Finite Group Databasebeta

← #34300461

Subgroups — #34300461

Order 20224.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098865C211yesmaximal, characteristic
2#23766137C211yesmaximal, characteristic, complemented
3#23766037C211yesmaximal, characteristic
4#23764356C211yesmaximal, characteristic, complemented
5#23764160C211yesmaximal, characteristic, complemented
6#23764137C211yesmaximal, characteristic, complemented
7#23764126C211yesmaximal, characteristic, complemented
8#23764094C211yesmaximal, characteristic
9#23764085C211yesmaximal, characteristic
10#23764024C211yesmaximal, characteristic
11#23763986C211yesmaximal, characteristic, complemented
12#23763971C211yesmaximal, characteristic, complemented
13#23763398C211yesmaximal, characteristic
14#23763256C211yesmaximal, characteristic, complemented
15C79×#1926C211yesmaximal, characteristic
16C2×#14579152C2211yescharacteristic
17C2×D4×Dic79C2211yescharacteristic
18#19098865C418yescomplemented
19C2×#14579094C2211yescharacteristic
20C2×#14579093C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×Dic79⋊C8C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579071C2211yescharacteristic
27C2×C8×Dic79C2211yescharacteristic
28#19098466C2211yescharacteristic
29#19098462C2211yescharacteristic
30#19098461—21no—
31#19098461C2211yescharacteristic
32#19098460—21no—
33#19098459C2211yescharacteristic
34#19098459C2211yescharacteristic
35#19098457—21no—
36#19098451—21no—
37#19098447—21no—
38#19098446—21no—
39#19098432C2211yescharacteristic
40#19098424—21no—
41#19098421—21no—
42C4⋊C4×Dic79C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44#19098409C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098318C2211yescharacteristic
53C2×#106×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56#390×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C23×Dic79—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63#14579152—22no—
64#14579152C2×C414yes—
65D4×Dic79—22no—
66D4×Dic79C2×C414yes—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79C2311yescharacteristic
85C2×C4×Dic79C2311yescharacteristic
86#14579094C2×C414yes—
87#14579093C2×C414yes—
88#14579092—41no—
89Dic79⋊C8—22no—
90Dic79⋊C8C2×C414yes—
91#14579082—41no—
92#14579082C2×C414yes—
93#14579071—41no—
94C8×Dic79—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95D411yescharacteristic
102C79⋊#95D411yescharacteristic
103C79⋊#95D411yescharacteristic
104C79⋊#95D411yescharacteristic
105C22×D4×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C22⋊C4×C79—21no—
112C2×C22⋊C4×C79—21no—
113C2×C42×C79—21no—
114C2×C42×C79C2311yescharacteristic
115#106×C79—41no—
116#102×C79—41no—
117#102×C79C2×C414yes—
118#95×C79—21no—
119C22×Dic79—24no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79—81no—
125C22×Dic79—81no—
126C22×Dic79C2×D411yescharacteristic
127C22×Dic79C2×D411yescharacteristic
128C22×Dic79C2×D411yescharacteristic
129C22×Dic79C2×D411yescharacteristic
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8C22×C412yes—
136C4⋊Dic79—24no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—81no—
144C4⋊Dic79C22×C412yes—
145C4⋊Dic79C22×C412yes—
146Dic79⋊C4—24no—
147Dic79⋊C4—81no—
148Dic79⋊C4—81no—
149Dic79⋊C4—81no—
150Dic79⋊C4—81no—
151Dic79⋊C4—81no—
152C22⋊Dic79—24no—
153C22⋊Dic79—81no—
154C22⋊Dic79—81no—
155C4×Dic79—21no—
156C4×Dic79—21no—
157C4×Dic79—81no—
158C4×Dic79—81no—
159C4×Dic79C22×C412yes—
160C4×Dic79#4112yes—
161C24×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79—21no—
166C2×D4×C79—21no—
167C2×D4×C79C22×C412yes—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2411yescharacteristic
174C2×C8×C79—21no—
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79C22×C412yes—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79C22×C412yes—
186C22⋊C4×C79—81no—
187C22⋊C4×C79—81no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—81no—
191C23×C79—22no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79C22×D411yescharacteristic
199D4×C79—22no—
200D4×C79—22no—
201D4×C79—41no—
202D4×C79—41no—
203C2×C4×C79—22no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79C23×C411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×C4×C79H(5,𝔽2)11yescharacteristic
221C2×C4×C792−1+411yescharacteristic
222C2×C4×C792−1+411yescharacteristic
223C2×C4×C792−1+411yescharacteristic
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—84no—
247C2×Dic79C4×D412yes—
248C2×Dic79C4×D412yes—
249C8×C79—22no—
250C8×C79—41no—
251C79⋊C8—22no—
252C79⋊C8—41no—
253C22×C79—42no—
254C22×C79—42no—
255C22×C79—42no—
256C22×C79—81no—
257C22×C79—81no—
258C22×C79—81no—
259C22×C79—81no—
260C22×C79C2×C4×D411yescharacteristic
261C22×C79#53611yescharacteristic
262C22×C79#53711yescharacteristic
263C22×C79#57611yescharacteristic
264C22×C79#57711yescharacteristic
265C22×C79#57711yescharacteristic
266C22×C79#57711yescharacteristic
267C4×C79—42no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79—81no—
271C4×C79—81no—
272C4×C79#51911yescharacteristic
273C4×C79#52011yescharacteristic
274C4×C79#54111yescharacteristic
275C4×C79#54211yescharacteristic
276Dic79—22no—
277Dic79—22no—
278Dic79—42no—
279Dic79—42no—
280Dic79—41no—
281Dic79—41no—
282Dic79—81no—
283Dic79—81no—
284#22122—791nomaximal, Sylow
285C2×C79—82no—
286C2×C79#232311yescharacteristic
287C2×C79#294611yescharacteristic
288C2×C79#295211yescharacteristic
289C2×C79#324111yescharacteristic
290C2×C79#324411yescharacteristic
291C2×C79#325711yescharacteristic
292C2×C79#326011yescharacteristic
293C2×#478—791no—
294#2330—791no—
295#2319—791no—
296#1989—791no—
297#1948—791no—
298#1943—791no—
299#1941—791no—
300#1936—791no—
301#1934—791no—
302#1926—791no—
303#1926D7911yescharacteristic
304#1920—791no—
305#1918—791no—
306#1802—791no—
307#1768—791no—
308C79#2212211yescharacteristic, complemented, Sylow
309C2×#125—791no—
310C2×C4×D4—791no—
311#478—798no—
312C2×#107—791no—
313C2×#106—791no—
314C2×#106—791no—
315C2×#106C2×D7911yescharacteristic
316C2×C4⋊C8—791no—
317C2×C4⋊C8—791no—
318C2×#102—791no—
319C2×#102—791no—
320C2×#102C2×D7911yescharacteristic
321C2×#102C2×D7911yescharacteristic
322C2×#97—791no—
323C2×C4×C8—791no—
324#390—791no—
325#390C2×D7911yescharacteristic
326#389—791no—
327#389—791no—
328#389—791no—
329#389—791no—
330#389—1581no—
331#389—1581no—
332#388—1581no—
333#388—1581no—
334#387—1581no—
335#387—1581no—
336#384—791no—
337#381—1581no—
338#380—1581no—
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4C2×D7911yescharacteristic
342#376—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340C2×D7911yescharacteristic
351#336—791no—
352#336C2×D7911yescharacteristic
353C22×D4C22×D7911yescharacteristic
354C23×C4—1581no—
355C22×C8—791no—
356C22×C8—791no—
357C22×C8C22×D7911yescharacteristic
358C22×C8C22×D7911yescharacteristic
359#125—794no—
360#125—1582no—
361C4×D4—794no—
362C4×D4—1582no—
363C2×C4⋊C4—791no—
364C2×C4⋊C4—791no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—1581no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4—1581no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C4⋊C4C22×D7911yescharacteristic
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1581no—
381C2×C22⋊C4—1581no—
382C2×C22⋊C4—1581no—
383C2×C42—21no—
384C2×C42—791no—
385C2×C42—791no—
386C2×C42—1581no—
387C2×C42—1581no—
388C2×C42C22×D7911yescharacteristic
389#107—794no—
390#106—41no—
391#106—3161no—
392#106—794no—
393C4⋊C8—794no—
394C4⋊C8—1582no—
395#102—41no—
396#102—3161no—
397#102—794no—
398#102C4×D7914yes—
399#97—3161no—
400C4×C8—3161no—
401#95—21no—
402#95—791no—
403#95—791no—
404#95—791no—
405#95—791no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C4×D7912yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—791no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—6321no—
427C22×C4—6321no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1584no—
433C22×C4D4×D7911yescharacteristic
434C22×C4D4×D7911yescharacteristic
435C22×C4C23×D7911yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—792no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8C2×C4×D7912yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—792no—
454C4⋊C4—792no—
455C4⋊C4—6321no—
456C4⋊C4—6321no—
457C4⋊C4—6321no—
458C4⋊C4—6321no—
459C4⋊C4—6321no—
460C4⋊C4—6321no—
461C4⋊C4—1584no—
462C4⋊C4—1584no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4C2×C4×D7912yes—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—6321no—
473C22⋊C4—6321no—
474C22⋊C4—1584no—
475C42—21no—
476C42—21no—
477C42—81no—
478C42—792no—
479C42—792no—
480C42—6321no—
481C42—6321no—
482C42—1581no—
483C42—1581no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D7911yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—792no—
522C2×C4—792no—
523C2×C4—6324no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1582no—
527C2×C4—1582no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4C22×C4×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C2×C4#1457920611yescharacteristic
539C2×C4#1457920811yescharacteristic
540C2×C4#1457920811yescharacteristic
541C2×C4#1457920811yescharacteristic
542C8—22no—
543C8—41no—
544C8—3161no—
545C8—1582no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C4×D4×D7911yescharacteristic
554C22#1909913511yescharacteristic
555C22#1909914211yescharacteristic
556C22#1909936711yescharacteristic
557C22#1909936911yescharacteristic
558C22#1909937011yescharacteristic
559C22#1909937311yescharacteristic
560C4—42no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—3162no—
566C4—3161no—
567C4—3162no—
568C4—3161no—
569C4—6321no—
570C4—6321no—
571C4—1582no—
572C4—1582no—
573C4#1909904011yescharacteristic
574C4#1909904211yescharacteristic
575C4#1909917011yescharacteristic
576C4#1909917711yescharacteristic
577C2—82no—
578C2#2376607911yescharacteristic
579C2#2377222911yescharacteristic
580C2#2377227811yescharacteristic
581C2#2377465711yescharacteristic
582C2#2377469611yescharacteristic
583C2#2377481611yescharacteristic
584C2#2377484411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1926C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1926D79
hypocenterC79#22122
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
2. #23766137C2semidirect
4. #23764356C2semidirect
5. #23764160C2semidirect
6. #23764137C2semidirect
7. #23764126C2semidirect
11. #23763986C2semidirect
12. #23763971C2semidirect
14. #23763256C2semidirect
18. #19098865C4semidirect
308. C79#22122semidirect

Maximal subgroups

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

subgroupindex
C2×#190988652
#237661372
#237660372
#237643562
#237641602
#237641372
#237641262
#237640942
#237640852
#237640242
#237639862
#237639712
#237633982
#237632562
C79×#19262
#2212279