Finite Group Databasebeta

← #34305953

Subgroups — #34305953

Order 20224.

766 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098600C211yesmaximal, characteristic
2#23766292C211yesmaximal, characteristic
3#23766281C211yesmaximal, characteristic
4#23764142C211yesmaximal, characteristic, complemented
5#23764140C211yesmaximal, characteristic, complemented
6#23763994C211yesmaximal, characteristic, complemented
7#23763972C211yesmaximal, characteristic, complemented
8#23763970C211yesmaximal, characteristic, complemented
9#23763493C211yesmaximal, characteristic
10#23763493C211yesmaximal, characteristic
11#23763428C211yesmaximal, characteristic
12#23763298C211yesmaximal, characteristic, complemented
13#23763296C211yesmaximal, characteristic, complemented
14#23763175C211yesmaximal, characteristic
15C79×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D79C2211yescharacteristic
17C2×#14579119C2211yescharacteristic
18C2×#14579119C2211yescharacteristic
19#19098600C418yescomplemented
20C2×#14579095C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×(C2×D79)⋊C8C2211yescharacteristic
26C2×#14579072C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28#19098462C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098459C2211yescharacteristic
31#19098455—21no—
32#19098454—21no—
33#19098453—21no—
34#19098452—21no—
35#19098430—21no—
36#19098427—21no—
37#19098420C2211yescharacteristic
38#19098420C2211yescharacteristic
39C4⋊C4×Dic79—21no—
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4×C4⋊Dic79C2211yescharacteristic
43C4×(C2×D79)⋊C4C2211yescharacteristic
44#19098409—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55C4×C4⋊C4×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×C4×D79C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C2×C8×D79—22no—
64C2×C8×D79—22no—
65C2×C8×D79C2×C414yes—
66#14579119—22no—
67#14579119—22no—
68#14579119C2×C414yes—
69#14579119C2×C414yes—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4D411yescharacteristic
82C2×(C2×D79)⋊C4D411yescharacteristic
83C2×C4×Dic79—21no—
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#14579094—22no—
92#14579092C2×C414yes—
93#14579092C2×C414yes—
94(C2×D79)⋊C8—22no—
95#14579072—22no—
96#14579071—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#107×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—21no—
128C2×C4×D79—21no—
129C2×C4×D79—21no—
130C2×C4×D79—21no—
131C2×C4×D79C22×C412yes—
132C2×C4×D79#4112yes—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8C22×C412yes—
137C2×C79⋊C8#4112yes—
138C2×C79⋊C8#4112yes—
139C8×D79—28no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—42no—
147C4⋊Dic79C22×C412yes—
148C4⋊Dic79C22×C412yes—
149Dic79⋊C4—81no—
150Dic79⋊C4—81no—
151(C2×D79)⋊C4—24no—
152(C2×D79)⋊C4—24no—
153(C2×D79)⋊C4—41no—
154(C2×D79)⋊C4—41no—
155(C2×D79)⋊C4—81no—
156(C2×D79)⋊C4—81no—
157C4×Dic79—24no—
158C4×Dic79—24no—
159C4×Dic79—21no—
160C4×Dic79—21no—
161C4×Dic79—42no—
162C4×Dic79—81no—
163C4×Dic79—81no—
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2411yescharacteristic
171C2×C8×C79—21no—
172C2×C8×C79—21no—
173C2×C8×C79—21no—
174C2×C8×C79—21no—
175C2×C8×C79—21no—
176C2×C8×C79C22×C412yes—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—42no—
184C4⋊C4×C79C22×C412yes—
185C4⋊C4×C79C22×C412yes—
186C42×C79—21no—
187C42×C79—21no—
188C42×C79—42no—
189C23×C79C22×D411yescharacteristic
190C22×D79—21no—
191C22×D79—21no—
192C22×D79—21no—
193C22×D79—21no—
194C22×D79—21no—
195C22×D79—21no—
196C22×D79C4×D412yes—
197C2×C4×C79—22no—
198C2×C4×C79—22no—
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—41no—
206C2×C4×C79—41no—
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×C79C23×C411yescharacteristic
216C2×C4×C79C2×#4111yescharacteristic
217C2×C4×C79C2×#4111yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×C4×C792−1+411yescharacteristic
220C2×C4×C792−1+411yescharacteristic
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—22no—
224C2×Dic79—22no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—41no—
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×Dic79C4×D412yes—
242C4×D79—22no—
243C4×D79—22no—
244C4×D79—22no—
245C4×D79—22no—
246C8×C79—22no—
247C8×C79—41no—
248C79⋊C8—41no—
249C79⋊C8#11714yes—
250C22×C79C2×C4×D411yescharacteristic
251C22×C79#54711yescharacteristic
252C22×C79#54711yescharacteristic
253C22×C79#57511yescharacteristic
254C22×C79#57511yescharacteristic
255C22×C79#57611yescharacteristic
256C22×C79#57811yescharacteristic
257C2×D79—22no—
258C2×D79—22no—
259C2×D79—22no—
260C2×D79—41no—
261C2×D79—41no—
262C2×D79—41no—
263C2×D79—41no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79—42no—
267C4×C79—42no—
268C4×C79—81no—
269C4×C79—81no—
270C4×C79C2×#11711yescharacteristic
271C4×C79#51911yescharacteristic
272C4×C79#55211yescharacteristic
273C4×C79#55211yescharacteristic
274Dic79—22no—
275Dic79—41no—
276Dic79—41no—
277Dic79—42no—
278Dic79—42no—
279Dic79—81no—
280Dic79—81no—
281#22616—791nomaximal, Sylow
282C2×C79#234411yescharacteristic
283C2×C79#294311yescharacteristic
284C2×C79#295111yescharacteristic
285C2×C79#334711yescharacteristic
286C2×C79#334711yescharacteristic
287C2×C79#334911yescharacteristic
288C2×C79#334911yescharacteristic
289D79—42no—
290C2×#427—791no—
291#2347—791no—
292#2346—791no—
293#1944—791no—
294#1944—791no—
295#1921—791no—
296#1921D7911yescharacteristic, complemented
297#1918—791no—
298#1918—791no—
299#1825—791no—
300#1825—791no—
301#1808—791no—
302#1776—791no—
303#1776—791no—
304#1752—791no—
305C79#2261611yescharacteristic, complemented, Sylow
306C23×C8—791no—
307C2×#117—791no—
308C2×#117—791no—
309#427—798no—
310C2×#107—791no—
311C2×#107—791no—
312C2×#107—791no—
313C2×#107C2×D7911yescharacteristic
314C2×#106—791no—
315C2×#106—791no—
316C2×C22⋊C8—791no—
317C2×#97—791no—
318C2×#97—791no—
319#389—791no—
320#389—791no—
321#389—791no—
322#389C2×D7911yescharacteristic
323#388—1581no—
324#388—1581no—
325#388—1581no—
326#388—1581no—
327#383—1581no—
328#382—1581no—
329#380—791no—
330#380—791no—
331C4×C4⋊C4—791no—
332C4×C4⋊C4—791no—
333C4×C4⋊C4—791no—
334C4×C4⋊C4—1581no—
335C4×C4⋊C4C2×D7911yescharacteristic
336C4×C22⋊C4—791no—
337#376—1581no—
338#340—791no—
339#340—791no—
340#340—791no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340C2×D7911yescharacteristic
346#340C2×D7911yescharacteristic
347#340C2×D7911yescharacteristic
348#336—791no—
349#336C2×D7911yescharacteristic
350C23×C4—791no—
351C22×C8—791no—
352C22×C8—794no—
353C22×C8—791no—
354C22×C8—1582no—
355C22×C8—1582no—
356C22×C8C22×D7911yescharacteristic
357C22×C8C22×D7911yescharacteristic
358#117—794no—
359#117—794no—
360#117—1582no—
361#117—1582no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—791no—
364C2×C4⋊C4—791no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—1581no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4—1581no—
370C2×C4⋊C4C22×D7911yescharacteristic
371C2×C4⋊C4C22×D7911yescharacteristic
372C2×C4⋊C4C22×D7911yescharacteristic
373C2×C4⋊C4C22×D7911yescharacteristic
374C2×C22⋊C4—791no—
375C2×C22⋊C4—791no—
376C2×C22⋊C4—1581no—
377C2×C22⋊C4—1581no—
378C2×C22⋊C4—1581no—
379C2×C22⋊C4—1581no—
380C2×C42—21no—
381C2×C42—791no—
382C2×C42—1581no—
383C2×C42—1581no—
384C2×C42—1581no—
385C2×C42—1581no—
386C2×C42—1581no—
387C2×C42C22×D7911yescharacteristic
388#107—794no—
389#107—1582no—
390#107—1582no—
391#107C4×D7914yes—
392#106—794no—
393#106—794no—
394C22⋊C8—1582no—
395#97—1582no—
396#97—1582no—
397#95—21no—
398#95—21no—
399#95—791no—
400#95—791no—
401#95—1581no—
402#95—1581no—
403#95—1581no—
404#95—1581no—
405#95—1581no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409C24—791no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—791no—
415C22×C4—792no—
416C22×C4—792no—
417C22×C4—791no—
418C22×C4—791no—
419C22×C4—1581no—
420C22×C4—1581no—
421C22×C4—1581no—
422C22×C4—1581no—
423C22×C4—1581no—
424C22×C4—1581no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4D4×D7911yescharacteristic
428C22×C4D4×D7911yescharacteristic
429C22×C4C23×D7911yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—792no—
436C2×C8—792no—
437C2×C8—792no—
438C2×C8—1588no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8C2×C4×D7912yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—3162no—
451C4⋊C4—792no—
452C4⋊C4—792no—
453C4⋊C4—6321no—
454C4⋊C4—6321no—
455C4⋊C4—1581no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4C2×C4×D7912yes—
462C4⋊C4C2×C4×D7912yes—
463C22⋊C4—3161no—
464C22⋊C4—3161no—
465C22⋊C4—6321no—
466C22⋊C4—6321no—
467C22⋊C4—1584no—
468C22⋊C4—1584no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—3162no—
473C42—6321no—
474C42—6321no—
475C42—1584no—
476C42—1584no—
477C42—1581no—
478C42—1581no—
479C23—792no—
480C23—1581no—
481C23—1581no—
482C23—1581no—
483C23—1581no—
484C23—1581no—
485C23—1581no—
486C23C2×D4×D7911yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—3161no—
506C2×C4—3161no—
507C2×C4—3161no—
508C2×C4—3161no—
509C2×C4—3161no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
515C2×C4—792no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1582no—
519C2×C4—1582no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1582no—
527C2×C4—1582no—
528C2×C4—1582no—
529C2×C4—1582no—
530C2×C4C22×C4×D7911yescharacteristic
531C2×C4#41×D7911yescharacteristic
532C2×C4#41×D7911yescharacteristic
533C2×C4#41×D7911yescharacteristic
534C2×C4#1457920811yescharacteristic
535C2×C4#1457920911yescharacteristic
536C8—22no—
537C8—41no—
538C8—3161no—
539C8—794no—
540C22—3161no—
541C22—3161no—
542C22—3161no—
543C22—3161no—
544C22—1582no—
545C22—1582no—
546C22—1582no—
547C22C4×D4×D7911yescharacteristic
548C22#1909921111yescharacteristic
549C22#1909921511yescharacteristic
550C22#1909935911yescharacteristic
551C22#1909935911yescharacteristic
552C22#1909936511yescharacteristic
553C22#1909937611yescharacteristic
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—3162no—
561C4—3162no—
562C4—3161no—
563C4—3161no—
564C4—6321no—
565C4—6321no—
566C4—1582no—
567C4#117×D7911yescharacteristic
568C4#1909903911yescharacteristic
569C4#1909924511yescharacteristic
570C4#1909924911yescharacteristic
571C2—3162no—
572C2#2376626611yescharacteristic
573C2#2377219711yescharacteristic
574C2#2377227511yescharacteristic
575C2#2377589311yescharacteristic
576C2#2377589711yescharacteristic
577C2#2377592311yescharacteristic
578C2#2377592711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1921C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1921D79
hypocenterC79#22616
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #23764142C2semidirect
5. #23764140C2semidirect
6. #23763994C2semidirect
7. #23763972C2semidirect
8. #23763970C2semidirect
12. #23763298C2semidirect
13. #23763296C2semidirect
15. C79×#1921C2semidirect
19. #19098600C4semidirect
296. #1921D79semidirect
305. C79#22616semidirect

Maximal subgroups

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

subgroupindex
C2×#190986002
#237662922
#237662812
#237641422
#237641402
#237639942
#237639722
#237639702
#237634932
#237634282
#237632982
#237632962
#237631752
C79×#19212
#2261679