Finite Group Databasebeta

← #34291271

Subgroups — #34291271

Order 20224.

934 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic79C211yesmaximal, characteristic
2#23765862C211yesmaximal, characteristic
3C4×Q8×Dic79C211yesmaximal, characteristic, complemented
4#23764023C211yesmaximal, characteristic
5#23764017C211yesmaximal, characteristic
6#23763986C211yesmaximal, characteristic, complemented
7#23763984C211yesmaximal, characteristic, complemented
8#23763757C211yesmaximal, characteristic
9#23763654C211yesmaximal, characteristic, complemented
10#23763652C211yesmaximal, characteristic, complemented
11#23763292C211yesmaximal, characteristic, complemented
12#103×Dic79C211yesmaximal, characteristic, complemented
13#102×Dic79C211yesmaximal, characteristic
14#23763144C211yesmaximal, characteristic, complemented
15C79×#1880C211yesmaximal, characteristic
16C2×Q8×Dic79C2211yescharacteristic
17C2×Q8×Dic79C2211yescharacteristic
18C2×D4×Dic79C2211yescharacteristic
19C2×#14579117—21no—
20SD16×Dic79C418yescomplemented
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098466C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098459C2211yescharacteristic
33#19098457—21no—
34#19098451—21no—
35C4⋊C4×Dic79—21no—
36C4⋊C4×Dic79—21no—
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4×C4⋊Dic79—21no—
41C4×C4⋊Dic79C2211yescharacteristic
42C22⋊C4×Dic79—21no—
43C42×Dic79—21no—
44C42×Dic79C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098318C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C22×SD16×C79C2211yescharacteristic
54C2×C4×Q8×C79C2211yescharacteristic
55C2×#103×C79C2211yescharacteristic
56C2×#102×C79C2211yescharacteristic
57#390×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C23×Dic79—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63Q8×Dic79—22no—
64Q8×Dic79—22no—
65Q8×Dic79—22no—
66Q8×Dic79C2×C414yes—
67D4×Dic79—22no—
68D4×Dic79C2×C414yes—
69#14579117—44no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×C22⋊Dic79—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79D411yescharacteristic
88C2×C4×Dic79D411yescharacteristic
89C2×C4×Dic79D411yescharacteristic
90C2×C4×Dic79D411yescharacteristic
91C2×C4×Dic79C2311yescharacteristic
92C2×C4×Dic79C2311yescharacteristic
93#14579094—22no—
94#14579094—22no—
95#14579093C2×C414yes—
96#14579085—22no—
97#14579085C2×C414yes—
98#14579082—22no—
99#14579082C2×C414yes—
100C8×Dic79—22no—
101C8×Dic79C2×C414yes—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C22×Q8×C79C2311yescharacteristic
109C22×D4×C79C2311yescharacteristic
110C2×SD16×C79—22no—
111C2×SD16×C79—22no—
112C2×SD16×C79C2×C414yes—
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115C4×Q8×C79—22no—
116C4×Q8×C79—22no—
117C2×C4⋊C4×C79—21no—
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C22⋊C4×C79—21no—
121C2×C22⋊C4×C79—21no—
122C2×C42×C79—21no—
123C2×C42×C79C2311yescharacteristic
124#103×C79—22no—
125#102×C79—22no—
126C22×Dic79—24no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—42no—
132C22×Dic79—42no—
133C22×Dic79C2×D411yescharacteristic
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8C22×C412yes—
143C4⋊Dic79—24no—
144C4⋊Dic79—24no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79C22×C412yes—
151C4⋊Dic79C22×C412yes—
152C4⋊Dic79#4112yes—
153Dic79⋊C4—42no—
154Dic79⋊C4—42no—
155C22⋊Dic79—24no—
156C22⋊Dic79—42no—
157C22⋊Dic79—42no—
158C4×Dic79—24no—
159C4×Dic79—24no—
160C4×Dic79—22no—
161C4×Dic79—22no—
162C4×Dic79—22no—
163C4×Dic79—22no—
164C4×Dic79—21no—
165C4×Dic79—21no—
166C4×Dic79—42no—
167C4×Dic79—42no—
168C4×Dic79—42no—
169C4×Dic79—42no—
170C4×Dic79C22×C412yes—
171C4×Dic79#4112yes—
172C24×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79—21no—
175C2×Q8×C79—21no—
176C2×Q8×C79C22×C412yes—
177C2×Q8×C79#4112yes—
178C2×Q8×C79#4112yes—
179C2×D4×C79—21no—
180C2×D4×C79—21no—
181C2×D4×C79—21no—
182C2×D4×C79—21no—
183C2×D4×C79—21no—
184C2×D4×C79C22×C412yes—
185C22×C4×C79—21no—
186C22×C4×C79—21no—
187C22×C4×C79—21no—
188C22×C4×C79C2×D411yescharacteristic
189C22×C4×C79C2×D411yescharacteristic
190C22×C4×C79C2411yescharacteristic
191SD16×C79—28no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79C22×C412yes—
198C4⋊C4×C79—24no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79#4112yes—
203C22⋊C4×C79—42no—
204C22⋊C4×C79—42no—
205C42×C79—24no—
206C42×C79—21no—
207C42×C79—21no—
208C23×C79—22no—
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79—41no—
213C23×C79—41no—
214C23×C79—41no—
215C23×C79C22×D411yescharacteristic
216Q8×C79—22no—
217Q8×C79—22no—
218Q8×C79—22no—
219Q8×C79#11714yescomplemented
220D4×C79—22no—
221D4×C79—22no—
222D4×C79—41no—
223D4×C79—41no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—21no—
227C2×C4×C79—22no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—22no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79#13512yes—
239C2×C4×C79C23×C411yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×C4×C79C2×#4111yescharacteristic
242C2×C4×C79C2×#4111yescharacteristic
243C2×C4×C79C2×#4111yescharacteristic
244C2×C4×C79C2×#4111yescharacteristic
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—22no—
250C2×Dic79—22no—
251C2×Dic79—21no—
252C2×Dic79—22no—
253C2×Dic79—21no—
254C2×Dic79—21no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—48no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79C4×D412yes—
266C2×Dic79C4×D412yes—
267C2×Dic79#13512yes—
268C2×Dic79AGL(1,ℤ/8)12yes—
269C2×Dic79#13712yes—
270C8×C79—22no—
271C8×C79—41no—
272C79⋊C8—22no—
273C79⋊C8—41no—
274C22×C79—42no—
275C22×C79—42no—
276C22×C79—42no—
277C22×C79—81no—
278C22×C79—81no—
279C22×C79—81no—
280C22×C79—81no—
281C22×C79C2×C4×D411yescharacteristic
282C22×C79#52511yescharacteristic
283C22×C79#52511yescharacteristic
284C22×C79C2×#13511yescharacteristic
285C22×C79C2×#13511yescharacteristic
286C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C79C2×#13711yescharacteristic
288C4×C79—22no—
289C4×C79—24no—
290C4×C79—41no—
291C4×C79—42no—
292C4×C79—81no—
293C4×C79C2×#11711yescharacteristic
294C4×C79C4×#4111yescharacteristic
295C4×C79#52911yescharacteristic
296C4×C79#52911yescharacteristic
297Dic79—22no—
298Dic79—22no—
299Dic79—41no—
300Dic79—42no—
301Dic79—42no—
302Dic79—42no—
303Dic79—81no—
304Dic79#44014yes—
305#21120—791nomaximal, Sylow
306C2×C79—82no—
307C2×C79#229011yescharacteristic
308C2×C79C4×#13511yescharacteristic
309C2×C79C2×#44011yescharacteristic
310C2×C79#310611yescharacteristic
311C2×C79#310611yescharacteristic
312C2×C79#310811yescharacteristic
313C2×C79#310811yescharacteristic
314C2×C4×SD16—791no—
315#2293—791no—
316C42×Q8—791no—
317#1925—791no—
318#1925—791no—
319#1920—791no—
320#1920—791no—
321#1880—791no—
322#1880D7911yescharacteristic
323#1863—791no—
324#1863—791no—
325C4×#107—791no—
326C4×#103—791no—
327C4×#102—791no—
328#1744—791no—
329C79#2112011yescharacteristic, complemented, Sylow
330C22×SD16C2×D7911yescharacteristic
331C2×C4×Q8—791no—
332C2×C4×Q8—791no—
333C2×C4×Q8C2×D7911yescharacteristic
334C2×C4×D4—791no—
335C2×#117—1581no—
336C4×SD16—798no—
337C2×#107—791no—
338C2×#107—791no—
339C2×#106—791no—
340C2×#103—791no—
341C2×#103—791no—
342C2×#103C2×D7911yescharacteristic
343C2×#102—791no—
344C2×#102—791no—
345C2×#102C2×D7911yescharacteristic
346C2×C4×C8—791no—
347C2×C4×C8—791no—
348#390—791no—
349#390C2×D7911yescharacteristic
350#389—791no—
351#389—791no—
352#388—1581no—
353#388—1581no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4—791no—
356C4×C4⋊C4—791no—
357C4×C4⋊C4—791no—
358C4×C4⋊C4—1581no—
359C4×C4⋊C4—1581no—
360C4×C4⋊C4—1581no—
361C4×C22⋊C4—1581no—
362C43—791no—
363C43—1581no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340C2×D7911yescharacteristic
370#336—791no—
371#336—791no—
372#336—791no—
373#336C2×D7911yescharacteristic
374C22×Q8C22×D7911yescharacteristic
375C22×D4C22×D7911yescharacteristic
376C23×C4—1581no—
377C2×SD16—22no—
378C2×SD16—22no—
379C2×SD16C4×D7914yes—
380C22×C8—791no—
381C22×C8—791no—
382C22×C8C22×D7911yescharacteristic
383C22×C8C22×D7911yescharacteristic
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—794no—
387C4×Q8—1582no—
388C4×Q8—1582no—
389C4×Q8—1582no—
390C4×D4—794no—
391C4×D4—1582no—
392#117—3164no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—791no—
396C2×C4⋊C4—791no—
397C2×C4⋊C4—791no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4—1581no—
402C2×C4⋊C4C22×D7911yescharacteristic
403C2×C4⋊C4C22×D7911yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1581no—
407C2×C22⋊C4—1581no—
408C2×C22⋊C4—1581no—
409C2×C42—21no—
410C2×C42—791no—
411C2×C42—791no—
412C2×C42—791no—
413C2×C42—791no—
414C2×C42—791no—
415C2×C42—791no—
416C2×C42—1581no—
417C2×C42—1581no—
418C2×C42—1581no—
419C2×C42—1581no—
420C2×C42—1581no—
421C2×C42—1581no—
422C2×C42C22×D7911yescharacteristic
423#107—1582no—
424#107—1582no—
425#106—794no—
426#103—22no—
427#103—794no—
428#103—1582no—
429#102—22no—
430#102—794no—
431#102—1582no—
432C4×C8—794no—
433C4×C8—1582no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437#95—1581no—
438#95—1581no—
439#95—1581no—
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D7912yes—
445C2×Q8#1192962312yes—
446C2×Q8#1192962312yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D7912yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—3162no—
457C22×C4—3162no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—791no—
462C22×C4—1581no—
463C22×C4—1584no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4D4×D7911yescharacteristic
468C22×C4D4×D7911yescharacteristic
469C22×C4C23×D7911yescharacteristic
470SD16—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—792no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8C2×C4×D7912yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—3162no—
488C4⋊C4—3162no—
489C4⋊C4—792no—
490C4⋊C4—792no—
491C4⋊C4—792no—
492C4⋊C4—1584no—
493C4⋊C4—1584no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4#1192962312yes—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—3162no—
503C22⋊C4—3162no—
504C22⋊C4—1584no—
505C42—24no—
506C42—21no—
507C42—21no—
508C42—3162no—
509C42—3162no—
510C42—3162no—
511C42—3162no—
512C42—792no—
513C42—792no—
514C42—1582no—
515C42—1582no—
516C42—1582no—
517C42—1584no—
518C42—1584no—
519C42—1581no—
520C42—1581no—
521C42—1582no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D7911yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1457911714yescomplemented
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3168no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—792no—
563C2×C4—792no—
564C2×C4—792no—
565C2×C4—792no—
566C2×C4—792no—
567C2×C4—1582no—
568C2×C4—1582no—
569C2×C4—1582no—
570C2×C4—1581no—
571C2×C4—1581no—
572C2×C4—1581no—
573C2×C4—1581no—
574C2×C4—1581no—
575C2×C4—1581no—
576C2×C4—1581no—
577C2×C4#1457918212yes—
578C2×C4C22×C4×D7911yescharacteristic
579C2×C4#41×D7911yescharacteristic
580C2×C4#41×D7911yescharacteristic
581C2×C4C2×#1192962311yescharacteristic
582C2×C4C2×#1192962311yescharacteristic
583C2×C4C2×#1192962311yescharacteristic
584C8—22no—
585C8—41no—
586C8—3161no—
587C8—1582no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D7911yescharacteristic
596C22#1909906911yescharacteristic
597C22#1909907311yescharacteristic
598C22#135×D7911yescharacteristic
599C22C2×#1457918211yescharacteristic
600C22AGL(1,ℤ/8)×D7911yescharacteristic
601C22#137×D7911yescharacteristic
602C4—22no—
603C4—24no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4—3162no—
608C4—3162no—
609C4—3162no—
610C4—3161no—
611C4—794no—
612C4—6321no—
613C4—1582no—
614C4—1582no—
615C4C2×#1457911711yescharacteristic
616C4C4×#1192962311yescharacteristic
617C4#1909909011yescharacteristic
618C4#1909909411yescharacteristic
619C2—82no—
620C2#2376584011yescharacteristic
621C2C4×#1457918211yescharacteristic
622C2#440×D7911yescharacteristic
623C2#2377345311yescharacteristic
624C2#2377345911yescharacteristic
625C2#2377347511yescharacteristic
626C2#2377347911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1880C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1880D79
hypocenterC79#21120
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. C4×Q8×Dic79C2semidirect
6. #23763986C2semidirect
7. #23763984C2semidirect
9. #23763654C2semidirect
10. #23763652C2semidirect
11. #23763292C2semidirect
12. #103×Dic79C2semidirect
14. #23763144C2semidirect
20. SD16×Dic79C4semidirect
219. Q8×C79#117semidirect
329. C79#21120semidirect
533. Q8#14579117semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic792
#237658622
C4×Q8×Dic792
#237640232
#237640172
#237639862
#237639842
#237637572
#237636542
#237636522
#237632922
#103×Dic792
#102×Dic792
#237631442
C79×#18802
#2112079