106116yescomplemented
183C22×D79C2×C4⋊C414yes—
184C22×D79C2×SD1614yes—
185C22×D79C2×SD1614yes—
186C2×C4×C79—24no—
187C2×C4×C79—42no—
188C2×C4×C79—42no—
189C2×C4×C79—48no—
190C2×C4×C79—42no—
191C2×C4×C79—42no—
192C2×C4×C79—42no—
193C2×C4×C79—84no—
194C2×C4×C79C2×C4⋊C412yes—
195C2×C4×C79C4×D414yes—
196C2×C4×C79C4×D414yes—
197C2×C4×C79#12012yes—
198C2×C4×C79C22⋊Q814yes—
199C2×C4×C79C22⋊Q814yes—
200C2×C4×C79#12312yes—
201C2×C4×C79C23×C411yescharacteristic
202C2×C4×C79C22×D411yescharacteristic
203C2×C4×C79C22×D411yescharacteristic
204C2×C4×C79C22×Q811yescharacteristic
205C2×C4×C79C2×#4111yescharacteristic
206C2×C4×C79C2×#4111yescharacteristic
207C2×Dic79—22no—
208C2×Dic79—22no—
209C2×Dic79—28no—
210C2×Dic79—24no—
211C2×Dic79—42no—
212C2×Dic79—42no—
213C2×Dic79—48no—
214C2×Dic79—42no—
215C2×Dic79—42no—
216C2×Dic79—42no—
217C2×Dic79—84no—
218C2×Dic79C2×C4⋊C414yes—
219C4×D79—24no—
220C4×D79—24no—
221C4×D79—24no—
222C4×D79—24no—
223C4×D79—216no—
224C4×D79—432no—
225C8×C79—24no—
226C79⋊C8—24no—
227C22×C79#38514yes—
228C22×C79C2×#10614yes—
229C22×C79#42814yes—
230C22×C79#44814yes—
231C22×C79#45014yes—
232C22×C79#48114yes—
233C22×C79#48314yes—
234C22×C79C22×C4⋊C411yescharacteristic
235C22×C79C2×#12011yescharacteristic
236C22×C79C2×#12311yescharacteristic
237C22×C79C22×SD1611yescharacteristic
238C22×C79C22×SD1611yescharacteristic
239C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
240C22×C79C2×#13711yescharacteristic
241C2×D79—24no—
242C2×D79—24no—
243C2×D79—232no—
244C2×D79—24no—
245C2×D79—24no—
246C2×D79—24no—
247C2×D79—24no—
248C2×D79C2×#10618yescomplemented
249C4×C79—44no—
250C4×C79—82no—
251C4×C79#38514yes—
252C4×C79C2×C4×D411yescharacteristic
253C4×C79C2×C4×D411yescharacteristic
254C4×C79C2×C22⋊Q811yescharacteristic
255C4×C79C2×C22⋊Q811yescharacteristic
256Dic79—24no—
257Dic79—44no—
258Dic79—82no—
259C2×#1853—791nomaximal, Sylow
260C2×C79#185318yescomplemented
261C2×C79C2×#38511yescharacteristic
262C2×C79C22×#10611yescharacteristic
263C2×C79C2×#42811yescharacteristic
264C2×C79C2×#45011yescharacteristic
265C2×C79C2×#44811yescharacteristic
266C2×C79C2×#48311yescharacteristic
267C2×C79C2×#48111yescharacteristic
268D79—28no—
269C23×C4⋊C4—791no—
270C22×#106—791no—
271C22×#106—791no—
272C22×C22⋊C8—791no—
273C2×#380—791no—
274#1853—798no—
275C2×#340—791no—
276C2×#340D7911yescharacteristic, complemented
277C79C2×#185311yescharacteristic, complemented, Sylow
278C24×C4—791no—
279C24×C4—1581no—
280C23×C8—791no—
281C23×C8C2×D7911yescharacteristic
282C22×C4⋊C4—791no—
283C22×C4⋊C4—791no—
284C22×C4⋊C4—794no—
285C22×C4⋊C4—1584no—
286C22×C4⋊C4—1582no—
287C22×C4⋊C4—1581no—
288C22×C4⋊C4—1584no—
289C22×C4⋊C4C2×D7911yescharacteristic
290C22×C4⋊C4C2×D7911yescharacteristic
291C22×C22⋊C4—1581no—
292C2×#106—794no—
293C2×#106—794no—
294C2×#106—1584no—
295C2×#106—1584no—
296C2×C22⋊C8—794no—
297C2×C22⋊C8—798no—
298#380—794no—
299#380—1582no—
300C2×#95—1581no—
301#340—794no—
302#340C2×D7914yescomplemented
303C25—791no—
304C23×C4—21no—
305C23×C4—21no—
306C23×C4—3164no—
307C23×C4—3164no—
308C23×C4—791no—
309C23×C4—794no—
310C23×C4—792no—
311C23×C4—794no—
312C23×C4—792no—
313C23×C4—1584no—
314C23×C4—1588no—
315C23×C4—1581no—
316C23×C4—1581no—
317C23×C4—1582no—
318C23×C4—1582no—
319C23×C4—1582no—
320C23×C4—1582no—
321C23×C4C22×D7911yescharacteristic
322C22×C8—22no—
323C22×C8—22no—
324C22×C8—794no—
325C22×C8—792no—
326C22×C8—1582no—
327C22×C8—1582no—
328C22×C8C4×D7914yescomplemented
329C22×C8C22×D7912yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—794no—
336C2×C4⋊C4—792no—
337C2×C4⋊C4—792no—
338C2×C4⋊C4—1588no—
339C2×C4⋊C4—15816no—
340C2×C4⋊C4—1581no—
341C2×C4⋊C4—1582no—
342C2×C4⋊C4—1582no—
343C2×C4⋊C4—1584no—
344C2×C4⋊C4—1588no—
345C2×C4⋊C4—1581no—
346C2×C4⋊C4—1582no—
347C2×C4⋊C4—1588no—
348C2×C4⋊C4D31614yes—
349C2×C4⋊C4C22×D7912yes—
350C2×C4⋊C4C22×D7912yes—
351C2×C22⋊C4—3164no—
352C2×C22⋊C4—1584no—
353#106—1588no—
354#106—1588no—
355C22⋊C8—7916no—
356#95—1584no—
357C24—794no—
358C24—794no—
359C24—794no—
360C24—792no—
361C24—1582no—
362C24—1582no—
363C24—1582no—
364C24—1582no—
365C24D4×D7911yescharacteristic
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—3168no—
375C22×C4—31632no—
376C22×C4—3168no—
377C22×C4—3162no—
378C22×C4—3162no—
379C22×C4—3162no—
380C22×C4—3162no—
381C22×C4—3162no—
382C22×C4—794no—
383C22×C4—792no—
384C22×C4—794no—
385C22×C4—798no—
386C22×C4—794no—
387C22×C4—15816no—
388C22×C4—1582no—
389C22×C4—1584no—
390C22×C4—1582no—
391C22×C4—1582no—
392C22×C4—1582no—
393C22×C4—1582no—
394C22×C4—1582no—
395C22×C4—1588no—
396C22×C4—1588no—
397C22×C4—1584no—
398C22×C4—1588no—
399C22×C4—1588no—
400C22×C4—1584no—
401C22×C4—1582no—
402C22×C4—1582no—
403C22×C4C2×C4×D7912yes—
404C22×C4D4×D7911yescharacteristic
405C22×C4D4×D7912yes—
406C22×C4C2×D31612yes—
407C22×C4Q8×D7912yes—
408C22×C4#1192962312yes—
409C22×C4#1192962412yes—
410C22×C4C23×D7911yescharacteristic
411C2×C8—22no—
412C2×C8—28no—
413C2×C8—22no—
414C2×C8—794no—
415C2×C8—1582no—
416C2×C8—1588no—
417C2×C8—1582no—
418C2×C8C2×C4×D7914yescomplemented
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—3162no—
425C4⋊C4—794no—
426C4⋊C4—15816no—
427C4⋊C4—1582no—
428C4⋊C4—1582no—
429C4⋊C4—1584no—
430C4⋊C4—1582no—
431C4⋊C4C2×D31614yes—
432C22⋊C4—3168no—
433C23—794no—
434C23—794no—
435C23—794no—
436C23—7916no—
437C23—1588no—
438C23—1588no—
439C23—1588no—
440C23—1588no—
441C23—1588no—
442C23—1582no—
443C23—1582no—
444C23—1582no—
445C23—1582no—
446C23—1588no—
447C23C4⋊C4×D7912yes—
448C23#1457913112yes—
449C23#1457914312yes—
450C23SD16×D7912yes—
451C23SD16×D7912yes—
452C23#1457918512yes—
453C23#1457919312yes—
454C23C2×D4×D7911yescharacteristic
455C2×C4—24no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—31632no—
464C2×C4—3168no—
465C2×C4—3162no—
466C2×C4—3162no—
467C2×C4—3162no—
468C2×C4—3162no—
469C2×C4—3162no—
470C2×C4—794no—
471C2×C4—6324no—
472C2×C4—1584no—
473C2×C4—1588no—
474C2×C4—1582no—
475C2×C4—1582no—
476C2×C4—15816no—
477C2×C4—1584no—
478C2×C4—1584no—
479C2×C4—1584no—
480C2×C4—1584no—
481C2×C4C4⋊C4×D7912yes—
482C2×C4(C22⋊D79)⋊C414yes—
483C2×C4C4×D31614yes—
484C2×C4#1457913112yes—
485C2×C4#1457913914yes—
486C2×C4#1457914114yes—
487C2×C4#1457914312yes—
488C2×C4C22×C4×D7911yescharacteristic
489C2×C4C2×D4×D7911yescharacteristic
490C2×C4C22×D31611yescharacteristic
491C2×C4C2×Q8×D7911yescharacteristic
492C2×C4C2×#1192962311yescharacteristic
493C2×C4C2×#1192962411yescharacteristic
494C8—24no—
495C8—1584no—
496C22—798no—
497C22—1584no—
498C22—1584no—
499C22—1584no—
500C22—1584no—
501C22—15832no—
502C22—1584no—
503C22—1584no—
504C22#1909843714yes—
505C22#106×D7914yes—
506C22#1909860414yes—
507C22#1909870314yes—
508C22#1909871514yes—
509C22#1909888014yes—
510C22#1909889214yes—
511C22C2×C4⋊C4×D7911yescharacteristic
512C22C2×#1457913111yescharacteristic
513C22C2×#1457914311yescharacteristic
514C22C2×SD16×D7911yescharacteristic
515C22C2×SD16×D7911yescharacteristic
516C22C2×#1457918511yescharacteristic
517C22C2×#1457919311yescharacteristic
518C4—44no—
519C4—82no—
520C4—3164no—
521C4—6322no—
522C4—1584no—
523C4#1909843714yes—
524C4C2×(C22⋊D79)⋊C411yescharacteristic
525C4C2×C4×D31611yescharacteristic
526C4C2×#1457913911yescharacteristic
527C4C2×#1457914111yescharacteristic
528C2—1588no—
529C2#2376359418yesdirect product factor
530C2C2×#1909843711yescharacteristic
531C2C2×#106×D7911yescharacteristic
532C2C2×#1909860411yescharacteristic
533C2C2×#1909871511yescharacteristic
534C2C2×#1909870311yescharacteristic
535C2C2×#1909889211yescharacteristic
536C2C2×#1909888011yescharacteristic
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×#1853
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
2. C22×#14579093C2semidirect
3. C22×#14579092C2semidirect
6. #23763594C2direct (N × H)
7. C2×#19098325C2semidirect
8. C2×#340×C79C2semidirect
21. C2×#14579093C22semidirect
23. C2×#14579092C22semidirect
24. C2×(C2×D79)⋊C8C4semidirect
29. #19098325C22semidirect
33. #340×C79C22semidirect
52. C22×C79⋊C8C2×C4semidirect
71. (C2×D79)⋊C8C2×C4semidirect
78. C22×C8×C79C2×C4semidirect
128. C2×C79⋊C8C22×C4semidirect
157. C2×C8×C79C22×C4semidirect
182. C22×D79#106semidirect
248. C2×D79C2×#106semidirect
260. C2×C79#1853semidirect
276. C2×#340D79semidirect
277. C79C2×#1853semidirect
302. #340C2×D79semidirect
328. C22×C8C4×D79semidirect
418. C2×C8C2×C4×D79semidirect
529. C2#23763594direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D792
C22×#145790932
C22×#145790922
C22×(C2×D79)⋊C82
C2×#190984202
#237635942
C2×#190983252
C2×#340×C792
C2×#185379