Finite Group Databasebeta

← #1805×D79 #34287730

Subgroups — #1805×D79 #34287730

Order 20224.

1510 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D79C211yesmaximal, characteristic
2C2×C4⋊C8×D79C211yesmaximal, characteristic
3C2×#103×D79C212yesmaximal
4#382×D79C211yesmaximal, characteristic
5#23763416C211yesmaximal, characteristic, complemented
6#23763415C211yesmaximal, characteristic, complemented
7#23763414C212yesmaximal, complemented
8#23763413C212yesmaximal, complemented
9#23763412C211yesmaximal, characteristic, complemented
10#340×D79C212yesmaximal
11C79×#1805C211yesmaximal, characteristic, complemented
12C22×Q8×D79C2211yescharacteristic
13C22×C8×D79C2212yes—
14C2×Q8×Dic79C2211yescharacteristic
15C2×C4×Dic158—21no—
16C2×C4×Dic158C2211yescharacteristic
17C2×#14579126—21no—
18C2×#14579126C2211yescharacteristic
19C4×Q8×D79—24no—
20C4×Q8×D79C418yescomplemented
21C2×C4⋊C4×D79—21no—
22C2×C4⋊C4×D79C2212yes—
23C2×C4⋊C4×D79C2211yescharacteristic
24C2×C42×D79—21no—
25C2×C42×D79C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×Dic79⋊C8C2212yes—
28C4⋊C8×D79—24no—
29C2×#14579086C2212yes—
30C2×#14579085C2212yes—
31C2×#14579084C2212yes—
32#103×D79—28no—
33#19098429C2211yescharacteristic
34#19098428C2212yes—
35#19098427—21no—
36#19098427—21no—
37#95×D79—21no—
38#95×D79—21no—
39#19098326C2212yes—
40#19098325C2212yes—
41#19098325C2212yes—
42C2×C4×Q8×C79C2211yescharacteristic
43C2×C4⋊C8×C79C2211yescharacteristic
44C2×#103×C79C2212yes—
45#382×C79C2211yescharacteristic
46#340×C79C2212yes—
47C22×Dic158—21no—
48C22×Dic158C2311yescharacteristic
49C2×Q8×D79—22no—
50C2×Q8×D79—22no—
51C2×Q8×D79—22no—
52C2×Q8×D79C2×C414yes—
53C2×Q8×D79D414yes—
54C2×Q8×D79Q814yes—
55C22×C4×D79—21no—
56C22×C4×D79—21no—
57C22×C4×D79—22no—
58C22×C4×D79D411yescharacteristic
59C22×C4×D79D411yescharacteristic
60C22×C4×D79C2311yescharacteristic
61C22×C79⋊C8C2312yes—
62C2×C8×D79—24no—
63C2×C8×D79—24no—
64C2×C8×D79—24no—
65Q8×Dic79—22no—
66Q8×Dic79C2×C414yescomplemented
67C4×Dic158—22no—
68C4×Dic158—28no—
69C4×Dic158C2×C414yescomplemented
70#14579126—28no—
71#14579126—22no—
72#14579126C2×C414yescomplemented
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79C2312yes—
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—22no—
80C4⋊C4×D79—28no—
81C4⋊C4×D79—22no—
82C4⋊C4×D79—24no—
83C4⋊C4×D79—24no—
84C4⋊C4×D79C2×C414yes—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79D411yescharacteristic
89C2×C4×Dic79D411yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91C42×D79—28no—
92C42×D79C2×C414yes—
93C42×D79D414yes—
94C4⋊(C79⋊C8)—22no—
95Dic79⋊C8—24no—
96#14579086—24no—
97#14579085—24no—
98#14579084—24no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—22no—
102C79⋊#95—22no—
103C22×Q8×C79C2311yescharacteristic
104C22×C8×C79C2312yes—
105C4×Q8×C79—22no—
106C4×Q8×C79C2×C414yescomplemented
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79C2312yes—
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C42×C79—21no—
111C2×C42×C79C2311yescharacteristic
112C4⋊C8×C79—22no—
113#103×C79—24no—
114#95×C79—21no—
115#95×C79—21no—
116C23×D79C2×D411yescharacteristic
117C2×Dic158—24no—
118C2×Dic158—24no—
119C2×Dic158—24no—
120C2×Dic158—21no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158C22×C412yes—
124C2×Dic158C2×D412yes—
125C2×Dic158C2×Q812yes—
126Q8×D79—28no—
127Q8×D79—28no—
128Q8×D79—28no—
129Q8×D79C4⋊C4116yes—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—22no—
133C22×Dic79C2×D411yescharacteristic
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C2×C4×D79—24no—
137C2×C4×D79—24no—
138C2×C4×D79—24no—
139C2×C4×D79—24no—
140C2×C4×D79—24no—
141C2×C4×D79—24no—
142C2×C4×D79—44no—
143C2×C4×D79—44no—
144C2×C4×D79—44no—
145C2×C4×D79—44no—
146C2×C4×D79C22⋊C414yes—
147C2×C4×D79C22⋊C414yes—
148C2×C4×D79C22×C412yes—
149C2×C4×D79C2×D412yes—
150C2×C4×D79C2×D412yes—
151C2×C4×D79C2×Q812yes—
152C2×C4×D79#4112yes—
153C2×C4×D79#4112yes—
154C2×C79⋊C8—22no—
155C2×C79⋊C8—22no—
156C2×C79⋊C8—22no—
157C8×D79—48no—
158C4⋊Dic79—24no—
159C4⋊Dic79—21no—
160C4⋊Dic79—22no—
161C4⋊Dic79—22no—
162C4⋊Dic79C22×C412yes—
163Dic79⋊C4—24no—
164Dic79⋊C4—24no—
165Dic79⋊C4—24no—
166Dic79⋊C4—44no—
167C4×Dic79—24no—
168C4×Dic79—24no—
169C4×Dic79—24no—
170C4×Dic79C22⋊C414yes—
171C4×Dic79C22⋊C414yes—
172C4×Dic79C22×C412yes—
173C4×Dic79C2×D412yes—
174C2×Q8×C79—21no—
175C2×Q8×C79—21no—
176C2×Q8×C79—21no—
177C2×Q8×C79C22×C412yes—
178C2×Q8×C79C2×D412yes—
179C2×Q8×C79C2×Q812yes—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—22no—
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2411yescharacteristic
186C2×C8×C79—22no—
187C2×C8×C79—22no—
188C2×C8×C79—22no—
189C4⋊C4×C79—24no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—22no—
192C4⋊C4×C79—22no—
193C4⋊C4×C79C22×C412yes—
194C42×C79—24no—
195C42×C79C22×C412yes—
196C42×C79C2×D412yes—
197C23×C79C22×D411yescharacteristic
198C22×D79C2×C22⋊C412yes—
199C22×D79#12112yes—
200C22×D79C22⋊Q812yes—
201C22×D79#13514yes—
202C22×D79#13714yes—
203Q8×C79—22no—
204Q8×C79—22no—
205Q8×C79—22no—
206Q8×C79C2×C4⋊C414yes—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
213C2×C4×C79—42no—
214C2×C4×C79—42no—
215C2×C4×C79—42no—
216C2×C4×C79—42no—
217C2×C4×C79C2×C22⋊C412yes—
218C2×C4×C79C2×C22⋊C412yes—
219C2×C4×C79C23×C411yescharacteristic
220C2×C4×C79C22×D411yescharacteristic
221C2×C4×C79C22×D411yescharacteristic
222C2×C4×C79C22×Q811yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C2×Dic79—22no—
226C2×Dic79—22no—
227C2×Dic79—22no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—42no—
232C2×Dic79—42no—
233C2×Dic79—42no—
234C2×Dic79—42no—
235C2×Dic79C2×C22⋊C412yes—
236C2×Dic79C2×C22⋊C412yes—
237C2×Dic79C2×C22⋊C412yes—
238C2×Dic79#12112yes—
239C2×Dic79C22⋊Q812yes—
240C4×D79—216no—
241C4×D79—28no—
242C4×D79—28no—
243C4×D79—48no—
244C4×D79—88no—
245C4×D79C2×C4⋊C414yes—
246C4×D79#11714yes—
247C4×D79#12114yes—
248C4×D79C22⋊Q814yes—
249Dic158—22no—
250Dic158—22no—
251Dic158—22no—
252Dic158—216no—
253Dic158C2×C4⋊C414yes—
254C8×C79—42no—
255C79⋊C8—42no—
256C22×C79C22×C22⋊C411yescharacteristic
257C22×C79C2×#12111yescharacteristic
258C22×C79C2×C22⋊Q811yescharacteristic
259C22×C79C2×#13512yes—
260C22×C79C2×#13712yes—
261C2×D79#38014yes—
262C2×D79#41614yes—
263C2×D79#41914yes—
264C2×D79#46418yes—
265C2×D79#47918yes—
266C4×C79—22no—
267C4×C79—22no—
268C4×C79—24no—
269C4×C79—42no—
270C4×C79—82no—
271C4×C79C22×C4⋊C411yescharacteristic
272C4×C79C2×#11711yescharacteristic
273C4×C79C2×#12111yescharacteristic
274C4×C79C2×C22⋊Q811yescharacteristic
275Dic79—22no—
276Dic79—22no—
277Dic79—24no—
278Dic79—42no—
279Dic79—82no—
280Dic79#38014yes—
281C2×#1805—791nomaximal, Sylow
282C2×C79C2×#38011yescharacteristic
283C2×C79C2×#41611yescharacteristic
284C2×C79C2×#41911yescharacteristic
285C2×C79C2×#46412yes—
286C2×C79C2×#47912yes—
287D79#180518yesdirect product factor
288C22×C4×Q8—791no—
289C22×C4⋊C8—791no—
290C22×#103—792no—
291C2×#382—791no—
292#1805—791no—
293#1805—792no—
294#1805—791no—
295#1805—792no—
296#1805—791no—
297#1805D7911yescharacteristic, direct product factor
298C2×#340—792no—
299C79C2×#180511yescharacteristic, complemented, Sylow
300C23×Q8—791no—
301C23×C8—792no—
302C2×C4×Q8—791no—
303C2×C4×Q8—798no—
304C2×C4×Q8—791no—
305C2×C4×Q8—791no—
306C2×C4×Q8—1581no—
307C2×C4×Q8—1581no—
308C2×C4×Q8—1584no—
309C2×C4×Q8C2×D7911yescharacteristic
310C22×C4⋊C4—791no—
311C22×C4⋊C4—792no—
312C22×C4⋊C4—1581no—
313C22×C42—791no—
314C22×C42—1581no—
315C2×C4⋊C8—792no—
316C2×C4⋊C8—791no—
317C2×C4⋊C8—1584no—
318C2×C4⋊C8C2×D7911yescharacteristic
319C2×#103—792no—
320C2×#103—792no—
321C2×#103—792no—
322C2×#103—1588no—
323C2×#103C2×D7912yes—
324#382—792no—
325#382—791no—
326#382—1581no—
327#382—1581no—
328#382C2×D7911yescharacteristic
329C2×#95—1581no—
330C2×#95—1581no—
331#340—792no—
332#340—792no—
333#340—792no—
334#340C2×D7912yes—
335C22×Q8—794no—
336C22×Q8—794no—
337C22×Q8—794no—
338C22×Q8—791no—
339C22×Q8—1581no—
340C22×Q8—1582no—
341C22×Q8—1582no—
342C22×Q8—1582no—
343C22×Q8C22×D7911yescharacteristic
344C23×C4—791no—
345C23×C4—791no—
346C23×C4—791no—
347C23×C4—1582no—
348C23×C4—1581no—
349C23×C4—1581no—
350C22×C8—792no—
351C22×C8—1584no—
352C22×C8—1584no—
353C22×C8—1584no—
354C22×C8C22×D7912yes—
355C4×Q8—22no—
356C4×Q8—794no—
357C4×Q8—794no—
358C4×Q8—794no—
359C4×Q8—1588no—
360C4×Q8—1588no—
361C4×Q8—1582no—
362C4×Q8—1582no—
363C4×Q8—1582no—
364C4×Q8C4×D7914yescomplemented
365C2×C4⋊C4—21no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—794no—
368C2×C4⋊C4—792no—
369C2×C4⋊C4—1584no—
370C2×C4⋊C4—1584no—
371C2×C4⋊C4—1582no—
372C2×C4⋊C4—1588no—
373C2×C4⋊C4—1582no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4C22×D7912yes—
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C42—21no—
381C2×C42—791no—
382C2×C42—791no—
383C2×C42—791no—
384C2×C42—794no—
385C2×C42—794no—
386C2×C42—1588no—
387C2×C42—1581no—
388C2×C42—1581no—
389C2×C42—1581no—
390C2×C42C22×D7911yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—1584no—
393C4⋊C8—1582no—
394#103—24no—
395#103—1584no—
396#103—1584no—
397#103—1584no—
398#95—21no—
399#95—21no—
400#95—1582no—
401#95—1582no—
402#95—1581no—
403#95—1581no—
404C24—791no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—7916no—
409C2×Q8—792no—
410C2×Q8—792no—
411C2×Q8—792no—
412C2×Q8—1584no—
413C2×Q8—1584no—
414C2×Q8—1584no—
415C2×Q8—1588no—
416C2×Q8—1588no—
417C2×Q8—1588no—
418C2×Q8—1581no—
419C2×Q8—1581no—
420C2×Q8—1581no—
421C2×Q8C2×C4×D7912yes—
422C2×Q8D4×D7912yes—
423C2×Q8Q8×D7912yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—3164no—
428C22×C4—3164no—
429C22×C4—3164no—
430C22×C4—3164no—
431C22×C4—791no—
432C22×C4—792no—
433C22×C4—792no—
434C22×C4—792no—
435C22×C4—792no—
436C22×C4—792no—
437C22×C4—792no—
438C22×C4—791no—
439C22×C4—791no—
440C22×C4—794no—
441C22×C4—794no—
442C22×C4—1584no—
443C22×C4—1584no—
444C22×C4—1584no—
445C22×C4—1584no—
446C22×C4—1584no—
447C22×C4—1584no—
448C22×C4—1582no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4D4×D7911yescharacteristic
452C22×C4D4×D7911yescharacteristic
453C22×C4C23×D7911yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—3168no—
458C2×C8—1582no—
459C2×C8—1582no—
460C2×C8—1582no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—3164no—
466C4⋊C4—792no—
467C4⋊C4—1584no—
468C4⋊C4—1584no—
469C4⋊C4—1584no—
470C4⋊C4—1582no—
471C4⋊C4—1582no—
472C4⋊C4—1584no—
473C4⋊C4—1581no—
474C4⋊C4C2×C4×D7912yes—
475C42—24no—
476C42—794no—
477C42—794no—
478C42—792no—
479C42—792no—
480C42—1584no—
481C42—1584no—
482C42—1584no—
483C42C2×C4×D7912yes—
484C42D4×D7912yes—
485C23—792no—
486C23—792no—
487C23—792no—
488C23—794no—
489C23—794no—
490C23C2×D4×D7911yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—22no—
494Q8—794no—
495Q8—15816no—
496Q8—1582no—
497Q8—1582no—
498Q8—1582no—
499Q8C4⋊C4×D7914yes—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—3168no—
511C2×C4—3162no—
512C2×C4—3162no—
513C2×C4—3162no—
514C2×C4—3162no—
515C2×C4—792no—
516C2×C4—792no—
517C2×C4—792no—
518C2×C4—794no—
519C2×C4—794no—
520C2×C4—794no—
521C2×C4—794no—
522C2×C4—792no—
523C2×C4—792no—
524C2×C4—6328no—
525C2×C4—1588no—
526C2×C4—1588no—
527C2×C4—15816no—
528C2×C4—1582no—
529C2×C4—1582no—
530C2×C4—1582no—
531C2×C4—1582no—
532C2×C4—1582no—
533C2×C4—1582no—
534C2×C4C22⋊C4×D7912yes—
535C2×C4C22⋊C4×D7912yes—
536C2×C4C22×C4×D7911yescharacteristic
537C2×C4C2×D4×D7911yescharacteristic
538C2×C4C2×D4×D7911yescharacteristic
539C2×C4C2×Q8×D7911yescharacteristic
540C2×C4#41×D7911yescharacteristic
541C2×C4#41×D7911yescharacteristic
542C8—42no—
543C8—3162no—
544C22—794no—
545C22—798no—
546C22—794no—
547C22—798no—
548C22—794no—
549C22C2×C22⋊C4×D7911yescharacteristic
550C22#121×D7911yescharacteristic
551C22C22⋊Q8×D7911yescharacteristic
552C22#135×D7912yes—
553C22#137×D7912yes—
554C4—24no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—82no—
559C4—3162no—
560C4—794no—
561C4—6322no—
562C4—1584no—
563C4—1582no—
564C4—1582no—
565C4C2×C4⋊C4×D7911yescharacteristic
566C4#117×D7911yescharacteristic
567C4#121×D7911yescharacteristic
568C4C22⋊Q8×D7911yescharacteristic
569C2—798no—
570C2#380×D7911yescharacteristic
571C2#416×D7911yescharacteristic
572C2#419×D7911yescharacteristic
573C2#464×D7912yes—
574C2#479×D7912yes—
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1805C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1805D79
hypocenterC79C2×#1805
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
5. #23763416C2semidirect
6. #23763415C2semidirect
7. #23763414C2semidirect
8. #23763413C2semidirect
9. #23763412C2semidirect
11. C79×#1805C2semidirect
20. C4×Q8×D79C4semidirect
66. Q8×Dic79C2×C4semidirect
69. C4×Dic158C2×C4semidirect
72. #14579126C2×C4semidirect
106. C4×Q8×C79C2×C4semidirect
287. D79#1805direct (N × H)
297. #1805D79direct (N × H)
299. C79C2×#1805semidirect
364. C4×Q8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D792
C2×C4⋊C8×D792
C2×#103×D792
#382×D792
#237634162
#237634152
#237634142
#237634132
#237634122
#340×D792
C79×#18052
C2×#180579