Finite Group Databasebeta

← #34360994

Subgroups — #34360994

Order 20224.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767813C211yesmaximal, characteristic
2#23766409C211yesmaximal, characteristic
3#23764821C211yesmaximal, characteristic, complemented
4#23764791C211yesmaximal, characteristic, complemented
5#23764600C211yesmaximal, characteristic
6#23764169C211yesmaximal, characteristic, complemented
7#23764132C211yesmaximal, characteristic, complemented
8#23764003C211yesmaximal, characteristic
9#23763987C211yesmaximal, characteristic, complemented
10#23763974C211yesmaximal, characteristic, complemented
11#23763652C211yesmaximal, characteristic, complemented
12#23763587C211yesmaximal, characteristic
13#23763487C211yesmaximal, characteristic
14#103×Dic79C211yesmaximal, characteristic, complemented
15C79×#2035C211yesmaximal, characteristic
16C2×Q8×Dic79C2211yescharacteristic
17C22×C4⋊Dic79C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×Dic79⋊C8C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×C22⋊(C79⋊C8)C2211yescharacteristic
25C2×C8×Dic79C2211yescharacteristic
26#19098503—21no—
27#19098491—21no—
28#19098470C2211yescharacteristic
29#19098462—21no—
30#19098462C2211yescharacteristic
31#19098462C2211yescharacteristic
32#19098459C2211yescharacteristic
33#19098456—21no—
34#19098449—21no—
35#19098438—21no—
36#19098430—21no—
37#19098428C2211yescharacteristic
38#19098423C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C22⋊C4×Dic79—21no—
42C4×C22⋊Dic79C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098326C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C2×C22⋊Q8×C79C2211yescharacteristic
53C2×#106×C79C2211yescharacteristic
54C2×#103×C79C2211yescharacteristic
55C2×#103×C79C2211yescharacteristic
56C2×C22⋊C8×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59C23×Dic79—21no—
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62Q8×Dic79—22no—
63Q8×Dic79—41no—
64C2×C4⋊Dic79—22no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—41no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×Dic79⋊C4—21no—
75C2×C22⋊Dic79—21no—
76C2×C22⋊Dic79D411yescharacteristic
77C2×C22⋊Dic79D411yescharacteristic
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79C2311yescharacteristic
82C2×C4×Dic79C2311yescharacteristic
83#14579094—22no—
84#14579094—22no—
85#14579094—41no—
86Dic79⋊C8—22no—
87#14579085—22no—
88#14579085—22no—
89C22⋊(C79⋊C8)—22no—
90C8×Dic79—41no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95D411yescharacteristic
102C79⋊#95D411yescharacteristic
103C22×Q8×C79C2311yescharacteristic
104C23×C4×C79C2311yescharacteristic
105C22×C8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22⋊Q8×C79—22no—
108C22⋊Q8×C79—22no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79—21no—
115#106×C79—22no—
116#103×C79—22no—
117#103×C79—41no—
118C22⋊C8×C79—41no—
119#95×C79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79—42no—
125C22×Dic79—42no—
126C22×Dic79—81no—
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—21no—
136C2×C79⋊C8#4112yes—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—42no—
145C4⋊Dic79—42no—
146C4⋊Dic79—81no—
147C4⋊Dic79—81no—
148C4⋊Dic79#4112yes—
149Dic79⋊C4—81no—
150C22⋊Dic79—22no—
151C22⋊Dic79—22no—
152C22⋊Dic79—81no—
153C4×Dic79—21no—
154C4×Dic79—21no—
155C4×Dic79—21no—
156C4×Dic79—21no—
157C4×Dic79—42no—
158C4×Dic79—42no—
159C4×Dic79—81no—
160C24×C79C2×D411yescharacteristic
161C2×Q8×C79—21no—
162C2×Q8×C79—21no—
163C2×Q8×C79—21no—
164C2×Q8×C79—21no—
165C2×Q8×C79—21no—
166C2×Q8×C79#4112yes—
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×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2411yescharacteristic
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C4⋊C4×C79—24no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79#4112yes—
192C22⋊C4×C79—24no—
193C22⋊C4×C79—42no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79—21no—
200C23×C79#13512yes—
201C23×C79C22×D411yescharacteristic
202Q8×C79—22no—
203Q8×C79—22no—
204Q8×C79—41no—
205Q8×C79—41no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
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×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79C2×#4111yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×C4×C79H(5,𝔽2)11yescharacteristic
230C2×C4×C792−1+411yescharacteristic
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—22no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—84no—
255C2×Dic79#13712yes—
256C8×C79—41no—
257C8×C79—41no—
258C79⋊C8—22no—
259C79⋊C8—41no—
260C22×C79—22no—
261C22×C79—22no—
262C22×C79—22no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79#52511yescharacteristic
268C22×C79#54611yescharacteristic
269C22×C79#54711yescharacteristic
270C22×C79C2×#13511yescharacteristic
271C22×C79C2×#13711yescharacteristic
272C22×C79#57611yescharacteristic
273C22×C79#57811yescharacteristic
274C4×C79—42no—
275C4×C79—41no—
276C4×C79—42no—
277C4×C79—81no—
278C4×C79—81no—
279C4×C79#52911yescharacteristic
280C4×C79#53911yescharacteristic
281C4×C79#55311yescharacteristic
282C4×C79#55611yescharacteristic
283Dic79—22no—
284Dic79—41no—
285Dic79—41no—
286Dic79—41no—
287Dic79—42no—
288Dic79—81no—
289Dic79—81no—
290Dic79—81no—
291#27035—791nomaximal, Sylow
292C2×C79—42no—
293C2×C79#246611yescharacteristic
294C2×C79#310811yescharacteristic
295C2×C79#312411yescharacteristic
296C2×C79#329711yescharacteristic
297C2×C79#331211yescharacteristic
298C2×C79#333611yescharacteristic
299C2×C79#335711yescharacteristic
300#2495—791no—
301#2360—791no—
302#2074—791no—
303#2068—791no—
304#2036—791no—
305#2035D7911yescharacteristic
306#1949—791no—
307#1942—791no—
308#1923—791no—
309#1920—791no—
310#1918—791no—
311#1863—791no—
312#1852—791no—
313#1823—791no—
314C4×#103—791no—
315C79#2703511yescharacteristic, complemented, Sylow
316C2×C22⋊Q8C2×D7911yescharacteristic
317C2×C4×Q8—791no—
318C22×C4⋊C4—791no—
319C2×#107—791no—
320C2×#107—791no—
321C2×#107—791no—
322C2×#106C2×D7911yescharacteristic
323C2×C4⋊C8—791no—
324C2×#103—791no—
325C2×#103—791no—
326C2×#103C2×D7911yescharacteristic
327C2×#103C2×D7911yescharacteristic
328C2×C22⋊C8—791no—
329C2×C22⋊C8C2×D7911yescharacteristic
330C2×C4×C8—791no—
331#400—1581no—
332#397—1581no—
333#391—791no—
334#389—791no—
335#389—791no—
336#389—791no—
337#389—1581no—
338#388—1581no—
339#387—1581no—
340#385—1581no—
341#383—1581no—
342#382—791no—
343#380—791no—
344#380C2×D7911yescharacteristic
345C4×C4⋊C4—791no—
346C4×C4⋊C4—791no—
347C4×C22⋊C4—791no—
348C4×C22⋊C4—1581no—
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340C2×D7911yescharacteristic
357#336—791no—
358#336—791no—
359C22×Q8C22×D7911yescharacteristic
360C23×C4—1581no—
361C23×C4C22×D7911yescharacteristic
362C22×C8—791no—
363C22×C8—791no—
364C22×C8C22×D7911yescharacteristic
365C22×C8C22×D7911yescharacteristic
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—3161no—
369C4×Q8—1582no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—3161no—
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⋊C4—1582no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—791no—
388C2×C22⋊C4—791no—
389C2×C22⋊C4—1581no—
390C2×C42—791no—
391C2×C42—791no—
392C2×C42—1581no—
393C2×C42—1581no—
394C2×C42—1581no—
395#107—3161no—
396#107—1582no—
397#107—1582no—
398#106—22no—
399C4⋊C8—1582no—
400#103—22no—
401#103—41no—
402#103—1582no—
403#103—1582no—
404C22⋊C8—41no—
405C22⋊C8—1582no—
406C4×C8—3161no—
407#95—21no—
408#95—791no—
409#95—791no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420C24D4×D7911yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1192962312yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—3162no—
438C22×C4—3162no—
439C22×C4—791no—
440C22×C4—791no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—6321no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4D4×D7911yescharacteristic
449C22×C4C23×D7911yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—792no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—3162no—
469C4⋊C4—3162no—
470C4⋊C4—792no—
471C4⋊C4—6321no—
472C4⋊C4—6321no—
473C4⋊C4—6321no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4#1192962312yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—6321no—
485C22⋊C4—1582no—
486C22⋊C4—1582no—
487C42—3162no—
488C42—3162no—
489C42—6321no—
490C42—1581no—
491C42—1581no—
492C42—1581no—
493C42—1581no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1457918212yes—
501C23C2×D4×D7911yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
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—41no—
524C2×C4—41no—
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—3161no—
536C2×C4—792no—
537C2×C4—6324no—
538C2×C4—1582no—
539C2×C4—1581no—
540C2×C4—1581no—
541C2×C4—1581no—
542C2×C4—1581no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4#41×D7911yescharacteristic
551C2×C4#41×D7911yescharacteristic
552C2×C4C2×#1192962311yescharacteristic
553C2×C4C2×#1192962311yescharacteristic
554C2×C4#1457920611yescharacteristic
555C2×C4#1457920811yescharacteristic
556C8—41no—
557C8—41no—
558C8—3161no—
559C8—1582no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1909906911yescharacteristic
568C22#1909920211yescharacteristic
569C22#1909921111yescharacteristic
570C22C2×#1457918211yescharacteristic
571C22#137×D7911yescharacteristic
572C22#1909936511yescharacteristic
573C22#1909937911yescharacteristic
574C4—41no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—3161no—
580C4—3162no—
581C4—3161no—
582C4—3161no—
583C4—6321no—
584C4—6321no—
585C4—6321no—
586C4—1582no—
587C4#1909909011yescharacteristic
588C4#1909915811yescharacteristic
589C4#1909926011yescharacteristic
590C4#1909928211yescharacteristic
591C2—42no—
592C2#2376742911yescharacteristic
593C2#2377347511yescharacteristic
594C2#2377366311yescharacteristic
595C2#2377525611yescharacteristic
596C2#2377544611yescharacteristic
597C2#2377576611yescharacteristic
598C2#2377602411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2035C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2035D79
hypocenterC79#27035
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #23764821C2semidirect
4. #23764791C2semidirect
6. #23764169C2semidirect
7. #23764132C2semidirect
9. #23763987C2semidirect
10. #23763974C2semidirect
11. #23763652C2semidirect
14. #103×Dic79C2semidirect
315. C79#27035semidirect

Maximal subgroups

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

subgroupindex
#237678132
#237664092
#237648212
#237647912
#237646002
#237641692
#237641322
#237640032
#237639872
#237639742
#237636522
#237635872
#237634872
#103×Dic792
C79×#20352
#2703579