Finite Group Databasebeta

← #34306000

Subgroups — #34306000

Order 20224.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098602C211yesmaximal, characteristic
2#23766281C211yesmaximal, characteristic
3#23766276C211yesmaximal, characteristic
4#23764135C211yesmaximal, characteristic, complemented
5#23764132C211yesmaximal, characteristic, complemented
6#23763991C211yesmaximal, characteristic, complemented
7#23763987C211yesmaximal, characteristic, complemented
8#23763972C211yesmaximal, characteristic, complemented
9#23763494C211yesmaximal, characteristic
10#23763485C211yesmaximal, characteristic
11#23763432C211yesmaximal, characteristic
12#23763298C211yesmaximal, characteristic, complemented
13#23763288C211yesmaximal, characteristic, complemented
14#23763182C211yesmaximal, characteristic
15C79×#1919C211yesmaximal, characteristic, complemented
16C22×#11929607C2211yescharacteristic
17C2×#14579119C2211yescharacteristic
18C2×C4⋊C4×D79C2211yescharacteristic
19#19098602C418yescomplemented
20C2×#14579095C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×(C2×D79)⋊C8C2211yescharacteristic
26C2×#14579072C2211yescharacteristic
27C2×C8×Dic79C2211yescharacteristic
28#19098462C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098459C2211yescharacteristic
31#19098454—21no—
32#19098452—21no—
33#19098450—21no—
34#19098444—21no—
35#19098427—21no—
36#19098420—21no—
37#19098420C2211yescharacteristic
38#19098420C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42C4×(C2×D79)⋊C4C2211yescharacteristic
43#19098410—21no—
44#19098409—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#106×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55C4×C4⋊C4×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×C4×D79—21no—
61C22×C4×D79C2311yescharacteristic
62C2×#11929607—22no—
63C2×#11929607—22no—
64C2×#11929607C2×C414yes—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579119—22no—
68#14579119C2×C414yes—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C4⋊C4×D79—22no—
78C4⋊C4×D79C2×C414yes—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4D411yescharacteristic
83C2×(C2×D79)⋊C4D411yescharacteristic
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79C2311yescharacteristic
89#14579095C2×C414yes—
90#14579094—22no—
91#14579094C2×C414yes—
92#14579092—22no—
93#14579092C2×C414yes—
94(C2×D79)⋊C8—22no—
95#14579072—22no—
96C8×Dic79—22no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C42×C79—21no—
115C2×C42×C79C2311yescharacteristic
116#106×C79C2×C414yes—
117#95×C79—21no—
118#95×C79—21no—
119C23×D79C2×D411yescharacteristic
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79C2×D411yescharacteristic
125C22×Dic79C2×D411yescharacteristic
126C22×Dic79C2×D411yescharacteristic
127C2×C4×D79—24no—
128C2×C4×D79—21no—
129C2×C4×D79—21no—
130C2×C4×D79—21no—
131C2×C4×D79—21no—
132C2×C4×D79—81no—
133C2×C4×D79—81no—
134C2×C4×D79C22×C412yes—
135C2×C4×D79#4112yes—
136#11929607—28no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8C22×C412yes—
141C2×C79⋊C8#4112yes—
142C2×C79⋊C8#4112yes—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—81no—
150C4⋊Dic79C22×C412yes—
151C4⋊Dic79C22×C412yes—
152Dic79⋊C4—24no—
153Dic79⋊C4—81no—
154Dic79⋊C4—81no—
155(C2×D79)⋊C4—22no—
156(C2×D79)⋊C4—24no—
157(C2×D79)⋊C4—41no—
158(C2×D79)⋊C4—42no—
159(C2×D79)⋊C4—81no—
160C4×Dic79—24no—
161C4×Dic79—21no—
162C4×Dic79—21no—
163C4×Dic79—42no—
164C4×Dic79—81no—
165C4×Dic79—81no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C2×C8×C79—21no—
174C2×C8×C79—21no—
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79C22×C412yes—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—81no—
186C4⋊C4×C79C22×C412yes—
187C4⋊C4×C79C22×C412yes—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—42no—
191C23×C79C22×D411yescharacteristic
192C22×D79—21no—
193C22×D79—21no—
194C22×D79—21no—
195C22×D79—21no—
196C22×D79—21no—
197C22×D79—21no—
198C22×D79C4×D412yes—
199C2×C4×C79—22no—
200C2×C4×C79—22no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C2×C4×C792−1+411yescharacteristic
222C2×C4×C792−1+411yescharacteristic
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—22no—
228C2×Dic79—22no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79C4×D412yes—
243C2×Dic79#13712yes—
244C4×D79—22no—
245C4×D79—22no—
246C4×D79—22no—
247C4×D79—22no—
248C4×D79—84no—
249C8×C79—22no—
250C8×C79—41no—
251C79⋊C8—41no—
252C79⋊C8#11714yes—
253C22×C79C2×C4×D411yescharacteristic
254C22×C79#54711yescharacteristic
255C22×C79#54711yescharacteristic
256C22×C79C2×#13711yescharacteristic
257C22×C79#57511yescharacteristic
258C22×C79#57611yescharacteristic
259C22×C79#57711yescharacteristic
260C2×D79—22no—
261C2×D79—22no—
262C2×D79—22no—
263C2×D79—41no—
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79—42no—
270C4×C79—42no—
271C4×C79—81no—
272C4×C79—81no—
273C4×C79C2×#11711yescharacteristic
274C4×C79#51911yescharacteristic
275C4×C79#55211yescharacteristic
276C4×C79#55211yescharacteristic
277Dic79—22no—
278Dic79—22no—
279Dic79—41no—
280Dic79—42no—
281Dic79—42no—
282Dic79—81no—
283Dic79—81no—
284#22621—791nomaximal, Sylow
285C2×C79#234411yescharacteristic
286C2×C79C4×#13711yescharacteristic
287C2×C79#294611yescharacteristic
288C2×C79#334411yescharacteristic
289C2×C79#334711yescharacteristic
290C2×C79#335111yescharacteristic
291C2×C79#335711yescharacteristic
292D79—42no—
293C2×#428—791no—
294#2346—791no—
295#2345—791no—
296#1943—791no—
297#1942—791no—
298#1921—791no—
299#1920—791no—
300#1919D7911yescharacteristic, complemented
301#1918—791no—
302#1825—791no—
303#1823—791no—
304#1809—791no—
305#1776—791no—
306C4×#106—791no—
307#1753—791no—
308C79#2262111yescharacteristic, complemented, Sylow
309C22×M16—791no—
310C2×#117—791no—
311C22×C4⋊C4—791no—
312#428—798no—
313C2×#107—791no—
314C2×#107—791no—
315C2×#107—791no—
316C2×#106—791no—
317C2×#106—791no—
318C2×#106C2×D7911yescharacteristic
319C2×C22⋊C8—791no—
320C2×#97—791no—
321C2×C4×C8—791no—
322#389—791no—
323#389—791no—
324#389—791no—
325#389C2×D7911yescharacteristic
326#388—1581no—
327#388—1581no—
328#387—1581no—
329#387—1581no—
330#382—1581no—
331#380—791no—
332#380—791no—
333#380—1581no—
334C4×C4⋊C4—791no—
335C4×C4⋊C4—791no—
336C4×C4⋊C4—791no—
337C4×C4⋊C4C2×D7911yescharacteristic
338C4×C22⋊C4—791no—
339C4×C22⋊C4—1581no—
340#376—1581no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340C2×D7911yescharacteristic
349#340C2×D7911yescharacteristic
350#340C2×D7911yescharacteristic
351#336—791no—
352#336C2×D7911yescharacteristic
353C23×C4—791no—
354C23×C4—1581no—
355C2×M16—794no—
356C2×M16—1582no—
357C2×M16—1582no—
358C22×C8—791no—
359C22×C8—791no—
360C22×C8C22×D7911yescharacteristic
361C22×C8C22×D7911yescharacteristic
362#117—794no—
363#117—1582no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—794no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—1582no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C4⋊C4C22×D7911yescharacteristic
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C4⋊C4C22×D7911yescharacteristic
379C2×C22⋊C4—791no—
380C2×C22⋊C4—791no—
381C2×C22⋊C4—1581no—
382C2×C22⋊C4—1581no—
383C2×C22⋊C4—1581no—
384C2×C42—21no—
385C2×C42—791no—
386C2×C42—1581no—
387C2×C42—1581no—
388C2×C42—1581no—
389C2×C42—1581no—
390C2×C42C22×D7911yescharacteristic
391#107—794no—
392#107—794no—
393#107—1582no—
394#106—794no—
395#106—1582no—
396#106C4×D7914yes—
397C22⋊C8—1582no—
398#97—1582no—
399C4×C8—1582no—
400#95—21no—
401#95—21no—
402#95—791no—
403#95—791no—
404#95—1581no—
405#95—1581no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412C24—791no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—791no—
418C22×C4—791no—
419C22×C4—792no—
420C22×C4—792no—
421C22×C4—791no—
422C22×C4—6321no—
423C22×C4—6321no—
424C22×C4—1584no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4D4×D7911yescharacteristic
434C22×C4D4×D7911yescharacteristic
435C22×C4C23×D7911yescharacteristic
436M16—1588no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—792no—
443C2×C8—792no—
444C2×C8—792no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8C2×C4×D7912yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—81no—
456C4⋊C4—792no—
457C4⋊C4—792no—
458C4⋊C4—6321no—
459C4⋊C4—6321no—
460C4⋊C4—6321no—
461C4⋊C4—1584no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4C2×C4×D7912yes—
469C4⋊C4C2×C4×D7912yes—
470C22⋊C4—3161no—
471C22⋊C4—3162no—
472C22⋊C4—6321no—
473C22⋊C4—1582no—
474C22⋊C4—1584no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42—3162no—
479C42—6321no—
480C42—6321no—
481C42—1584no—
482C42—1581no—
483C42—1581no—
484C23—792no—
485C23—1581no—
486C23—1581no—
487C23—1581no—
488C23—1581no—
489C23—1581no—
490C23—1581no—
491C23C2×D4×D7911yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—792no—
521C2×C4—792no—
522C2×C4—6324no—
523C2×C4—1581no—
524C2×C4—1582no—
525C2×C4—1582no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1582no—
533C2×C4—1582no—
534C2×C4—1582no—
535C2×C4—1582no—
536C2×C4C22×C4×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C2×C4#41×D7911yescharacteristic
539C2×C4#41×D7911yescharacteristic
540C2×C4#1457920811yescharacteristic
541C2×C4#1457920911yescharacteristic
542C8—22no—
543C8—41no—
544C8—3161no—
545C8—794no—
546C22—3161no—
547C22—3161no—
548C22—3161no—
549C22—3161no—
550C22—1582no—
551C22—1582no—
552C22—1582no—
553C22C4×D4×D7911yescharacteristic
554C22#1909921111yescharacteristic
555C22#1909921511yescharacteristic
556C22#137×D7911yescharacteristic
557C22#1909935911yescharacteristic
558C22#1909936511yescharacteristic
559C22#1909936911yescharacteristic
560C4—41no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—3162no—
567C4—3162no—
568C4—3161no—
569C4—6321no—
570C4—6321no—
571C4—1582no—
572C4—1582no—
573C4#117×D7911yescharacteristic
574C4#1909903911yescharacteristic
575C4#1909924511yescharacteristic
576C4#1909924911yescharacteristic
577C2—3162no—
578C2#2376626611yescharacteristic
579C2#2377222211yescharacteristic
580C2#2377222711yescharacteristic
581C2#2377585211yescharacteristic
582C2#2377589711yescharacteristic
583C2#2377596411yescharacteristic
584C2#2377602411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1919C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1919D79
hypocenterC79#22621
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #23764135C2semidirect
5. #23764132C2semidirect
6. #23763991C2semidirect
7. #23763987C2semidirect
8. #23763972C2semidirect
12. #23763298C2semidirect
13. #23763288C2semidirect
15. C79×#1919C2semidirect
19. #19098602C4semidirect
300. #1919D79semidirect
308. C79#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#190986022
#237662812
#237662762
#237641352
#237641322
#237639912
#237639872
#237639722
#237634942
#237634852
#237634322
#237632982
#237632882
#237631822
C79×#19192
#2262179