Finite Group Databasebeta

← #34295625

Subgroups — #34295625

Order 20224.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098901C211yesmaximal, characteristic
2#23765756C211yesmaximal, characteristic
3#23764031C211yesmaximal, characteristic, complemented
4#23764031C211yesmaximal, characteristic, complemented
5#23763989C211yesmaximal, characteristic, complemented
6#23763975C211yesmaximal, characteristic, complemented
7#23763708C211yesmaximal, characteristic
8#23763592C211yesmaximal, characteristic
9#23763492C211yesmaximal, characteristic
10#23763298C211yesmaximal, characteristic, complemented
11C4×#14579092C211yesmaximal, characteristic, complemented
12#23763230C211yesmaximal, characteristic, complemented
13C4×#14579085C211yesmaximal, characteristic, complemented
14#23763166C211yesmaximal, characteristic
15C79×#2323C211yesmaximal, characteristic
16C2×#14579120C2211yescharacteristic
17C22×C4⋊Dic79C2211yescharacteristic
18#19098901C418yescomplemented
19C2×#14579094C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×#14579085C2211yescharacteristic
27C2×C22⋊(C79⋊C8)C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31C4⋊C4×Dic79—21no—
32C4⋊C4×Dic79—21no—
33C4×C4⋊Dic79C2211yescharacteristic
34C4×C4⋊Dic79C2211yescharacteristic
35C4×C4⋊Dic79C2211yescharacteristic
36C4×C4⋊Dic79C2211yescharacteristic
37C22⋊C4×Dic79—21no—
38C22⋊C4×Dic79—21no—
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49C2×C22⋊Q8×C79C2211yescharacteristic
50C2×C4×Q8×C79C2211yescharacteristic
51C2×#117×C79C2211yescharacteristic
52#391×C79C2211yescharacteristic
53#389×C79—21no—
54#389×C79C2211yescharacteristic
55#388×C79—21no—
56#388×C79—21no—
57#387×C79—21no—
58#380×C79C2211yescharacteristic
59C4×C4⋊C4×C79C2211yescharacteristic
60C23×Dic79—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579120—22no—
66#14579120C2×C414yes—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—22no—
69C2×C4⋊Dic79—22no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C22⋊Dic79—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79—21no—
77C2×C4×Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80#14579094C2×C414yescomplemented
81#14579092—22no—
82#14579092—22no—
83#14579092C2×C414yescomplemented
84#14579085—22no—
85#14579085—22no—
86#14579085C2×C414yescomplemented
87#14579085C2×C414yescomplemented
88C22⋊(C79⋊C8)—22no—
89C22⋊(C79⋊C8)C2×C414yes—
90#14579072—22no—
91C4×C79⋊C8—22no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C22×Q8×C79C2311yescharacteristic
97C23×C4×C79C2311yescharacteristic
98C22⋊Q8×C79—22no—
99C22⋊Q8×C79C2×C414yes—
100C4×Q8×C79—22no—
101C4×Q8×C79—22no—
102#117×C79—22no—
103#117×C79—22no—
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79D411yescharacteristic
108C2×C4⋊C4×C79D411yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79D411yescharacteristic
115C2×C22⋊C4×C79D411yescharacteristic
116C2×C42×C79—21no—
117C2×C42×C79—21no—
118C2×C42×C79C2311yescharacteristic
119C2×C42×C79C2311yescharacteristic
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124#95×C79—21no—
125C22×Dic79—24no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—81no—
131C22×Dic79—81no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8C22×C412yes—
143C2×C79⋊C8C22×C412yes—
144C4⋊Dic79—24no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79C22×C412yes—
150C4⋊Dic79C22×C412yes—
151C4⋊Dic79#4112yes—
152C4⋊Dic79#4112yes—
153C22⋊Dic79—24no—
154C4×Dic79—24no—
155C4×Dic79—81no—
156C4×Dic79—81no—
157C4×Dic79—81no—
158C4×Dic79—81no—
159C24×C79C2×D411yescharacteristic
160C2×Q8×C79—21no—
161C2×Q8×C79—21no—
162C2×Q8×C79—21no—
163C2×Q8×C79—21no—
164C2×Q8×C79—21no—
165C2×Q8×C79C22×C412yes—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79C22×C412yes—
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79#4112yes—
181C22×C4×C79C2411yescharacteristic
182C4⋊C4×C79—24no—
183C4⋊C4×C79—22no—
184C4⋊C4×C79—22no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—42no—
191C4⋊C4×C79—81no—
192C4⋊C4×C79C22×C412yes—
193C22⋊C4×C79—24no—
194C22⋊C4×C79—22no—
195C22⋊C4×C79—22no—
196C22⋊C4×C79—81no—
197C42×C79—21no—
198C42×C79—21no—
199C42×C79—42no—
200C42×C79—81no—
201C42×C79#4112yes—
202C42×C79#4112yes—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79C4×D412yes—
210C23×C79C22×D411yescharacteristic
211Q8×C79—22no—
212Q8×C79—22no—
213Q8×C79—41no—
214Q8×C79—41no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—22no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79C4×D412yes—
244C2×C4×C79#12312yes—
245C2×C4×C79#12312yes—
246C2×C4×C79C23×C411yescharacteristic
247C2×C4×C79C2×#4111yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×C4×C79C2×#4111yescharacteristic
252C2×Dic79—22no—
253C2×Dic79—22no—
254C2×Dic79—22no—
255C2×Dic79—22no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—84no—
265C79⋊C8—22no—
266C79⋊C8—22no—
267C79⋊C8—41no—
268C79⋊C8—41no—
269C22×C79—22no—
270C22×C79—22no—
271C22×C79—22no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79C2×C4×D411yescharacteristic
277C22×C79C2×#12311yescharacteristic
278C22×C79#52511yescharacteristic
279C22×C79#57711yescharacteristic
280C22×C79#57711yescharacteristic
281C22×C79#57711yescharacteristic
282C22×C79#57811yescharacteristic
283C4×C79—22no—
284C4×C79—22no—
285C4×C79—22no—
286C4×C79—41no—
287C4×C79—41no—
288C4×C79—42no—
289C4×C79—42no—
290C4×C79—81no—
291C4×C79—81no—
292C4×C79C4×#4111yescharacteristic
293C4×C79C4×#4111yescharacteristic
294C4×C79#52911yescharacteristic
295C4×C79#52911yescharacteristic
296Dic79—42no—
297Dic79—42no—
298Dic79—42no—
299Dic79—42no—
300#21591—791nomaximal, Sylow
301C2×C79—42no—
302C2×C79C4×#12311yescharacteristic
303C2×C79#294711yescharacteristic
304C2×C79#295211yescharacteristic
305C2×C79#309711yescharacteristic
306C2×C79#309911yescharacteristic
307C2×C79#312111yescharacteristic
308C2×C79#312311yescharacteristic
309C2×#485—791no—
310#2323D7911yescharacteristic
311#2274—791no—
312#1927—791no—
313#1927—791no—
314#1921—791no—
315#1919—791no—
316#1872—791no—
317#1853—791no—
318#1825—791no—
319#1776—791no—
320C4×#106—791no—
321#1763—791no—
322C4×#103—791no—
323#1750—791no—
324C79#2159111yescharacteristic, complemented, Sylow
325C2×C22⋊Q8C2×D7911yescharacteristic
326C2×C4×Q8C2×D7911yescharacteristic
327C2×#117—791no—
328C2×#117C2×D7911yescharacteristic
329C22×C4⋊C4—791no—
330#485—798no—
331C2×#107—791no—
332C2×#106—791no—
333C2×#106—791no—
334C2×#106—791no—
335C2×#103—791no—
336C2×#103—791no—
337C2×#103—791no—
338C2×#103—791no—
339C2×C22⋊C8—791no—
340C2×C22⋊C8—791no—
341C2×#97—791no—
342C2×C4×C8—791no—
343#391C2×D7911yescharacteristic
344#389—21no—
345#389C2×D7911yescharacteristic
346#388—21no—
347#388—21no—
348#387—21no—
349#380C2×D7911yescharacteristic
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—1581no—
355C4×C4⋊C4—1581no—
356C4×C4⋊C4C2×D7911yescharacteristic
357C4×C22⋊C4—1581no—
358C4×C22⋊C4—1581no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#336—791no—
368#336—791no—
369C22×Q8C22×D7911yescharacteristic
370C23×C4—1581no—
371C23×C4C22×D7911yescharacteristic
372C22×C8—791no—
373C22×C8—791no—
374C22×C8—791no—
375C22×C8—791no—
376C22⋊Q8—22no—
377C22⋊Q8C4×D7914yes—
378C4×Q8—22no—
379C4×Q8—22no—
380#117—22no—
381#117—22no—
382#117—794no—
383#117—1582no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—1582no—
392C2×C4⋊C4—1582no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4C22⋊D7911yescharacteristic
395C2×C4⋊C4C22⋊D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4C22⋊D7911yescharacteristic
403C2×C22⋊C4C22⋊D7911yescharacteristic
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1581no—
407C2×C42—1581no—
408C2×C42—1581no—
409C2×C42—1581no—
410C2×C42—1581no—
411C2×C42C22×D7911yescharacteristic
412C2×C42C22×D7911yescharacteristic
413#107—794no—
414#106—794no—
415#106—1582no—
416#106—1582no—
417#103—794no—
418#103—794no—
419#103—1582no—
420#103—1582no—
421C22⋊C8—794no—
422C22⋊C8—1582no—
423#97—1582no—
424C4×C8—1582no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434C24C2×C22⋊D7911yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D7912yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—6321no—
450C22×C4—6321no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1584no—
456C22×C4C2×C4×D7912yes—
457C22×C4C2×C22⋊D7911yescharacteristic
458C22×C4C2×C22⋊D7911yescharacteristic
459C22×C4C2×C22⋊D7911yescharacteristic
460C22×C4C2×C22⋊D7911yescharacteristic
461C22×C4C2×C22⋊D7911yescharacteristic
462C22×C4#1192962412yes—
463C22×C4C23×D7911yescharacteristic
464C2×C8—792no—
465C2×C8—792no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—792no—
487C4⋊C4—792no—
488C4⋊C4—792no—
489C4⋊C4—792no—
490C4⋊C4—1584no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4C2×C4×D7912yes—
496C22⋊C4—24no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—81no—
500C22⋊C4—1584no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—81no—
505C42—6321no—
506C42—6321no—
507C42—6321no—
508C42—6321no—
509C42—1584no—
510C42#1192962312yes—
511C42#1192962312yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C4×C22⋊D7912yes—
519C23C22×C22⋊D7911yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—6324no—
561C2×C4—1582no—
562C2×C4—1582no—
563C2×C4—1582no—
564C2×C4—1582no—
565C2×C4C4×C22⋊D7912yes—
566C2×C4#1457914212yes—
567C2×C4#1457914212yes—
568C2×C4C22×C4×D7911yescharacteristic
569C2×C4#41×D7911yescharacteristic
570C2×C4#41×D7911yescharacteristic
571C2×C4C2×#1192962311yescharacteristic
572C2×C4C2×#1192962311yescharacteristic
573C2×C4C2×#1192962411yescharacteristic
574C8—3161no—
575C8—3161no—
576C8—1582no—
577C8—1582no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×C4×C22⋊D7911yescharacteristic
586C22C2×#1457914211yescharacteristic
587C22#1909907011yescharacteristic
588C22#1909937111yescharacteristic
589C22#1909937111yescharacteristic
590C22#1909937411yescharacteristic
591C22#1909937811yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—3162no—
602C4—3162no—
603C4—3162no—
604C4—3162no—
605C4C4×#1192962311yescharacteristic
606C4C4×#1192962311yescharacteristic
607C4#1909909311yescharacteristic
608C4#1909909311yescharacteristic
609C2—42no—
610C2C4×#1457914211yescharacteristic
611C2#2377224311yescharacteristic
612C2#2377228111yescharacteristic
613C2#2377336311yescharacteristic
614C2#2377338511yescharacteristic
615C2#2377362311yescharacteristic
616C2#2377364711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2323C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#2323D79
hypocenterC79#21591
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. #23764031C2semidirect
4. #23764031C2semidirect
5. #23763989C2semidirect
6. #23763975C2semidirect
10. #23763298C2semidirect
11. C4×#14579092C2semidirect
12. #23763230C2semidirect
13. C4×#14579085C2semidirect
18. #19098901C4semidirect
80. #14579094C2×C4semidirect
83. #14579092C2×C4semidirect
86. #14579085C2×C4semidirect
87. #14579085C2×C4semidirect
324. C79#21591semidirect

Maximal subgroups

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

subgroupindex
C2×#190989012
#237657562
#237640312
#237639892
#237639752
#237637082
#237635922
#237634922
#237632982
C4×#145790922
#237632302
C4×#145790852
#237631662
C79×#23232
#2159179