Finite Group Databasebeta

← #34347148

Subgroups — #34347148

Order 20224.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098971C211yesmaximal, characteristic, complemented
2C2×#19098603C211yesmaximal, characteristic
3#23767118C211yesmaximal, characteristic
4#23767092C211yesmaximal, characteristic
5#23764886C211yesmaximal, characteristic, complemented
6#23764884C211yesmaximal, characteristic, complemented
7#23764741C211yesmaximal, characteristic, complemented
8#23764726C211yesmaximal, characteristic, complemented
9#23764137C211yesmaximal, characteristic, complemented
10#23764132C211yesmaximal, characteristic, complemented
11#23763601C211yesmaximal, characteristic
12#23763563C211yesmaximal, characteristic, complemented
13#23763431C211yesmaximal, characteristic
14#23763186C211yesmaximal, characteristic
15C79×#1852C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579160C2211yescharacteristic
18C2×#14579152C2211yescharacteristic
19C2×#14579120C2211yescharacteristic
20#19098971—24no—
21#19098603—24no—
22C2×#14579095C2211yescharacteristic
23C2×#14579094C2211yescharacteristic
24C2×#14579094C2211yescharacteristic
25C2×#14579094C2211yescharacteristic
26C2×#14579093C2211yescharacteristic
27C2×#14579092C2211yescharacteristic
28C2×#14579092C2211yescharacteristic
29C2×#14579092C2211yescharacteristic
30C2×#14579071C2211yescharacteristic
31C2×C8×Dic79C2211yescharacteristic
32#19098502—21no—
33#19098496—21no—
34#19098486—21no—
35#19098475—21no—
36#19098467—21no—
37#19098462C2211yescharacteristic
38#19098462C2211yescharacteristic
39#19098459C2211yescharacteristic
40#19098459C2211yescharacteristic
41#19098446—21no—
42#19098436—21no—
43#19098428C2211yescharacteristic
44#19098428C2211yescharacteristic
45#19098423—21no—
46#19098423C2211yescharacteristic
47C4×C22⋊Dic79C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098317C2211yescharacteristic
53#19098317C2211yescharacteristic
54C22×C4⋊C4×C79C2211yescharacteristic
55C2×#107×C79C2211yescharacteristic
56C2×#107×C79C2211yescharacteristic
57C2×C22⋊C8×C79C2211yescharacteristic
58#380×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60#340×C79C2211yescharacteristic
61C22×Dic158—21no—
62C2×#11929608—22no—
63C2×#11929608—22no—
64C2×#11929608Q814yescomplemented
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579160—22no—
68#14579160—41no—
69#14579152—22no—
70#14579152—41no—
71#14579120—22no—
72#14579120—41no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79—21no—
75C2×C4⋊Dic79—21no—
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—21no—
84C2×C22⋊Dic79—21no—
85C2×C22⋊Dic79—21no—
86C2×C22⋊Dic79D411yescharacteristic
87C2×C22⋊Dic79D411yescharacteristic
88C2×C4×Dic79—21no—
89C2×C4×Dic79C2311yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91#14579095—41no—
92#14579094—22no—
93#14579094—22no—
94#14579094D414yes—
95#14579093—22no—
96#14579092—22no—
97#14579092—22no—
98#14579092—22no—
99#14579071—22no—
100C8×Dic79—22no—
101C79⋊#95—21no—
102C79⋊#95—21no—
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×C79—21no—
113C23×C4×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115C22×C8×C79C2311yescharacteristic
116C2×C4⋊C4×C79—22no—
117C2×C4⋊C4×C79—21no—
118C2×C4⋊C4×C79—41no—
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C22⋊C4×C79—21no—
124#107×C79—22no—
125#107×C79—41no—
126C22⋊C8×C79Q814yescomplemented
127#95×C79—21no—
128C2×Dic158—42no—
129C2×Dic158—42no—
130C2×Dic158—81no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
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—81no—
156C4⋊Dic79C2×D412yes—
157Dic79⋊C4—42no—
158Dic79⋊C4—42no—
159Dic79⋊C4—81no—
160Dic79⋊C4—81no—
161C22⋊Dic79—41no—
162C22⋊Dic79—41no—
163C22⋊Dic79—42no—
164C22⋊Dic79—42no—
165C4×Dic79—21no—
166C4×Dic79—21no—
167C4×Dic79—21no—
168C4×Dic79—21no—
169C4×Dic79—42no—
170C24×C79C2×D411yescharacteristic
171C22×C4×C79—21no—
172C22×C4×C79—21no—
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—42no—
180C22×C4×C79—42no—
181C22×C4×C79—81no—
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×Q812yes—
184C22×C4×C79C2×Q812yes—
185C22×C4×C79C2411yescharacteristic
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79C2×Q812yes—
191C2×C8×C79C2×Q812yes—
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×C79—21no—
199C4⋊C4×C79—42no—
200C4⋊C4×C79C2×D412yes—
201C22⋊C4×C79—81no—
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×C79—84no—
229C2×C4×C79C4⋊Q812yes—
230C2×C4×C79C22×D411yescharacteristic
231C2×C4×C79C22×Q811yescharacteristic
232C2×C4×C79C22×Q811yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79H(5,𝔽2)11yescharacteristic
235C2×C4×C792−1+411yescharacteristic
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—22no—
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—21no—
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—41no—
260Dic158—84no—
261C8×C79—22no—
262C8×C79—22no—
263C79⋊C8—22no—
264C79⋊C8C22⋊Q814yes—
265C22×C79—22no—
266C22×C79—22no—
267C22×C79—22no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79C2×C4⋊Q811yescharacteristic
273C22×C79D4211yescharacteristic
274C22×C79#54711yescharacteristic
275C22×C79#57511yescharacteristic
276C22×C79#57511yescharacteristic
277C22×C79#57611yescharacteristic
278C22×C79#57811yescharacteristic
279C4×C79—22no—
280C4×C79—42no—
281C4×C79—81no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79C2×C22⋊Q811yescharacteristic
285C4×C79#54411yescharacteristic
286C4×C79D4×Q811yescharacteristic
287C4×C79#55411yescharacteristic
288Dic79—41no—
289Dic79—41no—
290Dic79—41no—
291Dic79—42no—
292Dic79—41no—
293Dic79—81no—
294Dic79—81no—
295Dic79—81no—
296#26079—791nomaximal, Sylow
297C2×C79—42no—
298C2×C79#244011yescharacteristic
299C2×C79#316411yescharacteristic
300C2×C79#316611yescharacteristic
301C2×C79#327311yescharacteristic
302C2×C79#327711yescharacteristic
303C2×C79#334711yescharacteristic
304C2×C79#334911yescharacteristic
305C2×#501—791no—
306C2×#428—791no—
307#2442—791no—
308#2439—791no—
309#2085—791no—
310#2085—791no—
311#2059—791no—
312#2057—791no—
313#1943—791no—
314#1942—791no—
315#1854—791no—
316#1852D7911yescharacteristic
317#1846—791no—
318#1809—791no—
319#1753—791no—
320C79#2607911yescharacteristic, complemented, Sylow
321C22×M16—791no—
322C2×C4⋊Q8—791no—
323C2×#125—791no—
324C2×#117—791no—
325C22×C4⋊C4C2×D7911yescharacteristic
326#501—1584no—
327#428—1584no—
328C2×#107—791no—
329C2×#107—791no—
330C2×#107—791no—
331C2×#107—791no—
332C2×#107C2×D7911yescharacteristic
333C2×#107C2×D7911yescharacteristic
334C2×#106—791no—
335C2×#106—791no—
336C2×#106—791no—
337C2×#106—791no—
338C2×C22⋊C8C2×D7911yescharacteristic
339C2×#97—791no—
340C2×C4×C8—791no—
341#400—1581no—
342#398—1581no—
343#396—1581no—
344#393—1581no—
345#391—1581no—
346#389—791no—
347#389—791no—
348#389—791no—
349#389—791no—
350#387—1581no—
351#385—1581no—
352#382—791no—
353#382—791no—
354#380—791no—
355#380—1581no—
356#380C2×D7911yescharacteristic
357C4×C22⋊C4—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340C2×D7911yescharacteristic
363#340C2×D7911yescharacteristic
364#336—791no—
365#336—791no—
366C22×Q8—1581no—
367C23×C4—21no—
368C23×C4C22×D7911yescharacteristic
369C2×M16—794no—
370C2×M16—1582no—
371C2×M16—1582no—
372C22×C8—791no—
373C22×C8—791no—
374C22×C8C22×D7911yescharacteristic
375C22×C8C22×D7911yescharacteristic
376C4⋊Q8—3161no—
377C4⋊Q8—1582no—
378#125—3161no—
379#125—1582no—
380#117—3161no—
381#117—1582no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—41no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—791no—
402C2×C22⋊C4—791no—
403C2×C22⋊C4—1581no—
404C2×C22⋊C4—1581no—
405C2×C42—791no—
406C2×C42—791no—
407C2×C42—1581no—
408#107—22no—
409#107—41no—
410#107—3161no—
411#107—794no—
412#107—1582no—
413#107—1582no—
414#106—1582no—
415#106—1582no—
416#106—1582no—
417#106—1582no—
418C22⋊C8Dic15814yescomplemented
419#97—1582no—
420C4×C8—1582no—
421#95—21no—
422#95—791no—
423#95—791no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433C24D4×D7911yescharacteristic
434C2×Q8—3162no—
435C2×Q8—3162no—
436C2×Q8—6321no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—791no—
449C22×C4—791no—
450C22×C4—791no—
451C22×C4—791no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4D4×D7911yescharacteristic
457C22×C4Q8×D7912yes—
458C22×C4C2×Dic15812yes—
459C22×C4C23×D7911yescharacteristic
460M16—1588no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—792no—
466C2×C8—792no—
467C2×C8—792no—
468C2×C8—792no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8C2×Dic15812yes—
472C2×C8C2×Dic15812yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—3162no—
482C4⋊C4—3162no—
483C4⋊C4—3162no—
484C4⋊C4—3162no—
485C4⋊C4—792no—
486C4⋊C4—6321no—
487C4⋊C4—6321no—
488C4⋊C4—6321no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4D4×D7912yes—
497C22⋊C4—81no—
498C22⋊C4—3162no—
499C22⋊C4—3162no—
500C22⋊C4—3161no—
501C22⋊C4—3161no—
502C42—3162no—
503C42—1581no—
504C42—1581no—
505C42—1581no—
506C42—1581no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1457916012yes—
514C23C2×D4×D7911yescharacteristic
515Q8—6324no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
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—41no—
533C2×C4—41no—
534C2×C4—84no—
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—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
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—1581no—
558C2×C4—1581no—
559C2×C4#1457916012yes—
560C2×C4C2×D4×D7911yescharacteristic
561C2×C4C2×Q8×D7911yescharacteristic
562C2×C4C22×Dic15811yescharacteristic
563C2×C4#41×D7911yescharacteristic
564C2×C4#1457920611yescharacteristic
565C2×C4#1457920811yescharacteristic
566C8—22no—
567C8—22no—
568C8—794no—
569C8—1582no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1457916011yescharacteristic
578C22#1909919511yescharacteristic
579C22#1909921111yescharacteristic
580C22#1909935911yescharacteristic
581C22#1909936211yescharacteristic
582C22#1909936411yescharacteristic
583C22#1909937611yescharacteristic
584C4—22no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—3162no—
590C4—3161no—
591C4—3161no—
592C4—3161no—
593C4—3161no—
594C4—6321no—
595C4—6321no—
596C4—6321no—
597C4C22⋊Q8×D7911yescharacteristic
598C4#1909919211yescharacteristic
599C4D4×Dic15811yescharacteristic
600C4#1909927211yescharacteristic
601C2—42no—
602C2#2376710111yescharacteristic
603C2#2377393611yescharacteristic
604C2#2377395011yescharacteristic
605C2#2377501111yescharacteristic
606C2#2377505711yescharacteristic
607C2#2377590311yescharacteristic
608C2#2377592311yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1852C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1852D79
hypocenterC79#26079
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#19098971C2semidirect
5. #23764886C2semidirect
6. #23764884C2semidirect
7. #23764741C2semidirect
8. #23764726C2semidirect
9. #23764137C2semidirect
10. #23764132C2semidirect
12. #23763563C2semidirect
64. C2×#11929608Q8semidirect
126. C22⋊C8×C79Q8semidirect
320. C79#26079semidirect
418. C22⋊C8Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190989712
C2×#190986032
#237671182
#237670922
#237648862
#237648842
#237647412
#237647262
#237641372
#237641322
#237636012
#237635632
#237634312
#237631862
C79×#18522
#2607979