Finite Group Databasebeta

← #34300463

Subgroups — #34300463

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×#19098866C211yesmaximal, characteristic
2#23766137C211yesmaximal, characteristic, complemented
3#23766039C211yesmaximal, characteristic
4#23764356C211yesmaximal, characteristic, complemented
5#23764162C211yesmaximal, characteristic, complemented
6#23764139C211yesmaximal, characteristic, complemented
7#23764127C211yesmaximal, characteristic, complemented
8#23764095C211yesmaximal, characteristic
9#23764084C211yesmaximal, characteristic
10#23764028C211yesmaximal, characteristic
11#23764027C211yesmaximal, characteristic
12#23763982C211yesmaximal, characteristic, complemented
13#23763399C211yesmaximal, characteristic
14#23763256C211yesmaximal, characteristic, complemented
15C79×#1918C211yesmaximal, characteristic, complemented
16C2×#14579152C2211yescharacteristic
17C2×#14579123C2211yescharacteristic
18#19098866C418yescomplemented
19C2×#14579093C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×Dic79⋊C8C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×#14579083C2211yescharacteristic
25C2×#14579083C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579071C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098465C2211yescharacteristic
31#19098464C2211yescharacteristic
32#19098462C2211yescharacteristic
33#19098462C2211yescharacteristic
34#19098461—21no—
35#19098460—21no—
36#19098459C2211yescharacteristic
37#19098455—21no—
38#19098453—21no—
39#19098447—21no—
40#19098446—21no—
41#19098432C2211yescharacteristic
42#19098425—21no—
43#19098420—21no—
44C4⋊C4×Dic79C2211yescharacteristic
45C4×C4⋊Dic79C2211yescharacteristic
46#19098409C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098318C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55C4×C4⋊C4×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579152—22no—
65#14579152C2×C414yes—
66#14579123—22no—
67#14579123C2×C414yes—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79C2311yescharacteristic
86C2×C4×Dic79C2311yescharacteristic
87#14579093—41no—
88#14579092—41no—
89#14579092C2×C414yes—
90Dic79⋊C8—22no—
91Dic79⋊C8C2×C414yes—
92#14579083—41no—
93#14579083C2×C414yes—
94#14579081—41no—
95#14579081C2×C414yes—
96#14579071—41no—
97C8×Dic79—41no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95D411yescharacteristic
104C79⋊#95D411yescharacteristic
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C42×C79—21no—
115C2×C42×C79C2311yescharacteristic
116#107×C79C2×C414yes—
117#95×C79—21no—
118#95×C79—21no—
119C23×D79—21no—
120C2×D316—21no—
121C2×D316—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316C22×C412yes—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79C2×D411yescharacteristic
130C22×Dic79C2×D411yescharacteristic
131C22×Dic79C2×D411yescharacteristic
132C22×Dic79C2×D411yescharacteristic
133C2×C4×D79—24no—
134C2×C4×D79—81no—
135C2×C4×D79—81no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8C22×C412yes—
142C4⋊Dic79—24no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—81no—
149C4⋊Dic79—81no—
150C4⋊Dic79C22×C412yes—
151Dic79⋊C4—24no—
152Dic79⋊C4—81no—
153Dic79⋊C4—81no—
154Dic79⋊C4—81no—
155(C2×D79)⋊C4—24no—
156(C2×D79)⋊C4—81no—
157(C2×D79)⋊C4—81no—
158(C2×D79)⋊C4—81no—
159(C2×D79)⋊C4—81no—
160C4×Dic79—21no—
161C4×Dic79—21no—
162C4×Dic79—81no—
163C4×Dic79—81no—
164C4×Dic79C22×C412yes—
165C4×Dic79#4112yes—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C2×C8×C79—21no—
174C2×C8×C79—21no—
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79C22×C412yes—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—81no—
186C4⋊C4×C79C22×C412yes—
187C4⋊C4×C79C22×C412yes—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—81no—
191C23×C79C22×D411yescharacteristic
192C22×D79—22no—
193C22×D79—41no—
194C22×D79—41no—
195C22×D79—41no—
196C22×D79—41no—
197C22×D79—41no—
198C22×D79—41no—
199C2×C4×C79—22no—
200C2×C4×C79—22no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
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×C79C23×C411yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×C4×C79H(5,𝔽2)11yescharacteristic
220C2×C4×C792−1+411yescharacteristic
221C2×C4×C792−1+411yescharacteristic
222C2×C4×C792−1+411yescharacteristic
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—22no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
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×Dic79C4×D412yes—
243C2×Dic79C4×D412yes—
244D316—22no—
245D316—22no—
246D316—41no—
247D316—41no—
248C4×D79—84no—
249C8×C79—22no—
250C8×C79—41no—
251C79⋊C8—22no—
252C79⋊C8—41no—
253C22×C79C2×C4×D411yescharacteristic
254C22×C79#53611yescharacteristic
255C22×C79#53711yescharacteristic
256C22×C79#57611yescharacteristic
257C22×C79#57711yescharacteristic
258C22×C79#57711yescharacteristic
259C22×C79#57711yescharacteristic
260C2×D79—42no—
261C2×D79—42no—
262C2×D79—42no—
263C2×D79—81no—
264C2×D79—81no—
265C2×D79—81no—
266C2×D79—81no—
267C4×C79—42no—
268C4×C79—42no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79—81no—
272C4×C79—81no—
273C4×C79#51911yescharacteristic
274C4×C79#52011yescharacteristic
275C4×C79#54111yescharacteristic
276C4×C79#54211yescharacteristic
277Dic79—22no—
278Dic79—22no—
279Dic79—42no—
280Dic79—41no—
281Dic79—41no—
282Dic79—81no—
283Dic79—81no—
284#22122—791nomaximal, Sylow
285C2×C79#232311yescharacteristic
286C2×C79#294611yescharacteristic
287C2×C79#295211yescharacteristic
288C2×C79#324111yescharacteristic
289C2×C79#324411yescharacteristic
290C2×C79#325711yescharacteristic
291C2×C79#326011yescharacteristic
292D79—82no—
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#1926—791no—
304#1920—791no—
305#1918D7911yescharacteristic, complemented
306#1802—791no—
307#1768—791no—
308C79#2212211yescharacteristic, complemented, Sylow
309C2×#125—791no—
310C2×C4×D4—791no—
311#478—798no—
312C2×#107C2×D7911yescharacteristic
313C2×#106—791no—
314C2×#106—791no—
315C2×#106—791no—
316C2×C4⋊C8—791no—
317C2×C4⋊C8—791no—
318C2×#102—791no—
319C2×#102—791no—
320C2×#102—791no—
321C2×#102—791no—
322C2×#97—791no—
323C2×C4×C8—791no—
324#390—791no—
325#390—791no—
326#389—791no—
327#389—791no—
328#389—791no—
329#389—1581no—
330#389—1581no—
331#389C2×D7911yescharacteristic
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#340C2×D7911yescharacteristic
349#340C2×D7911yescharacteristic
350#340C2×D7911yescharacteristic
351#336—791no—
352#336C2×D7911yescharacteristic
353C22×D4—791no—
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—21no—
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⋊C4C22×D7911yescharacteristic
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C4⋊C4C22×D7911yescharacteristic
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C22⋊C4—1581no—
379C2×C22⋊C4—1581no—
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#107C4×D7914yes—
390#106—3161no—
391#106—3161no—
392#106—794no—
393C4⋊C8—794no—
394C4⋊C8—1582no—
395#102—3161no—
396#102—3161no—
397#102—794no—
398#102—794no—
399#97—3161no—
400C4×C8—3161no—
401#95—21no—
402#95—21no—
403#95—791no—
404#95—791no—
405#95—791no—
406#95—791no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412C24—1581no—
413C2×D4—792no—
414C2×D4—1581no—
415C2×D4—1581no—
416C2×D4—1581no—
417C2×D4—1581no—
418C2×D4—1581no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—791no—
427C22×C4—6321no—
428C22×C4—6321no—
429C22×C4—1584no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
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—21no—
454C4⋊C4—81no—
455C4⋊C4—792no—
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⋊C4C2×C4×D7912yes—
469C4⋊C4C2×C4×D7912yes—
470C22⋊C4—6321no—
471C22⋊C4—6321no—
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—3161no—
485C23—3161no—
486C23—3161no—
487C23—3161no—
488C23—3161no—
489C23—3161no—
490C23—1582no—
491C23C2×D4×D7911yescharacteristic
492D4—3161no—
493D4—3161no—
494D4—1582no—
495D4—1582no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—792no—
523C2×C4—792no—
524C2×C4—6324no—
525C2×C4—1581no—
526C2×C4—1581no—
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#1457920711yescharacteristic
539C2×C4#1457920811yescharacteristic
540C2×C4#1457920811yescharacteristic
541C2×C4#1457920911yescharacteristic
542C8—22no—
543C8—41no—
544C8—3161no—
545C8—1582no—
546C22—3162no—
547C22—3162no—
548C22—3162no—
549C22—6321no—
550C22—6321no—
551C22—6321no—
552C22—6321no—
553C22C4×D4×D7911yescharacteristic
554C22#1909913511yescharacteristic
555C22#1909914311yescharacteristic
556C22#1909936411yescharacteristic
557C22#1909937011yescharacteristic
558C22#1909937211yescharacteristic
559C22#1909937211yescharacteristic
560C4—42no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—3161no—
567C4—3162no—
568C4—3161no—
569C4—6321no—
570C4—6321no—
571C4—1582no—
572C4—1582no—
573C4#1909903911yescharacteristic
574C4#1909904311yescharacteristic
575C4#1909917211yescharacteristic
576C4#1909917911yescharacteristic
577C2—6322no—
578C2#2376608111yescharacteristic
579C2#2377222511yescharacteristic
580C2#2377228011yescharacteristic
581C2#2377464911yescharacteristic
582C2#2377470411yescharacteristic
583C2#2377481811yescharacteristic
584C2#2377484511yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1918C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1918D79
hypocenterC79#22122
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #23766137C2semidirect
4. #23764356C2semidirect
5. #23764162C2semidirect
6. #23764139C2semidirect
7. #23764127C2semidirect
12. #23763982C2semidirect
14. #23763256C2semidirect
15. C79×#1918C2semidirect
18. #19098866C4semidirect
305. #1918D79semidirect
308. C79#22122semidirect

Maximal subgroups

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

subgroupindex
C2×#190988662
#237661372
#237660392
#237643562
#237641622
#237641392
#237641272
#237640952
#237640842
#237640282
#237640272
#237639822
#237633992
#237632562
C79×#19182
#2212279