Finite Group Databasebeta

← #34259609

Subgroups — #34259609

Order 20224.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773404C211yesmaximal, characteristic
2#23772502C211yesmaximal, characteristic, complemented
3#23772437C211yesmaximal, characteristic
4#23762305C211yesmaximal, characteristic, complemented
5#23762287C211yesmaximal, characteristic, complemented
6#23762275C211yesmaximal, characteristic, complemented
7#23762260C211yesmaximal, characteristic, complemented
8#23762240C211yesmaximal, characteristic, complemented
9#23762239C211yesmaximal, characteristic, complemented
10#23762215C211yesmaximal, characteristic, complemented
11#23762197C211yesmaximal, characteristic, complemented
12#23762181C211yesmaximal, characteristic, complemented
13#23761792C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1497C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17#19098891C411yescharacteristic
18#19098891C411yescharacteristic
19#19098850C411yescharacteristic
20#19098850C411yescharacteristic
21#19098765C411yescharacteristic
22#19098765C411yescharacteristic
23#19098713C411yescharacteristic
24#19098713C411yescharacteristic
25#19098694—21no—
26#19098694C2211yescharacteristic, complemented
27#19098694C2211yescharacteristic, complemented
28Q8×C79⋊C8—21no—
29Q8×C79⋊C8C2211yescharacteristic, complemented
30Q8×C79⋊C8C2211yescharacteristic, complemented
31C4×#11929611C2211yescharacteristic
32C4×#11929611C2211yescharacteristic
33C4×#11929611C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35#19098640C2211yescharacteristic
36#19098640C2211yescharacteristic
37#19098636—21no—
38#19098636—21no—
39#19098628C2211yescharacteristic
40#19098628C2211yescharacteristic
41#19098624—21no—
42#19098624—21no—
43#19098616C2211yescharacteristic
44#19098616C2211yescharacteristic
45#19098616C2211yescharacteristic
46#19098525—21no—
47C22⋊C8×D79—21no—
48#19098315C2211yescharacteristic, complemented
49#19098315C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51#19098313C2211yescharacteristic, complemented
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098285C2211yescharacteristic, complemented
57#19098285C2211yescharacteristic, complemented
58#19098284C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#533×C79C2211yescharacteristic
61#432×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65Q8⋊C8×C79C2211yescharacteristic, complemented
66Q8⋊C8×C79C2211yescharacteristic, complemented
67C22×D316C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929611C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×#11929611C2×C411yescharacteristic
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C4×D316—21no—
78C4×D316—21no—
79C4×D316C2×C411yescharacteristic
80C4×D316C2×C411yescharacteristic
81C4×D316C2311yescharacteristic
82C4×D316C2311yescharacteristic
83C2×C4⋊Dic79C2×C411yescharacteristic
84C2×(C2×D79)⋊C4—21no—
85#14579092C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)—21no—
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)C2×C411yescharacteristic
94C4⋊(C79⋊C8)C2×C411yescharacteristic
95C4⋊(C79⋊C8)D411yescharacteristic
96C4⋊(C79⋊C8)D411yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99#14579085C2×C411yescharacteristic
100#14579085C2×C411yescharacteristic
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579081C2×C411yescharacteristic
104#14579081C2×C411yescharacteristic
105#14579081C2×C411yescharacteristic
106#14579081C2×C411yescharacteristic
107C22⋊(C79⋊C8)—21no—
108C22⋊(C79⋊C8)C2×C411yescharacteristic
109C22⋊(C79⋊C8)C2×C411yescharacteristic
110C22⋊(C79⋊C8)D411yescharacteristic
111C22⋊(C79⋊C8)D411yescharacteristic
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116#14579072—21no—
117#14579072—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8—21no—
120C4×C79⋊C8—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4⋊Q8×C79C2×C411yescharacteristic
126C4⋊Q8×C79C2×C411yescharacteristic
127#125×C79—21no—
128C22⋊Q8×C79—21no—
129C22⋊Q8×C79C2×C411yescharacteristic
130C22⋊Q8×C79C2×C411yescharacteristic
131C4×Q8×C79—21no—
132C4×Q8×C79C2311yescharacteristic
133C4×Q8×C79C2311yescharacteristic
134#117×C79—21no—
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79—21no—
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C23×D79—21no—
145C2×D316—22no—
146C2×D316—21no—
147C2×D316—41no—
148C2×D316C22×C411yescharacteristic
149C2×D316C22×C411yescharacteristic
150C22×Dic79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—81no—
156C2×C4×D79—81no—
157#11929611—22no—
158#11929611—22no—
159#11929611—41no—
160#11929611—41no—
161#11929607—81no—
162#11929607—81no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C8×D79—81no—
174C8×D79—81no—
175C4⋊Dic79—21no—
176C4⋊Dic79C22×C411yescharacteristic
177C4⋊Dic79C22×C411yescharacteristic
178(C2×D79)⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182C2×Q8×C79—21no—
183C2×Q8×C79C22×C411yescharacteristic
184C2×Q8×C79C22×C411yescharacteristic
185C22×C4×C79C22×C411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79#4111yescharacteristic
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79C22×C411yescharacteristic
200C4⋊C4×C79C22×C411yescharacteristic
201C22⋊C4×C79—21no—
202C22⋊C4×C79—21no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79C4×D411yescharacteristic
210C22×D79—21no—
211C22×D79—21no—
212C22×D79—42no—
213C22×D79—42no—
214C22×D79—81no—
215Q8×C79—21no—
216Q8×C79—21no—
217Q8×C79—21no—
218Q8×C79—21no—
219Q8×C79—42no—
220C2×C4×C79—21no—
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×C79C4×D411yescharacteristic
230C2×C4×C79C4×D411yescharacteristic
231C2×C4×C79C4×D411yescharacteristic
232C2×C4×C79C23×C411yescharacteristic
233C2×C4×C79C22×D411yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—81no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—42no—
242D316#13112yes—
243C4×D79—41no—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—81no—
248C4×D79—81no—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79—41no—
252C8×C79—41no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—21no—
264C22×C79—21no—
265C22×C79—21no—
266C22×C79C2×C4×D411yescharacteristic
267C2×D79—22no—
268C2×D79—41no—
269C2×D79—41no—
270C2×D79—41no—
271C2×D79—41no—
272C2×D79—41no—
273C2×D79—84no—
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79C2×#13111yescharacteristic
283C4×C79#56811yescharacteristic
284C4×C79#57711yescharacteristic
285C4×C79#57711yescharacteristic
286Dic79—41no—
287Dic79—41no—
288#18028—791nomaximal, Sylow
289C2×C79—41no—
290C2×C79#293311yescharacteristic
291C2×C79#294911yescharacteristic
292C2×C79#295511yescharacteristic
293D79—42no—
294D79—81no—
295#3101—791no—
296#2986—791no—
297#2974—791no—
298#1592—791no—
299#1590—791no—
300#1588—791no—
301#1585—791no—
302#1583—791no—
303#1582—791no—
304SD16⋊C8—791no—
305C8×SD16—791no—
306#1572—791no—
307#1497D7911yescharacteristic, complemented
308#1478—791no—
309#1454—791no—
310C79#1802811yescharacteristic, complemented, Sylow
311#533C2×D7911yescharacteristic
312C2×C4×D4—791no—
313#483—791no—
314#483—791no—
315#476—791no—
316#476—791no—
317#461—791no—
318#461—791no—
319#450—791no—
320#450—791no—
321#446—791no—
322#446—791no—
323#446—1581no—
324C8×Q8—791no—
325C8×Q8—791no—
326C8×Q8—1581no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330C4×SD16—791no—
331#436—791no—
332#436—791no—
333#435—1581no—
334#435—1581no—
335C8×D4—791no—
336C8×D4—791no—
337#433—1581no—
338#433—1581no—
339#432C2×D7911yescharacteristic
340#431—791no—
341#431—791no—
342#431—791no—
343#431C2×D7911yescharacteristic
344#407—1581no—
345C2×C22⋊C8—1581no—
346#335—791no—
347#335—791no—
348#334—791no—
349#334—791no—
350Q8⋊C8C2×D7911yescharacteristic, complemented
351Q8⋊C8C2×D7911yescharacteristic, complemented
352Q8⋊C8C2×D7911yescharacteristic, complemented
353Q8⋊C8C2×D7911yescharacteristic, complemented
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8—791no—
358#322—791no—
359#322—791no—
360#322—791no—
361C82—791no—
362C22×D4—791no—
363C23×C4—1581no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16—791no—
368C2×M16—1581no—
369C2×M16—1581no—
370C22×C8—1581no—
371C22×C8—1581no—
372C4⋊Q8C4×D7911yescharacteristic
373C4⋊Q8C4×D7911yescharacteristic
374#125—21no—
375C22⋊Q8—21no—
376C22⋊Q8C4×D7911yescharacteristic
377C22⋊Q8C4×D7911yescharacteristic
378C4×Q8—21no—
379C4×Q8C22×D7911yescharacteristic
380C4×Q8C22×D7911yescharacteristic
381C4×D4—791no—
382C4×D4—791no—
383C4×D4—791no—
384C4×D4—791no—
385C4×D4—1581no—
386C4×D4—1581no—
387#117—21no—
388C2×C4⋊C4—791no—
389C2×C22⋊C4—1581no—
390C2×C42C22×D7911yescharacteristic
391#106—791no—
392#106—791no—
393#106—791no—
394#106—791no—
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⋊C8—1581no—
406C4⋊C8C22×D7911yescharacteristic
407C4⋊C8C22×D7911yescharacteristic
408#103—791no—
409#103—791no—
410#103—791no—
411#103—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415#102—791no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—791no—
421C22⋊C8—791no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427#97—21no—
428#97—1581no—
429#97—1581no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—1581no—
435C4×C8—1581no—
436C4×C8—1581no—
437C4×C8C22×D7911yescharacteristic
438C4×C8C22×D7911yescharacteristic
439C24—1581no—
440C2×Q8—21no—
441C2×Q8C2×C4×D7911yescharacteristic
442C2×Q8C2×C4×D7911yescharacteristic
443C2×D4—3161no—
444C2×D4—791no—
445C2×D4—791no—
446C2×D4—1582no—
447C2×D4—1581no—
448C22×C4—6321no—
449C22×C4—6321no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4C2×C4×D7911yescharacteristic
456C22×C4D4×D7911yescharacteristic
457C22×C4#1192962311yescharacteristic
458SD16—3161no—
459SD16—3161no—
460SD16—1582no—
461SD16—1582no—
462M16—6321no—
463M16—6321no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—791no—
473C2×C8—791no—
474C2×C8—791no—
475C2×C8—791no—
476C2×C8—6321no—
477C2×C8—6321no—
478C2×C8—1581no—
479C2×C8—1581no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—791no—
488C4⋊C4—791no—
489C4⋊C4—1581no—
490C4⋊C4C2×C4×D7911yescharacteristic
491C4⋊C4C2×C4×D7911yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—1581no—
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C22⋊C4—1581no—
498C42—21no—
499C42—21no—
500C42C2×C4×D7911yescharacteristic
501C42D4×D7911yescharacteristic
502C42#1192962311yescharacteristic
503C42C23×D7911yescharacteristic
504C23—3162no—
505C23—3162no—
506C23—6321no—
507C23—1581no—
508C23—1581no—
509C23(C22⋊D79)⋊C411yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—42no—
515D4—3162no—
516D4—792no—
517D4—1581no—
518D4—1581no—
519D4—1581no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—6321no—
534C2×C4—6321no—
535C2×C4—6321no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4(C22⋊D79)⋊C411yescharacteristic
539C2×C4(C22⋊D79)⋊C411yescharacteristic
540C2×C4(C22⋊D79)⋊C411yescharacteristic
541C2×C4C22×C4×D7911yescharacteristic
542C2×C4C2×D4×D7911yescharacteristic
543C2×C4C2×#1192962311yescharacteristic
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—3161no—
549C8—3161no—
550C8—1581no—
551C8—1581no—
552C8—1581no—
553C8—1581no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C8—1581no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—3161no—
562C22—3161no—
563C22—3161no—
564C22—3161no—
565C22—3161no—
566C22—6324no—
567C22—1582no—
568C22C2×(C22⋊D79)⋊C411yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—3161no—
578C4—3161no—
579C4#131×D7911yescharacteristic
580C4#1909932811yescharacteristic
581C4#1909937211yescharacteristic
582C4#1909937211yescharacteristic
583C2—41no—
584C2—3162no—
585C2—6321no—
586C2#2377213011yescharacteristic
587C2#2377225511yescharacteristic
588C2#2377229911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1497C2
socleC22×C79C2×C4×D4
prime radicalC22×D316C2×C4
cosocleC42×C79C24
hypercenter#1497D79
hypocenterC79#18028
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
2. #23772502C2semidirect
4. #23762305C2semidirect
5. #23762287C2semidirect
6. #23762275C2semidirect
7. #23762260C2semidirect
8. #23762240C2semidirect
9. #23762239C2semidirect
10. #23762215C2semidirect
11. #23762197C2semidirect
12. #23762181C2semidirect
14. #23761700C2semidirect
15. C79×#1497C2semidirect
26. #19098694C22semidirect
27. #19098694C22semidirect
29. Q8×C79⋊C8C22semidirect
30. Q8×C79⋊C8C22semidirect
48. #19098315C22semidirect
49. #19098315C22semidirect
50. #19098313C22semidirect
51. #19098313C22semidirect
56. #19098285C22semidirect
57. #19098285C22semidirect
58. #19098284C22semidirect
59. C8×C79⋊C8C22semidirect
63. Q8⋊C8×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
65. Q8⋊C8×C79C22semidirect
66. Q8⋊C8×C79C22semidirect
307. #1497D79semidirect
310. C79#18028semidirect
350. Q8⋊C8C2×D79semidirect
351. Q8⋊C8C2×D79semidirect
352. Q8⋊C8C2×D79semidirect
353. Q8⋊C8C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734042
#237725022
#237724372
#237623052
#237622872
#237622752
#237622602
#237622402
#237622392
#237622152
#237621972
#237621812
#237617922
#237617002
C79×#14972
#1802879