Finite Group Databasebeta

← #34259599

Subgroups — #34259599

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#23772434C211yesmaximal, characteristic
3#23762300C211yesmaximal, characteristic, complemented
4#23762290C211yesmaximal, characteristic, complemented
5#23762278C211yesmaximal, characteristic, complemented
6#23762255C211yesmaximal, characteristic, complemented
7#23762243C211yesmaximal, characteristic, complemented
8#23762234C211yesmaximal, characteristic, complemented
9#23762210C211yesmaximal, characteristic, complemented
10C8×#11929611C211yesmaximal, characteristic, complemented
11#23762178C211yesmaximal, characteristic, complemented
12#23761873C211yesmaximal, characteristic, complemented
13#23761790C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic, complemented
15C79×#2986C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17#19098891C411yescharacteristic
18#19098891C411yescharacteristic
19#19098850C411yescharacteristic
20#19098850C411yescharacteristic
21#19098765C411yescharacteristic
22#19098765C411yescharacteristic
23#19098713C411yescharacteristic
24#19098713C411yescharacteristic
25C4×#11929611C2211yescharacteristic
26C4×#11929611C2211yescharacteristic
27C4×#11929611C2211yescharacteristic
28C4×#11929611C2211yescharacteristic
29#19098642C2211yescharacteristic
30#19098642C2211yescharacteristic
31#19098633—21no—
32#19098633—21no—
33C8×D316C2211yescharacteristic
34C8×D316C2211yescharacteristic
35#19098620C2211yescharacteristic
36#19098616C2211yescharacteristic
37#19098616C2211yescharacteristic
38#19098525—21no—
39C22⋊C8×D79—21no—
40#19098315C2211yescharacteristic, complemented
41#19098315C2211yescharacteristic, complemented
42#19098313C2211yescharacteristic, complemented
43#19098313C2211yescharacteristic, complemented
44Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
45Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
46Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
47Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
48#19098290C2211yescharacteristic
49#19098290C2211yescharacteristic
50#19098290C2211yescharacteristic
51#19098290C2211yescharacteristic
52#19098285C2211yescharacteristic, complemented
53#19098285C2211yescharacteristic, complemented
54#19098284C2211yescharacteristic, complemented
55C8×C79⋊C8C2211yescharacteristic, complemented
56#533×C79C2211yescharacteristic
57#446×C79—21no—
58#446×C79C2211yescharacteristic, complemented
59#446×C79C2211yescharacteristic, complemented
60C8×Q8×C79—21no—
61C8×Q8×C79C2211yescharacteristic, complemented
62C8×Q8×C79C2211yescharacteristic, complemented
63#433×C79—21no—
64#433×C79—21no—
65#431×C79C2211yescharacteristic
66#431×C79C2211yescharacteristic
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)C2×C411yescharacteristic
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94#14579085C2×C411yescharacteristic
95#14579085C2×C411yescharacteristic
96#14579085C2×C411yescharacteristic
97#14579085C2×C411yescharacteristic
98#14579081C2×C411yescharacteristic
99#14579081C2×C411yescharacteristic
100#14579081C2×C411yescharacteristic
101#14579081C2×C411yescharacteristic
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)C2×C411yescharacteristic
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106(C2×D79)⋊C8—21no—
107(C2×D79)⋊C8—21no—
108(C2×D79)⋊C8—21no—
109(C2×D79)⋊C8—21no—
110#14579072—21no—
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C4×C79⋊C8C2311yescharacteristic
114C4×C79⋊C8C2311yescharacteristic
115C4⋊Q8×C79C2×C411yescharacteristic
116C4⋊Q8×C79C2×C411yescharacteristic
117#125×C79—21no—
118C22⋊Q8×C79—21no—
119C22⋊Q8×C79C2×C411yescharacteristic
120C22⋊Q8×C79C2×C411yescharacteristic
121C4×Q8×C79—21no—
122C4×Q8×C79C2311yescharacteristic
123C4×Q8×C79C2311yescharacteristic
124#117×C79—21no—
125C2×C42×C79C2311yescharacteristic
126C4⋊C8×C79—21no—
127C4⋊C8×C79—21no—
128C4⋊C8×C79—21no—
129C4⋊C8×C79—21no—
130C4⋊C8×C79D411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79D411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137#97×C79—21no—
138#97×C79—21no—
139C4×C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×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⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C22×C411yescharacteristic
171C8×D79—81no—
172C8×D79—81no—
173C4⋊Dic79—21no—
174C4⋊Dic79C22×C411yescharacteristic
175C4⋊Dic79C22×C411yescharacteristic
176(C2×D79)⋊C4—21no—
177(C2×D79)⋊C4—21no—
178(C2×D79)⋊C4—21no—
179(C2×D79)⋊C4—21no—
180C2×Q8×C79—21no—
181C2×Q8×C79C22×C411yescharacteristic
182C2×Q8×C79C22×C411yescharacteristic
183C22×C4×C79C22×C411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79#4111yescharacteristic
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79C2×D411yescharacteristic
189C2×C8×C79C2×D411yescharacteristic
190C2×C8×C79C2×D411yescharacteristic
191C2×C8×C79C2×D411yescharacteristic
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—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—41no—
254C8×C79—41no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—41no—
260C79⋊C8—41no—
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#2986D7911yescharacteristic, complemented
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#1497—791no—
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—21no—
322#446C2×D7911yescharacteristic, complemented
323#446C2×D7911yescharacteristic, complemented
324C8×Q8—21no—
325C8×Q8C2×D7911yescharacteristic, complemented
326C8×Q8C2×D7911yescharacteristic, complemented
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—21no—
338#433—21no—
339#432—791no—
340#431—791no—
341#431—791no—
342#431C2×D7911yescharacteristic
343#431C2×D7911yescharacteristic
344#407—1581no—
345C2×C22⋊C8—1581no—
346#335—791no—
347#335—791no—
348#334—791no—
349#334—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353Q8⋊C8—791no—
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—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—791no—
402C4⋊C8—791no—
403C4⋊C8—1581no—
404C4⋊C8C22⋊D7911yescharacteristic
405C4⋊C8C22⋊D7911yescharacteristic
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—791no—
418C22⋊C8—791no—
419C22⋊C8—1581no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8C22⋊D7911yescharacteristic
426C22⋊C8C22⋊D7911yescharacteristic
427#97—21no—
428#97—21no—
429#97—1581no—
430C4×C8—21no—
431C4×C8—21no—
432C4×C8—21no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—791no—
436C4×C8—791no—
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×C4C2×C22⋊D7911yescharacteristic
457C22×C4#1192962411yescharacteristic
458SD16—3161no—
459SD16—3161no—
460SD16—1582no—
461SD16—1582no—
462M16—6321no—
463M16—6321no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—6321no—
471C2×C8—6321no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8C2×C22⋊D7911yescharacteristic
477C2×C8C2×C22⋊D7911yescharacteristic
478C2×C8C2×C22⋊D7911yescharacteristic
479C2×C8C2×C22⋊D7911yescharacteristic
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
501C42C2×C22⋊D7911yescharacteristic
502C42#1192962411yescharacteristic
503C42C23×D7911yescharacteristic
504C23—3162no—
505C23—3162no—
506C23—6321no—
507C23—1581no—
508C23—1581no—
509C23C4×C22⋊D7911yescharacteristic
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×C4C4×C22⋊D7911yescharacteristic
539C2×C4C4×C22⋊D7911yescharacteristic
540C2×C4C4×C22⋊D7911yescharacteristic
541C2×C4C22×C4×D7911yescharacteristic
542C2×C4C22×C22⋊D7911yescharacteristic
543C2×C4C2×#1192962411yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—3161no—
551C8—3161no—
552C8—3161no—
553C8—3161no—
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×C4×C22⋊D7911yescharacteristic
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#1909937111yescharacteristic
582C4#1909937111yescharacteristic
583C2—41no—
584C2—3162no—
585C2—6321no—
586C2#2377212211yescharacteristic
587C2#2377225911yescharacteristic
588C2#2377230111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2986C2
socleC22×C79C2×C4×D4
prime radicalC22×D316C2×C4
cosocleC42×C79C24
hypercenter#2986D79
hypocenterC79#18028
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2986

Complements

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

NH ≅ G/Ntype
3. #23762300C2semidirect
4. #23762290C2semidirect
5. #23762278C2semidirect
6. #23762255C2semidirect
7. #23762243C2semidirect
8. #23762234C2semidirect
9. #23762210C2semidirect
10. C8×#11929611C2semidirect
11. #23762178C2semidirect
12. #23761873C2semidirect
14. #23761700C2semidirect
15. C79×#2986C2semidirect
40. #19098315C22semidirect
41. #19098315C22semidirect
42. #19098313C22semidirect
43. #19098313C22semidirect
44. Q8⋊(C79⋊C8)C22semidirect
45. Q8⋊(C79⋊C8)C22semidirect
46. Q8⋊(C79⋊C8)C22semidirect
47. Q8⋊(C79⋊C8)C22semidirect
52. #19098285C22semidirect
53. #19098285C22semidirect
54. #19098284C22semidirect
55. C8×C79⋊C8C22semidirect
58. #446×C79C22semidirect
59. #446×C79C22semidirect
61. C8×Q8×C79C22semidirect
62. C8×Q8×C79C22semidirect
296. #2986D79semidirect
310. C79#18028semidirect
322. #446C2×D79semidirect
323. #446C2×D79semidirect
325. C8×Q8C2×D79semidirect
326. C8×Q8C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734042
#237724342
#237623002
#237622902
#237622782
#237622552
#237622432
#237622342
#237622102
C8×#119296112
#237621782
#237618732
#237617902
#237617002
C79×#29862
#1802879