Finite Group Databasebeta

← #34292477

Subgroups — #34292477

Order 20224.

898 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098654C211yesmaximal, characteristic
2#23765871C211yesmaximal, characteristic, complemented
3#23765862C211yesmaximal, characteristic
4#23764255C211yesmaximal, characteristic
5#23764253C211yesmaximal, characteristic
6#23764127C211yesmaximal, characteristic, complemented
7#23764125C211yesmaximal, characteristic, complemented
8#23764085C211yesmaximal, characteristic
9#23764076C211yesmaximal, characteristic, complemented
10#23764074C211yesmaximal, characteristic, complemented
11#23763288C211yesmaximal, characteristic, complemented
12#23763219C211yesmaximal, characteristic, complemented
13#102×Dic79C211yesmaximal, characteristic
14C4×C8×Dic79C211yesmaximal, characteristic, complemented
15C79×#1934C211yesmaximal, characteristic
16C22×#11929612C2211yescharacteristic
17C2×#14579126C2211yescharacteristic
18C2×D4×Dic79C2211yescharacteristic
19C2×#14579117—21no—
20#19098654C418yescomplemented
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579086C2211yescharacteristic
25C2×#14579086C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×C4×C79⋊C8C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30C2×C8×Dic79C2211yescharacteristic
31C2×C8×Dic79C2211yescharacteristic
32C2×C8×Dic79C2211yescharacteristic
33#19098469C2211yescharacteristic
34#19098468C2211yescharacteristic
35#19098466C2211yescharacteristic
36#19098462C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098459C2211yescharacteristic
39#19098457—21no—
40#19098451—21no—
41#19098450—21no—
42#19098444—21no—
43C4⋊C4×Dic79—21no—
44C4⋊C4×Dic79C2211yescharacteristic
45C4⋊C4×Dic79C2211yescharacteristic
46C22⋊C4×Dic79—21no—
47#19098408—21no—
48C42×Dic79C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C2×#102×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×C4×C8×C79C2211yescharacteristic
56#390×C79C2211yescharacteristic
57#389×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158C2311yescharacteristic
61C23×Dic79—21no—
62C2×#11929612—22no—
63C2×#11929612—22no—
64C2×#11929612C2×C414yes—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579126—22no—
68#14579126C2×C414yes—
69D4×Dic79—22no—
70D4×Dic79C2×C414yes—
71#14579117—44no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79D411yescharacteristic
86C2×C4×Dic79D411yescharacteristic
87C2×C4×Dic79D411yescharacteristic
88C2×C4×Dic79D411yescharacteristic
89C2×C4×Dic79C2311yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91#14579092—22no—
92#14579092—22no—
93#14579092C2×C414yes—
94#14579086—22no—
95#14579086C2×C414yes—
96#14579084—22no—
97#14579082—22no—
98C4×C79⋊C8—22no—
99C8×Dic79—22no—
100C8×Dic79—22no—
101C8×Dic79—22no—
102C8×Dic79C2×C414yes—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95—21no—
111C79⋊#95—21no—
112C22×D4×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115C2×C4⋊C4×C79—21no—
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C22⋊C4×C79—21no—
120C2×C22⋊C4×C79—21no—
121C2×C42×C79C2311yescharacteristic
122#102×C79—22no—
123#102×C79C2×C414yes—
124C4×C8×C79—22no—
125#95×C79—21no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158C22×C412yes—
132C22×Dic79—24no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C22×Dic79—42no—
138C22×Dic79—42no—
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C22×Dic79C2×D411yescharacteristic
142C22×Dic79C2×D411yescharacteristic
143#11929612—28no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C22×C412yes—
147C2×C79⋊C8#4112yes—
148C2×C79⋊C8#4112yes—
149C2×C79⋊C8#4112yes—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—42no—
156C4⋊Dic79C22×C412yes—
157Dic79⋊C4—24no—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160C22⋊Dic79—24no—
161C22⋊Dic79—42no—
162C22⋊Dic79—42no—
163C4×Dic79—24no—
164C4×Dic79—22no—
165C4×Dic79—22no—
166C4×Dic79—22no—
167C4×Dic79—21no—
168C4×Dic79—21no—
169C4×Dic79—42no—
170C4×Dic79—42no—
171C4×Dic79SD1614yescomplemented
172C4×Dic79C22×C412yes—
173C4×Dic79#4112yes—
174C24×C79—21no—
175C2×D4×C79—21no—
176C2×D4×C79—21no—
177C2×D4×C79—21no—
178C2×D4×C79—21no—
179C2×D4×C79—21no—
180C2×D4×C79C22×C412yes—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2411yescharacteristic
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79C22×C412yes—
192C2×C8×C79#4112yes—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—42no—
199C4⋊C4×C79C22×C412yes—
200C22⋊C4×C79—42no—
201C22⋊C4×C79—42no—
202C42×C79—21no—
203C42×C79—21no—
204C23×C79—22no—
205C23×C79—41no—
206C23×C79—41no—
207C23×C79—41no—
208C23×C79—41no—
209C23×C79—41no—
210C23×C79—41no—
211C23×C79C22×D411yescharacteristic
212D4×C79—22no—
213D4×C79—22no—
214D4×C79—41no—
215D4×C79—41no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—22no—
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×C79C2×SD1612yes—
230C2×C4×C79#13512yes—
231C2×C4×C79C23×C411yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—48no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79C4×D412yes—
257C2×Dic79C4×D412yes—
258C2×Dic79C2×SD1612yes—
259C2×Dic79C2×SD1612yes—
260C2×Dic79#13512yes—
261C2×Dic79#13512yes—
262Dic158—22no—
263Dic158—22no—
264Dic158—41no—
265Dic158—41no—
266C8×C79—22no—
267C8×C79—22no—
268C79⋊C8—22no—
269C79⋊C8#11714yescomplemented
270C22×C79—42no—
271C22×C79—42no—
272C22×C79—42no—
273C22×C79—81no—
274C22×C79—81no—
275C22×C79—81no—
276C22×C79—81no—
277C22×C79C2×C4×D411yescharacteristic
278C22×C79#53211yescharacteristic
279C22×C79#53211yescharacteristic
280C22×C79C22×SD1611yescharacteristic
281C22×C79C2×#13511yescharacteristic
282C22×C79C2×#13511yescharacteristic
283C22×C79C2×#13511yescharacteristic
284C4×C79—22no—
285C4×C79—22no—
286C4×C79—42no—
287C4×C79—81no—
288C4×C79—81no—
289C4×C79C2×#11711yescharacteristic
290C4×C79C4×#4111yescharacteristic
291C4×C79#52911yescharacteristic
292C4×C79#52911yescharacteristic
293Dic79—22no—
294Dic79—22no—
295Dic79—22no—
296Dic79—42no—
297Dic79—42no—
298Dic79—81no—
299Dic79—81no—
300Dic79C4×SD1614yes—
301#21274—791nomaximal, Sylow
302C2×C79—82no—
303C2×C79#229311yescharacteristic
304C2×C79C2×C4×SD1611yescharacteristic
305C2×C79C4×#13511yescharacteristic
306C2×C79#316711yescharacteristic
307C2×C79#316711yescharacteristic
308C2×C79#316811yescharacteristic
309C2×C79#316811yescharacteristic
310C2×C4×SD16—791no—
311#2295—791no—
312#2293—791no—
313#1968—791no—
314#1968—791no—
315#1941—791no—
316#1941—791no—
317#1934—791no—
318#1934D7911yescharacteristic
319#1933—791no—
320#1933—791no—
321C4×#106—791no—
322C4×#103—791no—
323C4×#102—791no—
324C42×C8—791no—
325C79#2127411yescharacteristic, complemented, Sylow
326C22×SD16—791no—
327C2×C4×Q8—791no—
328C2×C4×D4—791no—
329C2×#117—1581no—
330C4×SD16—798no—
331C2×#106—791no—
332C2×#106—791no—
333C2×#106—791no—
334C2×#103—791no—
335C2×#103—791no—
336C2×#103—791no—
337C2×#102—791no—
338C2×#102C2×D7911yescharacteristic
339C2×#102C2×D7911yescharacteristic
340C2×C4×C8—791no—
341C2×C4×C8—791no—
342C2×C4×C8—791no—
343C2×C4×C8—791no—
344C2×C4×C8—791no—
345C2×C4×C8C2×D7911yescharacteristic
346#391—791no—
347#391—791no—
348#390—791no—
349#390C2×D7911yescharacteristic
350#389—791no—
351#389—791no—
352#389—791no—
353#389C2×D7911yescharacteristic
354#388—1581no—
355#388—1581no—
356#387—1581no—
357#387—1581no—
358C4×C4⋊C4—791no—
359C4×C4⋊C4—791no—
360C4×C4⋊C4—1581no—
361C4×C22⋊C4—1581no—
362#376—1581no—
363C43—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340C2×D7911yescharacteristic
369#340C2×D7911yescharacteristic
370C22×Q8—791no—
371C22×D4C22×D7911yescharacteristic
372C23×C4—1581no—
373C2×SD16—794no—
374C2×SD16—1582no—
375C2×SD16—1582no—
376C22×C8—791no—
377C22×C8—791no—
378C22×C8C22×D7911yescharacteristic
379C22×C8C22×D7911yescharacteristic
380C4×Q8—794no—
381C4×Q8—1582no—
382C4×D4—794no—
383C4×D4—1582no—
384#117—3164no—
385C2×C4⋊C4—21no—
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⋊C4C22×D7911yescharacteristic
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C42—791no—
402C2×C42—791no—
403C2×C42—791no—
404C2×C42—791no—
405C2×C42—791no—
406C2×C42—791no—
407C2×C42—1581no—
408C2×C42—1581no—
409C2×C42—1581no—
410C2×C42C22×D7911yescharacteristic
411#106—794no—
412#106—1582no—
413#106—1582no—
414#103—794no—
415#103—1582no—
416#103—1582no—
417#102—22no—
418#102—1582no—
419#102C4×D7914yes—
420C4×C8—22no—
421C4×C8—794no—
422C4×C8—1582no—
423C4×C8—1582no—
424C4×C8—1582no—
425C4×C8—1582no—
426#95—21no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436C24—21no—
437C2×Q8—792no—
438C2×Q8—1581no—
439C2×Q8—1581no—
440C2×Q8—1581no—
441C2×Q8—1581no—
442C2×Q8—1581no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D7912yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—3162no—
453C22×C4—3162no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1584no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4D4×D7911yescharacteristic
464C22×C4D4×D7911yescharacteristic
465C22×C4C23×D7911yescharacteristic
466SD16—1588no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—792no—
472C2×C8—792no—
473C2×C8—792no—
474C2×C8—792no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8C2×C4×D7912yes—
478C2×C8#1192962312yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—3162no—
486C4⋊C4—3162no—
487C4⋊C4—3162no—
488C4⋊C4—792no—
489C4⋊C4—1584no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4C2×C4×D7912yes—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—3162no—
499C22⋊C4—3162no—
500C22⋊C4—1584no—
501C42—21no—
502C42—21no—
503C42—3162no—
504C42—3162no—
505C42—792no—
506C42—794no—
507C42—792no—
508C42—1582no—
509C42—1582no—
510C42—1584no—
511C42—1581no—
512C42—1581no—
513C42—1582no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D7911yescharacteristic
522Q8—3161no—
523Q8—3161no—
524Q8—1582no—
525Q8—1582no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3168no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—792no—
555C2×C4—792no—
556C2×C4—792no—
557C2×C4—792no—
558C2×C4—792no—
559C2×C4—792no—
560C2×C4—1582no—
561C2×C4—1582no—
562C2×C4—1581no—
563C2×C4—1581no—
564C2×C4—1581no—
565C2×C4—1581no—
566C2×C4—1581no—
567C2×C4—1581no—
568C2×C4SD16×D7912yes—
569C2×C4#1457918012yes—
570C2×C4C22×C4×D7911yescharacteristic
571C2×C4#41×D7911yescharacteristic
572C2×C4#41×D7911yescharacteristic
573C2×C4#41×D7911yescharacteristic
574C2×C4#41×D7911yescharacteristic
575C2×C4C2×#1192962311yescharacteristic
576C8—22no—
577C8—22no—
578C8—794no—
579C8—1582no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D7911yescharacteristic
588C22#1909910811yescharacteristic
589C22#1909911211yescharacteristic
590C22C2×SD16×D7911yescharacteristic
591C22#135×D7911yescharacteristic
592C22#135×D7911yescharacteristic
593C22C2×#1457918011yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—3162no—
600C4—3162no—
601C4—794no—
602C4—6321no—
603C4—6321no—
604C4—1582no—
605C4—1582no—
606C4—1582no—
607C4#117×D7911yescharacteristic
608C4C4×#1192962311yescharacteristic
609C4#1909909011yescharacteristic
610C4#1909909611yescharacteristic
611C2—82no—
612C2#2376586011yescharacteristic
613C2C4×SD16×D7911yescharacteristic
614C2C4×#1457918011yescharacteristic
615C2#2377395411yescharacteristic
616C2#2377395811yescharacteristic
617C2#2377396511yescharacteristic
618C2#2377396911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1934C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1934D79
hypocenterC79#21274
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #23765871C2semidirect
6. #23764127C2semidirect
7. #23764125C2semidirect
9. #23764076C2semidirect
10. #23764074C2semidirect
11. #23763288C2semidirect
12. #23763219C2semidirect
14. C4×C8×Dic79C2semidirect
20. #19098654C4semidirect
171. C4×Dic79SD16semidirect
269. C79⋊C8#117semidirect
325. C79#21274semidirect

Maximal subgroups

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

subgroupindex
C2×#190986542
#237658712
#237658622
#237642552
#237642532
#237641272
#237641252
#237640852
#237640762
#237640742
#237632882
#237632192
#102×Dic792
C4×C8×Dic792
C79×#19342
#2127479