Finite Group Databasebeta

← #34310236

Subgroups — #34310236

Order 20224.

878 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098852C211yesmaximal, characteristic
2#23766459C211yesmaximal, characteristic, complemented
3#23766309C211yesmaximal, characteristic
4#23764371C211yesmaximal, characteristic, complemented
5#23764171C211yesmaximal, characteristic, complemented
6#23764144C211yesmaximal, characteristic, complemented
7#23764125C211yesmaximal, characteristic, complemented
8#23763981C211yesmaximal, characteristic, complemented
9#23763662C211yesmaximal, characteristic
10#23763650C211yesmaximal, characteristic
11#23763271C211yesmaximal, characteristic, complemented
12#23763224C211yesmaximal, characteristic
13#23763224C211yesmaximal, characteristic
14C4×#14579081C211yesmaximal, characteristic
15C79×#1920C211yesmaximal, characteristic, complemented
16C2×#14579160C2211yescharacteristic
17C2×C4×Dic158—21no—
18C2×C4×D316C2211yescharacteristic
19C2×#14579123C2211yescharacteristic
20C2×#14579123C2211yescharacteristic
21#19098852C418yescomplemented
22C2×#14579095C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×Dic79⋊C8C2211yescharacteristic
26C2×#14579083C2211yescharacteristic
27C2×#14579083C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30C2×#14579071C2211yescharacteristic
31C2×#14579071C2211yescharacteristic
32C2×C4×C79⋊C8C2211yescharacteristic
33#19098467—21no—
34#19098462C2211yescharacteristic
35#19098462C2211yescharacteristic
36#19098461C2211yescharacteristic
37#19098450—21no—
38#19098447—21no—
39#19098437—21no—
40#19098434—21no—
41#19098428C2211yescharacteristic
42#19098420—21no—
43C4⋊C4×Dic79C2211yescharacteristic
44C4⋊C4×Dic79C2211yescharacteristic
45C4×Dic79⋊C4C2211yescharacteristic
46#95×D79—21no—
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098317C2211yescharacteristic
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
69C4×Dic158—44no—
70C4×D316—22no—
71C4×D316—41no—
72#14579123—22no—
73#14579123—22no—
74#14579123—41no—
75#14579123C2×C414yes—
76C2×C4⋊Dic79—21no—
77C2×C4⋊Dic79—21no—
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×C4⋊Dic79C2311yescharacteristic
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4D411yescharacteristic
84C2×Dic79⋊C4D411yescharacteristic
85C2×(C2×D79)⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79—21no—
90C2×C4×Dic79C2311yescharacteristic
91C2×C4×Dic79C2311yescharacteristic
92#14579095—41no—
93#14579092—41no—
94#14579092C2×C414yescomplemented
95Dic79⋊C8C2×C414yescomplemented
96#14579083—22no—
97#14579083C2×C414yes—
98#14579081C2×C414yes—
99#14579081Q814yescomplemented
100#14579071—22no—
101#14579071—41no—
102C4×C79⋊C8—41no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95D411yescharacteristic
111C79⋊#95D411yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C2×C4⋊C4×C79—21no—
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C42×C79—21no—
120C2×C42×C79C2311yescharacteristic
121#106×C79C2×C414yescomplemented
122#95×C79—21no—
123#95×C79—21no—
124C23×D79—21no—
125C2×Dic158—24no—
126C2×Dic158—42no—
127C2×Dic158—42no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316C22×C412yes—
132C2×D316C2×Q812yes—
133C2×D316#4112yes—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C2×C4×D79—24no—
142C2×C4×D79—42no—
143C2×C4×D79—42no—
144C2×C4×D79—42no—
145C2×C4×D79—42no—
146C2×C4×D79—81no—
147C2×C4×D79—81no—
148C2×C4×D79—81no—
149C2×C4×D79—81no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8C22×C412yes—
155C2×C79⋊C8C2×Q812yes—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—42no—
162C4⋊Dic79—81no—
163C4⋊Dic79C22×C412yes—
164Dic79⋊C4—24no—
165Dic79⋊C4—22no—
166Dic79⋊C4—41no—
167Dic79⋊C4—42no—
168(C2×D79)⋊C4—24no—
169(C2×D79)⋊C4—42no—
170(C2×D79)⋊C4—42no—
171C4×Dic79—21no—
172C4×Dic79—21no—
173C4×Dic79—42no—
174C4×Dic79—42no—
175C4×Dic79C22×C412yes—
176C4×Dic79#4112yes—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2411yescharacteristic
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79C22×C412yes—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—81no—
196C4⋊C4×C79C22×C412yes—
197C4⋊C4×C79C22×C412yes—
198C4⋊C4×C79C2×Q812yes—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—42no—
202C23×C79C22×D411yescharacteristic
203C22×D79—22no—
204C22×D79—22no—
205C22×D79—22no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
209C22×D79—41no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—21no—
215C2×C4×C79—22no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
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×C79AGL(1,ℤ/8)12yes—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×Q811yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79H(5,𝔽2)11yescharacteristic
233C2×C4×C792−1+411yescharacteristic
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—22no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79C4×D412yes—
254C2×Dic79C4×D412yes—
255D316—22no—
256D316—22no—
257D316—22no—
258D316C4×Q814yescomplemented
259C4×D79—84no—
260C4×D79—84no—
261C4×D79—84no—
262Dic158—48no—
263C8×C79—22no—
264C8×C79—41no—
265C79⋊C8—22no—
266C79⋊C8—22no—
267C22×C79C2×C4×D411yescharacteristic
268C22×C79#54611yescharacteristic
269C22×C79D4×Q811yescharacteristic
270C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C79#57511yescharacteristic
272C22×C79#57711yescharacteristic
273C22×C79#57711yescharacteristic
274C2×D79—24no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—42no—
279C2×D79—42no—
280C2×D79—42no—
281C4×C79—22no—
282C4×C79—42no—
283C4×C79—42no—
284C4×C79—42no—
285C4×C79—41no—
286C4×C79—81no—
287C4×C79C2×C4×Q811yescharacteristic
288C4×C79#52011yescharacteristic
289C4×C79#55211yescharacteristic
290C4×C79#55411yescharacteristic
291Dic79—22no—
292Dic79—22no—
293Dic79—42no—
294Dic79—41no—
295Dic79—41no—
296Dic79—81no—
297Dic79—81no—
298#22975—791nomaximal, Sylow
299C2×C79#236011yescharacteristic
300C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C79#294911yescharacteristic
302C2×C79#330611yescharacteristic
303C2×C79#331111yescharacteristic
304C2×C79#338411yescharacteristic
305C2×C79#338811yescharacteristic
306D79—44no—
307C2×#476—791no—
308#2365—791no—
309#2349—791no—
310#1993—791no—
311#1949—791no—
312#1944—791no—
313#1941—791no—
314#1920D7911yescharacteristic, complemented
315#1919—791no—
316#1864—791no—
317#1862—791no—
318#1771—791no—
319#1762—791no—
320#1762—791no—
321C4×#102—791no—
322C79#2297511yescharacteristic, complemented, Sylow
323C2×C4⋊Q8—791no—
324C2×C4×Q8—1581no—
325C2×C4×D4—791no—
326C2×C4×D4—791no—
327C2×C4×D4—791no—
328#476—798no—
329C2×#107—791no—
330C2×#106—791no—
331C2×#106—791no—
332C2×#106C2×D7911yescharacteristic
333C2×C4⋊C8—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#102—791no—
337C2×#102—791no—
338C2×#97—791no—
339C2×#97—791no—
340C2×C4×C8—791no—
341#391—1581no—
342#389—791no—
343#389—791no—
344#389—791no—
345#389C2×D7911yescharacteristic
346#387—1581no—
347#387—1581no—
348#385—1581no—
349#385—1581no—
350#382—791no—
351#380—1581no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4C2×D7911yescharacteristic
356C2×#95—1581no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340C2×D7911yescharacteristic
363#340C2×D7911yescharacteristic
364#340C2×D7911yescharacteristic
365#336—791no—
366#336C2×D7911yescharacteristic
367C22×Q8—1581no—
368C22×D4—791no—
369C23×C4—1581no—
370C23×C4—1581no—
371C23×C4—1581no—
372C22×C8—791no—
373C22×C8—791no—
374C22×C8C22×D7911yescharacteristic
375C22×C8C22×D7911yescharacteristic
376C4⋊Q8—794no—
377C4⋊Q8—1582no—
378C4×Q8—3164no—
379C4×D4—3161no—
380C4×D4—3161no—
381C4×D4—794no—
382C4×D4—1582no—
383C4×D4—1582no—
384C4×D4—1582no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C42—21no—
403C2×C42—791no—
404C2×C42—791no—
405C2×C42—1581no—
406C2×C42—1581no—
407C2×C42C22×D7911yescharacteristic
408#107—3161no—
409#106—3161no—
410#106—794no—
411#106C4×D7914yescomplemented
412C4⋊C8—794no—
413#102—794no—
414#102—794no—
415#102—794no—
416#102—1582no—
417#97—3161no—
418#97—1582no—
419C4×C8—3161no—
420#95—21no—
421#95—21no—
422#95—791no—
423#95—791no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431C24—1581no—
432C2×Q8—3162no—
433C2×Q8—3162no—
434C2×Q8—1584no—
435C2×D4—792no—
436C2×D4—792no—
437C2×D4—792no—
438C2×D4—1581no—
439C2×D4—1581no—
440C2×D4—1581no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—3162no—
446C22×C4—3162no—
447C22×C4—3162no—
448C22×C4—3162no—
449C22×C4—791no—
450C22×C4—791no—
451C22×C4—791no—
452C22×C4—791no—
453C22×C4—6321no—
454C22×C4—6321no—
455C22×C4—6321no—
456C22×C4—6321no—
457C22×C4—1584no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4D4×D7911yescharacteristic
462C22×C4D4×D7911yescharacteristic
463C22×C4C23×D7911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—792no—
470C2×C8—792no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8C2×C4×D7912yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—81no—
482C4⋊C4—3161no—
483C4⋊C4—3162no—
484C4⋊C4—3162no—
485C4⋊C4—792no—
486C4⋊C4—6321no—
487C4⋊C4—1584no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1582no—
494C4⋊C4C2×C4×D7912yes—
495C4⋊C4C2×C4×D7912yes—
496C4⋊C4Q8×D7912yes—
497C22⋊C4—3162no—
498C22⋊C4—3162no—
499C22⋊C4—1584no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—3162no—
504C42—3162no—
505C42—792no—
506C42—792no—
507C42—1581no—
508C42—1581no—
509C23—3161no—
510C23—3161no—
511C23—3161no—
512C23—3161no—
513C23—1582no—
514C23—1582no—
515C23—1582no—
516C23C2×D4×D7911yescharacteristic
517Q8—3168no—
518D4—794no—
519D4—1582no—
520D4—1582no—
521D4—1582no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—792no—
548C2×C4—792no—
549C2×C4—6324no—
550C2×C4—6324no—
551C2×C4—6324no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1582no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4—1581no—
563C2×C4#1457918812yes—
564C2×C4C22×C4×D7911yescharacteristic
565C2×C4C2×Q8×D7911yescharacteristic
566C2×C4#41×D7911yescharacteristic
567C2×C4#41×D7911yescharacteristic
568C2×C4#1457920711yescharacteristic
569C2×C4#1457920811yescharacteristic
570C8—22no—
571C8—41no—
572C8—1582no—
573C8—1582no—
574C22—3162no—
575C22—3162no—
576C22—3162no—
577C22—3162no—
578C22—3162no—
579C22—3162no—
580C22—1584no—
581C22C4×D4×D7911yescharacteristic
582C22#1909920511yescharacteristic
583C22#1909922611yescharacteristic
584C22C2×#1457918811yescharacteristic
585C22#1909936011yescharacteristic
586C22#1909937211yescharacteristic
587C22#1909937311yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—3162no—
595C4—3161no—
596C4—3161no—
597C4—6321no—
598C4—6321no—
599C4—1582no—
600C4—1582no—
601C4C4×Q8×D7911yescharacteristic
602C4#1909904311yescharacteristic
603C4#1909924911yescharacteristic
604C4#1909926911yescharacteristic
605C2—3164no—
606C2#2376640811yescharacteristic
607C2C4×#1457918811yescharacteristic
608C2#2377225611yescharacteristic
609C2#2377535911yescharacteristic
610C2#2377543411yescharacteristic
611C2#2377629811yescharacteristic
612C2#2377634311yescharacteristic
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
hypocenterC79#22975
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. #23766459C2semidirect
4. #23764371C2semidirect
5. #23764171C2semidirect
6. #23764144C2semidirect
7. #23764125C2semidirect
8. #23763981C2semidirect
11. #23763271C2semidirect
15. C79×#1920C2semidirect
21. #19098852C4semidirect
68. #14579160C2×C4semidirect
94. #14579092C2×C4semidirect
95. Dic79⋊C8C2×C4semidirect
99. #14579081Q8semidirect
121. #106×C79C2×C4semidirect
258. D316C4×Q8semidirect
314. #1920D79semidirect
322. C79#22975semidirect
411. #106C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988522
#237664592
#237663092
#237643712
#237641712
#237641442
#237641252
#237639812
#237636622
#237636502
#237632712
#237632242
C4×#145790812
C79×#19202
#2297579