Finite Group Databasebeta

← #34295432

Subgroups — #34295432

Order 20224.

888 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×#19098905C211yesmaximal, characteristic
2#23765928C211yesmaximal, characteristic
3#117×Dic79C211yesmaximal, characteristic
4#23764040C211yesmaximal, characteristic, complemented
5#23764033C211yesmaximal, characteristic, complemented
6#23763977C211yesmaximal, characteristic, complemented
7#23763970C211yesmaximal, characteristic, complemented
8#23763712C211yesmaximal, characteristic
9#23763492C211yesmaximal, characteristic
10#107×Dic79C211yesmaximal, characteristic, complemented
11#106×Dic79C211yesmaximal, characteristic, complemented
12#23763219C211yesmaximal, characteristic, complemented
13#23763219C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic79C211yesmaximal, characteristic
15C79×#1854C211yesmaximal, characteristic
16C2×#14579140C2211yescharacteristic
17C2×#14579126C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19C22×C4×Dic79C2211yescharacteristic
20#19098905C418yescomplemented
21C2×#14579095C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579086C2211yescharacteristic
24C2×#14579086C2211yescharacteristic
25C2×#14579084C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×C22⋊(C79⋊C8)C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098468C2211yescharacteristic
31#19098459C2211yescharacteristic
32#19098445—21no—
33#19098435—21no—
34C4⋊C4×Dic79C2211yescharacteristic
35C4⋊C4×Dic79C2211yescharacteristic
36C4⋊C4×Dic79C2211yescharacteristic
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39#19098414—21no—
40C22⋊C4×Dic79—21no—
41C22⋊C4×Dic79—21no—
42C22⋊C4×Dic79—21no—
43#19098409—21no—
44C42×Dic79—21no—
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#117×C79C2211yescharacteristic
54C2×#107×C79C2211yescharacteristic
55C2×#106×C79C2211yescharacteristic
56C2×C22⋊C8×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158C2311yescharacteristic
61C23×Dic79D411yescharacteristic
62C23×Dic79D411yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579140—22no—
66#14579140C2×C414yes—
67#14579126—22no—
68#14579126—22no—
69#14579120—22no—
70#14579120C2×C414yes—
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—22no—
81C2×C4×Dic79—22no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79D411yescharacteristic
89C2×C4×Dic79D411yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91C2×C4×Dic79C2311yescharacteristic
92#14579095C2×C414yescomplemented
93#14579093C2×C414yescomplemented
94#14579086C2×C414yescomplemented
95#14579086C2×C414yescomplemented
96#14579084—22no—
97#14579084—22no—
98C22⋊(C79⋊C8)—22no—
99C8×Dic79—22no—
100C8×Dic79—22no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C23×C4×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C22×C8×C79C2311yescharacteristic
112#117×C79—22no—
113#117×C79—22no—
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C22⋊C4×C79—21no—
119C2×C22⋊C4×C79—21no—
120C2×C42×C79—21no—
121#107×C79—22no—
122#106×C79—22no—
123C22⋊C8×C79C2×C414yes—
124#95×C79—21no—
125C2×Dic158—21no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158C22×C412yes—
131C22×Dic79—22no—
132C22×Dic79—22no—
133C22×Dic79—22no—
134C22×Dic79—22no—
135C22×Dic79—22no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C22×Dic79—41no—
141C22×Dic79C2×D411yescharacteristic
142C22×Dic79C2×D411yescharacteristic
143C22×Dic79C2×D411yescharacteristic
144C22×Dic79C2×D411yescharacteristic
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79C22×C412yes—
155C4⋊Dic79C22×C412yes—
156C4⋊Dic79C22×C412yes—
157Dic79⋊C4—24no—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160C22⋊Dic79—24no—
161C22⋊Dic79—24no—
162C4×Dic79—22no—
163C4×Dic79—22no—
164C4×Dic79—24no—
165C4×Dic79—21no—
166C4×Dic79—21no—
167C4×Dic79—42no—
168C4×Dic79—42no—
169C4×Dic79—42no—
170C4×Dic79—42no—
171C4×Dic79—42no—
172C4×Dic79—42no—
173C4×Dic79#4112yes—
174C4×Dic79#4112yes—
175C24×C79C2×D411yescharacteristic
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79C22×C412yes—
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79#4112yes—
187C22×C4×C79C2411yescharacteristic
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79C22×C412yes—
193C2×C8×C79C22×C412yes—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79#4112yes—
201C4⋊C4×C79#4112yes—
202C22⋊C4×C79—24no—
203C22⋊C4×C79—42no—
204C42×C79—24no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79C4×D412yes—
212C23×C79C22×D411yescharacteristic
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—22no—
216C2×C4×C79—22no—
217C2×C4×C79—22no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79C4×D412yes—
232C2×C4×C79C23×C411yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—22no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—44no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—44no—
259C2×Dic79#12312yes—
260C2×Dic79#12312yes—
261C2×Dic79#13512yes—
262C2×Dic79#13512yes—
263C2×Dic79#13712yes—
264Dic158—22no—
265Dic158—22no—
266Dic158—41no—
267Dic158—41no—
268C8×C79—22no—
269C8×C79—22no—
270C79⋊C8—41no—
271C79⋊C8—41no—
272C22×C79—22no—
273C22×C79—22no—
274C22×C79—22no—
275C22×C79—41no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79—41no—
279C22×C79C2×C4×D411yescharacteristic
280C22×C79C2×#12311yescharacteristic
281C22×C79#52511yescharacteristic
282C22×C79C2×#13511yescharacteristic
283C22×C79C2×#13511yescharacteristic
284C22×C79C2×#13711yescharacteristic
285C22×C79#57511yescharacteristic
286C4×C79—22no—
287C4×C79—42no—
288C4×C79—42no—
289C4×C79—81no—
290C4×C79—81no—
291C4×C79C4×#4111yescharacteristic
292C4×C79C4×#4111yescharacteristic
293C4×C79#52911yescharacteristic
294C4×C79#52911yescharacteristic
295Dic79—22no—
296Dic79—22no—
297Dic79—41no—
298Dic79—42no—
299Dic79—42no—
300Dic79—42no—
301Dic79—42no—
302Dic79#48514yescomplemented
303C4×#485—791nomaximal, Sylow
304C2×C79—42no—
305C2×C79C4×#12311yescharacteristic
306C2×C79C4×#13511yescharacteristic
307C2×C79C4×#13711yescharacteristic
308C2×C79C2×#48511yescharacteristic
309C2×C79#309111yescharacteristic
310C2×C79#310211yescharacteristic
311C2×C79#310711yescharacteristic
312C2×#485—791no—
313C4×C22⋊Q8—791no—
314C4×#117—791no—
315#1928—791no—
316#1927—791no—
317#1919—791no—
318#1918—791no—
319#1873—791no—
320#1854D7911yescharacteristic
321#1825—791no—
322C4×#107—791no—
323C4×#106—791no—
324C4×#103—791no—
325C4×#103—791no—
326C4×C22⋊C8—791no—
327C79C4×#48511yescharacteristic, complemented, Sylow
328C2×C22⋊Q8—791no—
329C2×C4×Q8—791no—
330C2×#117—791no—
331C2×#117C2×D7911yescharacteristic
332C22×C42—791no—
333#485—798no—
334C2×#107—791no—
335C2×#107C2×D7911yescharacteristic
336C2×#106—791no—
337C2×#106C2×D7911yescharacteristic
338C2×#103—791no—
339C2×#103—791no—
340C2×#103—791no—
341C2×#103—791no—
342C2×C22⋊C8—791no—
343C2×C22⋊C8C2×D7911yescharacteristic
344C2×C4×C8—791no—
345C2×C4×C8—791no—
346#391—791no—
347#389—791no—
348#387—1581no—
349#385—1581no—
350#380C2×D7911yescharacteristic
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4—791no—
356C4×C4⋊C4—1581no—
357C4×C22⋊C4—1581no—
358C4×C22⋊C4—1581no—
359C4×C22⋊C4—1581no—
360#376—1581no—
361C43—1581no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340C2×D7911yescharacteristic
369#340C2×D7911yescharacteristic
370#336—791no—
371#336—791no—
372C22×Q8—791no—
373C23×C4—791no—
374C23×C4—791no—
375C23×C4C22×D7911yescharacteristic
376C22×C8—791no—
377C22×C8—791no—
378C22×C8C22×D7911yescharacteristic
379C22×C8C22×D7911yescharacteristic
380C22⋊Q8—794no—
381C22⋊Q8—1582no—
382C4×Q8—1582no—
383C4×Q8—1582no—
384#117—22no—
385#117—22no—
386#117—794no—
387#117—1582no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C42—21no—
403C2×C42—791no—
404C2×C42—791no—
405C2×C42—791no—
406C2×C42—791no—
407C2×C42—1581no—
408C2×C42—1581no—
409C2×C42—1581no—
410C2×C42—1581no—
411C2×C42—1581no—
412C2×C42—1581no—
413C2×C42—1582no—
414C2×C42—1582no—
415C2×C42—1581no—
416#107—22no—
417#107—794no—
418#106—22no—
419#106—794no—
420#103—794no—
421#103—794no—
422#103—1582no—
423#103—1582no—
424C22⋊C8—1582no—
425C22⋊C8C4×D7914yes—
426C4×C8—1582no—
427C4×C8—1582no—
428#95—21no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437C24D4×D7911yescharacteristic
438C2×Q8—792no—
439C2×Q8—1581no—
440C2×Q8—1581no—
441C2×Q8—1581no—
442C2×Q8—1581no—
443C2×Q8—1581no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—3161no—
453C22×C4—791no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—1582no—
458C22×C4—1582no—
459C22×C4—1582no—
460C22×C4—1582no—
461C22×C4—1582no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4C2×C4×D7912yes—
467C22×C4D4×D7911yescharacteristic
468C22×C4#1192962512yes—
469C22×C4C23×D7911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8C2×C4×D7912yes—
481C2×C8C2×C4×D7912yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—3162no—
489C4⋊C4—3162no—
490C4⋊C4—792no—
491C4⋊C4—792no—
492C4⋊C4—792no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1584no—
497C4⋊C4#1192962312yes—
498C4⋊C4#1192962312yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1584no—
502C22⋊C4—1584no—
503C42—24no—
504C42—3162no—
505C42—3162no—
506C42—3162no—
507C42—3162no—
508C42—3162no—
509C42—3162no—
510C42—792no—
511C42—792no—
512C42—1582no—
513C42—1582no—
514C42—1581no—
515C42—1581no—
516C42—1584no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1457912312yes—
524C23C2×D4×D7911yescharacteristic
525Q8—3161no—
526Q8—3161no—
527Q8—1582no—
528Q8—1582no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
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—3161no—
554C2×C4—3161no—
555C2×C4—3164no—
556C2×C4—3164no—
557C2×C4—792no—
558C2×C4—792no—
559C2×C4—792no—
560C2×C4—792no—
561C2×C4—792no—
562C2×C4—1582no—
563C2×C4—1582no—
564C2×C4—1582no—
565C2×C4—1582no—
566C2×C4—1581no—
567C2×C4—1581no—
568C2×C4—1581no—
569C2×C4—1581no—
570C2×C4—1581no—
571C2×C4—1581no—
572C2×C4—1581no—
573C2×C4#1457912312yes—
574C2×C4C22×C4×D7911yescharacteristic
575C2×C4#41×D7911yescharacteristic
576C2×C4#41×D7911yescharacteristic
577C2×C4C2×#1192962311yescharacteristic
578C2×C4C2×#1192962311yescharacteristic
579C2×C4C2×#1192962511yescharacteristic
580C8—22no—
581C8—22no—
582C8—3161no—
583C8—3161no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×#1457912311yescharacteristic
592C22#123×D7911yescharacteristic
593C22#1909906911yescharacteristic
594C22#135×D7911yescharacteristic
595C22#135×D7911yescharacteristic
596C22#137×D7911yescharacteristic
597C22#1909935911yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—3162no—
604C4—3162no—
605C4—3162no—
606C4—3161no—
607C4—3162no—
608C4—794no—
609C4—1582no—
610C4—1582no—
611C4C4×#1192962311yescharacteristic
612C4C4×#1192962311yescharacteristic
613C4#1909909911yescharacteristic
614C4#1909909911yescharacteristic
615C2—42no—
616C2#2376591911yescharacteristic
617C2#2377217111yescharacteristic
618C2#2377221811yescharacteristic
619C2#485×D7911yescharacteristic
620C2#2377331211yescharacteristic
621C2#2377341111yescharacteristic
622C2#2377346411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1854C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1854D79
hypocenterC79C4×#485
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #23764040C2semidirect
5. #23764033C2semidirect
6. #23763977C2semidirect
7. #23763970C2semidirect
10. #107×Dic79C2semidirect
11. #106×Dic79C2semidirect
12. #23763219C2semidirect
13. #23763219C2semidirect
20. #19098905C4semidirect
92. #14579095C2×C4semidirect
93. #14579093C2×C4semidirect
94. #14579086C2×C4semidirect
95. #14579086C2×C4semidirect
302. Dic79#485semidirect
327. C79C4×#485semidirect

Maximal subgroups

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

subgroupindex
C2×#190989052
#237659282
#117×Dic792
#237640402
#237640332
#237639772
#237639702
#237637122
#237634922
#107×Dic792
#106×Dic792
#237632192
C22⋊C8×Dic792
C79×#18542
C4×#48579