Finite Group Databasebeta

← #34347088

Subgroups — #34347088

Order 20224.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098963C211yesmaximal, characteristic, complemented
2C2×#19098606C211yesmaximal, characteristic
3#23767125C211yesmaximal, characteristic
4#23767107C211yesmaximal, characteristic
5#23764885C211yesmaximal, characteristic, complemented
6#23764880C211yesmaximal, characteristic, complemented
7#23764737C211yesmaximal, characteristic, complemented
8#23764736C211yesmaximal, characteristic, complemented
9#23764137C211yesmaximal, characteristic, complemented
10#23764127C211yesmaximal, characteristic, complemented
11#23763595C211yesmaximal, characteristic
12#23763567C211yesmaximal, characteristic, complemented
13#23763431C211yesmaximal, characteristic
14#23763186C211yesmaximal, characteristic
15C79×#1854C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579152C2211yescharacteristic
18C2×#14579152C2211yescharacteristic
19C22×C4⋊Dic79C2211yescharacteristic
20#19098963—24no—
21#19098606—24no—
22C2×#14579094C2211yescharacteristic
23C2×#14579094C2211yescharacteristic
24C2×#14579094C2211yescharacteristic
25C2×#14579093C2211yescharacteristic
26C2×#14579093C2211yescharacteristic
27C2×#14579092C2211yescharacteristic
28C2×#14579092C2211yescharacteristic
29C2×#14579092C2211yescharacteristic
30C2×#14579071C2211yescharacteristic
31C2×C8×Dic79C2211yescharacteristic
32#19098505—21no—
33#19098499—21no—
34#19098495—21no—
35#19098486—21no—
36#19098462C2211yescharacteristic
37#19098462C2211yescharacteristic
38#19098461—21no—
39#19098459C2211yescharacteristic
40#19098459C2211yescharacteristic
41#19098447—21no—
42#19098438—21no—
43#19098432C2211yescharacteristic
44#19098428C2211yescharacteristic
45#19098423C2211yescharacteristic
46#19098421—21no—
47C4×C22⋊Dic79C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098317C2211yescharacteristic
53#19098317C2211yescharacteristic
54C2×#117×C79C2211yescharacteristic
55C2×#107×C79C2211yescharacteristic
56C2×#106×C79C2211yescharacteristic
57C2×C22⋊C8×C79C2211yescharacteristic
58#380×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60#340×C79C2211yescharacteristic
61C23×Dic79—21no—
62C2×#11929608—22no—
63C2×#11929608—22no—
64C2×#11929608Q814yes—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579152—22no—
68#14579152—22no—
69#14579152—41no—
70#14579152—41no—
71C2×C4⋊Dic79—22no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79—21no—
75C2×C4⋊Dic79—21no—
76C2×C4⋊Dic79—21no—
77C2×C4⋊Dic79—41no—
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×C4⋊Dic79C2311yescharacteristic
81C2×C4⋊Dic79C2311yescharacteristic
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—21no—
84C2×Dic79⋊C4—21no—
85C2×Dic79⋊C4—21no—
86C2×Dic79⋊C4—21no—
87C2×Dic79⋊C4—21no—
88C2×C22⋊Dic79—21no—
89C2×C22⋊Dic79D411yescharacteristic
90C2×C22⋊Dic79D411yescharacteristic
91C2×C4×Dic79C2311yescharacteristic
92C2×C4×Dic79C2311yescharacteristic
93#14579094—22no—
94#14579094—22no—
95#14579094D414yes—
96#14579093—22no—
97#14579093—41no—
98#14579092—22no—
99#14579092—22no—
100#14579092—22no—
101#14579071—22no—
102C8×Dic79—22no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95D411yescharacteristic
111C79⋊#95D411yescharacteristic
112C23×C4×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115#117×C79—22no—
116#117×C79—41no—
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C22⋊C4×C79—21no—
122C2×C22⋊C4×C79—21no—
123C2×C42×C79—21no—
124#107×C79—41no—
125#106×C79—22no—
126C22⋊C8×C79Q814yes—
127#95×C79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—42no—
133C22×Dic79—42no—
134C22×Dic79—81no—
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139#11929608—28no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8C2×D412yes—
143C2×C79⋊C8C2×Q812yes—
144C2×C79⋊C8C2×Q812yes—
145C2×C79⋊C8#4112yes—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—42no—
154C4⋊Dic79—42no—
155C4⋊Dic79—42no—
156C4⋊Dic79—42no—
157C4⋊Dic79—81no—
158C4⋊Dic79C2×D412yes—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161Dic79⋊C4—42no—
162Dic79⋊C4—81no—
163Dic79⋊C4—81no—
164Dic79⋊C4—81no—
165C22⋊Dic79—41no—
166C22⋊Dic79—41no—
167C22⋊Dic79—81no—
168C4×Dic79—21no—
169C4×Dic79—21no—
170C4×Dic79—21no—
171C4×Dic79—21no—
172C24×C79C2×D411yescharacteristic
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×Q812yes—
183C22×C4×C79C2×Q812yes—
184C22×C4×C79C2411yescharacteristic
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79C2×Q812yes—
190C2×C8×C79C2×Q812yes—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79C2×D412yes—
199C22⋊C4×C79—42no—
200C22⋊C4×C79—42no—
201C42×C79—42no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79C4⋊Q812yes—
209C23×C79C22×D411yescharacteristic
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
216C2×C4×C79—22no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79C4⋊Q812yes—
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C22×Q811yescharacteristic
231C2×C4×C79C22×Q811yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79H(5,𝔽2)11yescharacteristic
234C2×C4×C792−1+411yescharacteristic
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—22no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—84no—
260C8×C79—22no—
261C8×C79—22no—
262C79⋊C8—22no—
263C79⋊C8C22⋊Q814yes—
264C22×C79—22no—
265C22×C79—22no—
266C22×C79—22no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79C2×C4⋊Q811yescharacteristic
272C22×C79D4211yescharacteristic
273C22×C79#54711yescharacteristic
274C22×C79#57511yescharacteristic
275C22×C79#57511yescharacteristic
276C22×C79#57711yescharacteristic
277C22×C79#57711yescharacteristic
278C4×C79—22no—
279C4×C79—42no—
280C4×C79—81no—
281C4×C79—81no—
282C4×C79—81no—
283C4×C79C2×C22⋊Q811yescharacteristic
284C4×C79#54411yescharacteristic
285C4×C79D4×Q811yescharacteristic
286C4×C79#55411yescharacteristic
287Dic79—41no—
288Dic79—41no—
289Dic79—41no—
290Dic79—42no—
291Dic79—41no—
292Dic79—81no—
293Dic79—81no—
294Dic79—81no—
295#26075—791nomaximal, Sylow
296C2×C79—42no—
297C2×C79#244011yescharacteristic
298C2×C79#316311yescharacteristic
299C2×C79#316611yescharacteristic
300C2×C79#326711yescharacteristic
301C2×C79#327111yescharacteristic
302C2×C79#335111yescharacteristic
303C2×C79#335111yescharacteristic
304C2×#499—791no—
305C2×#428—791no—
306#2442—791no—
307#2441—791no—
308#2085—791no—
309#2084—791no—
310#2059—791no—
311#2059—791no—
312#1943—791no—
313#1941—791no—
314#1854D7911yescharacteristic
315#1853—791no—
316#1847—791no—
317#1809—791no—
318#1753—791no—
319C79#2607511yescharacteristic, complemented, Sylow
320C22×M16—791no—
321C2×#125—791no—
322C2×#125—791no—
323C2×#117C2×D7911yescharacteristic
324C22×C4⋊C4—791no—
325#499—1584no—
326#428—1584no—
327C2×#107—791no—
328C2×#107—791no—
329C2×#107—791no—
330C2×#107C2×D7911yescharacteristic
331C2×#106—791no—
332C2×#106—791no—
333C2×#106—791no—
334C2×#106—791no—
335C2×#106—791no—
336C2×#106C2×D7911yescharacteristic
337C2×C22⋊C8C2×D7911yescharacteristic
338C2×#97—791no—
339C2×C4×C8—791no—
340#400—1581no—
341#399—1581no—
342#398—1581no—
343#396—1581no—
344#389—791no—
345#389—791no—
346#389—791no—
347#389—791no—
348#389—1581no—
349#387—1581no—
350#385—1581no—
351#384—791no—
352#382—791no—
353#380—791no—
354#380—1581no—
355#380C2×D7911yescharacteristic
356C4×C22⋊C4—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340C2×D7911yescharacteristic
362#340C2×D7911yescharacteristic
363#336—791no—
364#336—791no—
365C23×C4—1581no—
366C23×C4C22×D7911yescharacteristic
367C2×M16—794no—
368C2×M16—1582no—
369C2×M16—1582no—
370C22×C8—791no—
371C22×C8—791no—
372C22×C8C22×D7911yescharacteristic
373C22×C8C22×D7911yescharacteristic
374#125—3161no—
375#125—3161no—
376#125—1582no—
377#125—1582no—
378#117—22no—
379#117—41no—
380C2×C4⋊C4—3161no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4—1581no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1582no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C4⋊C4C22×D7911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—791no—
404C2×C22⋊C4—791no—
405C2×C22⋊C4—1581no—
406C2×C42—21no—
407C2×C42—791no—
408C2×C42—791no—
409#107—41no—
410#107—794no—
411#107—1582no—
412#107—1582no—
413#106—22no—
414#106—3161no—
415#106—1582no—
416#106—1582no—
417#106—1582no—
418#106—1582no—
419C22⋊C8Dic15814yes—
420#97—1582no—
421C4×C8—1582no—
422#95—21no—
423#95—791no—
424#95—791no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432C24D4×D7911yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—3162no—
442C22×C4—3162no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—791no—
447C22×C4—6321no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4D4×D7911yescharacteristic
453C22×C4Q8×D7912yes—
454C22×C4C2×Dic15812yes—
455C22×C4C23×D7911yescharacteristic
456M16—1588no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—792no—
462C2×C8—792no—
463C2×C8—792no—
464C2×C8—792no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8C2×Dic15812yes—
468C2×C8C2×Dic15812yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—3162no—
477C4⋊C4—3162no—
478C4⋊C4—3162no—
479C4⋊C4—3162no—
480C4⋊C4—3162no—
481C4⋊C4—3162no—
482C4⋊C4—3162no—
483C4⋊C4—792no—
484C4⋊C4—6321no—
485C4⋊C4—6321no—
486C4⋊C4—6321no—
487C4⋊C4—6321no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4D4×D7912yes—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—3161no—
499C22⋊C4—3161no—
500C22⋊C4—6321no—
501C42—42no—
502C42—1581no—
503C42—1581no—
504C42—1581no—
505C42—1581no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1457916012yes—
513C23C2×D4×D7911yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—6324no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1582no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4#1457916012yes—
558C2×C4C2×D4×D7911yescharacteristic
559C2×C4C2×Q8×D7911yescharacteristic
560C2×C4C22×Dic15811yescharacteristic
561C2×C4#41×D7911yescharacteristic
562C2×C4#1457920611yescharacteristic
563C2×C4#1457920811yescharacteristic
564C8—22no—
565C8—22no—
566C8—794no—
567C8—1582no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1457916011yescharacteristic
576C22#1909919511yescharacteristic
577C22#1909921111yescharacteristic
578C22#1909935911yescharacteristic
579C22#1909936211yescharacteristic
580C22#1909937011yescharacteristic
581C22#1909937011yescharacteristic
582C4—22no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—3161no—
588C4—3162no—
589C4—3161no—
590C4—3161no—
591C4—3161no—
592C4—6321no—
593C4—6321no—
594C4—6321no—
595C4C22⋊Q8×D7911yescharacteristic
596C4#1909919211yescharacteristic
597C4D4×Dic15811yescharacteristic
598C4#1909927211yescharacteristic
599C2—42no—
600C2#2376710111yescharacteristic
601C2#2377393211yescharacteristic
602C2#2377395011yescharacteristic
603C2#2377492811yescharacteristic
604C2#2377498311yescharacteristic
605C2#2377595311yescharacteristic
606C2#2377595811yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1854C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1854D79
hypocenterC79#26075
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#19098963C2semidirect
5. #23764885C2semidirect
6. #23764880C2semidirect
7. #23764737C2semidirect
8. #23764736C2semidirect
9. #23764137C2semidirect
10. #23764127C2semidirect
12. #23763567C2semidirect
319. C79#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#190989632
C2×#190986062
#237671252
#237671072
#237648852
#237648802
#237647372
#237647362
#237641372
#237641272
#237635952
#237635672
#237634312
#237631862
C79×#18542
#2607579