Finite Group Databasebeta

← #34346322

Subgroups — #34346322

Order 20224.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767107C211yesmaximal, characteristic
2#23766277C211yesmaximal, characteristic
3#23764966C211yesmaximal, characteristic, complemented
4#23764900C211yesmaximal, characteristic, complemented
5#23764892C211yesmaximal, characteristic, complemented
6#23764741C211yesmaximal, characteristic, complemented
7#23764354C211yesmaximal, characteristic, complemented
8#23764157C211yesmaximal, characteristic, complemented
9#23763984C211yesmaximal, characteristic, complemented
10#23763981C211yesmaximal, characteristic, complemented
11#23763599C211yesmaximal, characteristic
12#23763583C211yesmaximal, characteristic
13#23763486C211yesmaximal, characteristic
14#23763433C211yesmaximal, characteristic
15C79×#1736C211yesmaximal, characteristic
16C2×#14579152C2211yescharacteristic
17C2×#14579095C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579093C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×Dic79⋊C8C2211yescharacteristic
22C2×Dic79⋊C8C2211yescharacteristic
23C2×C22⋊(C79⋊C8)C2211yescharacteristic
24#19098499—21no—
25#19098486—21no—
26#19098462C2211yescharacteristic
27#19098462C2211yescharacteristic
28#19098462C2211yescharacteristic
29#19098461—21no—
30#19098459C2211yescharacteristic
31#19098449—21no—
32#19098449—21no—
33#19098447—21no—
34#19098432C2211yescharacteristic
35#19098423—21no—
36#19098423C2211yescharacteristic
37#19098423C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4×C22⋊Dic79—21no—
41C4×C22⋊Dic79C2211yescharacteristic
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C22×M16×C79C2211yescharacteristic
53C2×#117×C79C2211yescharacteristic
54C22×C4⋊C4×C79C2211yescharacteristic
55#340×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61#14579152—22no—
62#14579152—41no—
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×C22⋊Dic79—21no—
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79D411yescharacteristic
78C2×C22⋊Dic79D411yescharacteristic
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79C2311yescharacteristic
82C2×C4×Dic79C2311yescharacteristic
83#14579095—41no—
84#14579094—41no—
85#14579093—22no—
86#14579092—22no—
87Dic79⋊C8—22no—
88Dic79⋊C8—41no—
89C22⋊(C79⋊C8)—22no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95D411yescharacteristic
99C79⋊#95D411yescharacteristic
100C23×C4×C79—21no—
101C23×C4×C79C2311yescharacteristic
102C2×M16×C79—22no—
103C2×M16×C79—22no—
104C2×M16×C79—41no—
105C22×C8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107#117×C79—22no—
108#117×C79—41no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79—22no—
111C2×C4⋊C4×C79Q814yes—
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C22⋊C4×C79—21no—
117C2×C42×C79—21no—
118C22×Dic79—21no—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79C2×D411yescharacteristic
123C22×Dic79C2×D411yescharacteristic
124C22×Dic79C2×D411yescharacteristic
125C22×Dic79C2×D411yescharacteristic
126C2×C79⋊C8—21no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C4⋊Dic79—21no—
133C4⋊Dic79—21no—
134C4⋊Dic79—21no—
135C4⋊Dic79—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—42no—
141C4⋊Dic79—42no—
142C4⋊Dic79—81no—
143C4⋊Dic79—81no—
144Dic79⋊C4—42no—
145Dic79⋊C4—42no—
146Dic79⋊C4—81no—
147Dic79⋊C4—81no—
148C22⋊Dic79—22no—
149C22⋊Dic79—41no—
150C22⋊Dic79—42no—
151C22⋊Dic79—81no—
152C4×Dic79—21no—
153C4×Dic79—21no—
154C4×Dic79—21no—
155C4×Dic79—21no—
156C4×Dic79—81no—
157C4×Dic79—81no—
158C24×C79C2×D411yescharacteristic
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—24no—
167C22×C4×C79—21no—
168C22×C4×C79—81no—
169C22×C4×C79—81no—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×Q812yes—
172C22×C4×C79#4112yes—
173C22×C4×C79C2411yescharacteristic
174M16×C79—44no—
175C2×C8×C79—21no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79#4112yes—
181C4⋊C4×C79—24no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79C2×Q812yes—
188C4⋊C4×C79C2×Q812yes—
189C4⋊C4×C79#4112yes—
190C22⋊C4×C79—42no—
191C42×C79—42no—
192C23×C79—21no—
193C23×C79—21no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79C22⋊Q812yes—
199C23×C79C22×D411yescharacteristic
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—22no—
203C2×C4×C79—22no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
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—84no—
219C2×C4×C79C22⋊Q812yes—
220C2×C4×C79C22×Q811yescharacteristic
221C2×C4×C79C2×#4111yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79H(5,𝔽2)11yescharacteristic
225C2×C4×C792−1+411yescharacteristic
226C2×Dic79—21no—
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—21no—
236C2×Dic79—21no—
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×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79#13712yes—
250C8×C79—22no—
251C8×C79—41no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—22no—
255C22×C79—22no—
256C22×C79—22no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79C2×C22⋊Q811yescharacteristic
262C22×C79#54611yescharacteristic
263C22×C79#54711yescharacteristic
264C22×C79C2×#13711yescharacteristic
265C22×C79#57511yescharacteristic
266C22×C79#57611yescharacteristic
267C22×C79#57711yescharacteristic
268C4×C79—22no—
269C4×C79—42no—
270C4×C79—42no—
271C4×C79—42no—
272C4×C79—81no—
273C4×C79#53311yescharacteristic
274C4×C79#54411yescharacteristic
275C4×C79#55111yescharacteristic
276C4×C79#55211yescharacteristic
277Dic79—22no—
278Dic79—41no—
279Dic79—41no—
280Dic79—41no—
281Dic79—81no—
282Dic79—81no—
283Dic79—81no—
284Dic79—81no—
285#26018—791nomaximal, Sylow
286C2×C79—42no—
287C2×C79#243811yescharacteristic
288C2×C79#307811yescharacteristic
289C2×C79#308111yescharacteristic
290C2×C79#330911yescharacteristic
291C2×C79#333111yescharacteristic
292C2×C79#334611yescharacteristic
293C2×C79#335111yescharacteristic
294#2441—791no—
295#2345—791no—
296#2100—791no—
297#2088—791no—
298#2086—791no—
299#2059—791no—
300#1989—791no—
301#1947—791no—
302#1920—791no—
303#1919—791no—
304#1854—791no—
305#1851—791no—
306#1823—791no—
307#1809—791no—
308#1736D7911yescharacteristic
309C79#2601811yescharacteristic, complemented, Sylow
310C22×M16C2×D7911yescharacteristic
311C2×#125—791no—
312C2×#117C2×D7911yescharacteristic
313C22×C4⋊C4C2×D7911yescharacteristic
314C2×#107—791no—
315C2×#107—791no—
316C2×#106—791no—
317C2×#106—791no—
318C2×C4⋊C8—791no—
319C2×C4⋊C8—791no—
320C2×C22⋊C8—791no—
321#399—1581no—
322#396—1581no—
323#389—791no—
324#389—791no—
325#389—791no—
326#389—791no—
327#389—1581no—
328#387—1581no—
329#387—1581no—
330#387—1581no—
331#384—791no—
332#380—791no—
333#380—791no—
334#380—1581no—
335C4×C4⋊C4—791no—
336C4×C4⋊C4—791no—
337C4×C22⋊C4—791no—
338C4×C22⋊C4—1581no—
339#340—791no—
340#340—791no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340C2×D7911yescharacteristic
348#340C2×D7911yescharacteristic
349#340C2×D7911yescharacteristic
350#340C2×D7911yescharacteristic
351#336—791no—
352#336—791no—
353C23×C4—21no—
354C23×C4C22×D7911yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—41no—
358C22×C8—791no—
359C22×C8—791no—
360C22×C8C22×D7911yescharacteristic
361C22×C8C22×D7911yescharacteristic
362#125—3161no—
363#125—1582no—
364#117—22no—
365#117—41no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4Dic15814yes—
381C2×C4⋊C4C22×D7911yescharacteristic
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—791no—
387C2×C22⋊C4—791no—
388C2×C22⋊C4—1581no—
389C2×C22⋊C4—1581no—
390C2×C42—21no—
391C2×C42—791no—
392C2×C42—791no—
393C2×C42—1581no—
394C2×C42—1581no—
395#107—3161no—
396#107—3161no—
397#106—1582no—
398#106—1582no—
399C4⋊C8—3161no—
400C4⋊C8—1582no—
401C22⋊C8—1582no—
402#95—791no—
403#95—791no—
404#95—1581no—
405#95—1581no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412C24D4×D7911yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—791no—
427C22×C4—791no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4D4×D7911yescharacteristic
433C22×C4C2×Dic15812yes—
434C22×C4#1192962312yes—
435C22×C4C23×D7911yescharacteristic
436M16—44no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8#1192962412yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—3162no—
456C4⋊C4—3162no—
457C4⋊C4—3162no—
458C4⋊C4—3162no—
459C4⋊C4—6321no—
460C4⋊C4—6321no—
461C4⋊C4—6321no—
462C4⋊C4—6321no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4C2×Dic15812yes—
472C4⋊C4C2×Dic15812yes—
473C4⋊C4#1192962412yes—
474C22⋊C4—42no—
475C22⋊C4—3162no—
476C22⋊C4—3161no—
477C22⋊C4—6321no—
478C22⋊C4—1582no—
479C42—42no—
480C42—6321no—
481C42—6321no—
482C42—1581no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1457913812yes—
493C23C2×D4×D7911yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
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—84no—
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—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4#1457913812yes—
538C2×C4C22×Dic15811yescharacteristic
539C2×C4C2×#1192962311yescharacteristic
540C2×C4C2×#1192962411yescharacteristic
541C2×C4C2×#1192962411yescharacteristic
542C2×C4#1457920611yescharacteristic
543C2×C4#1457920911yescharacteristic
544C8—22no—
545C8—41no—
546C8—3161no—
547C8—3161no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1457913811yescharacteristic
556C22#1909920311yescharacteristic
557C22#1909921411yescharacteristic
558C22#137×D7911yescharacteristic
559C22#1909936211yescharacteristic
560C22#1909936711yescharacteristic
561C22#1909937311yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—3161no—
568C4—3161no—
569C4—3161no—
570C4—6321no—
571C4—6321no—
572C4—6321no—
573C4—6321no—
574C4—1582no—
575C4#1909912011yescharacteristic
576C4#1909919411yescharacteristic
577C4#1909923911yescharacteristic
578C4#1909924811yescharacteristic
579C2—42no—
580C2#2376707611yescharacteristic
581C2#2377320211yescharacteristic
582C2#2377323311yescharacteristic
583C2#2377540711yescharacteristic
584C2#2377571211yescharacteristic
585C2#2377588611yescharacteristic
586C2#2377596111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1736C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1736D79
hypocenterC79#26018
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. #23764966C2semidirect
4. #23764900C2semidirect
5. #23764892C2semidirect
6. #23764741C2semidirect
7. #23764354C2semidirect
8. #23764157C2semidirect
9. #23763984C2semidirect
10. #23763981C2semidirect
309. C79#26018semidirect

Maximal subgroups

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

subgroupindex
#237671072
#237662772
#237649662
#237649002
#237648922
#237647412
#237643542
#237641572
#237639842
#237639812
#237635992
#237635832
#237634862
#237634332
C79×#17362
#2601879