Finite Group Databasebeta

← #34296123

Subgroups — #34296123

Order 20224.

730 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#23773190C211yesmaximal, characteristic
2#23772340C211yesmaximal, characteristic, complemented
3#23771825C211yesmaximal, characteristic
4#23764343C211yesmaximal, characteristic, complemented
5#23764176C211yesmaximal, characteristic, complemented
6#23764000C211yesmaximal, characteristic, complemented
7#23764000C211yesmaximal, characteristic, complemented
8#23763678C211yesmaximal, characteristic
9#23763678C211yesmaximal, characteristic
10#23763421C211yesmaximal, characteristic
11#23763284C211yesmaximal, characteristic, complemented
12#23763279C211yesmaximal, characteristic, complemented
13#23763244C211yesmaximal, characteristic
14#23763237C211yesmaximal, characteristic
15C79×#1771C211yesmaximal, characteristic, complemented
16C2×#14579167C2211yescharacteristic
17C2×#14579167C2211yescharacteristic
18#19099120C2211yescharacteristic
19C4×#11929624C2211yescharacteristic
20#19098871C411yescharacteristic, complemented
21#19098871C411yescharacteristic, complemented
22#19098864C411yescharacteristic, complemented
23#19098864C411yescharacteristic, complemented
24#19098859C411yescharacteristic, complemented
25#19098854C411yescharacteristic, complemented
26#19098849C411yescharacteristic, complemented
27#19098844C411yescharacteristic, complemented
28#19098691—21no—
29#19098691—21no—
30#19098691—21no—
31#19098691—21no—
32C2×#14579097C2211yescharacteristic
33C2×#14579097C2211yescharacteristic
34#19098601C2211yescharacteristic
35#19098601C2211yescharacteristic
36#19098589C2211yescharacteristic
37#19098589C2211yescharacteristic
38#19098587C2211yescharacteristic
39C2×#14579089C2211yescharacteristic
40C2×#14579089C2211yescharacteristic
41#19098560C2211yescharacteristic
42#19098560C2211yescharacteristic
43#19098520—21no—
44#19098520—21no—
45#19098518C2211yescharacteristic
46#19098518C2211yescharacteristic
47#19098518C2211yescharacteristic
48#19098518C2211yescharacteristic
49C4×#11929608C2211yescharacteristic
50#19098515—21no—
51#97×D79—21no—
52#19098328C2211yescharacteristic
53#19098328C2211yescharacteristic
54#19098323C2211yescharacteristic
55#19098323C2211yescharacteristic
56#19098323C2211yescharacteristic
57#19098323C2211yescharacteristic
58#19098323C2211yescharacteristic
59#19098323C2211yescharacteristic
60C2×C4⋊C8×C79C2211yescharacteristic
61C2×#97×C79C2211yescharacteristic
62C2×#97×C79C2211yescharacteristic
63C2×C4×C8×C79C2211yescharacteristic
64C4×C4⋊C4×C79C2211yescharacteristic
65#336×C79C2211yescharacteristic
66#336×C79C2211yescharacteristic
67C2×#11929624C2311yescharacteristic
68#14579167—22no—
69#14579167—22no—
70#14579167—22no—
71#14579167Q814yescomplemented
72C2×#11929608C2311yescharacteristic
73C2×#11929608C2311yescharacteristic
74C2×#11929608C2311yescharacteristic
75C2×#11929608C2311yescharacteristic
76C2×#11929607—21no—
77C2×#11929607—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80#14579161C2×C411yescharacteristic
81#14579161C2×C411yescharacteristic
82#14579153C2×C411yescharacteristic
83#14579153C2×C411yescharacteristic
84#14579140—21no—
85#14579140—21no—
86C4×Dic158—21no—
87C4×Dic158—21no—
88C4×Dic158C2×C411yescharacteristic
89C4×Dic158C2×C411yescharacteristic
90C4×D316C2×C411yescharacteristic
91C4×D316C2×C411yescharacteristic
92C4×C22⋊D79—21no—
93C4×C22⋊D79—21no—
94#14579120C2311yescharacteristic
95#14579120C2311yescharacteristic
96#14579120C2311yescharacteristic
97#14579118—21no—
98C42×D79—21no—
99#14579097—22no—
100#14579097—22no—
101#14579095C2×C411yescharacteristic
102#14579095C2×C411yescharacteristic
103#14579095C2×C411yescharacteristic
104#14579095C2×C411yescharacteristic
105#14579093C2×C411yescharacteristic
106#14579093C2×C411yescharacteristic
107#14579093C2×C411yescharacteristic
108#14579093C2×C411yescharacteristic
109C4⋊(C79⋊C8)—21no—
110C4⋊(C79⋊C8)—21no—
111Dic79⋊C8—21no—
112Dic79⋊C8—21no—
113Dic79⋊C8—21no—
114Dic79⋊C8—21no—
115#14579089—22no—
116#14579089—22no—
117#14579086C2×C411yescharacteristic
118#14579086C2×C411yescharacteristic
119#14579086C2×C411yescharacteristic
120#14579086C2×C411yescharacteristic
121#14579083C2×C411yescharacteristic
122#14579083C2×C411yescharacteristic
123#14579083C2×C411yescharacteristic
124#14579083C2×C411yescharacteristic
125#14579072—21no—
126#14579071—21no—
127#14579071—21no—
128#14579071—21no—
129#14579071—21no—
130C4×C79⋊C8—21no—
131C8×Dic79—21no—
132C8×Dic79—21no—
133C8×Dic79—21no—
134C8×Dic79—21no—
135C22×C8×C79C2311yescharacteristic
136C22×C8×C79C2311yescharacteristic
137C22×C8×C79C2311yescharacteristic
138C22×C8×C79C2311yescharacteristic
139C2×C4⋊C4×C79—21no—
140C2×C42×C79C2311yescharacteristic
141C2×C42×C79C2311yescharacteristic
142C2×C42×C79C2311yescharacteristic
143C4⋊C8×C79C2×C411yescharacteristic
144C4⋊C8×C79C2×C411yescharacteristic
145C4⋊C8×C79C2×C411yescharacteristic
146C4⋊C8×C79C2×C411yescharacteristic
147#97×C79—22no—
148#97×C79D411yescharacteristic
149#97×C79D411yescharacteristic
150#97×C79D411yescharacteristic
151#97×C79D411yescharacteristic
152C4×C8×C79—22no—
153#95×C79—21no—
154#11929624—21no—
155#11929624—21no—
156#11929624C2×Q812yes—
157#11929624#4112yes—
158C2×Dic158—21no—
159C2×Dic158—21no—
160C2×Dic158C22×C411yescharacteristic
161C2×D316C22×C411yescharacteristic
162C2×C22⋊D79—21no—
163C2×C4×D79—21no—
164C2×C4×D79—21no—
165C2×C4×D79—21no—
166#11929608—21no—
167#11929608—21no—
168#11929608—21no—
169#11929608—21no—
170#11929608—21no—
171#11929608—21no—
172#11929608C2×Q812yes—
173#11929608#4112yes—
174#11929607—24no—
175#11929607—24no—
176C2×C79⋊C8—21no—
177C2×C79⋊C8—21no—
178C2×C79⋊C8—21no—
179C2×C79⋊C8—21no—
180C8×D79—24no—
181C8×D79—24no—
182C4⋊Dic79—21no—
183C4⋊Dic79—21no—
184C4⋊Dic79C22×C411yescharacteristic
185C4⋊Dic79C22×C411yescharacteristic
186C4⋊Dic79C22×C411yescharacteristic
187C4⋊Dic79C22×C411yescharacteristic
188C4⋊Dic79C22×C411yescharacteristic
189C4⋊Dic79C22×C411yescharacteristic
190Dic79⋊C4—21no—
191Dic79⋊C4—21no—
192Dic79⋊C4—21no—
193Dic79⋊C4—21no—
194Dic79⋊C4—21no—
195Dic79⋊C4—21no—
196C22⋊Dic79—21no—
197C22⋊Dic79—21no—
198C22⋊Dic79—21no—
199(C2×D79)⋊C4—21no—
200(C2×D79)⋊C4—21no—
201C4×Dic79—21no—
202C4×Dic79—21no—
203C4×Dic79—21no—
204C4×Dic79—21no—
205C4×Dic79—21no—
206C22×C4×C79—21no—
207C22×C4×C79—21no—
208C22×C4×C79#4111yescharacteristic
209C22×C4×C79#4111yescharacteristic
210C22×C4×C79C2411yescharacteristic
211C2×C8×C79—21no—
212C2×C8×C79—21no—
213C2×C8×C79—21no—
214C2×C8×C79—21no—
215C2×C8×C79—21no—
216C2×C8×C79—21no—
217C2×C8×C79C22×C411yescharacteristic
218C2×C8×C79C22×C411yescharacteristic
219C2×C8×C79C22×C411yescharacteristic
220C2×C8×C79C22×C411yescharacteristic
221C2×C8×C79C2×D411yescharacteristic
222C2×C8×C79C2×D411yescharacteristic
223C2×C8×C79C2×D411yescharacteristic
224C2×C8×C79C2×D411yescharacteristic
225C2×C8×C79C2×Q812yes—
226C2×C8×C79#4112yes—
227C4⋊C4×C79—42no—
228C42×C79—21no—
229C42×C79—21no—
230C42×C79—21no—
231C42×C79—21no—
232C42×C79C22×C411yescharacteristic
233C42×C79C22×C411yescharacteristic
234C42×C79C2×D411yescharacteristic
235C42×C79C2×D411yescharacteristic
236C23×C79C2×#4111yescharacteristic
237C22×D79—21no—
238C2×C4×C79—22no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79C4×D411yescharacteristic
247C2×C4×C79C4×D411yescharacteristic
248C2×C4×C79C4×D411yescharacteristic
249C2×C4×C79C4×D411yescharacteristic
250C2×C4×C79C23×C411yescharacteristic
251C2×C4×C79C22×D411yescharacteristic
252C2×C4×C79C22×Q811yescharacteristic
253C2×C4×C79C2×#4111yescharacteristic
254C2×C4×C79C2×#4111yescharacteristic
255C2×C4×C79C2×#4111yescharacteristic
256C22⋊D79—22no—
257C22⋊D79—22no—
258C2×Dic79—21no—
259C2×Dic79—21no—
260C2×Dic79—21no—
261C2×Dic79—21no—
262C2×Dic79—21no—
263C2×Dic79—21no—
264C2×Dic79—21no—
265D316—21no—
266D316C4×Q812yes—
267C4×D79—22no—
268C4×D79—22no—
269C4×D79—81no—
270C4×D79—81no—
271Dic158—21no—
272Dic158—81no—
273Dic158—81no—
274Dic158C4×Q812yes—
275C8×C79—21no—
276C8×C79—21no—
277C8×C79—21no—
278C8×C79—21no—
279C8×C79—21no—
280C8×C79—21no—
281C8×C79C22⋊Q812yes—
282C8×C79C22⋊Q812yes—
283C79⋊C8—22no—
284C79⋊C8—22no—
285C79⋊C8—41no—
286C79⋊C8—41no—
287C22×C79—21no—
288C22×C79—21no—
289C22×C79C2×C4×D411yescharacteristic
290C22×C79#52911yescharacteristic
291C22×C79#53311yescharacteristic
292C2×D79—22no—
293C2×D79—41no—
294C2×D79—41no—
295C4×C79—21no—
296C4×C79—21no—
297C4×C79—21no—
298C4×C79—21no—
299C4×C79—42no—
300C4×C79—81no—
301C4×C79C2×C4×Q811yescharacteristic
302C4×C79C4×#4111yescharacteristic
303C4×C79C2×C22⋊Q811yescharacteristic
304C4×C79#52511yescharacteristic
305Dic79—22no—
306Dic79—41no—
307Dic79—41no—
308Dic79—41no—
309Dic79—41no—
310Dic79—41no—
311Dic79—41no—
312#21629—791nomaximal, Sylow
313C2×C79—21no—
314C2×C79—21no—
315C2×C79C4×C22⋊Q811yescharacteristic
316C2×C79#295611yescharacteristic
317C2×C79#295711yescharacteristic
318D79—42no—
319#3076—791no—
320#2961—791no—
321#2875—791no—
322#1986—791no—
323#1950—791no—
324#1922—791no—
325#1922—791no—
326#1866—791no—
327#1866—791no—
328#1806—791no—
329#1773—791no—
330#1772—791no—
331#1771D7911yescharacteristic, complemented
332#1765—791no—
333#1764—791no—
334C79#2162911yescharacteristic, complemented, Sylow
335C2×#131—791no—
336C2×#131—791no—
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—
349#446—1581no—
350#446—1581no—
351C2×#108—791no—
352C2×#108—791no—
353#427—791no—
354#427—791no—
355#424—791no—
356#424—791no—
357#423—791no—
358C2×C4⋊C8C2×D7911yescharacteristic
359C2×#104—791no—
360C2×#104—791no—
361#416—791no—
362#416—791no—
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—1581no—
372C2×#97C2×D7911yescharacteristic
373C2×#97C2×D7911yescharacteristic
374C2×C4×C8C2×D7911yescharacteristic
375C4×C4⋊C4C2×D7911yescharacteristic
376#341—791no—
377#341—791no—
378#339—791no—
379#339—791no—
380#339—791no—
381#339—791no—
382#339—791no—
383#339—791no—
384#336C2×D7911yescharacteristic
385#336C2×D7911yescharacteristic
386C2×#41—791no—
387#131—794no—
388#131—1582no—
389#131—1582no—
390#131—1582no—
391C2×M16—791no—
392C2×M16—791no—
393C2×M16—791no—
394C2×M16—791no—
395C2×M16—1581no—
396C2×M16—1581no—
397C22×C8—1581no—
398C22×C8—1581no—
399C22×C8C22×D7911yescharacteristic
400C22×C8C22×D7911yescharacteristic
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—791no—
420#117—1581no—
421C2×C4⋊C4—21no—
422C2×C42—1581no—
423C2×C42C22×D7911yescharacteristic
424C2×C42C22×D7911yescharacteristic
425C2×C42C22×D7911yescharacteristic
426#108—1582no—
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—1581no—
437C4⋊C8—1581no—
438C4⋊C8—1581no—
439C4⋊C8—1581no—
440C4⋊C8—1581no—
441C4⋊C8—1581no—
442C4⋊C8C4×D7911yescharacteristic
443C4⋊C8C4×D7911yescharacteristic
444C4⋊C8C4×D7911yescharacteristic
445C4⋊C8C4×D7911yescharacteristic
446#104—1582no—
447#104—1582no—
448#103—791no—
449#103—791no—
450#103—791no—
451#103—791no—
452#102—791no—
453#102—791no—
454#102—791no—
455#102—791no—
456#97—22no—
457#97—1581no—
458#97—1581no—
459#97—1581no—
460#97—1581no—
461#97—1581no—
462#97D31611yescharacteristic
463#97D31611yescharacteristic
464#97D31611yescharacteristic
465#97D31611yescharacteristic
466C4×C8—22no—
467C4×C8—1581no—
468C4×C8—1581no—
469C4×C8—1581no—
470C4×C8—1581no—
471C4×C8—1581no—
472#95—21no—
473#41—792no—
474#41—792no—
475#41—1581no—
476#41—1581no—
477C2×Q8—791no—
478C2×Q8—1581no—
479C2×Q8—1581no—
480C2×D4—791no—
481C2×D4—1581no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1581no—
485C22×C4—1581no—
486C22×C4—1581no—
487C22×C4#1192962411yescharacteristic
488C22×C4#1192962411yescharacteristic
489C22×C4C23×D7911yescharacteristic
490M16—792no—
491M16—792no—
492M16—1584no—
493M16—1581no—
494M16—1581no—
495M16—1581no—
496M16—1581no—
497M16—1581no—
498M16—1581no—
499M16—1584no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1584no—
507C2×C8—1581no—
508C2×C8—1581no—
509C2×C8—1581no—
510C2×C8—1581no—
511C2×C8—1584no—
512C2×C8C2×C4×D7911yescharacteristic
513C2×C8C2×C4×D7911yescharacteristic
514C2×C8C2×C4×D7911yescharacteristic
515C2×C8C2×C4×D7911yescharacteristic
516C2×C8C2×D31611yescharacteristic
517C2×C8C2×D31611yescharacteristic
518C2×C8C2×D31611yescharacteristic
519C2×C8C2×D31611yescharacteristic
520C2×C8Q8×D7912yes—
521C2×C8#1192962312yes—
522C4⋊C4—42no—
523C4⋊C4—791no—
524C4⋊C4—791no—
525C4⋊C4—791no—
526C4⋊C4—791no—
527C4⋊C4—791no—
528C4⋊C4—791no—
529C4⋊C4—1581no—
530C4⋊C4—1581no—
531C4⋊C4—1581no—
532C4⋊C4—1581no—
533C4⋊C4—1581no—
534C4⋊C4—1581no—
535C4⋊C4—1581no—
536C4⋊C4—1581no—
537C22⋊C4—1581no—
538C22⋊C4—1581no—
539C22⋊C4—1581no—
540C22⋊C4—1581no—
541C22⋊C4—1581no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—1581no—
547C42—1581no—
548C42—1581no—
549C42—1581no—
550C42—1581no—
551C42C2×C4×D7911yescharacteristic
552C42C2×C4×D7911yescharacteristic
553C42C2×D31611yescharacteristic
554C42C2×D31611yescharacteristic
555C23—1581no—
556C23C2×#1192962411yescharacteristic
557Q8—792no—
558Q8—6321no—
559Q8—6321no—
560Q8—1581no—
561D4—792no—
562D4—1582no—
563D4—1582no—
564D4—1581no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—6321no—
574C2×C4—6321no—
575C2×C4—1581no—
576C2×C4—1581no—
577C2×C4—1581no—
578C2×C4—1581no—
579C2×C4—1581no—
580C2×C4—1581no—
581C2×C4—1581no—
582C2×C4—1582no—
583C2×C4—1582no—
584C2×C4C4×D31611yescharacteristic
585C2×C4C4×D31611yescharacteristic
586C2×C4C4×D31611yescharacteristic
587C2×C4C4×D31611yescharacteristic
588C2×C4C22×C4×D7911yescharacteristic
589C2×C4C22×D31611yescharacteristic
590C2×C4C2×Q8×D7911yescharacteristic
591C2×C4#41×D7911yescharacteristic
592C2×C4#41×D7911yescharacteristic
593C2×C4C2×#1192962311yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—3161no—
601C8—3161no—
602C8—1582no—
603C8—1582no—
604C8#1457914112yes—
605C8#1457914112yes—
606C22—21no—
607C22—21no—
608C22—3161no—
609C22—3161no—
610C22—1582no—
611C22C2×C4×D31611yescharacteristic
612C22#1909909511yescharacteristic
613C22#1909911711yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—81no—
620C4—3161no—
621C4—3161no—
622C4—3161no—
623C4—3161no—
624C4—3161no—
625C4—3161no—
626C4—1582no—
627C4C4×Q8×D7911yescharacteristic
628C4C4×#1192962311yescharacteristic
629C4C2×#1457914111yescharacteristic
630C4#1909907511yescharacteristic
631C2—21no—
632C2—21no—
633C2—3162no—
634C2C4×#1457914111yescharacteristic
635C2#2377231111yescharacteristic
636C2#2377231811yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1771C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929624C23
cosocleC22×C4×C79C24
hypercenter#1771D79
hypocenterC79#21629
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #23772340C2semidirect
4. #23764343C2semidirect
5. #23764176C2semidirect
6. #23764000C2semidirect
7. #23764000C2semidirect
11. #23763284C2semidirect
12. #23763279C2semidirect
15. C79×#1771C2semidirect
20. #19098871C4semidirect
21. #19098871C4semidirect
22. #19098864C4semidirect
23. #19098864C4semidirect
24. #19098859C4semidirect
25. #19098854C4semidirect
26. #19098849C4semidirect
27. #19098844C4semidirect
71. #14579167Q8semidirect
331. #1771D79semidirect
334. C79#21629semidirect

Maximal subgroups

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

subgroupindex
#237731902
#237723402
#237718252
#237643432
#237641762
#237640002
#237636782
#237634212
#237632842
#237632792
#237632442
#237632372
C79×#17712
#2162979