Finite Group Databasebeta

← #34288847

Subgroups — #34288847

Order 20224.

810 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
1C2×#19098520C211yesmaximal, characteristic
2#23766010C211yesmaximal, characteristic
3#23766008C211yesmaximal, characteristic
4#23763568C211yesmaximal, characteristic, complemented
5#23763530C211yesmaximal, characteristic, complemented
6#23763527C211yesmaximal, characteristic, complemented
7#23763434C211yesmaximal, characteristic
8#23763430C211yesmaximal, characteristic
9#23763426C211yesmaximal, characteristic
10#23763422C211yesmaximal, characteristic
11#23763300C211yesmaximal, characteristic, complemented
12#23763299C211yesmaximal, characteristic, complemented
13#107×Dic79C211yesmaximal, characteristic, complemented
14#106×Dic79C211yesmaximal, characteristic, complemented
15C79×#1847C211yesmaximal, characteristic, complemented
16C22×#11929607C2211yescharacteristic
17C22×C8×D79C2211yescharacteristic
18C2×#14579118C2211yescharacteristic
19C2×#14579095C2211yescharacteristic
20C2×#14579095C2211yescharacteristic
21C2×#14579093C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23#19098520C418yescomplemented
24C2×#14579071C2211yescharacteristic
25C2×C4×C79⋊C8C2211yescharacteristic
26C2×C8×Dic79C2211yescharacteristic
27#19098455—21no—
28#19098455—21no—
29#19098453—21no—
30#19098453—21no—
31#19098450—21no—
32#19098450—21no—
33#19098444—21no—
34#19098444—21no—
35#19098433C2211yescharacteristic
36#19098429C2211yescharacteristic
37#19098420C2211yescharacteristic
38#19098420C2211yescharacteristic
39#19098420C2211yescharacteristic
40#19098420C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4⋊C4×Dic79C2211yescharacteristic
44C4⋊C4×Dic79C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54C2×#107×C79C2211yescharacteristic
55C2×#106×C79C2211yescharacteristic
56C2×#106×C79C2211yescharacteristic
57C2×#97×C79C2211yescharacteristic
58#384×C79C2211yescharacteristic
59#382×C79C2211yescharacteristic
60C22×C4×D79C2311yescharacteristic
61C2×#11929607—22no—
62C2×#11929607—22no—
63C2×#11929607C2×C414yes—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C2×C8×D79—22no—
67C2×C8×D79—22no—
68C2×C8×D79C2×C414yes—
69#14579118C2×C414yes—
70#14579118Q814yes—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4D411yescharacteristic
78C2×Dic79⋊C4D411yescharacteristic
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×(C2×D79)⋊C4D411yescharacteristic
84C2×(C2×D79)⋊C4D411yescharacteristic
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79C2311yescharacteristic
90#14579095—22no—
91#14579095—22no—
92#14579093—22no—
93#14579093—22no—
94#14579071C2×C414yes—
95C4×C79⋊C8C2×C414yes—
96C8×Dic79C2×C414yes—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C22×C8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
116#107×C79—22no—
117#107×C79—22no—
118#106×C79—22no—
119#106×C79—22no—
120#97×C79C2×C414yes—
121#95×C79—21no—
122#95×C79—21no—
123C23×D79C2×D411yescharacteristic
124C22×Dic79—21no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79C2×D411yescharacteristic
129C22×Dic79C2×D411yescharacteristic
130C22×Dic79C2×D411yescharacteristic
131C2×C4×D79—21no—
132C2×C4×D79—21no—
133C2×C4×D79—21no—
134C2×C4×D79—21no—
135C2×C4×D79C22×C412yes—
136C2×C4×D79C2×Q812yes—
137#11929607—28no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8C22×C412yes—
143C2×C79⋊C8C22×C412yes—
144C8×D79—28no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—81no—
154C4⋊Dic79—81no—
155Dic79⋊C4C4⋊C414yes—
156Dic79⋊C4C4⋊C414yes—
157(C2×D79)⋊C4—81no—
158(C2×D79)⋊C4—81no—
159(C2×D79)⋊C4—81no—
160(C2×D79)⋊C4—81no—
161(C2×D79)⋊C4C4⋊C414yes—
162(C2×D79)⋊C4C4⋊C414yes—
163C4×Dic79—81no—
164C4×Dic79—81no—
165C4×Dic79—81no—
166C4×Dic79—81no—
167C4×Dic79C22×C412yes—
168C4×Dic79C2×Q812yes—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2411yescharacteristic
176C2×C8×C79C22×C412yes—
177C2×C8×C79C22×C412yes—
178C2×C8×C79#4112yes—
179C2×C8×C79#4112yes—
180C2×C8×C79#4112yes—
181C2×C8×C79#4112yes—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—81no—
191C4⋊C4×C79—81no—
192C42×C79C22×C412yes—
193C42×C79C2×Q812yes—
194C23×C79C22×D411yescharacteristic
195C22×D79—21no—
196C22×D79—21no—
197C22×D79—21no—
198C22×D79—21no—
199C22×D79—21no—
200C22×D79—21no—
201C22×D79C2×C4⋊C412yes—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—41no—
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×C79—41no—
218C2×C4×C79C2×C4⋊C412yes—
219C2×C4×C79C2×C4⋊C412yes—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C22×Q811yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—41no—
233C2×Dic79—41no—
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×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79C2×C4⋊C412yes—
245C2×Dic79C2×C4⋊C412yes—
246C2×Dic79C2×C4⋊C412yes—
247C4×D79—22no—
248C4×D79—22no—
249C4×D79—22no—
250C4×D79—22no—
251C8×C79#11714yes—
252C8×C79#11714yes—
253C79⋊C8—22no—
254C79⋊C8—22no—
255C22×C79C22×C4⋊C411yescharacteristic
256C22×C79#53211yescharacteristic
257C22×C79#53211yescharacteristic
258C22×C79#57611yescharacteristic
259C22×C79#57711yescharacteristic
260C22×C79#57711yescharacteristic
261C22×C79#57811yescharacteristic
262C2×D79—22no—
263C2×D79—22no—
264C2×D79—22no—
265C2×D79—41no—
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—22no—
270C4×C79—22no—
271C4×C79—81no—
272C4×C79—81no—
273C4×C79—81no—
274C4×C79—81no—
275C4×C79C2×#11711yescharacteristic
276C4×C79C2×#11711yescharacteristic
277C4×C79#53311yescharacteristic
278C4×C79#53311yescharacteristic
279Dic79—22no—
280Dic79—22no—
281Dic79—22no—
282Dic79—81no—
283Dic79—81no—
284Dic79—81no—
285Dic79—81no—
286#20822—791nomaximal, Sylow
287C2×C79#228511yescharacteristic
288C2×C79#291211yescharacteristic
289C2×C79#291311yescharacteristic
290C2×C79#317811yescharacteristic
291C2×C79#317811yescharacteristic
292C2×C79#317911yescharacteristic
293C2×C79#317911yescharacteristic
294D79—42no—
295C2×#405—791no—
296#2314—791no—
297#2314—791no—
298#1847—791no—
299#1847D7911yescharacteristic, complemented
300#1835—791no—
301#1834—791no—
302#1809—791no—
303#1809—791no—
304#1808—791no—
305#1807—791no—
306#1776—791no—
307#1776—791no—
308C4×#107—791no—
309C4×#106—791no—
310C79#2082211yescharacteristic, complemented, Sylow
311C22×M16—791no—
312C23×C8—791no—
313C2×#117—791no—
314C2×#107—791no—
315C2×#107—791no—
316C2×#107C2×D7911yescharacteristic
317C2×#107C2×D7911yescharacteristic
318C2×#106—791no—
319C2×#106—791no—
320C2×#106C2×D7911yescharacteristic
321C2×#106C2×D7911yescharacteristic
322#405—798no—
323C2×#97—791no—
324C2×#97C2×D7911yescharacteristic
325C2×C4×C8—791no—
326C2×C4×C8—791no—
327#388—1581no—
328#388—1581no—
329#388—1581no—
330#388—1581no—
331#387—1581no—
332#387—1581no—
333#387—1581no—
334#387—1581no—
335#384—791no—
336#384C2×D7911yescharacteristic
337#382—791no—
338#382C2×D7911yescharacteristic
339#380—791no—
340#380—791no—
341#380—791no—
342#380—791no—
343C4×C4⋊C4—791no—
344C4×C4⋊C4—791no—
345C4×C4⋊C4—791no—
346C4×C4⋊C4—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355C23×C4—791no—
356C2×M16—794no—
357C2×M16—1582no—
358C2×M16—1582no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8—794no—
362C22×C8—1582no—
363C22×C8—1582no—
364C22×C8C22×D7911yescharacteristic
365C22×C8C22×D7911yescharacteristic
366#117—794no—
367#117—794no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4C22×D7911yescharacteristic
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C4⋊C4C22×D7911yescharacteristic
382C2×C22⋊C4—791no—
383C2×C22⋊C4—791no—
384C2×C22⋊C4—1581no—
385C2×C22⋊C4—1581no—
386C2×C22⋊C4—1581no—
387C2×C22⋊C4—1581no—
388C2×C42—791no—
389C2×C42—1581no—
390C2×C42—1581no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42C22×D7911yescharacteristic
394#107—22no—
395#107—22no—
396#107—1582no—
397#107—1582no—
398#106—22no—
399#106—22no—
400#106—1582no—
401#106—1582no—
402#97—794no—
403#97C4×D7914yes—
404C4×C8—794no—
405C4×C8—794no—
406#95—21no—
407#95—21no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418C24—791no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—792no—
427C22×C4—792no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4D4×D7911yescharacteristic
437C22×C4D4×D7911yescharacteristic
438C22×C4C23×D7911yescharacteristic
439M16—1588no—
440C2×C8—792no—
441C2×C8—792no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1588no—
447C2×C8C2×C4×D7912yes—
448C2×C8C2×C4×D7912yes—
449C2×C8#1192962312yes—
450C2×C8#1192962312yes—
451C2×C8#1192962512yes—
452C2×C8#1192962512yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—794no—
464C4⋊C4—794no—
465C4⋊C4—6321no—
466C4⋊C4—6321no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C22⋊C4—794no—
476C22⋊C4—794no—
477C22⋊C4—6321no—
478C22⋊C4—6321no—
479C22⋊C4—6321no—
480C22⋊C4—6321no—
481C42—792no—
482C42—792no—
483C42—6321no—
484C42—6321no—
485C42—6321no—
486C42—6321no—
487C42C2×C4×D7912yes—
488C42Q8×D7912yes—
489C23—792no—
490C23—1581no—
491C23—1581no—
492C23—1581no—
493C23—1581no—
494C23—1581no—
495C23—1581no—
496C23C2×D4×D7911yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
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—41no—
512C2×C4—41no—
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—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—792no—
526C2×C4—792no—
527C2×C4—792no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1582no—
535C2×C4—1582no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4C4⋊C4×D7912yes—
539C2×C4C4⋊C4×D7912yes—
540C2×C4C22×C4×D7911yescharacteristic
541C2×C4C2×Q8×D7911yescharacteristic
542C2×C4C2×#1192962311yescharacteristic
543C2×C4C2×#1192962311yescharacteristic
544C2×C4C2×#1192962511yescharacteristic
545C2×C4C2×#1192962511yescharacteristic
546C8—1582no—
547C8—1582no—
548C8#1457911914yes—
549C8#1457911914yes—
550C22—3161no—
551C22—3161no—
552C22—3161no—
553C22—3161no—
554C22—1582no—
555C22—1582no—
556C22—1582no—
557C22C2×C4⋊C4×D7911yescharacteristic
558C22#1909911111yescharacteristic
559C22#1909911411yescharacteristic
560C22#1909936711yescharacteristic
561C22#1909937211yescharacteristic
562C22#1909937311yescharacteristic
563C22#1909937911yescharacteristic
564C4—22no—
565C4—22no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—6321no—
571C4—6321no—
572C4—6321no—
573C4—6321no—
574C4—1582no—
575C4—1582no—
576C4—1582no—
577C4C2×#1457911911yescharacteristic
578C4C2×#1457911911yescharacteristic
579C4#1909911811yescharacteristic
580C4#1909911811yescharacteristic
581C2—3162no—
582C2#2376580511yescharacteristic
583C2#2377199411yescharacteristic
584C2#2377199911yescharacteristic
585C2#2377406511yescharacteristic
586C2#2377407311yescharacteristic
587C2#2377408011yescharacteristic
588C2#2377408811yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1847C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1847D79
hypocenterC79#20822
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #23763568C2semidirect
5. #23763530C2semidirect
6. #23763527C2semidirect
11. #23763300C2semidirect
12. #23763299C2semidirect
13. #107×Dic79C2semidirect
14. #106×Dic79C2semidirect
15. C79×#1847C2semidirect
23. #19098520C4semidirect
299. #1847D79semidirect
310. C79#20822semidirect

Maximal subgroups

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

subgroupindex
C2×#190985202
#237660102
#237660082
#237635682
#237635302
#237635272
#237634342
#237634302
#237634262
#237634222
#237633002
#237632992
#107×Dic792
#106×Dic792
C79×#18472
#2082279