Finite Group Databasebeta

← #34346316

Subgroups — #34346316

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#23767113C211yesmaximal, characteristic
2#23764963C211yesmaximal, characteristic, complemented
3#23764902C211yesmaximal, characteristic, complemented
4#23764891C211yesmaximal, characteristic, complemented
5#23764738C211yesmaximal, characteristic, complemented
6#23764357C211yesmaximal, characteristic, complemented
7#23764158C211yesmaximal, characteristic, complemented
8#23763985C211yesmaximal, characteristic, complemented
9#23763978C211yesmaximal, characteristic, complemented
10#23763597C211yesmaximal, characteristic
11#23763585C211yesmaximal, characteristic
12#23763486C211yesmaximal, characteristic
13#23763431C211yesmaximal, characteristic
14#23763119C211yesmaximal, characteristic
15C79×#2345C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C22⋊(C79⋊C8)C2211yescharacteristic
26#19098499—21no—
27#19098486—21no—
28#19098461C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098460—21no—
31#19098446—21no—
32#19098433C2211yescharacteristic
33#19098423C2211yescharacteristic
34#19098325C2211yescharacteristic
35#19098325C2211yescharacteristic
36#19098325C2211yescharacteristic
37#19098325C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C22×C4⋊C4×C79C2211yescharacteristic
49#389×C79C2211yescharacteristic
50#389×C79C2211yescharacteristic
51#387×C79—21no—
52#387×C79—21no—
53#380×C79—21no—
54#380×C79C2211yescharacteristic
55C4×C4⋊C4×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C22⋊C4×C79—21no—
58C4×C22⋊C4×C79C2211yescharacteristic
59C2×#11929608—22no—
60C2×#11929608—22no—
61C2×#11929608—41no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66#14579153—22no—
67#14579153—41no—
68#14579120—22no—
69#14579120—41no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83#14579094—41no—
84#14579094—41no—
85#14579092—22no—
86#14579092—22no—
87C4⋊(C79⋊C8)—22no—
88C4⋊(C79⋊C8)—41no—
89C22⋊(C79⋊C8)—22no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C23×C4×C79—21no—
94C23×C4×C79C2311yescharacteristic
95C2×C4⋊C4×C79—21no—
96C2×C4⋊C4×C79—22no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79Q814yes—
100C2×C4⋊C4×C79C2311yescharacteristic
101C2×C4⋊C4×C79C2311yescharacteristic
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C22⋊C4×C79—21no—
105C2×C22⋊C4×C79D411yescharacteristic
106C2×C22⋊C4×C79D411yescharacteristic
107C2×C42×C79—21no—
108C2×C42×C79—21no—
109C2×C42×C79C2311yescharacteristic
110C2×C42×C79C2311yescharacteristic
111#95×C79—21no—
112#95×C79—21no—
113#95×C79—21no—
114#95×C79—21no—
115#95×C79—21no—
116#95×C79D411yescharacteristic
117#95×C79D411yescharacteristic
118C22×Dic79—21no—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122#11929608—44no—
123C2×C79⋊C8—21no—
124C2×C79⋊C8—21no—
125C2×C79⋊C8—21no—
126C2×C79⋊C8—21no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8#4112yes—
135C4⋊Dic79—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—42no—
143C4⋊Dic79—81no—
144C4⋊Dic79#4112yes—
145Dic79⋊C4—42no—
146Dic79⋊C4—42no—
147Dic79⋊C4—42no—
148Dic79⋊C4—81no—
149C22⋊Dic79—42no—
150C22⋊Dic79—81no—
151C4×Dic79—42no—
152C24×C79C2×D411yescharacteristic
153C22×C4×C79—21no—
154C22×C4×C79—21no—
155C22×C4×C79—21no—
156C22×C4×C79—21no—
157C22×C4×C79—21no—
158C22×C4×C79—21no—
159C22×C4×C79—24no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—81no—
163C22×C4×C79—81no—
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×Q812yes—
170C22×C4×C79#4112yes—
171C22×C4×C79C2411yescharacteristic
172C4⋊C4×C79—24no—
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—81no—
180C4⋊C4×C79—81no—
181C4⋊C4×C79C2×Q812yes—
182C4⋊C4×C79C2×Q812yes—
183C22⋊C4×C79—22no—
184C22⋊C4×C79—41no—
185C22⋊C4×C79—42no—
186C42×C79—21no—
187C42×C79—21no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—81no—
191C42×C79—81no—
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—21no—
203C2×C4×C79—21no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—84no—
230C2×C4×C79C22⋊Q812yes—
231C2×C4×C79#13712yes—
232C2×C4×C79C22×Q811yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79H(5,𝔽2)11yescharacteristic
237C2×C4×C792−1+411yescharacteristic
238C2×Dic79—22no—
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—41no—
250C79⋊C8—22no—
251C79⋊C8—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—22no—
270C4×C79—41no—
271C4×C79—42no—
272C4×C79—42no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—81no—
276C4×C79—81no—
277C4×C79#53311yescharacteristic
278C4×C79#54411yescharacteristic
279C4×C79#55111yescharacteristic
280C4×C79#55211yescharacteristic
281Dic79—42no—
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#2345D7911yescharacteristic
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#1736—791no—
309C79#2601811yescharacteristic, complemented, Sylow
310C22×M16—791no—
311C2×#125—791no—
312C2×#117—791no—
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—1581no—
326#389C2×D7911yescharacteristic
327#389C2×D7911yescharacteristic
328#387—21no—
329#387—21no—
330#387—1581no—
331#384—791no—
332#380—21no—
333#380—791no—
334#380C2×D7911yescharacteristic
335C4×C4⋊C4C2×D7911yescharacteristic
336C4×C4⋊C4C2×D7911yescharacteristic
337C4×C22⋊C4—21no—
338C4×C22⋊C4C2×D7911yescharacteristic
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#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#336—791no—
352#336—791no—
353C23×C4—21no—
354C23×C4C22×D7911yescharacteristic
355C2×M16—3161no—
356C2×M16—1582no—
357C2×M16—1582no—
358C22×C8—791no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8—791no—
362#125—3161no—
363#125—1582no—
364#117—3161no—
365#117—1582no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
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—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—1581no—
387C2×C22⋊C4—1581no—
388C2×C22⋊C4C22⋊D7911yescharacteristic
389C2×C22⋊C4C22⋊D7911yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1581no—
393C2×C42C22×D7911yescharacteristic
394C2×C42C22×D7911yescharacteristic
395#107—3161no—
396#107—3161no—
397#106—1582no—
398#106—1582no—
399C4⋊C8—3161no—
400C4⋊C8—1582no—
401C22⋊C8—1582no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95C22⋊D7911yescharacteristic
411#95C22⋊D7911yescharacteristic
412C24C2×C22⋊D7911yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—1581no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4C2×C22⋊D7911yescharacteristic
429C22×C4C2×C22⋊D7911yescharacteristic
430C22×C4C2×C22⋊D7911yescharacteristic
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4C2×Dic15812yes—
434C22×C4#1192962412yes—
435C22×C4C23×D7911yescharacteristic
436M16—3164no—
437C2×C8—792no—
438C2×C8—1581no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1581no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—3162no—
459C4⋊C4—3162no—
460C4⋊C4—3162no—
461C4⋊C4—3162no—
462C4⋊C4—792no—
463C4⋊C4—6321no—
464C4⋊C4—6321no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×Dic15812yes—
473C4⋊C4C2×Dic15812yes—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—3162no—
478C22⋊C4—6321no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—81no—
484C42—81no—
485C42—3162no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1457914012yes—
493C23C22×C22⋊D7911yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—84no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—1582no—
536C2×C4#1457914012yes—
537C2×C4#1457919212yes—
538C2×C4C22×Dic15811yescharacteristic
539C2×C4#41×D7911yescharacteristic
540C2×C4#41×D7911yescharacteristic
541C2×C4C2×#1192962411yescharacteristic
542C2×C4#1457920711yescharacteristic
543C2×C4#1457920811yescharacteristic
544C8—3161no—
545C8—3161no—
546C8—3161no—
547C8—1582no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1457914011yescharacteristic
556C22#1909920111yescharacteristic
557C22#1909921211yescharacteristic
558C22C2×#1457919211yescharacteristic
559C22#1909936111yescharacteristic
560C22#1909936611yescharacteristic
561C22#1909937411yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—3162no—
572C4—6321no—
573C4—6321no—
574C4—6321no—
575C4#1909911611yescharacteristic
576C4#1909919111yescharacteristic
577C4#1909923711yescharacteristic
578C4#1909924611yescharacteristic
579C2—42no—
580C2#2376707011yescharacteristic
581C2#2377320511yescharacteristic
582C2#2377323111yescharacteristic
583C2#2377540011yescharacteristic
584C2#2377571011yescharacteristic
585C2#2377587911yescharacteristic
586C2#2377595411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2345C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2345D79
hypocenterC79#26018
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #23764963C2semidirect
3. #23764902C2semidirect
4. #23764891C2semidirect
5. #23764738C2semidirect
6. #23764357C2semidirect
7. #23764158C2semidirect
8. #23763985C2semidirect
9. #23763978C2semidirect
309. C79#26018semidirect

Maximal subgroups

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

subgroupindex
#237671132
#237649632
#237649022
#237648912
#237647382
#237643572
#237641582
#237639852
#237639782
#237635972
#237635852
#237634862
#237634312
#237631192
C79×#23452
#2601879