Finite Group Databasebeta

← #34295902

Subgroups — #34295902

Order 20224.

988 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098841C211yesmaximal, characteristic
2#23765971C211yesmaximal, characteristic, complemented
3C4×D4×Dic79C211yesmaximal, characteristic
4#23764350C211yesmaximal, characteristic, complemented
5#23764160C211yesmaximal, characteristic, complemented
6#23763993C211yesmaximal, characteristic, complemented
7#23763989C211yesmaximal, characteristic, complemented
8#23763661C211yesmaximal, characteristic
9#23763398C211yesmaximal, characteristic
10#23763300C211yesmaximal, characteristic, complemented
11#23763296C211yesmaximal, characteristic, complemented
12#23763252C211yesmaximal, characteristic, complemented
13#23763227C211yesmaximal, characteristic
14#23763227C211yesmaximal, characteristic
15C79×#1864C211yesmaximal, characteristic
16C2×#14579160C2211yescharacteristic
17C2×#14579126—21no—
18C2×D4×Dic79C2211yescharacteristic
19C2×D4×Dic79C2211yescharacteristic
20C22×C4×Dic79—21no—
21#19098841C418yescomplemented
22C2×#14579095C2211yescharacteristic
23C2×#14579094C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×Dic79⋊C8C2211yescharacteristic
26C2×Dic79⋊C8C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×#14579071C2211yescharacteristic
30C2×#14579071C2211yescharacteristic
31#19098467—21no—
32#19098461C2211yescharacteristic
33#19098458—21no—
34#19098432C2211yescharacteristic
35C4⋊C4×Dic79C2211yescharacteristic
36C4⋊C4×Dic79C2211yescharacteristic
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40#19098414—21no—
41C22⋊C4×Dic79—21no—
42C22⋊C4×Dic79—21no—
43C4×C22⋊Dic79—21no—
44C42×Dic79C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098318C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54C2×#106×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56C2×#102×C79C2211yescharacteristic
57#389×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×Dic158—21no—
61C23×Dic79—21no—
62C23×Dic79—21no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579160—22no—
66#14579160C2×C414yescomplemented
67#14579126—44no—
68D4×Dic79—22no—
69D4×Dic79—22no—
70D4×Dic79C2×C414yes—
71D4×Dic79Q814yescomplemented
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—44no—
88C2×C4×Dic79D411yescharacteristic
89C2×C4×Dic79D411yescharacteristic
90C2×C4×Dic79D411yescharacteristic
91C2×C4×Dic79D411yescharacteristic
92C2×C4×Dic79C2311yescharacteristic
93C2×C4×Dic79C2311yescharacteristic
94#14579095C2×C414yescomplemented
95#14579094C2×C414yescomplemented
96#14579092—22no—
97Dic79⋊C8—22no—
98Dic79⋊C8C2×C414yescomplemented
99#14579082—22no—
100#14579082C2×C414yes—
101#14579071—22no—
102#14579071—22no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C22×D4×C79C2311yescharacteristic
110C23×C4×C79—21no—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C4×D4×C79—22no—
114C4×D4×C79—22no—
115C2×C4⋊C4×C79—21no—
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C22⋊C4×C79—21no—
120C2×C42×C79C2311yescharacteristic
121#106×C79—22no—
122#102×C79—22no—
123#102×C79C2×C414yes—
124#95×C79—21no—
125C2×Dic158—24no—
126C2×Dic158—42no—
127C2×Dic158—42no—
128C22×Dic79—24no—
129C22×Dic79—24no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—42no—
135C22×Dic79—42no—
136C22×Dic79—42no—
137C22×Dic79—42no—
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C22×Dic79C2×D411yescharacteristic
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8C22×C412yes—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79C22×C412yes—
154C4⋊Dic79C22×C412yes—
155C4⋊Dic79C2×Q812yes—
156Dic79⋊C4—24no—
157Dic79⋊C4—42no—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160C22⋊Dic79—24no—
161C22⋊Dic79—24no—
162C4×Dic79—22no—
163C4×Dic79—22no—
164C4×Dic79—22no—
165C4×Dic79—41no—
166C4×Dic79—42no—
167C4×Dic79—42no—
168C4×Dic79—42no—
169C4×Dic79—42no—
170C4×Dic79—42no—
171C4×Dic79C22×C412yes—
172C4×Dic79C2×Q812yes—
173C4×Dic79#4112yes—
174C4×Dic79#4112yes—
175C24×C79—21no—
176C2×D4×C79—21no—
177C2×D4×C79—21no—
178C2×D4×C79—21no—
179C2×D4×C79C22×C412yes—
180C2×D4×C79C2×Q812yes—
181C2×D4×C79#4112yes—
182C22×C4×C79—24no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79—42no—
187C22×C4×C79—42no—
188C22×C4×C79C2×D411yescharacteristic
189C22×C4×C79C2×D411yescharacteristic
190C22×C4×C79C2411yescharacteristic
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79C22×C412yes—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—42no—
202C4⋊C4×C79C22×C412yes—
203C4⋊C4×C79#4112yes—
204C22⋊C4×C79—24no—
205C42×C79—21no—
206C42×C79—21no—
207C23×C79—22no—
208C23×C79—22no—
209C23×C79—22no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79—41no—
213C23×C79—41no—
214C23×C79C22×D411yescharacteristic
215D4×C79—22no—
216D4×C79—22no—
217D4×C79—22no—
218D4×C79C4×Q814yescomplemented
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
222C2×C4×C79—21no—
223C2×C4×C79—22no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—48no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79#13712yes—
235C2×C4×C79C23×C411yescharacteristic
236C2×C4×C79C22×Q811yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—22no—
248C2×Dic79—22no—
249C2×Dic79—22no—
250C2×Dic79—48no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—48no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79C4×D412yes—
262C2×Dic79C4×D412yes—
263C2×Dic79C22⋊Q812yes—
264C2×Dic79C22⋊Q812yes—
265C2×Dic79AGL(1,ℤ/8)12yes—
266C2×Dic79AGL(1,ℤ/8)12yes—
267Dic158—48no—
268C8×C79—22no—
269C8×C79—41no—
270C79⋊C8—22no—
271C79⋊C8—41no—
272C22×C79—24no—
273C22×C79—42no—
274C22×C79—42no—
275C22×C79—42no—
276C22×C79—42no—
277C22×C79—42no—
278C22×C79—42no—
279C22×C79C2×C4×D411yescharacteristic
280C22×C79C2×C22⋊Q811yescharacteristic
281C22×C79#52511yescharacteristic
282C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C79C2×#13711yescharacteristic
285C22×C79#57511yescharacteristic
286C4×C79—22no—
287C4×C79—42no—
288C4×C79—41no—
289C4×C79—42no—
290C4×C79—81no—
291C4×C79C2×C4×Q811yescharacteristic
292C4×C79C4×#4111yescharacteristic
293C4×C79#52911yescharacteristic
294C4×C79#53311yescharacteristic
295Dic79—22no—
296Dic79—22no—
297Dic79—22no—
298Dic79—42no—
299Dic79—42no—
300Dic79—42no—
301Dic79—81no—
302Dic79#44214yes—
303#21613—791nomaximal, Sylow
304C2×C79—44no—
305C2×C79C4×C22⋊Q811yescharacteristic
306C2×C79C2×#44211yescharacteristic
307C2×C79C4×#13711yescharacteristic
308C2×C79#307711yescharacteristic
309C2×C79#307911yescharacteristic
310C2×C79#311011yescharacteristic
311C2×C79#311511yescharacteristic
312C2×#474—791no—
313C4×C4⋊Q8—791no—
314C42×D4—791no—
315#1988—791no—
316#1948—791no—
317#1921—791no—
318#1921—791no—
319#1864—791no—
320#1864D7911yescharacteristic
321#1802—791no—
322#1776—791no—
323#1776—791no—
324#1767—791no—
325#1762—791no—
326#1762—791no—
327C79#2161311yescharacteristic, complemented, Sylow
328C2×C4⋊Q8—791no—
329C2×C4×Q8—1581no—
330C2×C4×D4—791no—
331C2×C4×D4—791no—
332C2×C4×D4C2×D7911yescharacteristic
333C22×C42—1581no—
334#474—798no—
335C2×#107—791no—
336C2×#107—791no—
337C2×#106—791no—
338C2×#106C2×D7911yescharacteristic
339C2×C4⋊C8—791no—
340C2×C4⋊C8—791no—
341C2×#102—791no—
342C2×#102—791no—
343C2×#102C2×D7911yescharacteristic
344C2×#102C2×D7911yescharacteristic
345C2×#97—791no—
346C2×#97—791no—
347#391—1581no—
348#389—791no—
349#389—1581no—
350#389C2×D7911yescharacteristic
351#384—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4—791no—
356C4×C4⋊C4—791no—
357C4×C4⋊C4—1581no—
358C4×C22⋊C4—1581no—
359C4×C22⋊C4—1581no—
360C4×C22⋊C4—1581no—
361C43—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340C2×D7911yescharacteristic
370#336—791no—
371#336C2×D7911yescharacteristic
372C22×Q8—1581no—
373C22×D4C22×D7911yescharacteristic
374C23×C4—21no—
375C23×C4—1581no—
376C23×C4—1581no—
377C22×C8—791no—
378C22×C8—791no—
379C22×C8C22×D7911yescharacteristic
380C22×C8C22×D7911yescharacteristic
381C4⋊Q8—794no—
382C4⋊Q8—1582no—
383C4×Q8—3164no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—794no—
387C4×D4—794no—
388C4×D4—1582no—
389C4×D4—1582no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C4⋊C4C22×D7911yescharacteristic
401C2×C4⋊C4C22×D7911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1581no—
404C2×C22⋊C4—1581no—
405C2×C42—3164no—
406C2×C42—791no—
407C2×C42—791no—
408C2×C42—791no—
409C2×C42—791no—
410C2×C42—791no—
411C2×C42—791no—
412C2×C42—1581no—
413C2×C42—1581no—
414C2×C42—1581no—
415C2×C42—1581no—
416C2×C42—1581no—
417C2×C42C22×D7911yescharacteristic
418#107—794no—
419#107—794no—
420#106—22no—
421#106—1582no—
422C4⋊C8—794no—
423C4⋊C8—1582no—
424#102—22no—
425#102—794no—
426#102—1582no—
427#102C4×D7914yes—
428#97—1582no—
429#97—1582no—
430#95—21no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437C24—21no—
438C2×Q8—3162no—
439C2×Q8—3162no—
440C2×Q8—1584no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D7912yes—
445C2×D4Q8×D7912yes—
446C2×D4#1192962312yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—3162no—
454C22×C4—3162no—
455C22×C4—3162no—
456C22×C4—3162no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—1581no—
462C22×C4—1584no—
463C22×C4—1584no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4D4×D7911yescharacteristic
468C22×C4D4×D7911yescharacteristic
469C22×C4C23×D7911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—792no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8C2×C4×D7912yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—3162no—
488C4⋊C4—3162no—
489C4⋊C4—3162no—
490C4⋊C4—792no—
491C4⋊C4—792no—
492C4⋊C4—792no—
493C4⋊C4—1584no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4C2×C4×D7912yes—
500C4⋊C4#1192962312yes—
501C22⋊C4—24no—
502C22⋊C4—1584no—
503C22⋊C4—1584no—
504C42—21no—
505C42—21no—
506C42—3162no—
507C42—3161no—
508C42—3162no—
509C42—3162no—
510C42—3162no—
511C42—3162no—
512C42—792no—
513C42—792no—
514C42—792no—
515C42—792no—
516C42—1582no—
517C42—1582no—
518C42—1582no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D7911yescharacteristic
527Q8—3168no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4#1457912614yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3168no—
554C2×C4—3168no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—792no—
559C2×C4—792no—
560C2×C4—792no—
561C2×C4—792no—
562C2×C4—792no—
563C2×C4—792no—
564C2×C4—1582no—
565C2×C4—1582no—
566C2×C4—1582no—
567C2×C4—1581no—
568C2×C4—1581no—
569C2×C4—1581no—
570C2×C4—1581no—
571C2×C4—1581no—
572C2×C4—1581no—
573C2×C4#1457919012yes—
574C2×C4C22×C4×D7911yescharacteristic
575C2×C4C2×Q8×D7911yescharacteristic
576C2×C4#41×D7911yescharacteristic
577C2×C4#41×D7911yescharacteristic
578C2×C4C2×#1192962311yescharacteristic
579C2×C4C2×#1192962311yescharacteristic
580C8—22no—
581C8—41no—
582C8—3161no—
583C8—1582no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D7911yescharacteristic
592C22C22⋊Q8×D7911yescharacteristic
593C22#1909906911yescharacteristic
594C22AGL(1,ℤ/8)×D7911yescharacteristic
595C22AGL(1,ℤ/8)×D7911yescharacteristic
596C22C2×#1457919011yescharacteristic
597C22#1909936011yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—3162no—
604C4—3162no—
605C4—3162no—
606C4—794no—
607C4—6321no—
608C4—1582no—
609C4—1582no—
610C4—1582no—
611C4C2×#1457912611yescharacteristic
612C4C4×#1192962311yescharacteristic
613C4#1909909011yescharacteristic
614C4#1909911511yescharacteristic
615C2—44no—
616C2#2376593211yescharacteristic
617C2#442×D7911yescharacteristic
618C2C4×#1457919011yescharacteristic
619C2#2377319511yescharacteristic
620C2#2377321211yescharacteristic
621C2#2377349411yescharacteristic
622C2#2377354411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1864C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1864D79
hypocenterC79#21613
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #23765971C2semidirect
4. #23764350C2semidirect
5. #23764160C2semidirect
6. #23763993C2semidirect
7. #23763989C2semidirect
10. #23763300C2semidirect
11. #23763296C2semidirect
12. #23763252C2semidirect
21. #19098841C4semidirect
66. #14579160C2×C4semidirect
71. D4×Dic79Q8semidirect
94. #14579095C2×C4semidirect
95. #14579094C2×C4semidirect
98. Dic79⋊C8C2×C4semidirect
218. D4×C79C4×Q8semidirect
327. C79#21613semidirect
531. D4#14579126semidirect

Maximal subgroups

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

subgroupindex
C2×#190988412
#237659712
C4×D4×Dic792
#237643502
#237641602
#237639932
#237639892
#237636612
#237633982
#237633002
#237632962
#237632522
#237632272
C79×#18642
#2161379