Finite Group Databasebeta

← #34297948

Subgroups — #34297948

Order 20224.

1222 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099115C211yesmaximal, characteristic
2C2×#19098613C211yesmaximal, characteristic
3C2×#19098432C211yesmaximal, characteristic
4#23764359C212yesmaximal, complemented
5#23764200C212yesmaximal, complemented
6#23763708C211yesmaximal, characteristic
7#23763707C211yesmaximal, characteristic
8#23763705C211yesmaximal, characteristic
9#23763404C212yesmaximal, complemented
10#23763403C212yesmaximal, complemented
11C79×#1872C211yesmaximal, characteristic
12#19099115—24no—
13#19099115C418yescomplemented
14C2×#14579160C2211yescharacteristic
15C2×#14579159C2211yescharacteristic
16C2×#14579151—21no—
17C2×#14579140C2211yescharacteristic
18C2×#14579138—21no—
19C2×Q8×Dic79C2211yescharacteristic
20C2×#14579126—21no—
21C2×#14579126C2211yescharacteristic
22C2×#14579117—21no—
23C22×C4⋊Dic79C2211yescharacteristic
24C22×Dic79⋊C4—21no—
25C22×Dic79⋊C4—21no—
26C22×C4×Dic79C2211yescharacteristic
27#19098613—24no—
28C2×Dic79⋊C8C2212yes—
29C2×#14579086C2212yes—
30C2×#14579085C2212yes—
31C2×#14579084C2212yes—
32C2×C22⋊(C79⋊C8)C2211yescharacteristic
33C2×C8×Dic79C2212yes—
34#19098432—21no—
35#19098432—21no—
36#19098432C2212yes—
37#19098432C2212yes—
38#19098326C2212yes—
39#19098325C2212yes—
40#19098325C2212yes—
41C2×C22⋊Q8×C79C2211yescharacteristic
42C22×C4⋊C4×C79C2211yescharacteristic
43C2×#103×C79C2212yes—
44C2×C22⋊C8×C79C2211yescharacteristic
45#340×C79C2212yes—
46C22×Dic158—21no—
47C22×Dic158C2311yescharacteristic
48C23×Dic79—21no—
49C23×Dic79D411yescharacteristic
50C23×Dic79D411yescharacteristic
51C22×C79⋊C8C2312yes—
52#14579160—22no—
53#14579160C2×C414yescomplemented
54#14579159—22no—
55#14579159C2×C414yescomplemented
56#14579151—28no—
57#14579140—22no—
58#14579140C2×C414yes—
59#14579138—28no—
60Q8×Dic79—22no—
61Q8×Dic79C2×C414yescomplemented
62#14579126—28no—
63#14579126—22no—
64#14579126C2×C414yescomplemented
65#14579117—28no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—42no—
69C2×C4⋊Dic79C2312yes—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—22no—
77C2×Dic79⋊C4—22no—
78C2×Dic79⋊C4—22no—
79C2×Dic79⋊C4—22no—
80C2×Dic79⋊C4—44no—
81C2×Dic79⋊C4—44no—
82C2×C22⋊Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79C2×C414yes—
86C2×C4×Dic79D411yescharacteristic
87C2×C4×Dic79D411yescharacteristic
88C2×C4×Dic79D414yes—
89C2×C4×Dic79C2311yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91Dic79⋊C8—24no—
92#14579086—42no—
93#14579085—42no—
94#14579084—42no—
95C22⋊(C79⋊C8)—22no—
96C8×Dic79—24no—
97C22×Q8×C79C2311yescharacteristic
98C23×C4×C79—21no—
99C23×C4×C79C2311yescharacteristic
100C22×C8×C79C2312yes—
101C22⋊Q8×C79—22no—
102C22⋊Q8×C79C2×C414yes—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—42no—
106C2×C4⋊C4×C79C2312yes—
107C2×C4⋊C4×C79C2311yescharacteristic
108C2×C22⋊C4×C79—21no—
109#103×C79—42no—
110C22⋊C8×C79—22no—
111C2×Dic158—24no—
112C2×Dic158—21no—
113C2×Dic158—22no—
114C2×Dic158—22no—
115C2×Dic158—42no—
116C2×Dic158—42no—
117C2×Dic158C22×C412yes—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—22no—
123C22×Dic79—42no—
124C22×Dic79—42no—
125C22×Dic79—42no—
126C22×Dic79C22⋊C414yes—
127C22×Dic79C22⋊C414yes—
128C22×Dic79SD1614yes—
129C22×Dic79Q1614yes—
130C22×Dic79C2×D411yescharacteristic
131C22×Dic79C2×D411yescharacteristic
132C22×Dic79C2×D411yescharacteristic
133C22×Dic79C2×D411yescharacteristic
134C2×C79⋊C8—22no—
135C2×C79⋊C8—22no—
136C2×C79⋊C8—22no—
137C4⋊Dic79—24no—
138C4⋊Dic79—21no—
139C4⋊Dic79—22no—
140C4⋊Dic79—22no—
141C4⋊Dic79—42no—
142C4⋊Dic79C22×C412yes—
143Dic79⋊C4—24no—
144Dic79⋊C4—24no—
145Dic79⋊C4—24no—
146Dic79⋊C4—24no—
147Dic79⋊C4—24no—
148Dic79⋊C4—44no—
149Dic79⋊C4—44no—
150Dic79⋊C4—44no—
151Dic79⋊C4—44no—
152C22⋊Dic79—24no—
153C4×Dic79—24no—
154C4×Dic79—24no—
155C4×Dic79C22⋊C414yes—
156C4×Dic79C22⋊C414yes—
157C4×Dic79C22×C412yes—
158C4×Dic79C22×C412yes—
159C4×Dic79C2×D412yes—
160C4×Dic79C2×D412yes—
161C24×C79C2×D411yescharacteristic
162C2×Q8×C79—21no—
163C2×Q8×C79—22no—
164C2×Q8×C79—22no—
165C2×Q8×C79C22×C412yes—
166C22×C4×C79—22no—
167C22×C4×C79—22no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—22no—
171C22×C4×C79—42no—
172C22×C4×C79—42no—
173C22×C4×C79—42no—
174C22×C4×C79C22×C412yes—
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D412yes—
177C22×C4×C79C2411yescharacteristic
178C2×C8×C79—22no—
179C2×C8×C79—22no—
180C2×C8×C79—22no—
181C4⋊C4×C79—24no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—22no—
184C4⋊C4×C79—22no—
185C4⋊C4×C79—42no—
186C4⋊C4×C79C22×C412yes—
187C22⋊C4×C79—24no—
188C23×C79—21no—
189C23×C79—21no—
190C23×C79—21no—
191C23×C79—21no—
192C23×C79C2×C22⋊C412yes—
193C23×C79C2×SD1612yes—
194C23×C79C2×Q1612yes—
195C23×C79C22×D411yescharacteristic
196Q8×C79—24no—
197Q8×C79—42no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—24no—
201C2×C4×C79—24no—
202C2×C4×C79—22no—
203C2×C4×C79—22no—
204C2×C4×C79—41no—
205C2×C4×C79—41no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—84no—
212C2×C4×C79C2×C22⋊C412yes—
213C2×C4×C79C23×C411yescharacteristic
214C2×C4×C79C22×D411yescharacteristic
215C2×C4×C79H(5,𝔽2)12yes—
216C2×C4×C792−1+412yes—
217C2×Dic79—21no—
218C2×Dic79—21no—
219C2×Dic79—28no—
220C2×Dic79—22no—
221C2×Dic79—22no—
222C2×Dic79—41no—
223C2×Dic79—41no—
224C2×Dic79—41no—
225C2×Dic79—41no—
226C2×Dic79—42no—
227C2×Dic79—42no—
228C2×Dic79—42no—
229C2×Dic79—84no—
230C2×Dic79#103116yescomplemented
231C2×Dic79C2×C22⋊C412yes—
232C2×Dic79C2×C22⋊C412yes—
233C2×Dic79C2×C22⋊C412yes—
234C2×Dic79C2×C22⋊C412yes—
235C2×Dic79C2×SD1612yes—
236C2×Dic79C2×SD1612yes—
237C2×Dic79C2×Q1612yes—
238C2×Dic79C2×Q1612yes—
239C2×Dic79AGL(1,ℤ/8)12yes—
240C2×Dic79#13712yes—
241Dic158—24no—
242Dic158—42no—
243Dic158—48no—
244C8×C79—42no—
245C79⋊C8—42no—
246C22×C79—22no—
247C22×C79—22no—
248C22×C79—22no—
249C22×C79—22no—
250C22×C79—22no—
251C22×C79—22no—
252C22×C79C2×#10314yes—
253C22×C79C22×C22⋊C411yescharacteristic
254C22×C79#53511yescharacteristic
255C22×C79#53611yescharacteristic
256C22×C79C22×SD1611yescharacteristic
257C22×C79C22×Q1611yescharacteristic
258C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C79C2×#13711yescharacteristic
260C4×C79—22no—
261C4×C79—42no—
262C4×C79—42no—
263C4×C79—82no—
264C4×C79#52012yes—
265C4×C79#53712yes—
266Dic79—22no—
267Dic79—42no—
268Dic79—42no—
269Dic79—82no—
270Dic79C2×#10314yes—
271Dic79C2×#10314yes—
272Dic79#41714yes—
273#21860—791nomaximal, Sylow
274C2×C79—24no—
275C2×C79#231111yescharacteristic
276C2×C79C22×#10311yescharacteristic
277C2×C79C2×#41711yescharacteristic
278C2×C79#320811yescharacteristic
279C2×C79#320911yescharacteristic
280C2×C79#322911yescharacteristic
281C2×C79#323011yescharacteristic
282C2×#533—791no—
283C2×#431—791no—
284C2×#384—791no—
285#1990—792no—
286#1956—792no—
287#1872—791no—
288#1872—791no—
289#1872—791no—
290#1872D7911yescharacteristic
291#1803—792no—
292#1803—792no—
293C79#2186011yescharacteristic, complemented, Sylow
294#533—798no—
295#533—1584no—
296C2×C4⋊Q8—791no—
297C2×C4⋊Q8—791no—
298C2×#125—1581no—
299C2×C22⋊Q8—791no—
300C2×C22⋊Q8—1581no—
301C2×C22⋊Q8C2×D7911yescharacteristic
302C2×C4×Q8—791no—
303C2×C4×Q8—791no—
304C2×C4×Q8—1581no—
305C2×#117—1581no—
306C22×C4⋊C4—791no—
307C22×C4⋊C4—1581no—
308C22×C4⋊C4—1581no—
309C22×C4⋊C4C2×D7911yescharacteristic
310C22×C42—791no—
311#431—1584no—
312C2×C4⋊C8—792no—
313C2×#103—792no—
314C2×#103—792no—
315C2×#103—792no—
316C2×#103C2×D7912yes—
317C2×C22⋊C8—791no—
318C2×C22⋊C8C2×D7911yescharacteristic
319C2×C4×C8—792no—
320#384—792no—
321#384—792no—
322#384—1581no—
323#384—1581no—
324#340—792no—
325#340—792no—
326#340—792no—
327#340C2×D7912yes—
328C22×Q8—791no—
329C22×Q8—1581no—
330C22×Q8C22×D7911yescharacteristic
331C23×C4—21no—
332C23×C4—791no—
333C23×C4—791no—
334C23×C4—1581no—
335C23×C4C22×D7911yescharacteristic
336C22×C8—792no—
337C22×C8C22×D7912yes—
338C4⋊Q8—794no—
339C4⋊Q8—794no—
340C4⋊Q8—1582no—
341C4⋊Q8—1582no—
342#125—1588no—
343C22⋊Q8—22no—
344C22⋊Q8—794no—
345C22⋊Q8—1588no—
346C22⋊Q8—1582no—
347C22⋊Q8C4×D7914yes—
348C4×Q8—794no—
349C4×Q8—794no—
350C4×Q8—1582no—
351C4×Q8—1588no—
352C4×Q8—1582no—
353#117—1588no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—3164no—
358C2×C4⋊C4—3164no—
359C2×C4⋊C4—3162no—
360C2×C4⋊C4—791no—
361C2×C4⋊C4—792no—
362C2×C4⋊C4—1582no—
363C2×C4⋊C4—1582no—
364C2×C4⋊C4—1582no—
365C2×C4⋊C4—1582no—
366C2×C4⋊C4—1581no—
367C2×C4⋊C4—1581no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4—1581no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4C22×D7912yes—
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1581no—
377C2×C42—794no—
378C2×C42—791no—
379C2×C42—791no—
380C2×C42—794no—
381C2×C42—791no—
382C2×C42—791no—
383C2×C42—1581no—
384C2×C42—1581no—
385C4⋊C8—1584no—
386#103—42no—
387#103—3162no—
388#103—3162no—
389#103—3162no—
390C22⋊C8—22no—
391C22⋊C8—1582no—
392C4×C8—1584no—
393C24D4×D7911yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—3162no—
398C2×Q8—3162no—
399C2×Q8—792no—
400C2×Q8—1584no—
401C2×Q8—1582no—
402C2×Q8—1582no—
403C2×Q8—1581no—
404C2×Q8C2×C4×D7912yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—3162no—
414C22×C4—3162no—
415C22×C4—3162no—
416C22×C4—794no—
417C22×C4—794no—
418C22×C4—794no—
419C22×C4—791no—
420C22×C4—791no—
421C22×C4—791no—
422C22×C4—794no—
423C22×C4—791no—
424C22×C4—1582no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1582no—
428C22×C4—1582no—
429C22×C4C2×C4×D7912yes—
430C22×C4D4×D7911yescharacteristic
431C22×C4D4×D7912yes—
432C22×C4C23×D7911yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1582no—
437C2×C8—1582no—
438C2×C8—1582no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—3162no—
445C4⋊C4—3164no—
446C4⋊C4—3164no—
447C4⋊C4—3164no—
448C4⋊C4—3164no—
449C4⋊C4—792no—
450C4⋊C4—1584no—
451C4⋊C4—1584no—
452C4⋊C4—1584no—
453C4⋊C4—1584no—
454C4⋊C4—1584no—
455C4⋊C4—1582no—
456C4⋊C4—1582no—
457C4⋊C4—1584no—
458C4⋊C4—1581no—
459C4⋊C4C2×C4×D7912yes—
460C22⋊C4—24no—
461C22⋊C4—1584no—
462C42—794no—
463C42—794no—
464C42—792no—
465C42—792no—
466C42—792no—
467C42—792no—
468C42—1584no—
469C42—1584no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22⋊C4×D7912yes—
475C23SD16×D7912yes—
476C23Q16×D7912yes—
477C23C2×D4×D7911yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—3168no—
481Q8—3162no—
482Q8—1584no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—3161no—
498C2×C4—3161no—
499C2×C4—3161no—
500C2×C4—3161no—
501C2×C4—3162no—
502C2×C4—3162no—
503C2×C4—3162no—
504C2×C4—792no—
505C2×C4—792no—
506C2×C4—792no—
507C2×C4—7916no—
508C2×C4—792no—
509C2×C4—792no—
510C2×C4—792no—
511C2×C4—792no—
512C2×C4—792no—
513C2×C4—792no—
514C2×C4—792no—
515C2×C4—6324no—
516C2×C4—1582no—
517C2×C4—1582no—
518C2×C4—1588no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4C22⋊C4×D7912yes—
522C2×C4C22×C4×D7911yescharacteristic
523C2×C4C2×D4×D7911yescharacteristic
524C2×C4#1457920612yes—
525C2×C4#1457920812yes—
526C8—42no—
527C8—3162no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22#103×D7914yes—
535C22C2×C22⋊C4×D7911yescharacteristic
536C22#1909912811yescharacteristic
537C22#1909913511yescharacteristic
538C22C2×SD16×D7911yescharacteristic
539C22C2×Q16×D7911yescharacteristic
540C22AGL(1,ℤ/8)×D7911yescharacteristic
541C22#137×D7911yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—3162no—
547C4—3162no—
548C4—794no—
549C4—794no—
550C4—794no—
551C4—6322no—
552C4—1582no—
553C4#1909904212yes—
554C4#1909914012yes—
555C2—24no—
556C2#2376598711yescharacteristic
557C2C2×#103×D7911yescharacteristic
558C2#417×D7911yescharacteristic
559C2#2377433911yescharacteristic
560C2#2377434611yescharacteristic
561C2#2377454211yescharacteristic
562C2#2377454911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1872C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1872D79
hypocenterC79#21860
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #23764359C2semidirect
5. #23764200C2semidirect
9. #23763404C2semidirect
10. #23763403C2semidirect
13. #19099115C4semidirect
53. #14579160C2×C4semidirect
55. #14579159C2×C4semidirect
61. Q8×Dic79C2×C4semidirect
64. #14579126C2×C4semidirect
230. C2×Dic79#103semidirect
293. C79#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#190991152
C2×#190986132
C2×#190984322
#237643592
#237642002
#237637082
#237637072
#237637052
#237634042
#237634032
C79×#18722
#2186079