Finite Group Databasebeta

← #34361057

Subgroups — #34361057

Order 20224.

762 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766436C211yesmaximal, characteristic
2#23764804C211yesmaximal, characteristic, complemented
3#23764771C211yesmaximal, characteristic, complemented
4#23764589C211yesmaximal, characteristic
5#23764589C211yesmaximal, characteristic
6#23764161C211yesmaximal, characteristic, complemented
7#23764125C211yesmaximal, characteristic, complemented
8#23764008C211yesmaximal, characteristic, complemented
9#23763989C211yesmaximal, characteristic, complemented
10#23763985C211yesmaximal, characteristic, complemented
11#23763646C211yesmaximal, characteristic, complemented
12#23763597C211yesmaximal, characteristic, complemented
13#23763492C211yesmaximal, characteristic, complemented
14C4×#14579081C211yesmaximal, characteristic, complemented
15C79×#2496C211yesmaximal, characteristic, complemented
16C2×#14579135C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×C22⋊(C79⋊C8)C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31#19098465C2211yescharacteristic
32#19098463—21no—
33#19098461C2211yescharacteristic
34#19098452—21no—
35#19098439—21no—
36C4×C4⋊Dic79C2211yescharacteristic
37#19098410—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#126×C79—21no—
49#400×C79—21no—
50#397×C79—21no—
51#389×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#387×C79—21no—
54#384×C79C2211yescharacteristic
55#380×C79C2211yescharacteristic
56#380×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×D316C2311yescharacteristic
60C22×C22⋊D79—21no—
61C22×C4×D79—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66#14579135—22no—
67#14579135—22no—
68C4×D316—22no—
69C4×D316—41no—
70#14579120—22no—
71#14579120—41no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—21no—
76C2×C22⋊Dic79—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83#14579094—22no—
84#14579092—22no—
85#14579092—22no—
86#14579092—41no—
87C4⋊(C79⋊C8)—22no—
88#14579081—22no—
89#14579081—22no—
90#14579081—22no—
91#14579081—41no—
92C22⋊(C79⋊C8)—22no—
93C22⋊(C79⋊C8)—41no—
94C4×C79⋊C8—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C23×C4×C79C2311yescharacteristic
98#126×C79—44no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C22⋊C4×C79—21no—
108C2×C22⋊C4×C79—21no—
109C2×C22⋊C4×C79D411yescharacteristic
110C2×C22⋊C4×C79D411yescharacteristic
111C2×C42×C79—21no—
112C2×C42×C79C2311yescharacteristic
113C2×C42×C79C2311yescharacteristic
114#95×C79—21no—
115#95×C79—21no—
116#95×C79—21no—
117#95×C79—21no—
118#95×C79—21no—
119#95×C79D411yescharacteristic
120#95×C79D411yescharacteristic
121C23×D79—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316—21no—
126C2×D316—21no—
127C2×D316#4112yes—
128C2×C22⋊D79—24no—
129C2×C22⋊D79—81no—
130C2×C22⋊D79—81no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C2×C4×D79—42no—
135C2×C4×D79—42no—
136C2×C4×D79—81no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8#4112yes—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79#4112yes—
154C4⋊Dic79#4112yes—
155Dic79⋊C4—81no—
156C22⋊Dic79—42no—
157(C2×D79)⋊C4—24no—
158(C2×D79)⋊C4—42no—
159(C2×D79)⋊C4—42no—
160(C2×D79)⋊C4—81no—
161C4×Dic79—42no—
162C4×Dic79—81no—
163C24×C79C2×D411yescharacteristic
164C22×C4×C79—21no—
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×C79—21no—
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2411yescharacteristic
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
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—42no—
189C4⋊C4×C79—42no—
190C4⋊C4×C79—42no—
191C4⋊C4×C79—42no—
192C22⋊C4×C79—22no—
193C22⋊C4×C79—22no—
194C22⋊C4×C79—42no—
195C22⋊C4×C79—42no—
196C42×C79—21no—
197C42×C79—21no—
198C42×C79—21no—
199C42×C79—21no—
200C42×C79—42no—
201C23×C79—21no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79C22×D411yescharacteristic
209C22×D79—22no—
210C22×D79—41no—
211C22×D79—41no—
212C22×D79—41no—
213C22×D79—41no—
214C22×D79—41no—
215C22×D79—41no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
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—21no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79#13512yes—
246C2×C4×C79AGL(1,ℤ/8)12yes—
247C2×C4×C79C2×#4111yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×C4×C79H(5,𝔽2)11yescharacteristic
252C2×C4×C792−1+411yescharacteristic
253C22⋊D79—84no—
254C2×Dic79—22no—
255C2×Dic79—22no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263D316—22no—
264D316—22no—
265D316—41no—
266D316—41no—
267C4×D79—84no—
268C79⋊C8—22no—
269C79⋊C8—41no—
270C79⋊C8—41no—
271C79⋊C8—41no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79—41no—
279C22×C79#52511yescharacteristic
280C22×C79#54611yescharacteristic
281C22×C79#54711yescharacteristic
282C22×C79C2×#13511yescharacteristic
283C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C79#57711yescharacteristic
285C22×C79#57711yescharacteristic
286C2×D79—42no—
287C2×D79—42no—
288C2×D79—42no—
289C2×D79—81no—
290C2×D79—81no—
291C2×D79—81no—
292C2×D79—81no—
293C4×C79—22no—
294C4×C79—22no—
295C4×C79—41no—
296C4×C79—41no—
297C4×C79—41no—
298C4×C79—81no—
299C4×C79—81no—
300C4×C79—81no—
301C4×C79—81no—
302C4×C79#52911yescharacteristic
303C4×C79#53911yescharacteristic
304C4×C79#55311yescharacteristic
305C4×C79#55611yescharacteristic
306Dic79—42no—
307Dic79—42no—
308Dic79—81no—
309#27039—791nomaximal, Sylow
310C2×C79—81no—
311C2×C79#246611yescharacteristic
312C2×C79#310611yescharacteristic
313C2×C79#312311yescharacteristic
314C2×C79#330411yescharacteristic
315C2×C79#330611yescharacteristic
316C2×C79#333711yescharacteristic
317C2×C79#335411yescharacteristic
318D79—82no—
319#2496D7911yescharacteristic, complemented
320#2363—791no—
321#2071—791no—
322#2065—791no—
323#2034—791no—
324#2034—791no—
325#1948—791no—
326#1941—791no—
327#1923—791no—
328#1921—791no—
329#1920—791no—
330#1862—791no—
331#1854—791no—
332#1825—791no—
333C4×#102—791no—
334C79#2703911yescharacteristic, complemented, Sylow
335C2×#126—21no—
336C2×#121—791no—
337C2×C4×D4—791no—
338C2×#117—791no—
339C2×#107—791no—
340C2×#106—791no—
341C2×#106—791no—
342C2×#106—791no—
343C2×C4⋊C8—791no—
344C2×#102—791no—
345C2×#102—791no—
346C2×#102—791no—
347C2×#102—791no—
348C2×C22⋊C8—791no—
349C2×C22⋊C8—791no—
350C2×C4×C8—791no—
351#400—21no—
352#397—21no—
353#390—791no—
354#390—1581no—
355#389—791no—
356#389C2×D7911yescharacteristic
357#389C2×D7911yescharacteristic
358#388—1581no—
359#387—21no—
360#386—1581no—
361#384C2×D7911yescharacteristic
362#380C2×D7911yescharacteristic
363#380C2×D7911yescharacteristic
364C4×C4⋊C4—791no—
365C4×C4⋊C4C2×D7911yescharacteristic
366C4×C22⋊C4—1581no—
367C4×C22⋊C4C2×D7911yescharacteristic
368#340—791no—
369#340—791no—
370#340—791no—
371#340—791no—
372#340—791no—
373#340—791no—
374#340—791no—
375#340—791no—
376#336—791no—
377#336—791no—
378C22×D4—791no—
379C22×D4—1581no—
380C23×C4—1581no—
381C23×C4C22×D7911yescharacteristic
382C22×C8—791no—
383C22×C8—791no—
384C22×C8—791no—
385C22×C8—791no—
386#126—44no—
387#121—1582no—
388#121—1582no—
389C4×D4—3161no—
390C4×D4—1582no—
391#117—3161no—
392#117—1582no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—791no—
398C2×C4⋊C4—791no—
399C2×C4⋊C4—791no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4C22×D7911yescharacteristic
402C2×C4⋊C4C22×D7911yescharacteristic
403C2×C4⋊C4C22×D7911yescharacteristic
404C2×C4⋊C4C22×D7911yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1581no—
408C2×C22⋊C4—1581no—
409C2×C22⋊C4—1581no—
410C2×C22⋊C4—1581no—
411C2×C22⋊C4—1581no—
412C2×C22⋊C4C22⋊D7911yescharacteristic
413C2×C22⋊C4C22⋊D7911yescharacteristic
414C2×C42—21no—
415C2×C42—1581no—
416C2×C42—1581no—
417C2×C42C22×D7911yescharacteristic
418C2×C42C22×D7911yescharacteristic
419#107—1582no—
420#106—3161no—
421#106—1582no—
422#106—1582no—
423C4⋊C8—1582no—
424#102—3161no—
425#102—1582no—
426#102—1582no—
427#102—1582no—
428C22⋊C8—3161no—
429C22⋊C8—1582no—
430C4×C8—3161no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1581no—
437#95—1581no—
438#95C22⋊D7911yescharacteristic
439#95C22⋊D7911yescharacteristic
440C24—1581no—
441C24C2×C22⋊D7911yescharacteristic
442C2×D4—792no—
443C2×D4—6321no—
444C2×D4—6321no—
445C2×D4—1584no—
446C2×D4—1581no—
447C2×D4—1581no—
448C2×D4—1581no—
449C2×D4—1581no—
450C2×D4—1581no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—3162no—
462C22×C4—3162no—
463C22×C4—6321no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4C2×C22⋊D7911yescharacteristic
468C22×C4C2×C22⋊D7911yescharacteristic
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C2×C22⋊D7911yescharacteristic
471C22×C4C2×C22⋊D7911yescharacteristic
472C22×C4C23×D7911yescharacteristic
473C2×C8—792no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—792no—
498C4⋊C4—792no—
499C4⋊C4—6321no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—3162no—
509C22⋊C4—3162no—
510C22⋊C4—3162no—
511C22⋊C4—6321no—
512C22⋊C4—1584no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—3162no—
519C42—6321no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—3161no—
528C23—3161no—
529C23—3161no—
530C23—3161no—
531C23—3161no—
532C23—3161no—
533C23—1582no—
534C23C22×C22⋊D7911yescharacteristic
535D4—3161no—
536D4—3161no—
537D4—6324no—
538D4—1582no—
539D4—1582no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—3161no—
574C2×C4—3161no—
575C2×C4—3161no—
576C2×C4—6324no—
577C2×C4—1582no—
578C2×C4—1582no—
579C2×C4#1457918112yes—
580C2×C4#1457918712yes—
581C2×C4#41×D7911yescharacteristic
582C2×C4#41×D7911yescharacteristic
583C2×C4#41×D7911yescharacteristic
584C2×C4#41×D7911yescharacteristic
585C2×C4#1457920711yescharacteristic
586C2×C4#1457920811yescharacteristic
587C8—3161no—
588C8—3161no—
589C8—3161no—
590C8—1582no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—3162no—
599C22—3162no—
600C22—3162no—
601C22—6321no—
602C22—6321no—
603C22—6321no—
604C22—6321no—
605C22#1909907011yescharacteristic
606C22#1909920111yescharacteristic
607C22#1909921211yescharacteristic
608C22C2×#1457918111yescharacteristic
609C22C2×#1457918711yescharacteristic
610C22#1909937111yescharacteristic
611C22#1909937111yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—3162no—
622C4—3162no—
623C4—6321no—
624C4#1909909611yescharacteristic
625C4#1909915411yescharacteristic
626C4#1909926411yescharacteristic
627C4#1909928011yescharacteristic
628C2—81no—
629C2—6322no—
630C2#2376743211yescharacteristic
631C2#2377345411yescharacteristic
632C2#2377364711yescharacteristic
633C2#2377533311yescharacteristic
634C2#2377535511yescharacteristic
635C2#2377577911yescharacteristic
636C2#2377599511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2496C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenter#2496D79
hypocenterC79#27039
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #23764804C2semidirect
3. #23764771C2semidirect
6. #23764161C2semidirect
7. #23764125C2semidirect
8. #23764008C2semidirect
9. #23763989C2semidirect
10. #23763985C2semidirect
11. #23763646C2semidirect
12. #23763597C2semidirect
13. #23763492C2semidirect
14. C4×#14579081C2semidirect
15. C79×#2496C2semidirect
319. #2496D79semidirect
334. C79#27039semidirect

Maximal subgroups

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

subgroupindex
#237664362
#237648042
#237647712
#237645892
#237641612
#237641252
#237640082
#237639892
#237639852
#237636462
#237635972
#237634922
C4×#145790812
C79×#24962
#2703979