Finite Group Databasebeta

← #34298680

Subgroups — #34298680

Order 20224.

754 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×#19098591C211yesmaximal, characteristic
2#23766007C211yesmaximal, characteristic
3#23766007C211yesmaximal, characteristic
4#23764171C211yesmaximal, characteristic, complemented
5#23764169C211yesmaximal, characteristic, complemented
6#23764167C211yesmaximal, characteristic, complemented
7#23764165C211yesmaximal, characteristic, complemented
8#23764162C211yesmaximal, characteristic, complemented
9#23764160C211yesmaximal, characteristic, complemented
10#23764157C211yesmaximal, characteristic, complemented
11#23764155C211yesmaximal, characteristic, complemented
12#23763433C211yesmaximal, characteristic
13#23763427C211yesmaximal, characteristic
14#23763423C211yesmaximal, characteristic
15C79×#1809C211yesmaximal, characteristic
16C23×C79⋊C8C2211yescharacteristic
17C2×#14579120C2211yescharacteristic
18C2×#14579095C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579093C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24#19098591C418yescomplemented
25C2×Dic79⋊C8C2211yescharacteristic
26C2×Dic79⋊C8C2211yescharacteristic
27C2×Dic79⋊C8C2211yescharacteristic
28C2×Dic79⋊C8C2211yescharacteristic
29#19098457—21no—
30#19098457—21no—
31#19098451—21no—
32#19098451—21no—
33#19098448—21no—
34#19098448—21no—
35#19098446—21no—
36#19098446—21no—
37#19098432C2211yescharacteristic
38#19098432C2211yescharacteristic
39#19098428C2211yescharacteristic
40#19098428C2211yescharacteristic
41#19098423C2211yescharacteristic
42#19098423C2211yescharacteristic
43C4⋊C4×Dic79C2211yescharacteristic
44C4⋊C4×Dic79C2211yescharacteristic
45C4⋊C4×Dic79C2211yescharacteristic
46C4⋊C4×Dic79C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C22×M16×C79C2211yescharacteristic
54C2×#107×C79C2211yescharacteristic
55C2×#106×C79C2211yescharacteristic
56#380×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×C79⋊C8—41no—
61C22×C79⋊C8—41no—
62C22×C79⋊C8C2×C414yes—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579120C2×C414yes—
66#14579120Q814yes—
67C2×C4⋊Dic79D411yescharacteristic
68C2×C4⋊Dic79D411yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79—21no—
79C2×C22⋊Dic79D411yescharacteristic
80C2×C22⋊Dic79D411yescharacteristic
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79C2311yescharacteristic
86C2×C4×Dic79C2311yescharacteristic
87#14579095—41no—
88#14579094—41no—
89#14579094—41no—
90#14579093—41no—
91#14579092—41no—
92#14579092—41no—
93Dic79⋊C8C2×C414yes—
94Dic79⋊C8C2×C414yes—
95Dic79⋊C8C2×C414yes—
96Dic79⋊C8C2×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—
107C23×C4×C79C2311yescharacteristic
108C2×M16×C79—41no—
109C2×M16×C79—41no—
110C2×M16×C79C2×C414yes—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C22⋊C4×C79—21no—
118C2×C22⋊C4×C79—21no—
119#107×C79—41no—
120#106×C79—41no—
121#95×C79—21no—
122#95×C79—21no—
123C22×Dic79—21no—
124C22×Dic79—21no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79C2×D411yescharacteristic
128C22×Dic79C2×D411yescharacteristic
129C22×Dic79C2×D411yescharacteristic
130C22×Dic79C2×D411yescharacteristic
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—44no—
136C2×C79⋊C8C22×C412yes—
137C2×C79⋊C8C22×C412yes—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79C4⋊C414yes—
147C4⋊Dic79C4⋊C414yes—
148Dic79⋊C4—81no—
149Dic79⋊C4—81no—
150Dic79⋊C4—81no—
151Dic79⋊C4—81no—
152C22⋊Dic79—81no—
153C22⋊Dic79—81no—
154C22⋊Dic79C4⋊C414yes—
155C22⋊Dic79C4⋊C414yes—
156C4×Dic79—81no—
157C4×Dic79—81no—
158C4×Dic79—81no—
159C4×Dic79—81no—
160C4×Dic79C22×C412yes—
161C4×Dic79C22×C412yes—
162C4×Dic79C2×Q812yes—
163C4×Dic79C2×Q812yes—
164C24×C79C2×D411yescharacteristic
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79C22×C412yes—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×Q812yes—
176C22×C4×C79C2411yescharacteristic
177M16×C79—44no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79C22×C412yes—
183C2×C8×C79C22×C412yes—
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—21no—
191C4⋊C4×C79—21no—
192C22⋊C4×C79—81no—
193C22⋊C4×C79—81no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79—21no—
200C23×C79C2×C4⋊C412yes—
201C23×C79C22×D411yescharacteristic
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
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×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79C2×C4⋊C412yes—
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×Q811yescharacteristic
223C2×C4×C79H(5,𝔽2)11yescharacteristic
224C2×C4×C79H(5,𝔽2)11yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×C4×C792−1+411yescharacteristic
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
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×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79C2×C4⋊C412yes—
248C2×Dic79C2×C4⋊C412yes—
249C2×Dic79C2×C4⋊C412yes—
250C2×Dic79C2×C4⋊C412yes—
251C8×C79—22no—
252C8×C79—22no—
253C79⋊C8—22no—
254C79⋊C8—22no—
255C22×C79—22no—
256C22×C79—22no—
257C22×C79—22no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C22×C4⋊C411yescharacteristic
263C22×C79#53711yescharacteristic
264C22×C79#53711yescharacteristic
265C22×C79#57611yescharacteristic
266C22×C79#57711yescharacteristic
267C22×C79#57711yescharacteristic
268C22×C79#57811yescharacteristic
269C4×C79—22no—
270C4×C79—81no—
271C4×C79—81no—
272C4×C79—81no—
273C4×C79—81no—
274C4×C79#51811yescharacteristic
275C4×C79#51911yescharacteristic
276C4×C79#54411yescharacteristic
277C4×C79#54411yescharacteristic
278Dic79—22no—
279Dic79—22no—
280Dic79—22no—
281Dic79—22no—
282Dic79—81no—
283Dic79—81no—
284Dic79—81no—
285Dic79—81no—
286#21966—791nomaximal, Sylow
287C2×C79—42no—
288C2×C79#231611yescharacteristic
289C2×C79#291211yescharacteristic
290C2×C79#291311yescharacteristic
291C2×C79#325711yescharacteristic
292C2×C79#325711yescharacteristic
293C2×C79#325811yescharacteristic
294C2×C79#325811yescharacteristic
295C2×#424—791no—
296#2314—791no—
297#2314—791no—
298#1949—791no—
299#1949—791no—
300#1949—791no—
301#1949—791no—
302#1948—791no—
303#1948—791no—
304#1947—791no—
305#1947—791no—
306#1809—791no—
307#1809D7911yescharacteristic
308#1808—791no—
309#1807—791no—
310C79#2196611yescharacteristic, complemented, Sylow
311C22×M16C2×D7911yescharacteristic
312C23×C8—791no—
313C2×#117—791no—
314C2×#107—791no—
315C2×#107—791no—
316C2×#107—791no—
317C2×#107C2×D7911yescharacteristic
318C2×#106—791no—
319C2×#106—791no—
320C2×#106—791no—
321C2×#106C2×D7911yescharacteristic
322#424—798no—
323C2×C4⋊C8—791no—
324C2×C4⋊C8—791no—
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#384—791no—
337#382—791no—
338#382—791no—
339#380—791no—
340#380—791no—
341#380C2×D7911yescharacteristic
342#380C2×D7911yescharacteristic
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#340C2×D7911yescharacteristic
354#340C2×D7911yescharacteristic
355C23×C4C22×D7911yescharacteristic
356C2×M16—41no—
357C2×M16—41no—
358C2×M16C4×D7914yes—
359C22×C8—3161no—
360C22×C8—3161no—
361C22×C8—794no—
362C22×C8—791no—
363C22×C8—791no—
364C22×C8C22×D7911yescharacteristic
365C22×C8C22×D7911yescharacteristic
366#117—794no—
367#117—794no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
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—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—791no—
385C2×C22⋊C4—791no—
386C2×C22⋊C4—1581no—
387C2×C22⋊C4—1581no—
388C2×C42—791no—
389C2×C42—791no—
390C2×C42—1581no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42—1581no—
394#107—41no—
395#107—3161no—
396#107—3161no—
397#107—3161no—
398#106—41no—
399#106—3161no—
400#106—3161no—
401#106—3161no—
402C4⋊C8—794no—
403C4⋊C8—794no—
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—
418C24D4×D7911yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—791no—
428C22×C4—791no—
429C22×C4—791no—
430C22×C4—791no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4—1581no—
434C22×C4—1581no—
435C22×C4C2×C4×D7912yes—
436C22×C4D4×D7911yescharacteristic
437C22×C4Q8×D7912yes—
438C22×C4C23×D7911yescharacteristic
439M16—44no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—3164no—
445C2×C8—792no—
446C2×C8—792no—
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C2×C8C2×C4×D7912yes—
452C2×C8C2×C4×D7912yes—
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—794no—
462C4⋊C4—794no—
463C4⋊C4—6321no—
464C4⋊C4—6321no—
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—81no—
476C22⋊C4—81no—
477C22⋊C4—794no—
478C22⋊C4—794no—
479C22⋊C4—6321no—
480C22⋊C4—6321no—
481C42—792no—
482C42—792no—
483C42—792no—
484C42—792no—
485C42—6321no—
486C42—6321no—
487C42—6321no—
488C42—6321no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C4⋊C4×D7912yes—
496C23C2×D4×D7911yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
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—41no—
514C2×C4—41no—
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—3161no—
526C2×C4—3161no—
527C2×C4—792no—
528C2×C4—792no—
529C2×C4—792no—
530C2×C4—792no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4C4⋊C4×D7912yes—
540C2×C4C22×C4×D7911yescharacteristic
541C2×C4C2×Q8×D7911yescharacteristic
542C2×C4#1457920611yescharacteristic
543C2×C4#1457920611yescharacteristic
544C2×C4#1457920811yescharacteristic
545C2×C4#1457920811yescharacteristic
546C8—22no—
547C8—22no—
548C8—1582no—
549C8—1582no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C4⋊C4×D7911yescharacteristic
558C22#1909914011yescharacteristic
559C22#1909914011yescharacteristic
560C22#1909936711yescharacteristic
561C22#1909937211yescharacteristic
562C22#1909937311yescharacteristic
563C22#1909937911yescharacteristic
564C4—22no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—6321no—
570C4—6321no—
571C4—6321no—
572C4—6321no—
573C4—1582no—
574C4—1582no—
575C4—1582no—
576C4—1582no—
577C4#1909903611yescharacteristic
578C4#1909904011yescharacteristic
579C4#1909919211yescharacteristic
580C4#1909919211yescharacteristic
581C2—42no—
582C2#2376601711yescharacteristic
583C2#2377199411yescharacteristic
584C2#2377199911yescharacteristic
585C2#2377480611yescharacteristic
586C2#2377481211yescharacteristic
587C2#2377482111yescharacteristic
588C2#2377482711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1809C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1809D79
hypocenterC79#21966
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #23764171C2semidirect
5. #23764169C2semidirect
6. #23764167C2semidirect
7. #23764165C2semidirect
8. #23764162C2semidirect
9. #23764160C2semidirect
10. #23764157C2semidirect
11. #23764155C2semidirect
24. #19098591C4semidirect
310. C79#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#190985912
#237660072
#237641712
#237641692
#237641672
#237641652
#237641622
#237641602
#237641572
#237641552
#237634332
#237634272
#237634232
C79×#18092
#2196679