Finite Group Databasebeta

← #34296107

Subgroups — #34296107

Order 20224.

738 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773184C211yesmaximal, characteristic
2#23772333C211yesmaximal, characteristic, complemented
3#131×Dic79C211yesmaximal, characteristic
4#23764342C211yesmaximal, characteristic, complemented
5#23764172C211yesmaximal, characteristic, complemented
6#23764000C211yesmaximal, characteristic, complemented
7#23763996C211yesmaximal, characteristic, complemented
8#23763675C211yesmaximal, characteristic
9#23763418C211yesmaximal, characteristic
10#23763287C211yesmaximal, characteristic, complemented
11#23763283C211yesmaximal, characteristic, complemented
12#23763266C211yesmaximal, characteristic, complemented
13#23763247C211yesmaximal, characteristic
14#23763245C211yesmaximal, characteristic
15C79×#1866C211yesmaximal, characteristic
16C2×#14579168C2211yescharacteristic
17#19099116C2211yescharacteristic
18#41×Dic79C2211yescharacteristic
19#19098872C411yescharacteristic, complemented
20#19098868C411yescharacteristic, complemented
21#19098865C411yescharacteristic, complemented
22#19098861C411yescharacteristic, complemented
23#19098856C411yescharacteristic, complemented
24#19098851C411yescharacteristic, complemented
25#19098846C411yescharacteristic, complemented
26#19098841C411yescharacteristic, complemented
27#19098692—21no—
28#19098692—21no—
29#19098687—21no—
30#19098687—21no—
31C2×#14579096C2211yescharacteristic
32#19098601C2211yescharacteristic
33#19098599C2211yescharacteristic
34#19098589C2211yescharacteristic
35#19098586C2211yescharacteristic
36C2×Dic79⋊C8C2211yescharacteristic
37C2×#14579088C2211yescharacteristic
38#19098559C2211yescharacteristic
39#19098521—21no—
40#19098521C2211yescharacteristic
41#19098521C2211yescharacteristic
42#19098519C2211yescharacteristic
43#19098518—21no—
44#19098518C2211yescharacteristic
45M16×Dic79—21no—
46M16×Dic79C2211yescharacteristic
47C2×#14579072C2211yescharacteristic
48C2×#14579071C2211yescharacteristic
49C2×C8×Dic79—21no—
50C2×C8×Dic79C2211yescharacteristic
51C4×Dic79⋊C4C2211yescharacteristic
52#19098328C2211yescharacteristic
53#19098327C2211yescharacteristic
54#19098324C2211yescharacteristic
55#19098324C2211yescharacteristic
56#19098323C2211yescharacteristic
57#19098323C2211yescharacteristic
58#19098322C2211yescharacteristic
59#19098317C2211yescharacteristic
60C2×#131×C79C2211yescharacteristic
61C2×#108×C79C2211yescharacteristic
62#424×C79C2211yescharacteristic
63C2×#104×C79C2211yescharacteristic
64#416×C79C2211yescharacteristic
65#339×C79C2211yescharacteristic
66#336×C79C2211yescharacteristic
67#14579168—22no—
68#14579168Q814yescomplemented
69C2×#11929608—21no—
70C2×#11929608C2311yescharacteristic
71C2×#11929608C2311yescharacteristic
72C22×C79⋊C8—21no—
73C22×C79⋊C8C2311yescharacteristic
74C22×C79⋊C8C2311yescharacteristic
75#14579160C2×C411yescharacteristic
76#14579160C2×C411yescharacteristic
77#14579152C2×C411yescharacteristic
78#14579152C2×C411yescharacteristic
79#14579138—21no—
80#14579138—21no—
81Q8×Dic79C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158—21no—
85D4×Dic79—21no—
86D4×Dic79—21no—
87D4×Dic79C2×C411yescharacteristic
88D4×Dic79C2×C411yescharacteristic
89#14579120—21no—
90#14579120C2311yescharacteristic
91#14579120C2311yescharacteristic
92C2×Dic79⋊C4—21no—
93C2×C4×Dic79—21no—
94C2×C4×Dic79C2311yescharacteristic
95C2×C4×Dic79C2311yescharacteristic
96#14579096—22no—
97#14579095C2×C411yescharacteristic
98#14579095C2×C411yescharacteristic
99#14579094C2×C411yescharacteristic
100#14579094C2×C411yescharacteristic
101#14579093C2×C411yescharacteristic
102#14579093C2×C411yescharacteristic
103#14579092C2×C411yescharacteristic
104#14579092C2×C411yescharacteristic
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109Dic79⋊C8C2×C411yescharacteristic
110Dic79⋊C8C2×C411yescharacteristic
111Dic79⋊C8C2×C411yescharacteristic
112Dic79⋊C8C2×C411yescharacteristic
113#14579088—22no—
114#14579085C2×C411yescharacteristic
115#14579085C2×C411yescharacteristic
116#14579082C2×C411yescharacteristic
117#14579082C2×C411yescharacteristic
118#14579072—21no—
119#14579072—22no—
120#14579071—21no—
121#14579071—21no—
122#14579071—21no—
123#14579071—22no—
124#14579071—21no—
125C4×C79⋊C8—21no—
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
129C8×Dic79—21no—
130C8×Dic79D411yescharacteristic
131C8×Dic79D411yescharacteristic
132C8×Dic79D411yescharacteristic
133C8×Dic79D411yescharacteristic
134C79⋊#95—21no—
135C2×#41×C79C2311yescharacteristic
136#131×C79—22no—
137#131×C79—22no—
138C2×M16×C79—21no—
139C2×M16×C79C2311yescharacteristic
140C2×M16×C79C2311yescharacteristic
141C22×C8×C79—21no—
142C22×C8×C79C2311yescharacteristic
143C22×C8×C79C2311yescharacteristic
144#117×C79C2311yescharacteristic
145C2×C42×C79C2311yescharacteristic
146#108×C79—22no—
147C4⋊C8×C79—21no—
148C4⋊C8×C79—21no—
149#104×C79—22no—
150#103×C79C2×C411yescharacteristic
151#103×C79C2×C411yescharacteristic
152#102×C79C2×C411yescharacteristic
153#102×C79C2×C411yescharacteristic
154C2×Dic158—21no—
155C2×Dic158—21no—
156C22×Dic79—21no—
157C22×Dic79—21no—
158C22×Dic79—21no—
159C22×Dic79#4111yescharacteristic
160C22×Dic79#4111yescharacteristic
161#11929608—24no—
162#11929608—21no—
163#11929608—21no—
164#11929608—21no—
165#11929608C2×Q812yes—
166C2×C79⋊C8—24no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C22×C411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C2×C79⋊C8C2×Q812yes—
176C2×C79⋊C8#4112yes—
177C4⋊Dic79—21no—
178C4⋊Dic79—21no—
179C4⋊Dic79—21no—
180C4⋊Dic79—21no—
181C4⋊Dic79C22×C411yescharacteristic
182C4⋊Dic79C22×C411yescharacteristic
183C4⋊Dic79C22×C411yescharacteristic
184C4⋊Dic79C22×C411yescharacteristic
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187Dic79⋊C4—21no—
188Dic79⋊C4—21no—
189Dic79⋊C4—42no—
190C22⋊Dic79—21no—
191C22⋊Dic79—21no—
192C22⋊Dic79—21no—
193C22⋊Dic79—21no—
194C4×Dic79—21no—
195C4×Dic79—21no—
196C4×Dic79—21no—
197C4×Dic79—21no—
198C4×Dic79—21no—
199C4×Dic79—21no—
200C4×Dic79C22×C411yescharacteristic
201C4×Dic79C22×C411yescharacteristic
202C4×Dic79C2×D411yescharacteristic
203C4×Dic79C2×D411yescharacteristic
204#41×C79—21no—
205#41×C79—21no—
206#41×C79C2×Q812yes—
207#41×C79#4112yes—
208C2×Q8×C79C22×C411yescharacteristic
209C2×D4×C79—21no—
210C2×D4×C79C22×C411yescharacteristic
211C22×C4×C79—21no—
212C22×C4×C79—21no—
213C22×C4×C79C2411yescharacteristic
214M16×C79—24no—
215M16×C79—21no—
216M16×C79—21no—
217M16×C79—21no—
218M16×C79#4112yes—
219C2×C8×C79—24no—
220C2×C8×C79—21no—
221C2×C8×C79—21no—
222C2×C8×C79—21no—
223C2×C8×C79—21no—
224C2×C8×C79—21no—
225C2×C8×C79—21no—
226C2×C8×C79C22×C411yescharacteristic
227C2×C8×C79C22×C411yescharacteristic
228C2×C8×C79C2×D411yescharacteristic
229C2×C8×C79C2×D411yescharacteristic
230C4⋊C4×C79C22×C411yescharacteristic
231C4⋊C4×C79C22×C411yescharacteristic
232C22⋊C4×C79—21no—
233C42×C79—21no—
234C42×C79—21no—
235C42×C79—21no—
236C23×C79—21no—
237C23×C79C2×#4111yescharacteristic
238Q8×C79—21no—
239Q8×C79C4×Q812yes—
240D4×C79—22no—
241D4×C79—22no—
242D4×C79—21no—
243D4×C79C4×Q812yes—
244C2×C4×C79—21no—
245C2×C4×C79—22no—
246C2×C4×C79—22no—
247C2×C4×C79—21no—
248C2×C4×C79—21no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79C23×C411yescharacteristic
253C2×C4×C79C22×D411yescharacteristic
254C2×C4×C79C22×Q811yescharacteristic
255C2×C4×C79C2×#4111yescharacteristic
256C2×C4×C79C2×#4111yescharacteristic
257C2×C4×C79C2×#4111yescharacteristic
258C2×Dic79—22no—
259C2×Dic79—21no—
260C2×Dic79—21no—
261C2×Dic79—21no—
262C2×Dic79—21no—
263C2×Dic79—41no—
264C2×Dic79—42no—
265C2×Dic79—42no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—41no—
269C2×Dic79C4×D411yescharacteristic
270C2×Dic79C4×D411yescharacteristic
271C2×Dic79C4×D411yescharacteristic
272C2×Dic79C4×D411yescharacteristic
273Dic158—42no—
274Dic158—42no—
275C8×C79—21no—
276C8×C79—21no—
277C8×C79—21no—
278C8×C79—22no—
279C8×C79—21no—
280C8×C79—41no—
281C79⋊C8—21no—
282C79⋊C8—21no—
283C79⋊C8—22no—
284C79⋊C8—41no—
285C79⋊C8C22⋊Q812yes—
286C79⋊C8C22⋊Q812yes—
287C22×C79—21no—
288C22×C79—21no—
289C22×C79—22no—
290C22×C79—41no—
291C22×C79—41no—
292C22×C79C2×C4×D411yescharacteristic
293C22×C79#52911yescharacteristic
294C22×C79#53311yescharacteristic
295C4×C79—22no—
296C4×C79—41no—
297C4×C79—41no—
298C4×C79—81no—
299C4×C79C2×C4×Q811yescharacteristic
300C4×C79C4×#4111yescharacteristic
301C4×C79C2×C22⋊Q811yescharacteristic
302C4×C79#52511yescharacteristic
303Dic79—21no—
304Dic79—21no—
305Dic79—21no—
306Dic79—41no—
307Dic79—41no—
308Dic79—41no—
309Dic79—41no—
310Dic79—81no—
311Dic79#44412yes—
312#21628—791nomaximal, Sylow
313C2×C79—21no—
314C2×C79—21no—
315C2×C79—42no—
316C2×C79C4×C22⋊Q811yescharacteristic
317C2×C79C2×#44411yescharacteristic
318C2×C79#295511yescharacteristic
319#3076—791no—
320#2960—791no—
321C4×#131—791no—
322#1986—791no—
323#1950—791no—
324#1922—791no—
325#1922—791no—
326#1866—791no—
327#1866D7911yescharacteristic
328#1806—791no—
329#1773—791no—
330#1773—791no—
331#1770—791no—
332#1765—791no—
333#1765—791no—
334C79#2162811yescharacteristic, complemented, Sylow
335C2×#131—791no—
336C2×#131C2×D7911yescharacteristic
337#533—791no—
338C4×#41—791no—
339#479—791no—
340#479—791no—
341#478—791no—
342#478—791no—
343#477—791no—
344#476—791no—
345#475—791no—
346#474—791no—
347#446—1581no—
348#446—1581no—
349C8×Q8—1581no—
350C8×Q8—1581no—
351C2×#108—791no—
352C2×#108C2×D7911yescharacteristic
353#427—791no—
354#427—791no—
355#424—791no—
356#424C2×D7911yescharacteristic
357#423—791no—
358C2×C4⋊C8—791no—
359C2×#104—791no—
360C2×#104C2×D7911yescharacteristic
361#416—791no—
362#416C2×D7911yescharacteristic
363#405—791no—
364#405—791no—
365#405—791no—
366#405—791no—
367#405—1581no—
368#405—1581no—
369C4×M16—791no—
370C4×M16—1581no—
371C2×#97—791no—
372C2×#97—791no—
373C2×C4×C8—791no—
374C2×C4×C8—1581no—
375C4×C4⋊C4—791no—
376#341—791no—
377#341—791no—
378#339—791no—
379#339—791no—
380#339—791no—
381#339—791no—
382#339—791no—
383#339C2×D7911yescharacteristic
384#336—791no—
385#336C2×D7911yescharacteristic
386C2×#41C22×D7911yescharacteristic
387#131—22no—
388#131—22no—
389#131—794no—
390#131—1582no—
391C2×M16—21no—
392C2×M16—791no—
393C2×M16—791no—
394C2×M16—1581no—
395C2×M16C22×D7911yescharacteristic
396C2×M16C22×D7911yescharacteristic
397C22×C8—21no—
398C22×C8—791no—
399C22×C8—791no—
400C22×C8—1581no—
401C22×C8C22×D7911yescharacteristic
402C22×C8C22×D7911yescharacteristic
403C4⋊Q8—791no—
404C4⋊Q8—791no—
405#125—791no—
406#125—791no—
407C22⋊Q8—1581no—
408C22⋊Q8—1581no—
409C4×Q8—791no—
410C4×Q8—791no—
411C4×Q8—1581no—
412C4×Q8—1581no—
413C4×D4—791no—
414C4×D4—791no—
415C4×D4—1581no—
416C4×D4—1581no—
417#117—791no—
418#117—791no—
419#117—1581no—
420#117C22×D7911yescharacteristic
421C2×C4⋊C4—1581no—
422C2×C42—791no—
423C2×C42—791no—
424C2×C42—1581no—
425C2×C42C22×D7911yescharacteristic
426#108—22no—
427#108—1582no—
428#107—791no—
429#107—791no—
430#107—791no—
431#107—791no—
432#106—791no—
433#106—791no—
434#106—791no—
435#106—791no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—791no—
439C4⋊C8—791no—
440C4⋊C8—791no—
441C4⋊C8—791no—
442C4⋊C8—1581no—
443C4⋊C8—1581no—
444C4⋊C8—1581no—
445C4⋊C8—1581no—
446#104—22no—
447#104—1582no—
448#103—791no—
449#103—791no—
450#103C4×D7911yescharacteristic
451#103C4×D7911yescharacteristic
452#102—791no—
453#102—791no—
454#102C4×D7911yescharacteristic
455#102C4×D7911yescharacteristic
456#97—1582no—
457#97—1581no—
458#97—1581no—
459#97—1582no—
460#97—1581no—
461#97—1581no—
462#97—1581no—
463C4×C8—791no—
464C4×C8—791no—
465C4×C8—791no—
466C4×C8—791no—
467C4×C8—1581no—
468C4×C8—1581no—
469C4×C8—1581no—
470C4×C8—1581no—
471C4×C8—1581no—
472#95—1581no—
473#41—21no—
474#41—21no—
475#41Q8×D7912yes—
476#41#1192962312yes—
477C2×Q8—1581no—
478C2×Q8—1581no—
479C2×Q8C2×C4×D7911yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D7911yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—791no—
485C22×C4—791no—
486C22×C4—1581no—
487C22×C4—1581no—
488C22×C4—1581no—
489C22×C4C23×D7911yescharacteristic
490M16—24no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—792no—
495M16—1581no—
496M16—1581no—
497M16—1584no—
498M16—1581no—
499M16#1192962312yes—
500C2×C8—24no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—792no—
508C2×C8—792no—
509C2×C8—791no—
510C2×C8—791no—
511C2×C8—791no—
512C2×C8—791no—
513C2×C8—1581no—
514C2×C8—1584no—
515C2×C8—1581no—
516C2×C8—1581no—
517C2×C8—1581no—
518C2×C8C2×C4×D7911yescharacteristic
519C2×C8C2×C4×D7911yescharacteristic
520C2×C8D4×D7911yescharacteristic
521C2×C8D4×D7911yescharacteristic
522C4⋊C4—3162no—
523C4⋊C4—791no—
524C4⋊C4—791no—
525C4⋊C4—791no—
526C4⋊C4—791no—
527C4⋊C4—1581no—
528C4⋊C4—1581no—
529C4⋊C4—1581no—
530C4⋊C4—1581no—
531C4⋊C4—1581no—
532C4⋊C4—1581no—
533C4⋊C4—1581no—
534C4⋊C4—1581no—
535C4⋊C4C2×C4×D7911yescharacteristic
536C4⋊C4C2×C4×D7911yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1581no—
539C22⋊C4—1581no—
540C22⋊C4—1581no—
541C22⋊C4—1581no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—791no—
546C42—791no—
547C42—791no—
548C42—791no—
549C42—1581no—
550C42—1581no—
551C42—1581no—
552C42—1581no—
553C42—1581no—
554C42—1581no—
555C23—21no—
556C23#41×D7911yescharacteristic
557Q8—21no—
558Q8—3162no—
559Q8—3162no—
560Q8#1457912612yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4#1457912612yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—3162no—
574C2×C4—3162no—
575C2×C4—3161no—
576C2×C4—3161no—
577C2×C4—3161no—
578C2×C4—3161no—
579C2×C4—791no—
580C2×C4—791no—
581C2×C4—791no—
582C2×C4—791no—
583C2×C4—1581no—
584C2×C4—1581no—
585C2×C4—1581no—
586C2×C4—1581no—
587C2×C4—1582no—
588C2×C4C22×C4×D7911yescharacteristic
589C2×C4C2×D4×D7911yescharacteristic
590C2×C4C2×Q8×D7911yescharacteristic
591C2×C4#41×D7911yescharacteristic
592C2×C4C2×#1192962311yescharacteristic
593C2×C4C2×#1192962311yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—22no—
598C8—21no—
599C8—41no—
600C8—3161no—
601C8—792no—
602C8—792no—
603C8—1581no—
604C8—1581no—
605C8—1582no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C4×D4×D7911yescharacteristic
612C22#1909909011yescharacteristic
613C22#1909911511yescharacteristic
614C4—22no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—3161no—
619C4—3161no—
620C4—3161no—
621C4—3161no—
622C4—792no—
623C4—6321no—
624C4—1581no—
625C4—1581no—
626C4—1581no—
627C4C2×#1457912611yescharacteristic
628C4C4×#1192962311yescharacteristic
629C4C22⋊Q8×D7911yescharacteristic
630C4#1909906911yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2#2376593211yescharacteristic
635C2#444×D7911yescharacteristic
636C2#2377229811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1866C2
socleC22×C79C2×C4×D4
prime radicalC2×#41×C79C23
cosocleC22×C4×C79C24
hypercenter#1866D79
hypocenterC79#21628
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #23772333C2semidirect
4. #23764342C2semidirect
5. #23764172C2semidirect
6. #23764000C2semidirect
7. #23763996C2semidirect
10. #23763287C2semidirect
11. #23763283C2semidirect
12. #23763266C2semidirect
19. #19098872C4semidirect
20. #19098868C4semidirect
21. #19098865C4semidirect
22. #19098861C4semidirect
23. #19098856C4semidirect
24. #19098851C4semidirect
25. #19098846C4semidirect
26. #19098841C4semidirect
68. #14579168Q8semidirect
334. C79#21628semidirect

Maximal subgroups

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

subgroupindex
#237731842
#237723332
#131×Dic792
#237643422
#237641722
#237640002
#237639962
#237636752
#237634182
#237632872
#237632832
#237632662
#237632472
#237632452
C79×#18662
#2162879