Finite Group Databasebeta

← #34275892

Subgroups — #34275892

Order 20224.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774618C211yesmaximal, characteristic, complemented
2#23774518C211yesmaximal, characteristic, complemented
3#23772560C211yesmaximal, characteristic, complemented
4#23762938C211yesmaximal, characteristic, complemented
5#23762931C211yesmaximal, characteristic, complemented
6#23762926C211yesmaximal, characteristic, complemented
7#23762921C211yesmaximal, characteristic, complemented
8#23762814C211yesmaximal, characteristic, complemented
9#23762807C211yesmaximal, characteristic, complemented
10#23762802C211yesmaximal, characteristic, complemented
11#23762797C211yesmaximal, characteristic, complemented
12#23762183C211yesmaximal, characteristic, complemented
13#23762095C211yesmaximal, characteristic, complemented
14C2×#19098311C211yesmaximal, characteristic, complemented
15C79×#3247C211yesmaximal, characteristic
16C2×#14579161C2211yescharacteristic
17#19098973—21no—
18#19098966C2211yescharacteristic
19#19098966C2211yescharacteristic
20#19098922—21no—
21#19098909C2211yescharacteristic
22#19098909C2211yescharacteristic
23#19098891—21no—
24#19098874—21no—
25#19098874—21no—
26#19098874—21no—
27#19098858C2211yescharacteristic
28#19098858C2211yescharacteristic
29#19098853C2211yescharacteristic
30#19098853C2211yescharacteristic
31#19098843—21no—
32#19098843—21no—
33#19098694C2211yescharacteristic
34Q8×C79⋊C8C2211yescharacteristic
35#19098641C2211yescharacteristic
36#19098635—21no—
37D4×C79⋊C8—21no—
38D4×C79⋊C8C2211yescharacteristic
39#19098624—21no—
40#19098620C2211yescharacteristic
41#19098616C2211yescharacteristic
42#19098590—21no—
43C2×C4⋊(C79⋊C8)C2211yescharacteristic
44#19098315C2211yescharacteristic
45#19098315C2211yescharacteristic
46#19098313C2211yescharacteristic
47#19098313C2211yescharacteristic
48#19098312C2211yescharacteristic
49#19098312C2211yescharacteristic
50#19098311C2211yescharacteristic
51#19098311C2211yescharacteristic
52#19098311C2211yescharacteristic
53#19098311C2211yescharacteristic
54#19098307C2211yescharacteristic
55#19098307C2211yescharacteristic
56#532×C79C2211yescharacteristic
57#529×C79C2211yescharacteristic
58#478×C79—21no—
59#477×C79C2211yescharacteristic
60#476×C79C2211yescharacteristic
61#466×C79—21no—
62#465×C79—21no—
63#463×C79—21no—
64#460×C79C2211yescharacteristic
65#459×C79C2211yescharacteristic
66C2×C4⋊C8×C79C2211yescharacteristic
67C22×Dic158—21no—
68C2×#11929608—21no—
69C22×C79⋊C8—21no—
70C22×C79⋊C8—21no—
71#14579161—21no—
72#14579161—21no—
73#14579161—21no—
74#14579161C2311yescharacteristic
75#14579161C2311yescharacteristic
76C2×C4⋊Dic79—21no—
77C2×C4⋊Dic79—21no—
78#14579094—21no—
79#14579094—21no—
80#14579094—21no—
81#14579094—21no—
82#14579094—21no—
83#14579094—21no—
84#14579092—21no—
85#14579092—21no—
86C4⋊(C79⋊C8)—21no—
87C4⋊(C79⋊C8)—21no—
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)D411yescharacteristic
90C4⋊(C79⋊C8)D411yescharacteristic
91C4⋊(C79⋊C8)C2311yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95#14579085—21no—
96#14579085—21no—
97#14579082—21no—
98#14579082—21no—
99#14579082—21no—
100#14579082—21no—
101#14579082—21no—
102#14579082—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)—21no—
105C22⋊(C79⋊C8)—21no—
106C22⋊(C79⋊C8)D411yescharacteristic
107C22⋊(C79⋊C8)D411yescharacteristic
108#14579072—21no—
109#14579072—21no—
110C4×C79⋊C8—21no—
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C2×#41×C79—21no—
114C2×SD16×C79—21no—
115C2×D8×C79—21no—
116C22×C8×C79—21no—
117C4⋊Q8×C79C2311yescharacteristic
118#127×C79C2311yescharacteristic
119#126×C79—21no—
120#125×C79D411yescharacteristic
121#124×C79—21no—
122#124×C79D411yescharacteristic
123#123×C79—21no—
124C22⋊Q8×C79D411yescharacteristic
125#121×C79—21no—
126#121×C79—21no—
127#121×C79D411yescharacteristic
128C4×Q8×C79C2311yescharacteristic
129C4×D4×C79—21no—
130C4×D4×C79—21no—
131C4×D4×C79—21no—
132C4×D4×C79C2311yescharacteristic
133#117×C79—21no—
134C2×C42×C79C2311yescharacteristic
135#107×C79—21no—
136#106×C79—21no—
137C4⋊C8×C79—21no—
138C4⋊C8×C79C2311yescharacteristic
139C4⋊C8×C79C2311yescharacteristic
140#103×C79—21no—
141#102×C79—21no—
142#102×C79—21no—
143#102×C79—21no—
144C2×Dic158—21no—
145C2×Dic158—21no—
146C2×Dic158—41no—
147C2×Dic158—82no—
148C22×Dic79—41no—
149#11929608—81no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—81no—
155C2×C79⋊C8—81no—
156C2×C79⋊C8C2×D411yescharacteristic
157C2×C79⋊C8C2×D411yescharacteristic
158C2×C79⋊C8C2×D411yescharacteristic
159C2×C79⋊C8C2×D411yescharacteristic
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—41no—
165C4⋊Dic79—41no—
166#41×C79—82no—
167C2×Q8×C79—21no—
168C2×Q8×C79C2×D411yescharacteristic
169C2×D4×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79—41no—
174C2×D4×C79C2×D411yescharacteristic
175C22×C4×C79—21no—
176C22×C4×C79—41no—
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79#4111yescharacteristic
180SD16×C79—81no—
181D8×C79—81no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—81no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79C2×D411yescharacteristic
192C4⋊C4×C79C2×D411yescharacteristic
193C22⋊C4×C79—21no—
194C22⋊C4×C79—21no—
195C22⋊C4×C79—21no—
196C22⋊C4×C79—21no—
197C22⋊C4×C79—21no—
198C22⋊C4×C79—41no—
199C22⋊C4×C79—41no—
200C42×C79—21no—
201C42×C79C2×D411yescharacteristic
202C42×C79C2×D411yescharacteristic
203C42×C79#4111yescharacteristic
204C42×C79C2411yescharacteristic
205C23×C79—21no—
206C23×C79—41no—
207C23×C79#12111yescharacteristic
208Q8×C79—21no—
209Q8×C79—21no—
210Q8×C79—81no—
211D4×C79—21no—
212D4×C79—21no—
213D4×C79—42no—
214D4×C79—41no—
215D4×C79—41no—
216D4×C79—41no—
217D4×C79—41no—
218D4×C79—81no—
219D4×C79—82no—
220C2×C4×C79—22no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—82no—
232C2×C4×C79#12111yescharacteristic
233C2×C4×C79#12111yescharacteristic
234C2×C4×C79#12111yescharacteristic
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C22×D411yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—82no—
241Dic158—82no—
242Dic158—81no—
243Dic158—81no—
244C8×C79—41no—
245C8×C79—41no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—41no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—81no—
260C22×C79—81no—
261C22×C79—81no—
262C22×C79#47212yes—
263C22×C79C2×#12111yescharacteristic
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—21no—
267C4×C79—21no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79—81no—
272C4×C79#57711yescharacteristic
273C4×C79#57711yescharacteristic
274C4×C79#57811yescharacteristic
275C4×C79#57811yescharacteristic
276Dic79—81no—
277Dic79—81no—
278#19591—791nomaximal, Sylow
279C2×C79—22no—
280C2×C79—81no—
281C2×C79—161no—
282C2×C79#304611yescharacteristic
283C2×C79#306311yescharacteristic
284C2×C79C2×#47211yescharacteristic
285#3247D7911yescharacteristic
286#3238—791no—
287#3226—791no—
288#2994—791no—
289#1698—791no—
290#1697—791no—
291#1696—791no—
292#1695—791no—
293#1678—791no—
294#1677—791no—
295#1676—791no—
296#1675—791no—
297#1573—791no—
298#1551—791no—
299C2×#333—791no—
300C79#1959111yescharacteristic, complemented, Sylow
301#532C2×D7911yescharacteristic
302C2×C4⋊Q8—791no—
303#529C2×D7911yescharacteristic
304#501—1581no—
305#500—791no—
306#500—791no—
307#489—1581no—
308#486—791no—
309#486—791no—
310#483—1581no—
311#480—1581no—
312#480—1581no—
313#480—1581no—
314#478—21no—
315#477—791no—
316#477—791no—
317#477C2×D7911yescharacteristic
318#476—791no—
319#476—791no—
320#476C2×D7911yescharacteristic
321#474—1581no—
322#474—1581no—
323#466—21no—
324#465—21no—
325#463—21no—
326#460C2×D7911yescharacteristic
327#459C2×D7911yescharacteristic
328#446—791no—
329C8×Q8—791no—
330#436—791no—
331#435—1581no—
332C8×D4—791no—
333C8×D4—1581no—
334#433—1581no—
335#432—791no—
336#431—791no—
337#424—1581no—
338C2×C4⋊C8—791no—
339C2×C4⋊C8C2×D7911yescharacteristic
340#335—791no—
341#335—791no—
342#334—791no—
343#334—791no—
344#333—791no—
345#333—791no—
346#333—791no—
347#333—791no—
348#333—791no—
349#333—791no—
350#331—791no—
351#331—791no—
352C2×#41—21no—
353C22×Q8—1581no—
354C2×SD16—21no—
355C2×D8—21no—
356C2×M16—1581no—
357C22×C8—21no—
358C22×C8—1581no—
359C22×C8—1581no—
360C4⋊Q8—791no—
361C4⋊Q8—791no—
362C4⋊Q8—1581no—
363C4⋊Q8—1581no—
364C4⋊Q8—1581no—
365C4⋊Q8C22×D7911yescharacteristic
366#127C22×D7911yescharacteristic
367#126—21no—
368#125C22⋊D7911yescharacteristic
369#124—21no—
370#124C22⋊D7911yescharacteristic
371#123—21no—
372C22⋊Q8C22⋊D7911yescharacteristic
373#121—21no—
374#121—21no—
375#121C22⋊D7911yescharacteristic
376C4×Q8C22×D7911yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22×D7911yescharacteristic
381#117—21no—
382C2×C4⋊C4—1581no—
383C2×C4⋊C4—1581no—
384C2×C42C22×D7911yescharacteristic
385#107—21no—
386#107—1581no—
387#107—1581no—
388#107—1581no—
389#107—1581no—
390#107—1581no—
391#107—1581no—
392#106—21no—
393#106—1581no—
394#106—1581no—
395C4⋊C8—21no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—791no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8—1581no—
405C4⋊C8C22×D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407#103—21no—
408#103—1581no—
409#103—1581no—
410#102—21no—
411#102—21no—
412#102—21no—
413#102—1581no—
414#102—1581no—
415#102—1581no—
416#102—1581no—
417#102—1581no—
418#102—1581no—
419C22⋊C8—791no—
420C22⋊C8—791no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424#97—1581no—
425#97—1581no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—1581no—
429#41—82no—
430C2×Q8—21no—
431C2×Q8—3161no—
432C2×Q8—6322no—
433C2×Q8—1581no—
434C2×Q8—1581no—
435C2×Q8C2×C22⋊D7911yescharacteristic
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4C2×C22⋊D7911yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—3161no—
445C22×C4D4×D7911yescharacteristic
446C22×C4C2×C22⋊D7911yescharacteristic
447C22×C4#1192962311yescharacteristic
448SD16—81no—
449D8—81no—
450M16—6321no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—81no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—6321no—
459C2×C8—6321no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—3161no—
471C4⋊C4—3161no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4C2×C22⋊D7911yescharacteristic
477C4⋊C4C2×C22⋊D7911yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C42—21no—
486C42D4×D7911yescharacteristic
487C42C2×C22⋊D7911yescharacteristic
488C42#1192962311yescharacteristic
489C42C23×D7911yescharacteristic
490C23—21no—
491C23—41no—
492C23#1457913311yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—81no—
496Q8—6321no—
497Q8—6321no—
498Q8—6322no—
499D4—21no—
500D4—21no—
501D4—42no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—81no—
507D4—82no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—6322no—
523C2×C4#1457913311yescharacteristic
524C2×C4#1457913311yescharacteristic
525C2×C4#1457913311yescharacteristic
526C2×C4C2×D4×D7911yescharacteristic
527C2×C4C22×C22⋊D7911yescharacteristic
528C2×C4C2×#1192962311yescharacteristic
529C8—41no—
530C8—41no—
531C8—3161no—
532C8—3161no—
533C8—3161no—
534C8—3161no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C8—1581no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#1909883112yes—
548C22C2×#1457913311yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—6321no—
558C4—6321no—
559C4#1909937011yescharacteristic
560C4#1909937411yescharacteristic
561C4#1909937711yescharacteristic
562C4#1909937811yescharacteristic
563C2—22no—
564C2—81no—
565C2—161no—
566C2#2377294811yescharacteristic
567C2#2377309311yescharacteristic
568C2C2×#1909883111yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
Fitting, prime radicalC79×#3247C2
socleC22×C79C2×#121
hypercenter#3247D79
hypocenterC79#19591
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3247

Complements

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

NH ≅ G/Ntype
1. #23774618C2semidirect
2. #23774518C2semidirect
3. #23772560C2semidirect
4. #23762938C2semidirect
5. #23762931C2semidirect
6. #23762926C2semidirect
7. #23762921C2semidirect
8. #23762814C2semidirect
9. #23762807C2semidirect
10. #23762802C2semidirect
11. #23762797C2semidirect
12. #23762183C2semidirect
13. #23762095C2semidirect
14. C2×#19098311C2semidirect
300. C79#19591semidirect

Maximal subgroups

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

subgroupindex
#237746182
#237745182
#237725602
#237629382
#237629312
#237629262
#237629212
#237628142
#237628072
#237628022
#237627972
#237621832
#237620952
C2×#190983112
C79×#32472
#1959179