Finite Group Databasebeta

← #34296045

Subgroups — #34296045

Order 20224.

876 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×#19098852C211yesmaximal, characteristic
2#23766144C211yesmaximal, characteristic, complemented
3#23765760C211yesmaximal, characteristic
4#23764350C211yesmaximal, characteristic, complemented
5#23764169C211yesmaximal, characteristic, complemented
6#23763987C211yesmaximal, characteristic, complemented
7#23763976C211yesmaximal, characteristic, complemented
8#23763658C211yesmaximal, characteristic
9#23763648C211yesmaximal, characteristic
10#23763409C211yesmaximal, characteristic
11#23763298C211yesmaximal, characteristic, complemented
12#23763258C211yesmaximal, characteristic, complemented
13#23763228C211yesmaximal, characteristic
14C4×#14579083C211yesmaximal, characteristic
15C4×#106×C79C211yesmaximal, characteristic, complemented
16C2×#14579160C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×#14579123C2211yescharacteristic
19C2×#14579123C2211yescharacteristic
20#19098852C418yescomplemented
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×Dic79⋊C8C2211yescharacteristic
25C2×#14579083C2211yescharacteristic
26C2×#14579083C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30#19098469—21no—
31#19098467—21no—
32#19098462C2211yescharacteristic
33#19098459C2211yescharacteristic
34#19098450—21no—
35#19098447—21no—
36#19098428C2211yescharacteristic
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40C4×(C2×D79)⋊C4—21no—
41C4×(C2×D79)⋊C4—21no—
42#19098410—21no—
43#19098409C2211yescharacteristic
44#95×D79—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#106×C79C2211yescharacteristic
54C2×#106×C79C2211yescharacteristic
55C2×C4×C8×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158—21no—
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C22×C4×D79—21no—
64C22×C4×D79—21no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579160—22no—
68#14579160C2×C414yescomplemented
69C4×D316—22no—
70C4×D316Q814yescomplemented
71#14579123—22no—
72#14579123—22no—
73#14579123—22no—
74#14579123C2×C414yes—
75C2×C4⋊Dic79—21no—
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×(C2×D79)⋊C4—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79C2311yescharacteristic
88C2×C4×Dic79C2311yescharacteristic
89#14579094—22no—
90#14579092C2×C414yescomplemented
91Dic79⋊C8—22no—
92Dic79⋊C8C2×C414yescomplemented
93#14579083—22no—
94#14579083C2×C414yes—
95#14579081—22no—
96#14579081C2×C414yes—
97#14579072—22no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C79⋊#95D411yescharacteristic
108C79⋊#95D411yescharacteristic
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79—21no—
117C2×C42×C79C2311yescharacteristic
118#106×C79—22no—
119#106×C79C2×C414yescomplemented
120C4×C8×C79—22no—
121#95×C79—21no—
122#95×C79—21no—
123C23×D79—21no—
124C2×Dic158—24no—
125C2×Dic158—81no—
126C2×Dic158—81no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316C22×C412yes—
131C2×D316C2×Q812yes—
132C2×D316#4112yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C2×C4×D79—24no—
141C2×C4×D79—24no—
142C2×C4×D79—24no—
143C2×C4×D79—81no—
144C2×C4×D79—81no—
145C2×C4×D79—81no—
146C2×C4×D79—81no—
147C2×C4×D79—81no—
148C2×C4×D79—81no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8C22×C412yes—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—81no—
160C4⋊Dic79C22×C412yes—
161C4⋊Dic79C2×Q812yes—
162Dic79⋊C4—24no—
163Dic79⋊C4—81no—
164(C2×D79)⋊C4—24no—
165(C2×D79)⋊C4—24no—
166(C2×D79)⋊C4—24no—
167C4×Dic79—21no—
168C4×Dic79—21no—
169C4×Dic79—81no—
170C4×Dic79—81no—
171C4×Dic79—81no—
172C4×Dic79C22×C412yes—
173C4×Dic79#4112yes—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2411yescharacteristic
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79C22×C412yes—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79C22×C412yes—
193C4⋊C4×C79C22×C412yes—
194C4⋊C4×C79#4112yes—
195C42×C79—81no—
196C42×C79—81no—
197C42×C79C2×Q812yes—
198C42×C79#4112yes—
199C23×C79C22×D411yescharacteristic
200C22×D79—22no—
201C22×D79—22no—
202C22×D79—22no—
203C22×D79—41no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
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×C79C22⋊Q812yes—
224C2×C4×C79C22⋊Q812yes—
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×Q811yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—22no—
236C2×Dic79—22no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79C4×D412yes—
251C2×Dic79C4×D412yes—
252D316—22no—
253D316—22no—
254D316—22no—
255D316C4×Q814yes—
256C4×D79—84no—
257C4×D79—84no—
258C4×D79—84no—
259Dic158—84no—
260C8×C79—22no—
261C8×C79—41no—
262C79⋊C8—22no—
263C79⋊C8—41no—
264C22×C79C2×C4×D411yescharacteristic
265C22×C79C2×C22⋊Q811yescharacteristic
266C22×C79#52511yescharacteristic
267C22×C79#57611yescharacteristic
268C22×C79#57711yescharacteristic
269C22×C79#57711yescharacteristic
270C22×C79#57711yescharacteristic
271C2×D79—24no—
272C2×D79—42no—
273C2×D79—42no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C4×C79—22no—
279C4×C79—22no—
280C4×C79—42no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79—81no—
284C4×C79C2×C4×Q811yescharacteristic
285C4×C79C4×#4111yescharacteristic
286C4×C79#52911yescharacteristic
287C4×C79#53311yescharacteristic
288Dic79—22no—
289Dic79—22no—
290Dic79—42no—
291Dic79—41no—
292Dic79—41no—
293Dic79—42no—
294Dic79—81no—
295#21623—791nomaximal, Sylow
296C2×C79C4×C22⋊Q811yescharacteristic
297C2×C79#294611yescharacteristic
298C2×C79#294911yescharacteristic
299C2×C79#308111yescharacteristic
300C2×C79#308411yescharacteristic
301C2×C79#311811yescharacteristic
302C2×C79#312511yescharacteristic
303D79—44no—
304C2×#476—791no—
305#2331—791no—
306#2275—791no—
307#1988—791no—
308#1949—791no—
309#1920—791no—
310#1919—791no—
311#1864—791no—
312#1862—791no—
313#1804—791no—
314#1776—791no—
315C4×#106D7911yescharacteristic, complemented
316#1768—791no—
317#1762—791no—
318C4×#102—791no—
319C79#2162311yescharacteristic, complemented, Sylow
320C2×C4⋊Q8—791no—
321C2×C4×D4—791no—
322C2×C4×D4—791no—
323C2×C4×D4—791no—
324#476—798no—
325C2×#107—791no—
326C2×#106—791no—
327C2×#106C2×D7911yescharacteristic
328C2×#106C2×D7911yescharacteristic
329C2×C4⋊C8—791no—
330C2×C4⋊C8—791no—
331C2×#102—791no—
332C2×#102—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#97—791no—
336C2×C4×C8C2×D7911yescharacteristic
337#391—1581no—
338#391—1581no—
339#389—791no—
340#389—791no—
341#387—1581no—
342#387—1581no—
343#382—791no—
344C4×C4⋊C4—791no—
345C4×C4⋊C4—791no—
346C4×C4⋊C4—791no—
347C4×C4⋊C4C2×D7911yescharacteristic
348C4×C4⋊C4C2×D7911yescharacteristic
349C4×C22⋊C4—1581no—
350C4×C22⋊C4—1581no—
351C4×C22⋊C4—1581no—
352#376—791no—
353C2×#95—1581no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340C2×D7911yescharacteristic
361#340C2×D7911yescharacteristic
362#336—791no—
363#336—791no—
364C22×Q8—1581no—
365C22×D4—791no—
366C23×C4—1581no—
367C23×C4—1581no—
368C23×C4—1581no—
369C22×C8—791no—
370C22×C8—791no—
371C22×C8C22×D7911yescharacteristic
372C22×C8C22×D7911yescharacteristic
373C4⋊Q8—794no—
374C4⋊Q8—1582no—
375C4×D4—794no—
376C4×D4—794no—
377C4×D4—1582no—
378C4×D4—1582no—
379C4×D4—1582no—
380C4×D4—1582no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—1581no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4C22×D7911yescharacteristic
388C2×C4⋊C4C22×D7911yescharacteristic
389C2×C4⋊C4C22×D7911yescharacteristic
390C2×C4⋊C4C22×D7911yescharacteristic
391C2×C22⋊C4—1581no—
392C2×C22⋊C4—1581no—
393C2×C22⋊C4—1581no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—791no—
397C2×C42—791no—
398C2×C42—1581no—
399C2×C42—1581no—
400C2×C42—1581no—
401C2×C42C22×D7911yescharacteristic
402#107—1582no—
403#106—22no—
404#106—794no—
405#106C4×D7914yescomplemented
406C4⋊C8—794no—
407C4⋊C8—1582no—
408#102—794no—
409#102—794no—
410#102—1582no—
411#102—1582no—
412#97—1582no—
413C4×C8—22no—
414#95—21no—
415#95—21no—
416#95—791no—
417#95—791no—
418#95—791no—
419#95—791no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427C24—1581no—
428C2×Q8—6321no—
429C2×Q8—6321no—
430C2×Q8—1584no—
431C2×D4—792no—
432C2×D4—792no—
433C2×D4—792no—
434C2×D4—1581no—
435C2×D4—1581no—
436C2×D4—1581no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—6321no—
446C22×C4—6321no—
447C22×C4—6321no—
448C22×C4—6321no—
449C22×C4—6321no—
450C22×C4—6321no—
451C22×C4—1584no—
452C22×C4—1584no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4—1584no—
457C22×C4D4×D7911yescharacteristic
458C22×C4D4×D7911yescharacteristic
459C22×C4C23×D7911yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—792no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8C2×C4×D7912yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—792no—
478C4⋊C4—792no—
479C4⋊C4—6321no—
480C4⋊C4—6321no—
481C4⋊C4—1584no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4C2×C4×D7912yes—
487C4⋊C4C2×C4×D7912yes—
488C4⋊C4#1192962412yes—
489C22⋊C4—1584no—
490C22⋊C4—1584no—
491C22⋊C4—1584no—
492C42—81no—
493C42—81no—
494C42—792no—
495C42—792no—
496C42—6321no—
497C42—6321no—
498C42—6321no—
499C42—1581no—
500C42—1581no—
501C42Q8×D7912yes—
502C42#1192962412yes—
503C23—3161no—
504C23—3161no—
505C23—3161no—
506C23—3161no—
507C23—1582no—
508C23—1582no—
509C23—1582no—
510C23C2×D4×D7911yescharacteristic
511Q8—6324no—
512D4—794no—
513D4—1582no—
514D4—1582no—
515D4—1582no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
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—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—792no—
540C2×C4—792no—
541C2×C4—6324no—
542C2×C4—6324no—
543C2×C4—6324no—
544C2×C4—1582no—
545C2×C4—1582no—
546C2×C4—1581no—
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#1457913912yes—
557C2×C4#1457913912yes—
558C2×C4C22×C4×D7911yescharacteristic
559C2×C4C2×Q8×D7911yescharacteristic
560C2×C4#41×D7911yescharacteristic
561C2×C4#41×D7911yescharacteristic
562C2×C4C2×#1192962411yescharacteristic
563C2×C4C2×#1192962411yescharacteristic
564C8—22no—
565C8—41no—
566C8—3161no—
567C8—1582no—
568C22—3162no—
569C22—3162no—
570C22—3162no—
571C22—3162no—
572C22—3162no—
573C22—3162no—
574C22—1584no—
575C22C4×D4×D7911yescharacteristic
576C22C2×#1457913911yescharacteristic
577C22#1909907211yescharacteristic
578C22#1909936511yescharacteristic
579C22#1909936911yescharacteristic
580C22#1909936911yescharacteristic
581C22#1909937011yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—3162no—
589C4—3162no—
590C4—3161no—
591C4—3161no—
592C4—6321no—
593C4—1582no—
594C4—1582no—
595C4C4×Q8×D7911yescharacteristic
596C4C4×#1192962411yescharacteristic
597C4#1909909111yescharacteristic
598C4#1909911711yescharacteristic
599C2—3164no—
600C2C4×#1457913911yescharacteristic
601C2#2377222711yescharacteristic
602C2#2377225411yescharacteristic
603C2#2377323411yescharacteristic
604C2#2377326711yescharacteristic
605C2#2377359011yescharacteristic
606C2#2377368311yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC4×#106×C79C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenterC4×#106D79
hypocenterC79#21623
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #23766144C2semidirect
4. #23764350C2semidirect
5. #23764169C2semidirect
6. #23763987C2semidirect
7. #23763976C2semidirect
11. #23763298C2semidirect
12. #23763258C2semidirect
15. C4×#106×C79C2semidirect
20. #19098852C4semidirect
68. #14579160C2×C4semidirect
70. C4×D316Q8semidirect
90. #14579092C2×C4semidirect
92. Dic79⋊C8C2×C4semidirect
119. #106×C79C2×C4semidirect
315. C4×#106D79semidirect
319. C79#21623semidirect
405. #106C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988522
#237661442
#237657602
#237643502
#237641692
#237639872
#237639762
#237636582
#237636482
#237634092
#237632982
#237632582
#237632282
C4×#145790832
C4×#106×C792
#2162379