Finite Group Databasebeta

← #34299511

Subgroups — #34299511

Order 20224.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098714C211yesmaximal, characteristic
2C4×#14579140C211yesmaximal, characteristic, complemented
3#23764280C212yesmaximal, complemented
4#23764100C212yesmaximal, complemented
5#23764017C212yesmaximal, complemented
6#23763758C212yesmaximal, complemented
7#23763707C211yesmaximal, characteristic, complemented
8#23763588C211yesmaximal, characteristic, complemented
9#23763331C211yesmaximal, characteristic
10#23763162C211yesmaximal, characteristic, complemented
11C2×#390×C79C211yesmaximal, characteristic
12C22×#11929612C2212yes—
13C2×#14579140C2211yescharacteristic
14C2×C4×Dic158C2211yescharacteristic, complemented
15#19098714C418yescomplemented
16C2×#14579094C2212yescomplemented
17C2×#14579084C2212yescomplemented
18C2×#14579082C2212yes—
19C2×#14579082C2212yes—
20C2×#14579082C2212yes—
21C2×C22⋊(C79⋊C8)C2211yescharacteristic
22C2×C22⋊(C79⋊C8)C2211yescharacteristic
23C2×#14579072C2212yescomplemented
24#19098469C2211yescharacteristic, complemented
25#19098461C2211yescharacteristic, complemented
26#19098447—21no—
27#19098436—21no—
28#19098423C2211yescharacteristic
29C4×Dic79⋊C4—21no—
30C4×C4⋊Dic79C2211yescharacteristic, complemented
31C4×C22⋊Dic79—21no—
32#19098325C2212yescomplemented
33#19098325C2212yescomplemented
34#19098318C2212yescomplemented
35C23×D4×C79C2211yescharacteristic
36C22×C4⋊C4×C79C2211yescharacteristic
37C22×C22⋊C4×C79—21no—
38C22×C22⋊C4×C79—21no—
39C22×C42×C79C2211yescharacteristic
40#390×C79—21no—
41#390×C79—21no—
42#390×C79C2212yes—
43#390×C79C2212yes—
44C22×Dic158C2311yescharacteristic
45C2×#11929612—42no—
46C2×#11929612—42no—
47C2×#11929612C2×C418yescomplemented
48C22×C79⋊C8C2312yes—
49C22×C79⋊C8C2312yes—
50#14579140—22no—
51#14579140C2×C414yes—
52C4×Dic158—42no—
53C2×C4⋊Dic79C2311yescharacteristic
54C2×C4⋊Dic79C2311yescharacteristic
55C2×C4⋊Dic79C2311yescharacteristic
56C2×Dic79⋊C4—21no—
57C2×Dic79⋊C4—21no—
58C2×Dic79⋊C4—21no—
59C2×C22⋊Dic79—21no—
60C2×C22⋊Dic79—21no—
61C2×C4×Dic79—21no—
62C2×C4×Dic79—21no—
63#14579094—42no—
64#14579084—42no—
65#14579082—42no—
66#14579082—42no—
67#14579082C2×C418yescomplemented
68C22⋊(C79⋊C8)—22no—
69C22⋊(C79⋊C8)C2×C414yes—
70#14579072—42no—
71C79⋊#95—21no—
72C79⋊#95—21no—
73C79⋊#95—21no—
74C79⋊#95—21no—
75C79⋊#95—21no—
76C25×C79—21no—
77C22×D4×C79—21no—
78C22×D4×C79—22no—
79C22×D4×C79—42no—
80C22×D4×C79—42no—
81C22×D4×C79C2×C414yes—
82C22×D4×C79C2312yes—
83C23×C4×C79—21no—
84C23×C4×C79D411yescharacteristic
85C23×C4×C79D411yescharacteristic
86C23×C4×C79C2311yescharacteristic
87C2×C4⋊C4×C79—21no—
88C2×C4⋊C4×C79—42no—
89C2×C4⋊C4×C79C2312yes—
90C2×C22⋊C4×C79—22no—
91C2×C22⋊C4×C79—22no—
92C2×C22⋊C4×C79—22no—
93C2×C22⋊C4×C79—22no—
94C2×C22⋊C4×C79—44no—
95C2×C22⋊C4×C79—44no—
96C2×C42×C79—42no—
97C2×C42×C79D411yescharacteristic
98C2×C42×C79D411yescharacteristic
99C2×C42×C79C2311yescharacteristic
100C2×C42×C79C2311yescharacteristic
101C2×Dic158—21no—
102C2×Dic158—22no—
103C2×Dic158—22no—
104C2×Dic158C22×C412yes—
105C22×Dic79—21no—
106C22×Dic79—21no—
107C22×Dic79—21no—
108C22×Dic79—21no—
109#11929612—48no—
110C2×C79⋊C8—22no—
111C2×C79⋊C8—22no—
112C2×C79⋊C8—22no—
113C2×C79⋊C8—22no—
114C2×C79⋊C8—22no—
115C2×C79⋊C8C22×C414yes—
116C4⋊Dic79—21no—
117C4⋊Dic79—22no—
118C4⋊Dic79—22no—
119C4⋊Dic79C22×C412yes—
120Dic79⋊C4—24no—
121Dic79⋊C4—42no—
122Dic79⋊C4—42no—
123C22⋊Dic79—24no—
124C22⋊Dic79—42no—
125C4×Dic79—42no—
126C4×Dic79—42no—
127C24×C79—24no—
128C24×C79—22no—
129C24×C79—42no—
130C24×C79—42no—
131C24×C79—42no—
132C24×C79—42no—
133C24×C79—42no—
134C24×C79—42no—
135C24×C79C2×D411yescharacteristic
136C2×D4×C79—24no—
137C2×D4×C79—22no—
138C2×D4×C79—22no—
139C2×D4×C79—22no—
140C2×D4×C79—22no—
141C2×D4×C79—22no—
142C2×D4×C79—48no—
143C2×D4×C79—42no—
144C2×D4×C79—42no—
145C2×D4×C79—48no—
146C2×D4×C79C22×C414yes—
147C22×C4×C79—22no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—22no—
151C22×C4×C79—22no—
152C22×C4×C79—22no—
153C22×C4×C79—41no—
154C22×C4×C79—41no—
155C22×C4×C79—41no—
156C22×C4×C79—42no—
157C22×C4×C79—42no—
158C22×C4×C79—42no—
159C22×C4×C79C22×C412yes—
160C22×C4×C79C2×D411yescharacteristic
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79#4112yes—
166C22×C4×C79C2411yescharacteristic
167C4⋊C4×C79—22no—
168C4⋊C4×C79—22no—
169C4⋊C4×C79—42no—
170C22⋊C4×C79—44no—
171C22⋊C4×C79—44no—
172C22⋊C4×C79—44no—
173C22⋊C4×C79—44no—
174C42×C79—22no—
175C42×C79—22no—
176C42×C79—41no—
177C42×C79—41no—
178C23×C79—21no—
179C23×C79—21no—
180C23×C79—21no—
181C23×C79—21no—
182C23×C79—24no—
183C23×C79—48no—
184C23×C79—48no—
185C23×C79—42no—
186C23×C79—42no—
187C23×C79—42no—
188C23×C79—42no—
189C23×C79—42no—
190C23×C79—42no—
191C23×C79—48no—
192C23×C79—84no—
193C23×C79—84no—
194C23×C79—84no—
195C23×C79—84no—
196C23×C79C4×D412yes—
197C23×C79AGL(1,ℤ/8)12yes—
198C23×C79#13712yes—
199C23×C79C22×D411yescharacteristic
200D4×C79—24no—
201D4×C79—24no—
202D4×C79—48no—
203D4×C79—42no—
204D4×C79—42no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—24no—
210C2×C4×C79—24no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—44no—
221C2×C4×C79—44no—
222C2×C4×C79—42no—
223C2×C4×C79—42no—
224C2×C4×C79—42no—
225C2×C4×C79—84no—
226C2×C4×C79C4×D412yes—
227C2×C4×C79AGL(1,ℤ/8)12yes—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79H(5,𝔽2)12yes—
231C2×C4×C79H(5,𝔽2)12yes—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244Dic158—24no—
245Dic158—42no—
246C79⋊C8—24no—
247C79⋊C8—42no—
248C22×C79—22no—
249C22×C79—22no—
250C22×C79—22no—
251C22×C79—22no—
252C22×C79—22no—
253C22×C79—22no—
254C22×C79—44no—
255C22×C79—44no—
256C22×C79—44no—
257C22×C79—816no—
258C22×C79—82no—
259C22×C79—82no—
260C22×C79—82no—
261C22×C79—82no—
262C22×C79#44014yes—
263C22×C79C2×C4×D411yescharacteristic
264C22×C79#53411yescharacteristic
265C22×C79#53511yescharacteristic
266C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C79C2×#13711yescharacteristic
269C22×C79#57511yescharacteristic
270C4×C79—22no—
271C4×C79—22no—
272C4×C79—41no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—82no—
276C4×C79#51812yes—
277C4×C79#53812yes—
278Dic79—42no—
279Dic79—42no—
280Dic79—81no—
281Dic79—81no—
282#22049—791nomaximal, Sylow
283C2×C79—24no—
284C2×C79—84no—
285C2×C79#232111yescharacteristic
286C2×C79C2×#44011yescharacteristic
287C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
288C2×C79#319311yescharacteristic
289C2×C79#319411yescharacteristic
290C2×C79#321511yescharacteristic
291C2×C79#321711yescharacteristic
292C2×#450—791no—
293C4×C22⋊Q8—791no—
294C2×#390D7911yescharacteristic
295#1973—792no—
296#1936—792no—
297#1925—792no—
298#1880—792no—
299#1872—791no—
300#1852—791no—
301#1786—791no—
302#1749—791no—
303C79#2204911yescharacteristic, complemented, Sylow
304C23×D4C2×D7911yescharacteristic
305C22×SD16—792no—
306C2×C22⋊Q8—791no—
307C2×C4×Q8—791no—
308C22×C4⋊C4C2×D7911yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4—21no—
311C22×C42C2×D7911yescharacteristic
312#450—798no—
313C2×#107—792no—
314C2×#103—792no—
315C2×#102—792no—
316C2×#102—792no—
317C2×#102—792no—
318C2×C22⋊C8—791no—
319C2×C22⋊C8—791no—
320C2×#97—792no—
321#391—791no—
322#390—21no—
323#390—21no—
324#390C2×D7912yes—
325#390C2×D7912yes—
326#389—791no—
327#387—1581no—
328#385—1581no—
329#380—791no—
330C4×C4⋊C4—791no—
331C4×C4⋊C4—1581no—
332C4×C22⋊C4—1581no—
333#340—792no—
334#340—792no—
335#336—792no—
336C25—21no—
337C22×Q8—791no—
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C4×D7914yes—
343C22×D4C22×D7912yes—
344C23×C4—21no—
345C23×C4C22⋊D7911yescharacteristic
346C23×C4C22⋊D7911yescharacteristic
347C23×C4C22×D7911yescharacteristic
348C2×SD16—3162no—
349C2×SD16—3162no—
350C2×SD16—798no—
351C22×C8—792no—
352C22×C8—792no—
353C22⋊Q8—794no—
354C22⋊Q8—1582no—
355C4×Q8—3162no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—791no—
359C2×C4⋊C4—791no—
360C2×C4⋊C4—791no—
361C2×C4⋊C4—1581no—
362C2×C4⋊C4—1581no—
363C2×C4⋊C4—1581no—
364C2×C4⋊C4C22×D7912yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—1581no—
372C2×C22⋊C4—1581no—
373C2×C42—42no—
374C2×C42—1581no—
375C2×C42—1581no—
376C2×C42C22⋊D7911yescharacteristic
377C2×C42C22⋊D7911yescharacteristic
378C2×C42C22×D7911yescharacteristic
379C2×C42C22×D7911yescharacteristic
380#107—3162no—
381#103—3162no—
382#102—3162no—
383#102—3162no—
384#102—798no—
385C22⋊C8—794no—
386C22⋊C8—1582no—
387#97—3162no—
388#95—1581no—
389#95—1581no—
390#95—1581no—
391#95—1581no—
392#95—1581no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C2×C22⋊D7911yescharacteristic
402C2×Q8—792no—
403C2×Q8—1582no—
404C2×Q8—1582no—
405C2×Q8—1581no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C2×C4×D7914yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4C2×C4×D7912yes—
434C22×C4C2×C22⋊D7911yescharacteristic
435C22×C4C2×C22⋊D7911yescharacteristic
436C22×C4C2×C22⋊D7911yescharacteristic
437C22×C4C2×C22⋊D7911yescharacteristic
438C22×C4C2×C22⋊D7911yescharacteristic
439C22×C4#1192962412yes—
440C22×C4C23×D7911yescharacteristic
441SD16—3168no—
442C2×C8—794no—
443C2×C8—1582no—
444C2×C8—1582no—
445C2×C8—1582no—
446C2×C8—1582no—
447C2×C8—1582no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—3162no—
452C4⋊C4—3162no—
453C4⋊C4—792no—
454C4⋊C4—1582no—
455C4⋊C4—1582no—
456C4⋊C4—1581no—
457C4⋊C4—1584no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—3162no—
463C22⋊C4—1584no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—3162no—
469C42—3162no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23C4×C22⋊D7912yes—
489C23#1457918712yes—
490C23#1457919112yes—
491C23C22×C22⋊D7911yescharacteristic
492Q8—3162no—
493Q8—1584no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—1582no—
531C2×C4—1582no—
532C2×C4C4×C22⋊D7912yes—
533C2×C4#1457918712yes—
534C2×C4C22×C4×D7911yescharacteristic
535C2×C4C2×#1192962411yescharacteristic
536C2×C4#1457920612yes—
537C2×C4#1457920612yes—
538C8—3162no—
539C8—1584no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#1909866314yes—
555C22C2×C4×C22⋊D7911yescharacteristic
556C22#1909912211yescharacteristic
557C22#1909913011yescharacteristic
558C22C2×#1457918711yescharacteristic
559C22C2×#1457918711yescharacteristic
560C22C2×#1457919111yescharacteristic
561C22#1909936111yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—82no—
568C4—3162no—
569C4—3162no—
570C4—6321no—
571C4—6321no—
572C4#1909903612yes—
573C4#1909915112yes—
574C2—24no—
575C2—84no—
576C2#2376605811yescharacteristic
577C2C2×#1909866311yescharacteristic
578C2C4×#1457918711yescharacteristic
579C2#2377420611yescharacteristic
580C2#2377421411yescharacteristic
581C2#2377440311yescharacteristic
582C2#2377441811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#390×C79C2
socleC23×C79C22×D4
prime radicalC23×D4×C79C22
cosocleC22×C4×C79C24
hypercenterC2×#390D79
hypocenterC79#22049
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#14579140C2semidirect
3. #23764280C2semidirect
4. #23764100C2semidirect
5. #23764017C2semidirect
6. #23763758C2semidirect
7. #23763707C2semidirect
8. #23763588C2semidirect
10. #23763162C2semidirect
14. C2×C4×Dic158C22semidirect
15. #19098714C4semidirect
16. C2×#14579094C22semidirect
17. C2×#14579084C22semidirect
23. C2×#14579072C22semidirect
24. #19098469C22semidirect
25. #19098461C22semidirect
30. C4×C4⋊Dic79C22semidirect
32. #19098325C22semidirect
33. #19098325C22semidirect
34. #19098318C22semidirect
47. C2×#11929612C2×C4semidirect
67. #14579082C2×C4semidirect
303. C79#22049semidirect

Maximal subgroups

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

subgroupindex
C2×#190987142
C4×#145791402
#237642802
#237641002
#237640172
#237637582
#237637072
#237635882
#237633312
#237631622
C2×#390×C792
#2204979