Finite Group Databasebeta

← #34329334

Subgroups — #34329334

Order 20224.

808 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098863C211yesmaximal, characteristic
2C2×#19098663C211yesmaximal, characteristic
3#23768591C211yesmaximal, characteristic, complemented
4#23764813C211yesmaximal, characteristic
5#23764803C211yesmaximal, characteristic
6#23764717C211yesmaximal, characteristic
7#23764717C211yesmaximal, characteristic
8#23764368C211yesmaximal, characteristic, complemented
9#23764168C211yesmaximal, characteristic, complemented
10#23764158C211yesmaximal, characteristic, complemented
11#23764124C211yesmaximal, characteristic, complemented
12#23764041C211yesmaximal, characteristic, complemented
13#23764034C211yesmaximal, characteristic, complemented
14#23763402C211yesmaximal, characteristic, complemented
15C79×#2354C211yesmaximal, characteristic
16C22×#11929612C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19#19098863—24no—
20#19098663—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579084C2211yescharacteristic
28C2×#14579084C2211yescharacteristic
29C2×#14579084C2211yescharacteristic
30C2×#14579082C2211yescharacteristic
31C2×#14579082C2211yescharacteristic
32C2×#14579082C2211yescharacteristic
33C2×#14579072C2211yescharacteristic
34#19098493—21no—
35#19098493—21no—
36#19098469C2211yescharacteristic
37#19098469C2211yescharacteristic
38#19098448—21no—
39#19098448—21no—
40#19098433C2211yescharacteristic
41#19098429C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×C4×D4×C79C2211yescharacteristic
51C2×#117×C79—21no—
52#390×C79C2211yescharacteristic
53#390×C79C2211yescharacteristic
54#388×C79—21no—
55#388×C79—21no—
56#384×C79C2211yescharacteristic
57#380×C79—21no—
58C4×C4⋊C4×C79C2211yescharacteristic
59C4×C4⋊C4×C79C2211yescharacteristic
60C4×C4⋊C4×C79C2211yescharacteristic
61C22×Dic158C2311yescharacteristic
62C2×#11929612—22no—
63C2×#11929612—22no—
64C2×#11929612—22no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69#14579153—22no—
70#14579153—22no—
71C4×Dic158—22no—
72C4×Dic158—22no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79—21no—
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×C4×Dic79—21no—
82#14579094—22no—
83#14579094—22no—
84#14579092—22no—
85C4⋊(C79⋊C8)—22no—
86C4⋊(C79⋊C8)—22no—
87C4⋊(C79⋊C8)—22no—
88#14579084—22no—
89#14579084—22no—
90#14579084—22no—
91#14579082—22no—
92#14579082—22no—
93#14579082—22no—
94#14579072—22no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C22×D4×C79C2311yescharacteristic
102C23×C4×C79—21no—
103C4×D4×C79—22no—
104C4×D4×C79D414yes—
105#117×C79—44no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79D411yescharacteristic
109C2×C4⋊C4×C79D411yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79—21no—
115C2×C22⋊C4×C79—21no—
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79—21no—
118C2×C42×C79—21no—
119C2×C42×C79—21no—
120C2×C42×C79C2311yescharacteristic
121C2×C42×C79C2311yescharacteristic
122C2×C42×C79C2311yescharacteristic
123#95×C79—21no—
124#95×C79—21no—
125#95×C79D411yescharacteristic
126#95×C79D411yescharacteristic
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158—21no—
132C2×Dic158#4112yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137#11929612—28no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8#4112yes—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—42no—
157C4⋊Dic79—42no—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161C4×Dic79—42no—
162C24×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79—21no—
166C2×D4×C79—21no—
167C2×D4×C79—21no—
168C2×D4×C79C2×D412yes—
169C22×C4×C79—21no—
170C22×C4×C79—24no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—42no—
174C22×C4×C79—42no—
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2411yescharacteristic
182C4⋊C4×C79—22no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—41no—
189C4⋊C4×C79—42no—
190C4⋊C4×C79—81no—
191C4⋊C4×C79C2×D412yes—
192C22⋊C4×C79—24no—
193C22⋊C4×C79—42no—
194C22⋊C4×C79—42no—
195C22⋊C4×C79—81no—
196C22⋊C4×C79—81no—
197C42×C79—21no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—42no—
202C42×C79—42no—
203C42×C79C2×D412yes—
204C42×C79#4112yes—
205C23×C79—22no—
206C23×C79—41no—
207C23×C79—41no—
208C23×C79—41no—
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79C22×D411yescharacteristic
213D4×C79—22no—
214D4×C79—22no—
215D4×C79—41no—
216D4×C79—41no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—22no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—48no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79#12112yes—
243C2×C4×C79#12112yes—
244C2×C4×C79AGL(1,ℤ/8)12yes—
245C2×C4×C79C22×D411yescharacteristic
246C2×C4×C79C2×#4111yescharacteristic
247C2×C4×C79C2×#4111yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C792−1+411yescharacteristic
250C2×C4×C792−1+411yescharacteristic
251C2×Dic79—22no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263Dic158—22no—
264Dic158—22no—
265Dic158—41no—
266Dic158—41no—
267C79⋊C8—22no—
268C79⋊C8—41no—
269C79⋊C8—41no—
270C79⋊C8—41no—
271C22×C79—42no—
272C22×C79—42no—
273C22×C79—42no—
274C22×C79—81no—
275C22×C79—81no—
276C22×C79—81no—
277C22×C79—81no—
278C22×C79C2×#12111yescharacteristic
279C22×C79#54711yescharacteristic
280C22×C79#54811yescharacteristic
281C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C79#57511yescharacteristic
283C22×C79#57711yescharacteristic
284C22×C79#57811yescharacteristic
285C4×C79—22no—
286C4×C79—22no—
287C4×C79—22no—
288C4×C79—41no—
289C4×C79—41no—
290C4×C79—41no—
291C4×C79—42no—
292C4×C79—81no—
293C4×C79—81no—
294C4×C79#52511yescharacteristic
295C4×C79#53611yescharacteristic
296C4×C79#55211yescharacteristic
297C4×C79#55611yescharacteristic
298Dic79—42no—
299Dic79—81no—
300Dic79—81no—
301Dic79—81no—
302#24690—791nomaximal, Sylow
303C2×C79—82no—
304C2×C79#241511yescharacteristic
305C2×C79#303611yescharacteristic
306C2×C79#304611yescharacteristic
307C2×C79#334311yescharacteristic
308C2×C79#334811yescharacteristic
309C2×C79#336811yescharacteristic
310C2×C79#337711yescharacteristic
311C2×#478—791no—
312C2×#440—791no—
313#2565—791no—
314#2354D7911yescharacteristic
315#2073—791no—
316#2071—791no—
317#2055—791no—
318#2055—791no—
319#1992—791no—
320#1949—791no—
321#1947—791no—
322#1940—791no—
323#1928—791no—
324#1927—791no—
325#1803—791no—
326C79#2469011yescharacteristic, complemented, Sylow
327C22×SD16—791no—
328C2×#125—791no—
329C2×C4×Q8—791no—
330C2×C4×D4C2×D7911yescharacteristic
331C2×#117—21no—
332#478—1584no—
333#440—1584no—
334C2×#107—791no—
335C2×#107—791no—
336C2×#106—791no—
337C2×C4⋊C8—791no—
338C2×C4⋊C8—791no—
339C2×C4⋊C8—791no—
340C2×#103—791no—
341C2×#103—791no—
342C2×#103—791no—
343C2×#102—791no—
344C2×#102—791no—
345C2×#102—791no—
346C2×#97—791no—
347#398—1581no—
348#398—1581no—
349#391—791no—
350#391—791no—
351#390C2×D7911yescharacteristic
352#390C2×D7911yescharacteristic
353#388—21no—
354#388—21no—
355#387—1581no—
356#387—1581no—
357#384—791no—
358#384C2×D7911yescharacteristic
359#382—791no—
360#380—21no—
361C4×C4⋊C4C2×D7911yescharacteristic
362C4×C4⋊C4C2×D7911yescharacteristic
363C4×C4⋊C4C2×D7911yescharacteristic
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#336—791no—
371#336—791no—
372C22×Q8—791no—
373C22×D4C22×D7911yescharacteristic
374C23×C4—21no—
375C2×SD16—1582no—
376C2×SD16—1582no—
377C2×SD16—1582no—
378C22×C8—791no—
379C22×C8—791no—
380C22×C8—791no—
381C22×C8—791no—
382#125—1582no—
383#125—1582no—
384C4×Q8—1582no—
385C4×Q8—1582no—
386C4×D4—22no—
387C4×D4D31614yes—
388#117—44no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4C22⋊D7911yescharacteristic
400C2×C4⋊C4C22⋊D7911yescharacteristic
401C2×C4⋊C4C22×D7911yescharacteristic
402C2×C4⋊C4C22×D7911yescharacteristic
403C2×C4⋊C4C22×D7911yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—1581no—
412C2×C42C22×D7911yescharacteristic
413C2×C42C22×D7911yescharacteristic
414C2×C42C22×D7911yescharacteristic
415#107—1582no—
416#107—1582no—
417#106—1582no—
418C4⋊C8—1582no—
419C4⋊C8—1582no—
420C4⋊C8—1582no—
421#103—1582no—
422#103—1582no—
423#103—1582no—
424#102—1582no—
425#102—1582no—
426#102—1582no—
427#97—1582no—
428#95—21no—
429#95—21no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95C22⋊D7911yescharacteristic
437#95C22⋊D7911yescharacteristic
438C24—21no—
439C2×Q8—792no—
440C2×Q8—1581no—
441C2×Q8—1581no—
442C2×Q8—1581no—
443C2×Q8—1581no—
444C2×Q8—1581no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×D31612yes—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4C2×C22⋊D7911yescharacteristic
462C22×C4C2×C22⋊D7911yescharacteristic
463C22×C4C2×C22⋊D7911yescharacteristic
464C22×C4C2×C22⋊D7911yescharacteristic
465C22×C4C2×C22⋊D7911yescharacteristic
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4C23×D7911yescharacteristic
468SD16—1588no—
469C2×C8—792no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—3162no—
491C4⋊C4—3162no—
492C4⋊C4—3162no—
493C4⋊C4—3162no—
494C4⋊C4—3162no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4C2×D31612yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—3162no—
514C42C2×D31612yes—
515C42#1192962412yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C22×C22⋊D7911yescharacteristic
524Q8—3161no—
525Q8—3161no—
526Q8—1582no—
527Q8—1582no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—3161no—
563C2×C4—3161no—
564C2×C4—3161no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—3161no—
568C2×C4—1582no—
569C2×C4#1457913512yes—
570C2×C4#1457913512yes—
571C2×C4#1457918712yes—
572C2×C4C22×D31611yescharacteristic
573C2×C4#41×D7911yescharacteristic
574C2×C4#41×D7911yescharacteristic
575C2×C4C2×#1192962411yescharacteristic
576C2×C4#1457920811yescharacteristic
577C2×C4#1457920811yescharacteristic
578C8—3161no—
579C8—3161no—
580C8—3161no—
581C8—1582no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C2×#1457913511yescharacteristic
590C22#1909921211yescharacteristic
591C22#1909922011yescharacteristic
592C22C2×#1457918711yescharacteristic
593C22#1909936111yescharacteristic
594C22#1909937411yescharacteristic
595C22#1909937811yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—3162no—
606C4—6321no—
607C4—6321no—
608C4—6321no—
609C4#1909907511yescharacteristic
610C4#1909913911yescharacteristic
611C4#1909925211yescharacteristic
612C4#1909928111yescharacteristic
613C2—82no—
614C2#2376682311yescharacteristic
615C2#2377286411yescharacteristic
616C2#2377295011yescharacteristic
617C2#2377583411yescharacteristic
618C2#2377591511yescharacteristic
619C2#2377612511yescharacteristic
620C2#2377623611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2354C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#2354D79
hypocenterC79#24690
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
3. #23768591C2semidirect
8. #23764368C2semidirect
9. #23764168C2semidirect
10. #23764158C2semidirect
11. #23764124C2semidirect
12. #23764041C2semidirect
13. #23764034C2semidirect
14. #23763402C2semidirect
326. C79#24690semidirect

Maximal subgroups

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

subgroupindex
C2×#190988632
C2×#190986632
#237685912
#237648132
#237648032
#237647172
#237643682
#237641682
#237641582
#237641242
#237640412
#237640342
#237634022
C79×#23542
#2469079