Finite Group Databasebeta

← #34347081

Subgroups — #34347081

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×#19098965C211yesmaximal, characteristic, complemented
2C2×#19098605C211yesmaximal, characteristic
3#23767124C211yesmaximal, characteristic
4#23767113C211yesmaximal, characteristic
5#23764883C211yesmaximal, characteristic, complemented
6#23764878C211yesmaximal, characteristic, complemented
7#23764739C211yesmaximal, characteristic, complemented
8#23764739C211yesmaximal, characteristic, complemented
9#23764138C211yesmaximal, characteristic, complemented
10#23764128C211yesmaximal, characteristic, complemented
11#23763601C211yesmaximal, characteristic
12#23763595C211yesmaximal, characteristic
13#23763568C211yesmaximal, characteristic, complemented
14#23763433C211yesmaximal, characteristic
15C79×#1753C211yesmaximal, characteristic
16C2×#14579153C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19C22×C4⋊Dic79C2211yescharacteristic
20#19098965—24no—
21#19098605—24no—
22C2×#14579095C2211yescharacteristic
23C2×#14579095C2211yescharacteristic
24C2×#14579095C2211yescharacteristic
25C2×#14579095C2211yescharacteristic
26C2×#14579093C2211yescharacteristic
27C2×#14579093C2211yescharacteristic
28C2×#14579093C2211yescharacteristic
29C2×#14579093C2211yescharacteristic
30C2×#14579093C2211yescharacteristic
31C2×#14579093C2211yescharacteristic
32#19098505—21no—
33#19098499—21no—
34#19098497—21no—
35#19098486—21no—
36#19098461C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098460—21no—
41#19098446—21no—
42#19098435—21no—
43#19098433C2211yescharacteristic
44#19098429C2211yescharacteristic
45#19098423C2211yescharacteristic
46#19098423C2211yescharacteristic
47#19098421—21no—
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52#19098326C2211yescharacteristic
53#19098326C2211yescharacteristic
54C22×M16×C79C2211yescharacteristic
55C2×C22⋊C8×C79C2211yescharacteristic
56C2×#97×C79C2211yescharacteristic
57C2×C4×C8×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60#336×C79C2211yescharacteristic
61C23×Dic79—21no—
62#14579153—22no—
63#14579153—22no—
64#14579153—41no—
65#14579153—41no—
66#14579120—22no—
67#14579120—41no—
68C2×C4⋊Dic79—22no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79—41no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×C4⋊Dic79C2311yescharacteristic
81C2×Dic79⋊C4—21no—
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×Dic79⋊C4—21no—
89C2×C22⋊Dic79—21no—
90C2×C22⋊Dic79—21no—
91C2×C22⋊Dic79—21no—
92C2×C4×Dic79—21no—
93#14579095—22no—
94#14579095—22no—
95#14579095—41no—
96#14579095D414yes—
97#14579093—22no—
98#14579093—22no—
99#14579093—22no—
100#14579093—22no—
101#14579093—22no—
102#14579093—41no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95—21no—
111C23×C4×C79C2311yescharacteristic
112C2×M16×C79—22no—
113C2×M16×C79—22no—
114C2×M16×C79Q814yes—
115C22×C8×C79C2311yescharacteristic
116C22×C8×C79C2311yescharacteristic
117C22×C8×C79C2311yescharacteristic
118C22×C8×C79C2311yescharacteristic
119C2×C22⋊C4×C79D411yescharacteristic
120C2×C22⋊C4×C79D411yescharacteristic
121C2×C42×C79C2311yescharacteristic
122C2×C42×C79C2311yescharacteristic
123C22⋊C8×C79Q814yes—
124#97×C79—22no—
125C4×C8×C79—22no—
126#95×C79D411yescharacteristic
127#95×C79D411yescharacteristic
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—42no—
137C22×Dic79—42no—
138C22×Dic79—81no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
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—
157C4⋊Dic79C2×D412yes—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161Dic79⋊C4—42no—
162Dic79⋊C4—42no—
163Dic79⋊C4—81no—
164Dic79⋊C4—81no—
165Dic79⋊C4—81no—
166C22⋊Dic79—42no—
167C22⋊Dic79—42no—
168C22⋊Dic79—81no—
169C4×Dic79—42no—
170C24×C79C2×D411yescharacteristic
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×Q812yes—
181C22×C4×C79C2×Q812yes—
182C22×C4×C79C2411yescharacteristic
183M16×C79—28no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79C2×D412yes—
191C2×C8×C79C2×Q812yes—
192C2×C8×C79C2×Q812yes—
193C2×C8×C79C2×Q812yes—
194C2×C8×C79C2×Q812yes—
195C2×C8×C79#4112yes—
196C22⋊C4×C79—41no—
197C22⋊C4×C79—41no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
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—21no—
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—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
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—22no—
236C2×Dic79—22no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
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—
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—
262C8×C79—22no—
263C8×C79C22⋊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—41no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79C2×C22⋊Q811yescharacteristic
284C4×C79#54411yescharacteristic
285C4×C79D4×Q811yescharacteristic
286C4×C79#55411yescharacteristic
287Dic79—42no—
288Dic79—42no—
289Dic79—81no—
290Dic79—81no—
291Dic79—81no—
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#1854—791no—
315#1853—791no—
316#1847—791no—
317#1809—791no—
318#1753D7911yescharacteristic
319C79#2607511yescharacteristic, complemented, Sylow
320C22×M16C2×D7911yescharacteristic
321C2×#125—791no—
322C2×#125—791no—
323C2×#117—791no—
324C22×C4⋊C4—791no—
325#499—1584no—
326#428—1584no—
327C2×#107—791no—
328C2×#107—791no—
329C2×#107—791no—
330C2×#107—791no—
331C2×#106—791no—
332C2×#106—791no—
333C2×#106—791no—
334C2×#106—791no—
335C2×#106—791no—
336C2×#106—791no—
337C2×C22⋊C8C2×D7911yescharacteristic
338C2×#97C2×D7911yescharacteristic
339C2×C4×C8C2×D7911yescharacteristic
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—791no—
355#380—1581no—
356C4×C22⋊C4C2×D7911yescharacteristic
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#336C2×D7911yescharacteristic
364#336C2×D7911yescharacteristic
365C23×C4—1581no—
366C23×C4C22×D7911yescharacteristic
367C2×M16—22no—
368C2×M16—22no—
369C2×M16Dic15814yes—
370C22×C8C22×D7911yescharacteristic
371C22×C8C22×D7911yescharacteristic
372C22×C8C22×D7911yescharacteristic
373C22×C8C22×D7911yescharacteristic
374#125—3161no—
375#125—3161no—
376#125—1582no—
377#125—1582no—
378#117—3161no—
379#117—1582no—
380C2×C4⋊C4—3161no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
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⋊C4—1582no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C22⋊C4—1581no—
404C2×C22⋊C4D31611yescharacteristic
405C2×C22⋊C4D31611yescharacteristic
406C2×C42—1581no—
407C2×C42C22×D7911yescharacteristic
408C2×C42C22×D7911yescharacteristic
409#107—3161no—
410#107—794no—
411#107—1582no—
412#107—1582no—
413#106—3161no—
414#106—1582no—
415#106—1582no—
416#106—1582no—
417#106—1582no—
418#106—1582no—
419C22⋊C8Dic15814yes—
420#97—22no—
421C4×C8—22no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95D31611yescharacteristic
431#95D31611yescharacteristic
432C24C2×D31611yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—3162no—
438C22×C4—3162no—
439C22×C4—6321no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4C2×D31611yescharacteristic
449C22×C4C2×D31611yescharacteristic
450C22×C4C2×D31611yescharacteristic
451C22×C4C2×D31611yescharacteristic
452C22×C4C2×D31611yescharacteristic
453C22×C4C2×Dic15812yes—
454C22×C4C2×Dic15812yes—
455C22×C4C23×D7911yescharacteristic
456M16—28no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8D4×D7912yes—
464C2×C8C2×Dic15812yes—
465C2×C8C2×Dic15812yes—
466C2×C8C2×Dic15812yes—
467C2×C8C2×Dic15812yes—
468C2×C8#1192962312yes—
469C4⋊C4—3162no—
470C4⋊C4—3162no—
471C4⋊C4—3162no—
472C4⋊C4—3162no—
473C4⋊C4—3162no—
474C4⋊C4—3162no—
475C4⋊C4—3162no—
476C4⋊C4—792no—
477C4⋊C4—792no—
478C4⋊C4—6321no—
479C4⋊C4—6321no—
480C4⋊C4—6321no—
481C4⋊C4—6321no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—3162no—
499C22⋊C4—3162no—
500C22⋊C4—6321no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—3162no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1457916112yes—
513C23C22×D31611yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
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—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—6324no—
555C2×C4—1582no—
556C2×C4—1582no—
557C2×C4#1457916112yes—
558C2×C4C2×D4×D7911yescharacteristic
559C2×C4C22×Dic15811yescharacteristic
560C2×C4C22×Dic15811yescharacteristic
561C2×C4C2×#1192962311yescharacteristic
562C2×C4#1457920711yescharacteristic
563C2×C4#1457920811yescharacteristic
564C8—22no—
565C8—22no—
566C8—22no—
567C8#1457913814yes—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1457916111yescharacteristic
576C22D4×D31611yescharacteristic
577C22#1909921711yescharacteristic
578C22#1909936311yescharacteristic
579C22#1909936311yescharacteristic
580C22#1909937511yescharacteristic
581C22#1909937511yescharacteristic
582C4—22no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—3162no—
588C4—3162no—
589C4—6321no—
590C4—6321no—
591C4—6321no—
592C4—6321no—
593C4—6321no—
594C4—6321no—
595C4C2×#1457913811yescharacteristic
596C4#1909919111yescharacteristic
597C4D4×Dic15811yescharacteristic
598C4#1909927011yescharacteristic
599C2—42no—
600C2#2376710211yescharacteristic
601C2#2377393111yescharacteristic
602C2#2377395211yescharacteristic
603C2#2377493011yescharacteristic
604C2#2377499011yescharacteristic
605C2#2377596211yescharacteristic
606C2#2377596211yescharacteristic
subgroupG/H
centerC23C22×D316
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1753C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1753D79
hypocenterC79#26075
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#19098965C2semidirect
5. #23764883C2semidirect
6. #23764878C2semidirect
7. #23764739C2semidirect
8. #23764739C2semidirect
9. #23764138C2semidirect
10. #23764128C2semidirect
13. #23763568C2semidirect
319. C79#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#190989652
C2×#190986052
#237671242
#237671132
#237648832
#237648782
#237647392
#237641382
#237641282
#237636012
#237635952
#237635682
#237634332
C79×#17532
#2607579