Finite Group Databasebeta

← #34306017

Subgroups — #34306017

Order 20224.

810 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×#19098603C211yesmaximal, characteristic
2#23766291C211yesmaximal, characteristic
3#23764135C211yesmaximal, characteristic, complemented
4#23764130C211yesmaximal, characteristic, complemented
5#23763992C211yesmaximal, characteristic, complemented
6#23763982C211yesmaximal, characteristic, complemented
7#23763975C211yesmaximal, characteristic, complemented
8#23763971C211yesmaximal, characteristic, complemented
9#23763492C211yesmaximal, characteristic
10#23763490C211yesmaximal, characteristic
11#23763431C211yesmaximal, characteristic
12#23763298C211yesmaximal, characteristic, complemented
13C4×#14579092C211yesmaximal, characteristic, complemented
14#23763184C211yesmaximal, characteristic
15C79×#2344C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579120C2211yescharacteristic
18#19098603C418yescomplemented
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×C22⋊(C79⋊C8)C2211yescharacteristic
26C2×#14579072C2211yescharacteristic
27C2×C4×C79⋊C8C2211yescharacteristic
28#19098461C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098457—21no—
31#19098457—21no—
32#19098430—21no—
33#19098423C2211yescharacteristic
34C4×C4⋊Dic79C2211yescharacteristic
35C4×C4⋊Dic79C2211yescharacteristic
36#19098414—21no—
37#19098325C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49C2×#117×C79—21no—
50C22×C4⋊C4×C79C2211yescharacteristic
51#389×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#387×C79—21no—
54#387×C79—21no—
55#381×C79—21no—
56#380×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59C4×C22⋊C4×C79C2211yescharacteristic
60C2×#11929608—22no—
61C2×#11929608—22no—
62C2×#11929608C2×C414yes—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579120—22no—
68#14579120C2×C414yes—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×C22⋊Dic79—21no—
76C2×C22⋊Dic79—21no—
77C2×C4×Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80#14579094—22no—
81#14579094C2×C414yes—
82#14579094C2×C414yes—
83#14579092—22no—
84#14579092C2×C414yes—
85#14579092C2×C414yes—
86C22⋊(C79⋊C8)—22no—
87#14579072—22no—
88C4×C79⋊C8—22no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C23×C4×C79—21no—
95C23×C4×C79C2311yescharacteristic
96#117×C79—44no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—22no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79C2×C414yes—
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C22⋊C4×C79—21no—
107C2×C22⋊C4×C79D411yescharacteristic
108C2×C22⋊C4×C79D411yescharacteristic
109C2×C42×C79—21no—
110C2×C42×C79—21no—
111C2×C42×C79C2311yescharacteristic
112C2×C42×C79C2311yescharacteristic
113#95×C79—21no—
114#95×C79—21no—
115#95×C79—21no—
116#95×C79—21no—
117#95×C79—21no—
118#95×C79D411yescharacteristic
119#95×C79D411yescharacteristic
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124#11929608—28no—
125C2×C79⋊C8—21no—
126C2×C79⋊C8—21no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8C22×C412yes—
134C2×C79⋊C8C22×C412yes—
135C2×C79⋊C8#4112yes—
136C2×C79⋊C8#4112yes—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79C22×C412yes—
144C4⋊Dic79C22×C412yes—
145Dic79⋊C4—42no—
146Dic79⋊C4—42no—
147C22⋊Dic79—24no—
148C22⋊Dic79—81no—
149C4×Dic79—24no—
150C4×Dic79—81no—
151C4×Dic79—81no—
152C24×C79C2×D411yescharacteristic
153C22×C4×C79—21no—
154C22×C4×C79—21no—
155C22×C4×C79—21no—
156C22×C4×C79—21no—
157C22×C4×C79—24no—
158C22×C4×C79—21no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—42no—
163C22×C4×C79—42no—
164C22×C4×C79C22×C412yes—
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79#4112yes—
171C22×C4×C79C2411yescharacteristic
172C4⋊C4×C79—24no—
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—42no—
180C4⋊C4×C79—42no—
181C4⋊C4×C79C22×C412yes—
182C4⋊C4×C79C22×C412yes—
183C22⋊C4×C79—22no—
184C22⋊C4×C79—41no—
185C22⋊C4×C79—42no—
186C42×C79—21no—
187C42×C79—21no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—42no—
191C42×C79—42no—
192C23×C79—21no—
193C23×C79—21no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79C4×D412yes—
199C23×C79C22×D411yescharacteristic
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—48no—
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×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79C4×D412yes—
231C2×C4×C79AGL(1,ℤ/8)12yes—
232C2×C4×C79C23×C411yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C792−1+411yescharacteristic
237C2×C4×C792−1+411yescharacteristic
238C2×Dic79—22no—
239C2×Dic79—22no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C79⋊C8—22no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8#11714yescomplemented
254C22×C79—22no—
255C22×C79—22no—
256C22×C79—22no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79C2×C4×D411yescharacteristic
262C22×C79#54711yescharacteristic
263C22×C79#54711yescharacteristic
264C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C79#57511yescharacteristic
266C22×C79#57711yescharacteristic
267C22×C79#57811yescharacteristic
268C4×C79—22no—
269C4×C79—22no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79—42no—
274C4×C79—42no—
275C4×C79—81no—
276C4×C79—81no—
277C4×C79C2×#11711yescharacteristic
278C4×C79#51911yescharacteristic
279C4×C79#55211yescharacteristic
280C4×C79#55211yescharacteristic
281Dic79—42no—
282Dic79—42no—
283Dic79—81no—
284Dic79—81no—
285#22622—791nomaximal, Sylow
286C2×C79—42no—
287C2×C79#234411yescharacteristic
288C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C79#294711yescharacteristic
290C2×C79#334511yescharacteristic
291C2×C79#334911yescharacteristic
292C2×C79#335111yescharacteristic
293C2×C79#335311yescharacteristic
294C2×#428—791no—
295#2347—791no—
296#2344D7911yescharacteristic
297#1943—791no—
298#1942—791no—
299#1921—791no—
300#1920—791no—
301#1919—791no—
302#1918—791no—
303#1825—791no—
304#1824—791no—
305#1809—791no—
306#1776—791no—
307C4×#106—791no—
308#1753—791no—
309C79#2262211yescharacteristic, complemented, Sylow
310C22×M16—791no—
311C2×#117—21no—
312C2×#117—791no—
313C22×C4⋊C4C2×D7911yescharacteristic
314#428—798no—
315C2×#107—791no—
316C2×#107—791no—
317C2×#107—791no—
318C2×#106—791no—
319C2×#106—791no—
320C2×#106—791no—
321C2×C22⋊C8—791no—
322C2×#97—791no—
323C2×C4×C8—791no—
324#389—791no—
325#389—791no—
326#389C2×D7911yescharacteristic
327#389C2×D7911yescharacteristic
328#388—1581no—
329#388—1581no—
330#387—21no—
331#387—21no—
332#383—1581no—
333#381—21no—
334#380—791no—
335#380C2×D7911yescharacteristic
336C4×C4⋊C4—791no—
337C4×C4⋊C4—791no—
338C4×C4⋊C4—1581no—
339C4×C4⋊C4C2×D7911yescharacteristic
340C4×C4⋊C4C2×D7911yescharacteristic
341C4×C22⋊C4C2×D7911yescharacteristic
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#336—791no—
353#336—791no—
354C23×C4—21no—
355C23×C4C22×D7911yescharacteristic
356C2×M16—794no—
357C2×M16—1582no—
358C2×M16—1582no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8—791no—
362C22×C8—791no—
363#117—44no—
364#117—794no—
365#117—1582no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4C4×D7914yes—
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C4⋊C4C22×D7911yescharacteristic
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1581no—
383C2×C22⋊C4—1581no—
384C2×C22⋊C4C22⋊D7911yescharacteristic
385C2×C22⋊C4C22⋊D7911yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—1581no—
389C2×C42—1581no—
390C2×C42—1581no—
391C2×C42C22×D7911yescharacteristic
392C2×C42C22×D7911yescharacteristic
393#107—794no—
394#107—794no—
395#107—1582no—
396#106—794no—
397#106—794no—
398#106—1582no—
399C22⋊C8—1582no—
400#97—1582no—
401C4×C8—1582no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95C22⋊D7911yescharacteristic
413#95C22⋊D7911yescharacteristic
414C24C2×C22⋊D7911yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4C2×C4×D7912yes—
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4C2×C22⋊D7911yescharacteristic
434C22×C4C2×C22⋊D7911yescharacteristic
435C22×C4C2×C22⋊D7911yescharacteristic
436C22×C4#1192962412yes—
437C22×C4C23×D7911yescharacteristic
438M16—1588no—
439C2×C8—792no—
440C2×C8—792no—
441C2×C8—792no—
442C2×C8—792no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—3162no—
461C4⋊C4—3162no—
462C4⋊C4—792no—
463C4⋊C4—792no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4C2×C4×D7912yes—
471C4⋊C4C2×C4×D7912yes—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—42no—
475C22⋊C4—6321no—
476C22⋊C4—1584no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—42no—
482C42—42no—
483C42—6321no—
484C42—6321no—
485C42—1584no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D7912yes—
493C23C22×C22⋊D7911yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—1582no—
535C2×C4—1582no—
536C2×C4C4×C22⋊D7912yes—
537C2×C4#1457918712yes—
538C2×C4C22×C4×D7911yescharacteristic
539C2×C4#41×D7911yescharacteristic
540C2×C4#41×D7911yescharacteristic
541C2×C4C2×#1192962411yescharacteristic
542C2×C4#1457920811yescharacteristic
543C2×C4#1457920811yescharacteristic
544C8—3161no—
545C8—3161no—
546C8—794no—
547C8—1582no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D7911yescharacteristic
556C22#1909921211yescharacteristic
557C22#1909921211yescharacteristic
558C22C2×#1457918711yescharacteristic
559C22#1909936111yescharacteristic
560C22#1909937111yescharacteristic
561C22#1909937811yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—3162no—
572C4—3162no—
573C4—6321no—
574C4—6321no—
575C4#117×D7911yescharacteristic
576C4#1909904011yescharacteristic
577C4#1909924611yescharacteristic
578C4#1909924611yescharacteristic
579C2—42no—
580C2#2376627011yescharacteristic
581C2C4×#1457918711yescharacteristic
582C2#2377224311yescharacteristic
583C2#2377587011yescharacteristic
584C2#2377593011yescharacteristic
585C2#2377596011yescharacteristic
586C2#2377598411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2344C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#2344D79
hypocenterC79#22622
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #23764135C2semidirect
4. #23764130C2semidirect
5. #23763992C2semidirect
6. #23763982C2semidirect
7. #23763975C2semidirect
8. #23763971C2semidirect
12. #23763298C2semidirect
13. C4×#14579092C2semidirect
18. #19098603C4semidirect
253. C79⋊C8#117semidirect
309. C79#22622semidirect

Maximal subgroups

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

subgroupindex
C2×#190986032
#237662912
#237641352
#237641302
#237639922
#237639822
#237639752
#237639712
#237634922
#237634902
#237634312
#237632982
C4×#145790922
#237631842
C79×#23442
#2262279