Finite Group Databasebeta

← #34183437

Subgroups — #34183437

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098881C211yesmaximal, characteristic
2#23766103C211yesmaximal, characteristic
3#23763490C211yesmaximal, characteristic, complemented
4C2×#19098296C211yesmaximal, characteristic
5#23761279C211yesmaximal, characteristic, complemented
6#23761243C211yesmaximal, characteristic, complemented
7C79×#1445C211yesmaximal, characteristic, complemented
8C2×#14579135C2211yescharacteristic
9C2×C4×C22⋊D79—21no—
10#19098881—22no—
11#19098881C414yescomplemented
12C2×#14579092—21no—
13C2×#14579081—21no—
14C2×C22⋊(C79⋊C8)C2211yescharacteristic
15C2×C22⋊(C79⋊C8)C2211yescharacteristic
16#19098455—21no—
17#19098448—21no—
18#19098442—21no—
19#19098441—21no—
20#19098437—21no—
21#19098423C2211yescharacteristic
22#19098325—21no—
23#19098325—21no—
24#19098296—22no—
25C22×C4⋊C4×C79C2211yescharacteristic
26C2×#95×C79—21no—
27C2×#95×C79—21no—
28C2×#95×C79C2211yescharacteristic
29C2×#95×C79C2211yescharacteristic
30C22×D316D411yescharacteristic
31C22×C22⋊D79—21no—
32C22×C4×D79—21no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35#14579135—21no—
36#14579135—21no—
37#14579135C2×C412yes—
38#14579135D412yes—
39C4×C22⋊D79—42no—
40C4×C22⋊D79—42no—
41C2×C4⋊Dic79D411yescharacteristic
42C2×C4⋊Dic79D411yescharacteristic
43C2×C4⋊Dic79D411yescharacteristic
44C2×Dic79⋊C4—21no—
45C2×C22⋊Dic79—21no—
46C2×(C2×D79)⋊C4—21no—
47C2×(C2×D79)⋊C4—21no—
48C2×(C2×D79)⋊C4—21no—
49C2×(C2×D79)⋊C4—21no—
50C2×C4×Dic79—21no—
51#14579092—24no—
52#14579081—24no—
53C22⋊(C79⋊C8)—21no—
54C22⋊(C79⋊C8)—21no—
55C22⋊(C79⋊C8)—21no—
56C22⋊(C79⋊C8)C2×C412yescomplemented
57C79⋊#95—21no—
58C79⋊#95—21no—
59C79⋊#95—21no—
60C23×C4×C79—21no—
61C23×C4×C79—21no—
62C23×C4×C79D411yescharacteristic
63C23×C4×C79D411yescharacteristic
64C23×C4×C79C2311yescharacteristic
65C2×C4⋊C4×C79—21no—
66C2×C4⋊C4×C79—21no—
67C2×C4⋊C4×C79—21no—
68C2×C4⋊C4×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79C2×C412yescomplemented
71#95×C79—21no—
72#95×C79—21no—
73#95×C79—21no—
74#95×C79—21no—
75#95×C79—42no—
76#95×C79—42no—
77C23×D79—21no—
78C2×D316—21no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316C22⋊C412yes—
82C2×C22⋊D79—22no—
83C2×C22⋊D79—22no—
84C2×C22⋊D79—41no—
85C2×C22⋊D79—41no—
86C2×C22⋊D79—41no—
87C2×C22⋊D79—41no—
88C22×Dic79—21no—
89C22×Dic79—21no—
90C22×Dic79—21no—
91C2×C4×D79—42no—
92C2×C4×D79—81no—
93C2×C4×D79—81no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C4⋊Dic79—21no—
101C4⋊Dic79—41no—
102C4⋊Dic79—41no—
103C4⋊Dic79C22⋊C412yes—
104Dic79⋊C4—42no—
105C22⋊Dic79—41no—
106C22⋊Dic79—41no—
107(C2×D79)⋊C4—22no—
108(C2×D79)⋊C4—22no—
109(C2×D79)⋊C4—42no—
110(C2×D79)⋊C4—81no—
111(C2×D79)⋊C4—81no—
112C4×Dic79—42no—
113C24×C79C2×D411yescharacteristic
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—22no—
117C22×C4×C79—22no—
118C22×C4×C79—21no—
119C22×C4×C79—21no—
120C22×C4×C79—21no—
121C22×C4×C79—41no—
122C22×C4×C79—41no—
123C22×C4×C79—41no—
124C22×C4×C79—41no—
125C22×C4×C79—41no—
126C22×C4×C79—41no—
127C22×C4×C79—41no—
128C22×C4×C79—41no—
129C22×C4×C79—41no—
130C22×C4×C79—41no—
131C22×C4×C79—41no—
132C22×C4×C79—41no—
133C22×C4×C79—41no—
134C22×C4×C79—41no—
135C22×C4×C79—41no—
136C22×C4×C79—41no—
137C22×C4×C79—41no—
138C22×C4×C79—41no—
139C22×C4×C79SD1612yes—
140C22×C4×C79C22×C411yescharacteristic
141C22×C4×C79C2×D411yescharacteristic
142C22×C4×C79C2×D411yescharacteristic
143C22×C4×C79C2×D411yescharacteristic
144C22×C4×C79#4111yescharacteristic
145C22×C4×C79#4111yescharacteristic
146C4⋊C4×C79—22no—
147C4⋊C4×C79—22no—
148C4⋊C4×C79—41no—
149C4⋊C4×C79—41no—
150C23×C79—21no—
151C23×C79—21no—
152C23×C79—21no—
153C23×C79—21no—
154C23×C79C4×D411yescharacteristic
155C23×C79#12011yescharacteristic
156C23×C79#12111yescharacteristic
157C23×C79C2×SD1611yescharacteristic
158C23×C79#13511yescharacteristic
159C23×C79AGL(1,ℤ/8)11yescharacteristic
160C23×C79#13711yescharacteristic
161C22×D79—22no—
162C22×D79—41no—
163C22×D79—41no—
164C22×D79—41no—
165C22×D79—41no—
166C22×D79—41no—
167C22×D79—41no—
168C2×C4×C79—21no—
169C2×C4×C79—21no—
170C2×C4×C79—21no—
171C2×C4×C79—21no—
172C2×C4×C79—22no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79—41no—
178C2×C4×C79—41no—
179C2×C4×C79—42no—
180C2×C4×C79—42no—
181C2×C4×C79—42no—
182C2×C4×C79—42no—
183C2×C4×C79—42no—
184C2×C4×C79—42no—
185C2×C4×C79—42no—
186C2×C4×C79—42no—
187C2×C4×C79—41no—
188C2×C4×C79—41no—
189C2×C4×C79—41no—
190C2×C4×C79—81no—
191C2×C4×C79—81no—
192C2×C4×C79—81no—
193C2×C4×C79—81no—
194C2×C4×C79—81no—
195C2×C4×C79—81no—
196C2×C4×C79—81no—
197C2×C4×C79—81no—
198C2×C4×C79—81no—
199C2×C4×C79—81no—
200C2×C4×C79C2×C22⋊C411yescharacteristic
201C2×C4×C79C4×D411yescharacteristic
202C2×C4×C79#12111yescharacteristic
203C2×C4×C79#12311yescharacteristic
204C22⋊D79—42no—
205C22⋊D79—42no—
206C22⋊D79—42no—
207C22⋊D79—42no—
208C2×Dic79—22no—
209C2×Dic79—41no—
210C2×Dic79—41no—
211C2×Dic79—41no—
212C2×Dic79—41no—
213C2×Dic79—41no—
214C2×Dic79—41no—
215C2×Dic79—41no—
216C2×Dic79—41no—
217D316—42no—
218D316—81no—
219C4×D79—84no—
220C79⋊C8—42no—
221C79⋊C8—81no—
222C22×C79—21no—
223C22×C79—21no—
224C22×C79—21no—
225C22×C79—21no—
226C22×C79—21no—
227C22×C79—21no—
228C22×C79—21no—
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79#38611yescharacteristic
235C22×C79C4×SD1611yescharacteristic
236C22×C79#44011yescharacteristic
237C22×C79#45011yescharacteristic
238C22×C79#45211yescharacteristic
239C22×C79#45311yescharacteristic
240C22×C79#45411yescharacteristic
241C22×C79#45811yescharacteristic
242C22×C79#45811yescharacteristic
243C22×C79#46111yescharacteristic
244C22×C79#46311yescharacteristic
245C2×D79—42no—
246C2×D79—42no—
247C2×D79—42no—
248C2×D79—81no—
249C2×D79—81no—
250C2×D79—81no—
251C2×D79—81no—
252C4×C79—21no—
253C4×C79—21no—
254C4×C79—21no—
255C4×C79—21no—
256C4×C79—42no—
257C4×C79—82no—
258C4×C79—81no—
259C4×C79—82no—
260Dic79—42no—
261Dic79—81no—
262Dic79—81no—
263#9045—791nomaximal, Sylow
264C2×C79—21no—
265C2×C79—21no—
266C2×C79—21no—
267C2×C79—21no—
268C2×C79#161311yescharacteristic
269C2×C79#161511yescharacteristic
270C2×C79#161611yescharacteristic
271C2×C79#161811yescharacteristic
272C2×C79#187711yescharacteristic
273C2×C79#188311yescharacteristic
274C2×C79#189011yescharacteristic
275D79—82no—
276C2×#481—791no—
277#2325—791no—
278#1824—791no—
279C2×#327—791no—
280#1445D7911yescharacteristic, complemented
281#1294—791no—
282#1280—791no—
283C79#904511yescharacteristic, complemented, Sylow
284C2×#121—791no—
285C2×C4×D4—1581no—
286C22×C4⋊C4C2×D7911yescharacteristic
287#481—794no—
288#481—1582no—
289C2×#106—1581no—
290C2×#102—1581no—
291C2×C22⋊C8—791no—
292C2×C22⋊C8—791no—
293#388—1581no—
294#387—1581no—
295#386—1581no—
296#386—1581no—
297#385—1581no—
298#380—791no—
299C2×#95—21no—
300C2×#95—21no—
301C2×#95C2×D7911yescharacteristic
302C2×#95C2×D7911yescharacteristic
303#340—1581no—
304#340—1581no—
305#327—1582no—
306C22×D4—791no—
307C22×D4—1581no—
308C23×C4—21no—
309C23×C4—21no—
310C23×C4—1581no—
311C23×C4C22⋊D7911yescharacteristic
312C23×C4C22⋊D7911yescharacteristic
313C23×C4C22×D7911yescharacteristic
314C22×C8—1581no—
315C22×C8—1581no—
316#121—792no—
317#121—792no—
318#121—1581no—
319#121—1581no—
320C4×D4—3162no—
321C4×D4—3162no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—791no—
328C2×C4⋊C4—791no—
329C2×C4⋊C4—791no—
330C2×C4⋊C4—1581no—
331C2×C4⋊C4C4×D7912yescomplemented
332C2×C22⋊C4—1581no—
333C2×C22⋊C4—1581no—
334C2×C22⋊C4—1581no—
335C2×C22⋊C4—1581no—
336C2×C22⋊C4—1581no—
337C2×C42—1581no—
338#106—1584no—
339#102—1584no—
340C22⋊C8—792no—
341C22⋊C8—1581no—
342C22⋊C8—1581no—
343C22⋊C8—1581no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—42no—
349#95—42no—
350#95—1581no—
351#95—1581no—
352#95—1581no—
353C24—1581no—
354C24C2×C22⋊D7911yescharacteristic
355C2×D4—3161no—
356C2×D4—3161no—
357C2×D4—3161no—
358C2×D4—3161no—
359C2×D4—3161no—
360C2×D4—3161no—
361C2×D4—792no—
362C2×D4—1582no—
363C2×D4—1582no—
364C2×D4—1581no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—3162no—
391C22×C4—6321no—
392C22×C4—6321no—
393C22×C4—1581no—
394C22×C4—1581no—
395C22×C4—1581no—
396C22×C4#1192961112yes—
397C22×C4C2×C4×D7911yescharacteristic
398C22×C4D4×D7911yescharacteristic
399C22×C4D4×D7911yescharacteristic
400C22×C4D4×D7911yescharacteristic
401C22×C4#1192962311yescharacteristic
402C22×C4#1192962411yescharacteristic
403C2×C8—3161no—
404C2×C8—3161no—
405C2×C8—3161no—
406C2×C8—3161no—
407C2×C8—3161no—
408C2×C8—1582no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—3161no—
414C4⋊C4—3161no—
415C4⋊C4—3162no—
416C4⋊C4—792no—
417C4⋊C4—1581no—
418C22⋊C4—3162no—
419C22⋊C4—3161no—
420C22⋊C4—3161no—
421C22⋊C4—6321no—
422C22⋊C4—6321no—
423C22⋊C4—1582no—
424C22⋊C4—1582no—
425C42—3162no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—3161no—
431C23—3161no—
432C23—3161no—
433C23—3161no—
434C23—3161no—
435C23—3161no—
436C23—1582no—
437C23C4×C22⋊D7911yescharacteristic
438C23#1457912911yescharacteristic
439C23#1457913211yescharacteristic
440C23C2×#1192961111yescharacteristic
441C23#1457918111yescharacteristic
442C23#1457918611yescharacteristic
443C23#1457919211yescharacteristic
444D4—3162no—
445D4—3162no—
446D4—3162no—
447D4—3162no—
448D4—3162no—
449D4—6321no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—3161no—
483C2×C4—3161no—
484C2×C4—3161no—
485C2×C4—3161no—
486C2×C4—3161no—
487C2×C4—3161no—
488C2×C4—3161no—
489C2×C4—3161no—
490C2×C4—6324no—
491C2×C4—1582no—
492C2×C4C22⋊C4×D7911yescharacteristic
493C2×C4(C22⋊D79)⋊C411yescharacteristic
494C2×C4#1457913411yescharacteristic
495C2×C4#1457914311yescharacteristic
496C8—3162no—
497C8—6321no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—3162no—
511C22—3162no—
512C22—3162no—
513C22—6321no—
514C22—6321no—
515C22—6321no—
516C22—6321no—
517C22#1909843911yescharacteristic
518C22C4×#1192961111yescharacteristic
519C22#1909866111yescharacteristic
520C22#1909871311yescharacteristic
521C22#1909872311yescharacteristic
522C22#1909873011yescharacteristic
523C22#1909873511yescharacteristic
524C22#1909875211yescharacteristic
525C22#1909875211yescharacteristic
526C22#1909876511yescharacteristic
527C22#1909877511yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—82no—
534C4—81no—
535C4—82no—
536C4—3162no—
537C4—6321no—
538C4—6321no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—6322no—
544C2#2376241911yescharacteristic
545C2#2376242911yescharacteristic
546C2#2376243611yescharacteristic
547C2#2376244611yescharacteristic
548C2#2376373711yescharacteristic
549C2#2376377911yescharacteristic
550C2#2376381011yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1445C2
socleC23×C79#120
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#1445D79
hypocenterC79#9045
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
3. #23763490C2semidirect
5. #23761279C2semidirect
6. #23761243C2semidirect
7. C79×#1445C2semidirect
11. #19098881C4semidirect
56. C22⋊(C79⋊C8)C2×C4semidirect
70. C2×C4⋊C4×C79C2×C4semidirect
280. #1445D79semidirect
283. C79#9045semidirect
331. C2×C4⋊C4C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988812
#237661032
#237634902
C2×#190982962
#237612792
#237612432
C79×#14452
#904579