Finite Group Databasebeta

← #34295841

Subgroups — #34295841

Order 20224.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098852C211yesmaximal, characteristic
2#23765971C211yesmaximal, characteristic, complemented
3C4×#14579123C211yesmaximal, characteristic
4#23764346C211yesmaximal, characteristic, complemented
5#23764162C211yesmaximal, characteristic, complemented
6#23763985C211yesmaximal, characteristic, complemented
7#23763664C211yesmaximal, characteristic
8#23763660C211yesmaximal, characteristic
9#23763399C211yesmaximal, characteristic
10#106×Dic79C211yesmaximal, characteristic, complemented
11#23763288C211yesmaximal, characteristic, complemented
12#23763248C211yesmaximal, characteristic, complemented
13#23763214C211yesmaximal, characteristic
14#23763214C211yesmaximal, characteristic
15C79×#1920C211yesmaximal, characteristic, complemented
16C2×#14579160C2211yescharacteristic
17C2×#14579126—21no—
18C2×C4×D316C2211yescharacteristic
19C2×#14579123C2211yescharacteristic
20C2×#14579123C2211yescharacteristic
21C2×C42×D79—21no—
22#19098852C418yescomplemented
23C2×#14579093C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×Dic79⋊C8C2211yescharacteristic
27C2×Dic79⋊C8C2211yescharacteristic
28C2×#14579083C2211yescharacteristic
29C2×#14579083C2211yescharacteristic
30C2×#14579081C2211yescharacteristic
31C2×#14579081C2211yescharacteristic
32C2×C8×Dic79C2211yescharacteristic
33C2×C8×Dic79C2211yescharacteristic
34#19098467—21no—
35#19098461C2211yescharacteristic
36#19098458—21no—
37#19098432C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42#19098414—21no—
43C4×(C2×D79)⋊C4—21no—
44#19098410—21no—
45#19098410—21no—
46C42×Dic79C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098318C2211yescharacteristic
53C2×#106×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55C4×C4⋊C4×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×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
69#14579126—44no—
70C4×D316—22no—
71C4×D316—22no—
72#14579123—22no—
73#14579123—22no—
74#14579123C2×C414yes—
75#14579123Q814yescomplemented
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×(C2×D79)⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79—21no—
90C2×C4×Dic79D411yescharacteristic
91C2×C4×Dic79D411yescharacteristic
92C2×C4×Dic79D411yescharacteristic
93C2×C4×Dic79D411yescharacteristic
94C2×C4×Dic79C2311yescharacteristic
95C2×C4×Dic79C2311yescharacteristic
96C42×D79—44no—
97#14579093—22no—
98#14579092—22no—
99#14579092C2×C414yescomplemented
100Dic79⋊C8—22no—
101Dic79⋊C8C2×C414yescomplemented
102#14579083—22no—
103#14579083C2×C414yes—
104#14579081—22no—
105#14579081C2×C414yes—
106C8×Dic79—22no—
107C8×Dic79—22no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95—21no—
111C79⋊#95—21no—
112C79⋊#95—21no—
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115C2×C4⋊C4×C79—21no—
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C42×C79—21no—
121C2×C42×C79C2311yescharacteristic
122#106×C79C2×C414yescomplemented
123#95×C79—21no—
124#95×C79—21no—
125C23×D79—21no—
126C2×Dic158—24no—
127C2×Dic158—42no—
128C2×Dic158—42no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316C22×C412yes—
133C2×D316C2×Q812yes—
134C2×D316#4112yes—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C22×Dic79C2×D411yescharacteristic
142C2×C4×D79—24no—
143C2×C4×D79—24no—
144C2×C4×D79—24no—
145C2×C4×D79—42no—
146C2×C4×D79—42no—
147C2×C4×D79—42no—
148C2×C4×D79—42no—
149C2×C4×D79—42no—
150C2×C4×D79—42no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8C22×C412yes—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79C22×C412yes—
162C4⋊Dic79#4112yes—
163Dic79⋊C4—24no—
164Dic79⋊C4—42no—
165Dic79⋊C4—42no—
166Dic79⋊C4—42no—
167(C2×D79)⋊C4—24no—
168(C2×D79)⋊C4—24no—
169(C2×D79)⋊C4—24no—
170C4×Dic79—22no—
171C4×Dic79—22no—
172C4×Dic79—41no—
173C4×Dic79—42no—
174C4×Dic79—42no—
175C4×Dic79—42no—
176C4×Dic79—42no—
177C4×Dic79SD1614yes—
178C4×Dic79C22×C412yes—
179C4×Dic79C2×Q812yes—
180C4×Dic79#4112yes—
181C4×Dic79#4112yes—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2411yescharacteristic
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79C22×C412yes—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—42no—
201C4⋊C4×C79C22×C412yes—
202C4⋊C4×C79C22×C412yes—
203C4⋊C4×C79C2×Q812yes—
204C42×C79—21no—
205C42×C79—21no—
206C42×C79—42no—
207C23×C79C22×D411yescharacteristic
208C22×D79—22no—
209C22×D79—22no—
210C22×D79—22no—
211C22×D79—41no—
212C22×D79—41no—
213C22×D79—41no—
214C22×D79—41no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—22no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
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×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79C2×SD1612yes—
233C2×C4×C79C23×C411yescharacteristic
234C2×C4×C79C22×Q811yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
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—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79C4×D412yes—
253C2×Dic79C4×D412yes—
254C2×Dic79C22⋊Q812yes—
255C2×Dic79C22⋊Q812yes—
256C2×Dic79C2×SD1612yes—
257C2×Dic79C2×SD1612yes—
258C2×Dic79#13512yes—
259C2×Dic79AGL(1,ℤ/8)12yes—
260D316—22no—
261D316—22no—
262D316—22no—
263D316C4×Q814yescomplemented
264C4×D79—48no—
265C4×D79—48no—
266C4×D79—48no—
267Dic158—48no—
268C8×C79—22no—
269C8×C79—41no—
270C79⋊C8—22no—
271C79⋊C8—41no—
272C22×C79C2×C4×D411yescharacteristic
273C22×C79C2×C22⋊Q811yescharacteristic
274C22×C79#52511yescharacteristic
275C22×C79C22×SD1611yescharacteristic
276C22×C79C2×#13511yescharacteristic
277C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C79#57511yescharacteristic
279C2×D79—24no—
280C2×D79—42no—
281C2×D79—42no—
282C2×D79—42no—
283C2×D79—42no—
284C2×D79—42no—
285C2×D79—42no—
286C4×C79—22no—
287C4×C79—41no—
288C4×C79—42no—
289C4×C79—42no—
290C4×C79—42no—
291C4×C79—81no—
292C4×C79C2×C4×Q811yescharacteristic
293C4×C79C4×#4111yescharacteristic
294C4×C79#52911yescharacteristic
295C4×C79#53311yescharacteristic
296Dic79—22no—
297Dic79—22no—
298Dic79—42no—
299Dic79—42no—
300Dic79—81no—
301Dic79C4×SD1614yes—
302Dic79#47614yescomplemented
303C4×#476—791nomaximal, Sylow
304C2×C79C4×C22⋊Q811yescharacteristic
305C2×C79C2×C4×SD1611yescharacteristic
306C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C79C2×#47611yescharacteristic
308C2×C79#307611yescharacteristic
309C2×C79#310111yescharacteristic
310C2×C79#310611yescharacteristic
311D79—44no—
312C2×#476—791no—
313C4×C4⋊Q8—791no—
314C42×D4—791no—
315#1987—791no—
316#1948—791no—
317#1920—791no—
318#1920D7911yescharacteristic, complemented
319#1864—791no—
320#1864—791no—
321#1802—791no—
322C4×#106—791no—
323C4×#106—791no—
324C4×C4⋊C8—791no—
325C4×#102—791no—
326C4×#102—791no—
327C79C4×#47611yescharacteristic, complemented, Sylow
328C2×C4⋊Q8—791no—
329C2×C4×Q8—1581no—
330C2×C4×D4—791no—
331C2×C4×D4—791no—
332C2×C4×D4—791no—
333C22×C42—1581no—
334#476—798no—
335C2×#106—791no—
336C2×#106—791no—
337C2×#106—791no—
338C2×#106C2×D7911yescharacteristic
339C2×C4⋊C8—791no—
340C2×C4⋊C8—791no—
341C2×#102—791no—
342C2×#102—791no—
343C2×#102—791no—
344C2×#102—791no—
345C2×C4×C8—791no—
346C2×C4×C8—791no—
347#391—1581no—
348#389—791no—
349#389—1581no—
350#389C2×D7911yescharacteristic
351#384—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4—791no—
356C4×C4⋊C4—1581no—
357C4×C4⋊C4C2×D7911yescharacteristic
358C4×C22⋊C4—1581no—
359C4×C22⋊C4—1581no—
360C4×C22⋊C4—1581no—
361C43—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340C2×D7911yescharacteristic
368#340C2×D7911yescharacteristic
369#340C2×D7911yescharacteristic
370#336—791no—
371#336C2×D7911yescharacteristic
372C22×Q8—1581no—
373C22×D4—791no—
374C23×C4—1581no—
375C23×C4—1581no—
376C23×C4—1581no—
377C22×C8—791no—
378C22×C8—791no—
379C22×C8C22×D7911yescharacteristic
380C22×C8C22×D7911yescharacteristic
381C4⋊Q8—794no—
382C4⋊Q8—1582no—
383C4×Q8—3164no—
384C4×D4—794no—
385C4×D4—794no—
386C4×D4—1582no—
387C4×D4—1582no—
388C4×D4—1582no—
389C4×D4—1582no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C4⋊C4C22×D7911yescharacteristic
401C2×C4⋊C4C22×D7911yescharacteristic
402C2×C22⋊C4—1581no—
403C2×C22⋊C4—1581no—
404C2×C22⋊C4—1581no—
405C2×C42—21no—
406C2×C42—3164no—
407C2×C42—791no—
408C2×C42—791no—
409C2×C42—791no—
410C2×C42—791no—
411C2×C42—791no—
412C2×C42—791no—
413C2×C42—1581no—
414C2×C42—1581no—
415C2×C42—1581no—
416C2×C42—1581no—
417C2×C42C22×D7911yescharacteristic
418#106—794no—
419#106—1582no—
420#106—1582no—
421#106C4×D7914yescomplemented
422C4⋊C8—794no—
423C4⋊C8—1582no—
424#102—794no—
425#102—794no—
426#102—1582no—
427#102—1582no—
428C4×C8—1582no—
429C4×C8—1582no—
430#95—21no—
431#95—21no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437C24—1581no—
438C2×Q8—3162no—
439C2×Q8—3162no—
440C2×Q8—1584no—
441C2×D4—792no—
442C2×D4—792no—
443C2×D4—792no—
444C2×D4—1581no—
445C2×D4—1581no—
446C2×D4—1581no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—3162no—
452C22×C4—3162no—
453C22×C4—3162no—
454C22×C4—3162no—
455C22×C4—3162no—
456C22×C4—3162no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—1584no—
462C22×C4—1584no—
463C22×C4—1584no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4D4×D7911yescharacteristic
468C22×C4D4×D7911yescharacteristic
469C22×C4C23×D7911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—792no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8C2×C4×D7912yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—3162no—
489C4⋊C4—3162no—
490C4⋊C4—3162no—
491C4⋊C4—792no—
492C4⋊C4—792no—
493C4⋊C4—1584no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4C2×C4×D7912yes—
499C4⋊C4C2×C4×D7912yes—
500C4⋊C4Q8×D7912yes—
501C22⋊C4—1584no—
502C22⋊C4—1584no—
503C22⋊C4—1584no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—3161no—
508C42—3162no—
509C42—3162no—
510C42—3162no—
511C42—3162no—
512C42—792no—
513C42—792no—
514C42—792no—
515C42—794no—
516C42—792no—
517C42—1582no—
518C42—1582no—
519C23—3161no—
520C23—3161no—
521C23—3161no—
522C23—3161no—
523C23—1582no—
524C23—1582no—
525C23—1582no—
526C23C2×D4×D7911yescharacteristic
527Q8—3168no—
528D4—794no—
529D4—1582no—
530D4—1582no—
531D4—1582no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—3168no—
550C2×C4—3168no—
551C2×C4—3168no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—792no—
560C2×C4—792no—
561C2×C4—792no—
562C2×C4—792no—
563C2×C4—792no—
564C2×C4—792no—
565C2×C4—792no—
566C2×C4—792no—
567C2×C4—1582no—
568C2×C4—1582no—
569C2×C4—1581no—
570C2×C4—1581no—
571C2×C4—1581no—
572C2×C4—1581no—
573C2×C4SD16×D7912yes—
574C2×C4C22×C4×D7911yescharacteristic
575C2×C4C2×Q8×D7911yescharacteristic
576C2×C4#41×D7911yescharacteristic
577C2×C4#41×D7911yescharacteristic
578C2×C4#41×D7911yescharacteristic
579C2×C4#41×D7911yescharacteristic
580C8—22no—
581C8—41no—
582C8—3161no—
583C8—1582no—
584C22—3162no—
585C22—3162no—
586C22—3162no—
587C22—3162no—
588C22—3162no—
589C22—3162no—
590C22—1584no—
591C22C4×D4×D7911yescharacteristic
592C22C22⋊Q8×D7911yescharacteristic
593C22#1909907111yescharacteristic
594C22C2×SD16×D7911yescharacteristic
595C22#135×D7911yescharacteristic
596C22AGL(1,ℤ/8)×D7911yescharacteristic
597C22#1909936211yescharacteristic
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—3162no—
605C4—3162no—
606C4—794no—
607C4—794no—
608C4—6321no—
609C4—1582no—
610C4—1582no—
611C4C4×Q8×D7911yescharacteristic
612C4#1909903111yescharacteristic
613C4#1909909611yescharacteristic
614C4#1909911911yescharacteristic
615C2—3164no—
616C2#2376593411yescharacteristic
617C2C4×SD16×D7911yescharacteristic
618C2#2377220511yescharacteristic
619C2#476×D7911yescharacteristic
620C2#2377318711yescharacteristic
621C2#2377340511yescharacteristic
622C2#2377345511yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1920C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1920D79
hypocenterC79C4×#476
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
2. #23765971C2semidirect
4. #23764346C2semidirect
5. #23764162C2semidirect
6. #23763985C2semidirect
10. #106×Dic79C2semidirect
11. #23763288C2semidirect
12. #23763248C2semidirect
15. C79×#1920C2semidirect
22. #19098852C4semidirect
68. #14579160C2×C4semidirect
75. #14579123Q8semidirect
99. #14579092C2×C4semidirect
101. Dic79⋊C8C2×C4semidirect
122. #106×C79C2×C4semidirect
263. D316C4×Q8semidirect
302. Dic79#476semidirect
318. #1920D79semidirect
327. C79C4×#476semidirect
421. #106C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988522
#237659712
C4×#145791232
#237643462
#237641622
#237639852
#237636642
#237636602
#237633992
#106×Dic792
#237632882
#237632482
#237632142
C79×#19202
C4×#47679