Finite Group Databasebeta

← #34328832

Subgroups — #34328832

Order 20224.

980 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098765C211yesmaximal, characteristic, complemented
2C2×#19098616C211yesmaximal, characteristic, complemented
3C4×#14579135C211yesmaximal, characteristic
4#23764593C212yesmaximal, complemented
5#23764401C212yesmaximal
6#23764324C211yesmaximal, characteristic, complemented
7#23764257C212yesmaximal, complemented
8#23764161C211yesmaximal, characteristic, complemented
9#23764079C212yesmaximal, complemented
10#23763397C211yesmaximal, characteristic, complemented
11C79×#2857C211yesmaximal, characteristic, complemented
12C22×#11929611C2212yes—
13C2×#14579135C2211yescharacteristic
14C2×C4×D316C2211yescharacteristic
15C2×C4×C22⋊D79—21no—
16#19098765—24no—
17#19098616—24no—
18C2×#14579092C2212yes—
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579085C2212yes—
22C2×#14579085C2212yes—
23C2×#14579085C2212yes—
24C2×#14579081C2212yes—
25C2×C22⋊(C79⋊C8)C2212yes—
26C2×C4×C79⋊C8C2212yes—
27#19098465C2211yescharacteristic
28#19098461C2211yescharacteristic
29#19098454—21no—
30#19098442—21no—
31#19098423C2211yescharacteristic
32C4×C4⋊Dic79C2211yescharacteristic
33C4×(C2×D79)⋊C4—21no—
34#19098325C2212yes—
35#19098325C2212yes—
36C2×C4⋊Q8×C79C2211yescharacteristic
37C2×C22⋊Q8×C79C2212yes—
38C22×C42×C79C2211yescharacteristic
39#391×C79C2212yes—
40#389×C79—21no—
41#389×C79—21no—
42#384×C79C2211yescharacteristic
43#380×C79—21no—
44#380×C79—21no—
45C22×D316C2311yescharacteristic
46C22×C22⋊D79—21no—
47C22×C4×D79—21no—
48C2×#11929611—24no—
49C2×#11929611—24no—
50C2×#11929611—42no—
51C22×C79⋊C8C2312yes—
52C22×C79⋊C8C2312yes—
53#14579135—22no—
54#14579135—22no—
55C4×D316—22no—
56C4×D316—22no—
57C4×C22⋊D79—44no—
58C2×C4⋊Dic79C2311yescharacteristic
59C2×C4⋊Dic79C2311yescharacteristic
60C2×C4⋊Dic79C2311yescharacteristic
61C2×Dic79⋊C4—21no—
62C2×C22⋊Dic79—21no—
63C2×(C2×D79)⋊C4—21no—
64C2×(C2×D79)⋊C4—21no—
65C2×(C2×D79)⋊C4—21no—
66C2×(C2×D79)⋊C4—21no—
67C2×C4×Dic79—21no—
68#14579092—42no—
69C4⋊(C79⋊C8)—22no—
70C4⋊(C79⋊C8)D414yescomplemented
71#14579085—24no—
72#14579085—24no—
73#14579085—42no—
74#14579081—42no—
75C22⋊(C79⋊C8)—24no—
76C4×C79⋊C8—24no—
77C79⋊#95—21no—
78C79⋊#95—21no—
79C79⋊#95—21no—
80C22×Q8×C79C2312yes—
81C23×C4×C79D411yescharacteristic
82C23×C4×C79D411yescharacteristic
83C23×C4×C79C2311yescharacteristic
84C4⋊Q8×C79—22no—
85C4⋊Q8×C79—22no—
86C22⋊Q8×C79—42no—
87C22⋊Q8×C79—42no—
88C2×C4⋊C4×C79—22no—
89C2×C4⋊C4×C79—21no—
90C2×C4⋊C4×C79—21no—
91C2×C4⋊C4×C79—21no—
92C2×C4⋊C4×C79—21no—
93C2×C4⋊C4×C79C2312yes—
94C2×C22⋊C4×C79—22no—
95C2×C42×C79—22no—
96C2×C42×C79—22no—
97C2×C42×C79D411yescharacteristic
98C2×C42×C79D411yescharacteristic
99C2×C42×C79C2311yescharacteristic
100C2×C42×C79C2311yescharacteristic
101#95×C79—22no—
102#95×C79—22no—
103C23×D79—21no—
104C2×D316—21no—
105C2×D316—21no—
106C2×D316—21no—
107C2×D316—21no—
108C2×D316—21no—
109C2×D316#4112yes—
110C2×C22⋊D79—24no—
111C2×C22⋊D79—42no—
112C2×C22⋊D79—42no—
113C22×Dic79—21no—
114C22×Dic79—21no—
115C22×Dic79—21no—
116C2×C4×D79—42no—
117C2×C4×D79—42no—
118C2×C4×D79—42no—
119#11929611—48no—
120C2×C79⋊C8—22no—
121C2×C79⋊C8—22no—
122C2×C79⋊C8—22no—
123C2×C79⋊C8—22no—
124C2×C79⋊C8—22no—
125C2×C79⋊C8C2×D414yes—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—21no—
129C4⋊Dic79—21no—
130C4⋊Dic79—21no—
131C4⋊Dic79#4112yes—
132Dic79⋊C4—42no—
133C22⋊Dic79—42no—
134(C2×D79)⋊C4—24no—
135(C2×D79)⋊C4—42no—
136(C2×D79)⋊C4—42no—
137(C2×D79)⋊C4—42no—
138C4×Dic79—42no—
139C24×C79C2×D411yescharacteristic
140C2×Q8×C79—22no—
141C2×Q8×C79—22no—
142C2×Q8×C79—22no—
143C2×Q8×C79—22no—
144C2×Q8×C79—22no—
145C2×Q8×C79—22no—
146C22×C4×C79—22no—
147C22×C4×C79—22no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—21no—
151C22×C4×C79—21no—
152C22×C4×C79—21no—
153C22×C4×C79—21no—
154C22×C4×C79—21no—
155C22×C4×C79—21no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—41no—
159C22×C4×C79SD1614yescomplemented
160C22×C4×C79C2×D411yescharacteristic
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2411yescharacteristic
166C4⋊C4×C79—22no—
167C4⋊C4×C79—22no—
168C4⋊C4×C79—42no—
169C4⋊C4×C79—44no—
170C4⋊C4×C79—42no—
171C4⋊C4×C79—42no—
172C4⋊C4×C79—42no—
173C22⋊C4×C79—44no—
174C42×C79—22no—
175C42×C79—22no—
176C42×C79—21no—
177C42×C79—21no—
178C42×C79C2×D412yes—
179C42×C79#4112yes—
180C23×C79—21no—
181C23×C79—21no—
182C23×C79—21no—
183C23×C79—21no—
184C23×C79—21no—
185C23×C79—21no—
186C23×C79C2×SD1612yes—
187C23×C79C22×D411yescharacteristic
188C22×D79—22no—
189C22×D79—41no—
190C22×D79—41no—
191C22×D79—41no—
192C22×D79—41no—
193C22×D79—41no—
194C22×D79—41no—
195Q8×C79—42no—
196Q8×C79—42no—
197Q8×C79—42no—
198Q8×C79—42no—
199C2×C4×C79—21no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—28no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—44no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—42no—
215C2×C4×C79—42no—
216C2×C4×C79—42no—
217C2×C4×C79—42no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79#12112yes—
221C2×C4×C79#12112yes—
222C2×C4×C79C2×SD1612yes—
223C2×C4×C79C2×SD1612yes—
224C2×C4×C79#13512yes—
225C2×C4×C79#13712yes—
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79H(5,𝔽2)11yescharacteristic
231C2×C4×C79H(5,𝔽2)11yescharacteristic
232C22⋊D79—48no—
233C2×Dic79—22no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242D316—22no—
243D316—22no—
244D316—41no—
245D316—41no—
246C4×D79—84no—
247C79⋊C8—24no—
248C79⋊C8—42no—
249C22×C79—22no—
250C22×C79—22no—
251C22×C79—22no—
252C22×C79—41no—
253C22×C79—41no—
254C22×C79—41no—
255C22×C79—41no—
256C22×C79C2×#12111yescharacteristic
257C22×C79#52511yescharacteristic
258C22×C79#53511yescharacteristic
259C22×C79C22×SD1611yescharacteristic
260C22×C79C2×#13511yescharacteristic
261C22×C79C2×#13711yescharacteristic
262C22×C79#57511yescharacteristic
263C2×D79—42no—
264C2×D79—42no—
265C2×D79—42no—
266C2×D79—81no—
267C2×D79—81no—
268C2×D79—81no—
269C2×D79—81no—
270C4×C79—22no—
271C4×C79—22no—
272C4×C79—22no—
273C4×C79—41no—
274C4×C79—82no—
275C4×C79—82no—
276C4×C79#46514yes—
277C4×C79#54611yescharacteristic
278C4×C79#55011yescharacteristic
279C4×C79#55311yescharacteristic
280C4×C79#55511yescharacteristic
281Dic79—42no—
282Dic79—81no—
283Dic79—81no—
284#24651—791nomaximal, Sylow
285C2×C79—42no—
286C2×C79#241411yescharacteristic
287C2×C79C2×#46511yescharacteristic
288C2×C79#305611yescharacteristic
289C2×C79#310111yescharacteristic
290C2×C79#310711yescharacteristic
291C2×C79#320611yescharacteristic
292C2×C79#321211yescharacteristic
293D79—82no—
294C2×#461—791no—
295C2×#431—791no—
296#2857D7911yescharacteristic, complemented
297C4×#121—791no—
298#2035—792no—
299#2000—792no—
300#1981—791no—
301#1968—792no—
302#1948—791no—
303#1933—792no—
304#1802—791no—
305C79#2465111yescharacteristic, complemented, Sylow
306C22×SD16—792no—
307C2×C4⋊Q8C2×D7911yescharacteristic
308C2×C22⋊Q8C2×D7912yes—
309C2×#121—791no—
310C2×C4×D4—791no—
311C2×C4×D4—1581no—
312C22×C42C2×D7911yescharacteristic
313#461—1584no—
314#431—1584no—
315C2×#106—792no—
316C2×C4⋊C8—791no—
317C2×C4⋊C8—791no—
318C2×#103—792no—
319C2×#103—792no—
320C2×#103—792no—
321C2×#102—792no—
322C2×C22⋊C8—792no—
323C2×C4×C8—792no—
324#391C2×D7912yes—
325#390—791no—
326#389—21no—
327#389—21no—
328#389—791no—
329#388—1581no—
330#386—1581no—
331#384C2×D7911yescharacteristic
332#380—21no—
333#380—21no—
334#380—791no—
335C4×C4⋊C4—791no—
336C4×C22⋊C4—1581no—
337#340—792no—
338#340—792no—
339C22×Q8C22×D7912yes—
340C22×D4—791no—
341C22×D4—1581no—
342C23×C4—1581no—
343C23×C4C22⋊D7911yescharacteristic
344C23×C4C22⋊D7911yescharacteristic
345C23×C4C22×D7911yescharacteristic
346C2×SD16—3162no—
347C2×SD16—1584no—
348C2×SD16—1584no—
349C22×C8—792no—
350C22×C8—792no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353C22⋊Q8—42no—
354C22⋊Q8—42no—
355#121—1582no—
356#121—1582no—
357C4×D4—3164no—
358C4×D4—1582no—
359C4×D4—1582no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4C22×D7912yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—1581no—
372C2×C22⋊C4—1581no—
373C2×C22⋊C4—1581no—
374C2×C22⋊C4—1581no—
375C2×C22⋊C4—1581no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—1581no—
379C2×C42C22⋊D7911yescharacteristic
380C2×C42C22⋊D7911yescharacteristic
381C2×C42C22×D7911yescharacteristic
382C2×C42C22×D7911yescharacteristic
383#106—3162no—
384C4⋊C8—794no—
385C4⋊C8—1582no—
386#103—3162no—
387#103—1584no—
388#103—1584no—
389#102—3162no—
390C22⋊C8—1584no—
391C4×C8—1584no—
392#95—22no—
393#95—22no—
394#95—1581no—
395#95—1581no—
396#95—1581no—
397C24—1581no—
398C24C2×C22⋊D7911yescharacteristic
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×D4—3162no—
406C2×D4—3162no—
407C2×D4—792no—
408C2×D4—1584no—
409C2×D4—1581no—
410C2×D4—1581no—
411C2×D4—1581no—
412C2×D4—1581no—
413C2×D4—1581no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—3162no—
428C22×C4—3162no—
429C22×C4—3162no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4#1192961114yescomplemented
434C22×C4C2×C22⋊D7911yescharacteristic
435C22×C4C2×C22⋊D7911yescharacteristic
436C22×C4C2×C22⋊D7911yescharacteristic
437C22×C4C2×C22⋊D7911yescharacteristic
438C22×C4C2×C22⋊D7911yescharacteristic
439C22×C4C23×D7911yescharacteristic
440SD16—3168no—
441C2×C8—794no—
442C2×C8—1582no—
443C2×C8—1582no—
444C2×C8—1582no—
445C2×C8—1582no—
446C2×C8—1582no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—3162no—
455C4⋊C4—792no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C22⋊C4—44no—
462C22⋊C4—3162no—
463C22⋊C4—3162no—
464C22⋊C4—3162no—
465C22⋊C4—3162no—
466C22⋊C4—1584no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—3162no—
472C42D4×D7912yes—
473C42#1192962312yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—3161no—
481C23—3161no—
482C23—3161no—
483C23—3161no—
484C23—3161no—
485C23—3161no—
486C23—1582no—
487C23C2×#1192961112yes—
488C23C22×C22⋊D7911yescharacteristic
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493D4—3168no—
494D4—3161no—
495D4—3161no—
496D4—1582no—
497D4—1582no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—6324no—
528C2×C4—1582no—
529C2×C4#1457913312yes—
530C2×C4#1457913312yes—
531C2×C4C2×#1192961112yes—
532C2×C4C2×#1192961112yes—
533C2×C4#1457918112yes—
534C2×C4#1457919212yes—
535C2×C4C2×D4×D7911yescharacteristic
536C2×C4#41×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C2×C4C2×#1192962311yescharacteristic
539C2×C4#1457920611yescharacteristic
540C2×C4#1457920611yescharacteristic
541C8—3162no—
542C8—1584no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—3162no—
551C22—3162no—
552C22—3162no—
553C22—6321no—
554C22—6321no—
555C22—6321no—
556C22—6321no—
557C22C2×#1457913311yescharacteristic
558C22#1909907011yescharacteristic
559C22#1909913011yescharacteristic
560C22C22×#1192961111yescharacteristic
561C22C2×#1457918111yescharacteristic
562C22C2×#1457919211yescharacteristic
563C22#1909936111yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—82no—
569C4—82no—
570C4—3162no—
571C4—6321no—
572C4—6321no—
573C4#1909879414yes—
574C4#1909921011yescharacteristic
575C4#1909923111yescharacteristic
576C4#1909926011yescharacteristic
577C4#1909927711yescharacteristic
578C2—42no—
579C2—6322no—
580C2#2376681911yescharacteristic
581C2C2×#1909879411yescharacteristic
582C2#2377302611yescharacteristic
583C2#2377340411yescharacteristic
584C2#2377346511yescharacteristic
585C2#2377432511yescharacteristic
586C2#2377437611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2857C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenter#2857D79
hypocenterC79#24651
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#19098765C2semidirect
2. C2×#19098616C2semidirect
4. #23764593C2semidirect
6. #23764324C2semidirect
7. #23764257C2semidirect
8. #23764161C2semidirect
9. #23764079C2semidirect
10. #23763397C2semidirect
11. C79×#2857C2semidirect
70. C4⋊(C79⋊C8)D4semidirect
159. C22×C4×C79SD16semidirect
296. #2857D79semidirect
305. C79#24651semidirect
433. C22×C4#11929611semidirect

Maximal subgroups

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

subgroupindex
C2×#190987652
C2×#190986162
C4×#145791352
#237645932
#237644012
#237643242
#237642572
#237641612
#237640792
#237633972
C79×#28572
#2465179