Finite Group Databasebeta

← #34318318

Subgroups — #34318318

Order 20224.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765798C211yesmaximal, characteristic
2#23764968C211yesmaximal, characteristic, complemented
3#23764948C211yesmaximal, characteristic, complemented
4#23764881C211yesmaximal, characteristic, complemented
5#23764836C211yesmaximal, characteristic
6#23764826C211yesmaximal, characteristic
7#23764818C211yesmaximal, characteristic, complemented
8#23764600C211yesmaximal, characteristic
9#23764593C211yesmaximal, characteristic
10C4×#14579094C211yesmaximal, characteristic, complemented
11C4×#14579092C211yesmaximal, characteristic, complemented
12C4×#14579085C211yesmaximal, characteristic, complemented
13C4×#14579085C211yesmaximal, characteristic, complemented
14C4×C22⋊(C79⋊C8)C211yesmaximal, characteristic
15C4×C22⋊Q8×C79C211yesmaximal, characteristic
16C2×#14579094C2211yescharacteristic
17C2×#14579094C2211yescharacteristic
18C2×#14579092C2211yescharacteristic
19C2×#14579092C2211yescharacteristic
20C2×#14579085C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×C22⋊(C79⋊C8)C2211yescharacteristic
25C2×C22⋊(C79⋊C8)C2211yescharacteristic
26C2×C4×C79⋊C8C2211yescharacteristic
27C2×C4×C79⋊C8C2211yescharacteristic
28#19098431—21no—
29#19098429C2211yescharacteristic
30#19098429C2211yescharacteristic
31#19098426—21no—
32#19098423C2211yescharacteristic
33#19098423C2211yescharacteristic
34C4×C4⋊Dic79C2211yescharacteristic
35C4×C4⋊Dic79C2211yescharacteristic
36C4×C22⋊Dic79—21no—
37#19098409—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C22⋊Q8×C79C2211yescharacteristic
49C2×C4×Q8×C79C2211yescharacteristic
50C22×C42×C79C2211yescharacteristic
51#391×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#387×C79—21no—
54#385×C79—21no—
55#380×C79C2211yescharacteristic
56C4×C4⋊C4×C79—21no—
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79—21no—
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C2×C4⋊Dic79C2311yescharacteristic
64C2×C4⋊Dic79C2311yescharacteristic
65C2×C4⋊Dic79C2311yescharacteristic
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C22⋊Dic79—21no—
68C2×C22⋊Dic79—21no—
69C2×C4×Dic79—21no—
70C2×C4×Dic79—21no—
71#14579094—22no—
72#14579094—22no—
73#14579092—22no—
74#14579092—22no—
75#14579085—22no—
76#14579085—22no—
77#14579085—22no—
78#14579085—22no—
79C22⋊(C79⋊C8)—22no—
80C22⋊(C79⋊C8)—22no—
81C4×C79⋊C8—22no—
82C4×C79⋊C8—22no—
83C79⋊#95—21no—
84C79⋊#95—21no—
85C79⋊#95—21no—
86C79⋊#95—21no—
87C79⋊#95—21no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C22×Q8×C79C2311yescharacteristic
92C23×C4×C79D411yescharacteristic
93C23×C4×C79D411yescharacteristic
94C23×C4×C79C2311yescharacteristic
95C22⋊Q8×C79—22no—
96C22⋊Q8×C79—22no—
97C4×Q8×C79—22no—
98C4×Q8×C79—22no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C22⋊C4×C79—21no—
106C2×C22⋊C4×C79—21no—
107C2×C42×C79—22no—
108C2×C42×C79—22no—
109C2×C42×C79—21no—
110C2×C42×C79—21no—
111C2×C42×C79D411yescharacteristic
112C2×C42×C79D411yescharacteristic
113C2×C42×C79C2311yescharacteristic
114C2×C42×C79C2311yescharacteristic
115#95×C79—21no—
116#95×C79—21no—
117#95×C79—21no—
118#95×C79—21no—
119#95×C79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—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—
135C2×C79⋊C8#4112yes—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C22⋊Dic79—42no—
145C22⋊Dic79—42no—
146C4×Dic79—42no—
147C4×Dic79—42no—
148C24×C79C2×D411yescharacteristic
149C2×Q8×C79—21no—
150C2×Q8×C79—21no—
151C2×Q8×C79—21no—
152C2×Q8×C79—21no—
153C2×Q8×C79—21no—
154C2×Q8×C79#4112yes—
155C22×C4×C79—22no—
156C22×C4×C79—21no—
157C22×C4×C79—21no—
158C22×C4×C79—22no—
159C22×C4×C79—22no—
160C22×C4×C79—22no—
161C22×C4×C79—22no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
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—41no—
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2411yescharacteristic
177C4⋊C4×C79—24no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—42no—
184C4⋊C4×C79—42no—
185C4⋊C4×C79#4112yes—
186C22⋊C4×C79—24no—
187C22⋊C4×C79—42no—
188C42×C79—22no—
189C42×C79—22no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—42no—
193C42×C79—42no—
194C42×C79#4112yes—
195C42×C79#4112yes—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79—21no—
200C23×C79—21no—
201C23×C79—21no—
202C23×C79#13512yes—
203C23×C79C22×D411yescharacteristic
204Q8×C79—22no—
205Q8×C79—22no—
206Q8×C79—41no—
207Q8×C79—41no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—22no—
214C2×C4×C79—22no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—44no—
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—41no—
230C2×C4×C79—44no—
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#12312yes—
238C2×C4×C79#12312yes—
239C2×C4×C79#13512yes—
240C2×C4×C79#13712yes—
241C2×C4×C79C2×#4111yescharacteristic
242C2×C4×C79C2×#4111yescharacteristic
243C2×C4×C79C2×#4111yescharacteristic
244C2×C4×C79C2×#4111yescharacteristic
245C2×C4×C79C2×#4111yescharacteristic
246C2×C4×C79C2×#4111yescharacteristic
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C79⋊C8—22no—
260C79⋊C8—22no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—22no—
264C22×C79—22no—
265C22×C79—22no—
266C22×C79—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79C2×#12311yescharacteristic
271C22×C79#52511yescharacteristic
272C22×C79#53211yescharacteristic
273C22×C79C2×#13511yescharacteristic
274C22×C79C2×#13511yescharacteristic
275C22×C79C2×#13711yescharacteristic
276C22×C79#57511yescharacteristic
277C4×C79—22no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—42no—
281C4×C79—41no—
282C4×C79—42no—
283C4×C79—81no—
284C4×C79—81no—
285C4×C79#48514yes—
286C4×C79#52911yescharacteristic
287C4×C79#52911yescharacteristic
288C4×C79#52911yescharacteristic
289C4×C79#52911yescharacteristic
290Dic79—81no—
291Dic79—81no—
292Dic79—81no—
293Dic79—81no—
294#23759—791nomaximal, Sylow
295C2×C79—42no—
296C2×C79#239511yescharacteristic
297C2×C79C2×#48511yescharacteristic
298C2×C79#309111yescharacteristic
299C2×C79#310211yescharacteristic
300C2×C79#310811yescharacteristic
301C2×C79#316811yescharacteristic
302C2×C79#317211yescharacteristic
303C4×C22⋊Q8D7911yescharacteristic
304#2283—791no—
305#2101—791no—
306#2097—791no—
307#2085—791no—
308#2077—791no—
309#2075—791no—
310#2074—791no—
311#2036—791no—
312#2035—791no—
313C4×#107—791no—
314C4×#106—791no—
315C4×#103—791no—
316C4×#103—791no—
317C4×C22⋊C8—791no—
318C79#2375911yescharacteristic, complemented, Sylow
319C2×C22⋊Q8C2×D7911yescharacteristic
320C2×C4×Q8C2×D7911yescharacteristic
321C22×C42C2×D7911yescharacteristic
322C2×#107—791no—
323C2×#107—791no—
324C2×#106—791no—
325C2×#106—791no—
326C2×#103—791no—
327C2×#103—791no—
328C2×#103—791no—
329C2×#103—791no—
330C2×C22⋊C8—791no—
331C2×C22⋊C8—791no—
332C2×C4×C8—791no—
333C2×C4×C8—791no—
334#391C2×D7911yescharacteristic
335#389C2×D7911yescharacteristic
336#387—21no—
337#385—21no—
338#383—1581no—
339#382—791no—
340#382—791no—
341#381—1581no—
342#380—791no—
343#380—791no—
344#380C2×D7911yescharacteristic
345C4×C4⋊C4—21no—
346C4×C4⋊C4—791no—
347C4×C4⋊C4—791no—
348C4×C4⋊C4C2×D7911yescharacteristic
349C4×C22⋊C4—21no—
350C4×C22⋊C4—1581no—
351#376—1581no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#336—791no—
361#336—791no—
362C22×Q8C22×D7911yescharacteristic
363C23×C4C22⋊D7911yescharacteristic
364C23×C4C22⋊D7911yescharacteristic
365C23×C4C22×D7911yescharacteristic
366C22×C8—791no—
367C22×C8—791no—
368C22×C8—791no—
369C22×C8—791no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4C22×D7911yescharacteristic
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1581no—
387C2×C22⋊C4—1581no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1581no—
393C2×C42—1581no—
394C2×C42C22⋊D7911yescharacteristic
395C2×C42C22⋊D7911yescharacteristic
396C2×C42C22×D7911yescharacteristic
397C2×C42C22×D7911yescharacteristic
398#107—1582no—
399#107—1582no—
400#106—1582no—
401#106—1582no—
402#103—1582no—
403#103—1582no—
404#103—1582no—
405#103—1582no—
406C22⋊C8—1582no—
407C22⋊C8—1582no—
408C4×C8—1582no—
409C4×C8—1582no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423C24C2×C22⋊D7911yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1192962412yes—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4C2×C22⋊D7911yescharacteristic
451C22×C4C2×C22⋊D7911yescharacteristic
452C22×C4C2×C22⋊D7911yescharacteristic
453C22×C4C2×C22⋊D7911yescharacteristic
454C22×C4C2×C22⋊D7911yescharacteristic
455C22×C4C23×D7911yescharacteristic
456C2×C8—792no—
457C2×C8—792no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4#1192962412yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—3162no—
488C22⋊C4—3162no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—3162no—
496C42—3162no—
497C42#1192962312yes—
498C42#1192962312yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1457918112yes—
506C23C22×C22⋊D7911yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4#1457914212yes—
553C2×C4#1457914212yes—
554C2×C4#1457918112yes—
555C2×C4#1457919212yes—
556C2×C4#41×D7911yescharacteristic
557C2×C4#41×D7911yescharacteristic
558C2×C4C2×#1192962311yescharacteristic
559C2×C4C2×#1192962311yescharacteristic
560C2×C4C2×#1192962411yescharacteristic
561C2×C4C2×#1192962411yescharacteristic
562C8—3161no—
563C8—3161no—
564C8—1582no—
565C8—1582no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1457914211yescharacteristic
574C22#1909907411yescharacteristic
575C22#1909910911yescharacteristic
576C22C2×#1457918111yescharacteristic
577C22C2×#1457918111yescharacteristic
578C22C2×#1457919211yescharacteristic
579C22#1909936111yescharacteristic
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—6321no—
589C4—6321no—
590C4—6321no—
591C4—6321no—
592C4#1909890114yes—
593C4#1909909311yescharacteristic
594C4#1909909311yescharacteristic
595C4#1909909511yescharacteristic
596C4#1909909511yescharacteristic
597C2—42no—
598C2#2376665911yescharacteristic
599C2C2×#1909890111yescharacteristic
600C2#2377330911yescharacteristic
601C2#2377341811yescharacteristic
602C2#2377348011yescharacteristic
603C2#2377396411yescharacteristic
604C2#2377400011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC4×C22⋊Q8×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenterC4×C22⋊Q8D79
hypocenterC79#23759
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #23764968C2semidirect
3. #23764948C2semidirect
4. #23764881C2semidirect
7. #23764818C2semidirect
10. C4×#14579094C2semidirect
11. C4×#14579092C2semidirect
12. C4×#14579085C2semidirect
13. C4×#14579085C2semidirect
318. C79#23759semidirect

Maximal subgroups

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

subgroupindex
#237657982
#237649682
#237649482
#237648812
#237648362
#237648262
#237648182
#237646002
#237645932
C4×#145790942
C4×#145790922
C4×#145790852
C4×C22⋊(C79⋊C8)2
C4×C22⋊Q8×C792
#2375979