Finite Group Databasebeta

← #34180082

Subgroups — #34180082

Order 20224.

1020 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763708C212yesmaximal, complemented
2C2×#19098300C212yesmaximal, complemented
3C2×#19098287C211yesmaximal, characteristic
4#23761645C211yesmaximal, characteristic
5C2×#543×C79C211yesmaximal, characteristic
6C22×C4⋊Dic79C2211yescharacteristic
7C22×C22⋊Dic79—21no—
8C2×#14579085—22no—
9C2×C22⋊(C79⋊C8)C2212yes—
10C2×C79⋊#95—21no—
11C2×C79⋊#95C2211yescharacteristic
12#19098325—22no—
13#19098300—24no—
14#19098287—22no—
15#543×C79—24no—
16#543×C79—22no—
17#543×C79C414yescomplemented
18C2×C22⋊Q8×C79—22no—
19C2×C22⋊Q8×C79—21no—
20C2×C22⋊Q8×C79—22no—
21C2×C22⋊Q8×C79C2212yes—
22C22×C22⋊C4×C79—21no—
23C22×C22⋊C4×C79C2211yescharacteristic
24C23×Dic79—21no—
25C23×Dic79—21no—
26C22×C79⋊C8—22no—
27C2×C4⋊Dic79—22no—
28C2×C4⋊Dic79—42no—
29C2×C22⋊Dic79—42no—
30C2×C22⋊Dic79—42no—
31C2×C22⋊Dic79—42no—
32#14579085—44no—
33C22⋊(C79⋊C8)—22no—
34C22⋊(C79⋊C8)—22no—
35C79⋊#95—22no—
36C79⋊#95—42no—
37C25×C79D411yescharacteristic
38C22×Q8×C79—21no—
39C22×Q8×C79D412yes—
40C23×C4×C79—21no—
41C23×C4×C79—21no—
42C23×C4×C79D411yescharacteristic
43C23×C4×C79C2311yescharacteristic
44C22⋊Q8×C79—28no—
45C22⋊Q8×C79—28no—
46C22⋊Q8×C79—24no—
47C22⋊Q8×C79—28no—
48C22⋊Q8×C79—24no—
49C22⋊Q8×C79—22no—
50C22⋊Q8×C79—22no—
51C22⋊Q8×C79—28no—
52C22⋊Q8×C79C2×C414yes—
53C22⋊Q8×C79D414yes—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79—22no—
56C2×C4⋊C4×C79—22no—
57C2×C4⋊C4×C79D412yes—
58C2×C22⋊C4×C79—22no—
59C2×C22⋊C4×C79—24no—
60C2×C22⋊C4×C79—24no—
61C2×C22⋊C4×C79—22no—
62C2×C22⋊C4×C79—22no—
63C2×C22⋊C4×C79—22no—
64C2×C22⋊C4×C79C2×C412yes—
65C2×C22⋊C4×C79D412yes—
66C22×Dic79—22no—
67C22×Dic79—22no—
68C22×Dic79—41no—
69C22×Dic79—41no—
70C22×Dic79—41no—
71C22×Dic79—41no—
72C22×Dic79—41no—
73C22×Dic79—41no—
74C22×Dic79—41no—
75C22×Dic79—41no—
76C22×Dic79—41no—
77C22×Dic79—41no—
78C22×Dic79—41no—
79C22×Dic79—41no—
80C2×C79⋊C8—42no—
81C2×C79⋊C8—42no—
82C2×C79⋊C8—42no—
83C4⋊Dic79—24no—
84C4⋊Dic79—42no—
85C22⋊Dic79—48no—
86C24×C79—21no—
87C24×C79—21no—
88C24×C79—21no—
89C24×C79—21no—
90C24×C79—41no—
91C24×C79—41no—
92C24×C79—41no—
93C24×C79—41no—
94C24×C79C22⋊C412yes—
95C24×C79SD1612yes—
96C24×C79Q1612yes—
97C24×C79C2×D411yescharacteristic
98C2×Q8×C79—24no—
99C2×Q8×C79—22no—
100C2×Q8×C79—42no—
101C2×Q8×C79—42no—
102C2×Q8×C79—81no—
103C2×Q8×C79—81no—
104C2×Q8×C79C22⋊C414yescomplemented
105C22×C4×C79—22no—
106C22×C4×C79—21no—
107C22×C4×C79—22no—
108C22×C4×C79—24no—
109C22×C4×C79—24no—
110C22×C4×C79—21no—
111C22×C4×C79—41no—
112C22×C4×C79—41no—
113C22×C4×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79—81no—
116C22×C4×C79—81no—
117C22×C4×C79—81no—
118C22×C4×C79—81no—
119C22×C4×C79C22⋊C412yes—
120C22×C4×C79C22×C411yescharacteristic
121C22×C4×C79C2×D412yes—
122C22×C4×C79C2×D412yes—
123C22×C4×C79C2×D411yescharacteristic
124C4⋊C4×C79—24no—
125C4⋊C4×C79—28no—
126C4⋊C4×C79—24no—
127C4⋊C4×C79—24no—
128C4⋊C4×C79—22no—
129C4⋊C4×C79C22⋊C414yescomplemented
130C22⋊C4×C79—28no—
131C22⋊C4×C79—24no—
132C22⋊C4×C79—24no—
133C22⋊C4×C79—24no—
134C22⋊C4×C79—24no—
135C23×C79—21no—
136C23×C79—21no—
137C23×C79—21no—
138C23×C79—21no—
139C23×C79—22no—
140C23×C79—22no—
141C23×C79—22no—
142C23×C79—22no—
143C23×C79—22no—
144C23×C79—22no—
145C23×C79—42no—
146C23×C79—42no—
147C23×C79—42no—
148C23×C79—42no—
149C23×C79—42no—
150C23×C79—42no—
151C23×C79—41no—
152C23×C79—41no—
153C23×C79—41no—
154C23×C79—41no—
155C23×C79—81no—
156C23×C79—81no—
157C23×C79—81no—
158C23×C79—81no—
159C23×C79—81no—
160C23×C79—81no—
161C23×C79#10314yescomplemented
162C23×C79C2×C22⋊C411yescharacteristic
163C23×C79#12011yescharacteristic
164C23×C79#12011yescharacteristic
165C23×C79C2×SD1611yescharacteristic
166C23×C79C2×Q1611yescharacteristic
167C23×C79#13711yescharacteristic
168C23×C79#13711yescharacteristic
169Q8×C79—44no—
170Q8×C79—82no—
171Q8×C79—84no—
172C2×C4×C79—22no—
173C2×C4×C79—22no—
174C2×C4×C79—24no—
175C2×C4×C79—22no—
176C2×C4×C79—24no—
177C2×C4×C79—41no—
178C2×C4×C79—42no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—42no—
182C2×C4×C79—41no—
183C2×C4×C79—42no—
184C2×C4×C79—42no—
185C2×C4×C79—42no—
186C2×C4×C79—42no—
187C2×C4×C79—81no—
188C2×C4×C79—84no—
189C2×C4×C79—84no—
190C2×C4×C79C2×C22⋊C412yes—
191C2×C4×C79#12012yes—
192C2×Dic79—42no—
193C2×Dic79—42no—
194C2×Dic79—42no—
195C2×Dic79—42no—
196C2×Dic79—42no—
197C2×Dic79—42no—
198C2×Dic79—81no—
199C2×Dic79—81no—
200C2×Dic79—81no—
201C2×Dic79—81no—
202C2×Dic79—81no—
203C2×Dic79—81no—
204C2×Dic79—81no—
205C2×Dic79—81no—
206C79⋊C8—82no—
207C22×C79—21no—
208C22×C79—21no—
209C22×C79—21no—
210C22×C79—21no—
211C22×C79—21no—
212C22×C79—21no—
213C22×C79—21no—
214C22×C79—21no—
215C22×C79—21no—
216C22×C79—21no—
217C22×C79—21no—
218C22×C79—21no—
219C22×C79—24no—
220C22×C79—44no—
221C22×C79—42no—
222C22×C79—42no—
223C22×C79—42no—
224C22×C79—42no—
225C22×C79—42no—
226C22×C79—42no—
227C22×C79—82no—
228C22×C79—82no—
229C22×C79—82no—
230C22×C79#37911yescharacteristic
231C22×C79C2×#10311yescharacteristic
232C22×C79#41911yescharacteristic
233C22×C79#44811yescharacteristic
234C22×C79#44811yescharacteristic
235C22×C79#45111yescharacteristic
236C22×C79#45111yescharacteristic
237C22×C79#45311yescharacteristic
238C22×C79#45511yescharacteristic
239C22×C79B(4,𝔽2)11yescharacteristic
240C22×C79#45811yescharacteristic
241C4×C79—22no—
242C4×C79—22no—
243C4×C79—42no—
244C4×C79—44no—
245C4×C79—44no—
246Dic79—82no—
247Dic79—82no—
248#8405—791nomaximal, Sylow
249C2×C79—21no—
250C2×C79—21no—
251C2×C79—21no—
252C2×C79—21no—
253C2×C79—44no—
254C2×C79#159611yescharacteristic
255C2×C79#160211yescharacteristic
256C2×C79#160811yescharacteristic
257C2×C79#161411yescharacteristic
258C2×C79#178711yescharacteristic
259C2×C79#179011yescharacteristic
260C2×C79#179211yescharacteristic
261C2×#543D7911yescharacteristic
262#1872—792no—
263C2×#328—792no—
264C2×#323—791no—
265#1439—791no—
266C79#840511yescharacteristic, complemented, Sylow
267#543—24no—
268#543—22no—
269#543Dic7914yescomplemented
270C2×C22⋊Q8—22no—
271C2×C22⋊Q8—21no—
272C2×C22⋊Q8—22no—
273C2×C22⋊Q8C2×D7912yes—
274C22×C4⋊C4—791no—
275C22×C22⋊C4—21no—
276C22×C22⋊C4—1581no—
277C22×C22⋊C4C2×D7911yescharacteristic
278C2×#103—1582no—
279C2×C22⋊C8—792no—
280C2×#95—791no—
281C2×#95—1581no—
282#340—1582no—
283#328—1584no—
284#323—1582no—
285C25C22⋊D7911yescharacteristic
286C22×Q8—21no—
287C22×Q8C22⋊D7912yes—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—1581no—
291C23×C4—1581no—
292C23×C4C22⋊D7911yescharacteristic
293C23×C4C22×D7911yescharacteristic
294C22×C8—1582no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—24no—
298C22⋊Q8—28no—
299C22⋊Q8—24no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—28no—
303C22⋊Q8C2×Dic7914yes—
304C22⋊Q8C22⋊D7914yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—3162no—
309C2×C4⋊C4—1582no—
310C2×C4⋊C4C22⋊D7912yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—3162no—
318C2×C22⋊C4—3162no—
319C2×C22⋊C4—3162no—
320C2×C22⋊C4C2×Dic7912yes—
321C2×C22⋊C4C22⋊D7912yes—
322#103—3164no—
323C22⋊C8—1582no—
324C22⋊C8—1582no—
325#95—3162no—
326#95—1582no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—41no—
332C24—41no—
333C24—41no—
334C24—41no—
335C24C22⋊Dic7912yes—
336C24#1192961112yes—
337C24#1192961412yes—
338C24C2×C22⋊D7911yescharacteristic
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8C22⋊Dic7914yescomplemented
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—3161no—
361C22×C4—3161no—
362C22×C4—3161no—
363C22×C4—3161no—
364C22×C4—3161no—
365C22×C4—3161no—
366C22×C4—3161no—
367C22×C4—3161no—
368C22×C4—3161no—
369C22×C4—3161no—
370C22×C4—3161no—
371C22×C4—3161no—
372C22×C4—1582no—
373C22×C4—1582no—
374C22×C4C22⋊Dic7912yes—
375C22×C4C22×Dic7911yescharacteristic
376C22×C4C2×C22⋊D7912yes—
377C22×C4C2×C22⋊D7912yes—
378C22×C4C2×C22⋊D7911yescharacteristic
379C2×C8—3162no—
380C2×C8—3162no—
381C2×C8—3162no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—3162no—
388C4⋊C4—1584no—
389C4⋊C4C22⋊Dic7914yescomplemented
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—3168no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23#1457908514yescomplemented
423C23C2×C22⋊Dic7911yescharacteristic
424C23#1457913011yescharacteristic
425C23#1457913011yescharacteristic
426C23C2×#1192961111yescharacteristic
427C23C2×#1192961411yescharacteristic
428C23#1457919211yescharacteristic
429C23#1457919211yescharacteristic
430Q8—44no—
431Q8—82no—
432Q8—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—3162no—
452C2×C4—3162no—
453C2×C4—3162no—
454C2×C4—3162no—
455C2×C4—3162no—
456C2×C4—3162no—
457C2×C4—6321no—
458C2×C4—6321no—
459C2×C4—6321no—
460C2×C4—6321no—
461C2×C4—6321no—
462C2×C4—6321no—
463C2×C4—6321no—
464C2×C4—6321no—
465C2×C4C2×C22⋊Dic7912yes—
466C2×C4#1457913012yes—
467C8—6322no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22#1909841911yescharacteristic
492C22C2×#1457908511yescharacteristic
493C22#1909857211yescharacteristic
494C22#1909870211yescharacteristic
495C22#1909870211yescharacteristic
496C22#1909871911yescharacteristic
497C22#1909871911yescharacteristic
498C22#1909873111yescharacteristic
499C22#1909874111yescharacteristic
500C22#1909875011yescharacteristic
501C22#1909875311yescharacteristic
502C4—22no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—44no—
507C4—6322no—
508C4—6322no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—44no—
514C2#2376232311yescharacteristic
515C2#2376235711yescharacteristic
516C2#2376239111yescharacteristic
517C2#2376242511yescharacteristic
518C2#2376333511yescharacteristic
519C2#2376334811yescharacteristic
520C2#2376335411yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#543×C79C2
socleC23×C79#120
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenterC2×#543D79
hypocenterC79#8405
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #23763708C2semidirect
2. C2×#19098300C2semidirect
17. #543×C79C4semidirect
104. C2×Q8×C79C22⋊C4semidirect
129. C4⋊C4×C79C22⋊C4semidirect
161. C23×C79#103semidirect
266. C79#8405semidirect
269. #543Dic79semidirect
345. C2×Q8C22⋊Dic79semidirect
389. C4⋊C4C22⋊Dic79semidirect
422. C23#14579085semidirect

Maximal subgroups

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

subgroupindex
#237637082
C2×#190983002
C2×#190982872
#237616452
C2×#543×C792
#840579