Finite Group Databasebeta

← #34448421

Subgroups — #34448421

Order 20224.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099031C211yesmaximal, characteristic
2#23766370C214yesmaximal, complemented
3#23766362C214yesmaximal, complemented
4#23766339C214yesmaximal, complemented
5#23766335C214yesmaximal, complemented
6#23766010C211yesmaximal, characteristic, complemented
7#23766009C211yesmaximal, characteristic, complemented
8#23766008C211yesmaximal, characteristic, complemented
9#23765989C212yesmaximal
10#23765805C211yesmaximal, characteristic, complemented
11#23765796C212yesmaximal, complemented
12#23765795C212yesmaximal, complemented
13#382×D79C211yesmaximal, characteristic, complemented
14#23765754C212yesmaximal, complemented
15C79×#2314C211yesmaximal, characteristic, complemented
16C22×#11929624C2211yescharacteristic
17#19099031C4116yescomplemented
18C2×#14579126C2212yes—
19C2×#14579123C2212yes—
20C2×(C22⋊D79)⋊C4C2212yes—
21C2×(C22⋊D79)⋊C4C2212yes—
22C2×#14579119C2212yes—
23C2×#14579118C2211yescharacteristic
24C2×#14579118C2211yescharacteristic
25C2×C4⋊C4×D79C2212yes—
26C22×C4×Dic79C2211yescharacteristic
27C2×C42×D79C2211yescharacteristic
28C2×C42×D79C2211yescharacteristic
29#19098467C2214yes—
30#19098463C2214yes—
31#19098456C2214yes—
32#19098455C2214yes—
33#19098453C2214yes—
34#19098450C2214yes—
35#19098449C2214yes—
36#19098444C2214yes—
37#19098441C2214yes—
38#19098439C2214yes—
39#19098433C2211yescharacteristic
40#19098430C2214yes—
41#19098430C2214yes—
42#19098429C2211yescharacteristic
43#19098428C2212yes—
44#19098428C2212yes—
45#19098427C2214yes—
46#19098425C2214yes—
47#19098424C2214yes—
48#19098423C2212yes—
49#19098420C2212yes—
50#19098420C2212yes—
51#19098420C2214yes—
52C4⋊C4×Dic79C2212yes—
53C4⋊C4×Dic79C2212yes—
54C4⋊C4×Dic79C2214yes—
55C4×Dic79⋊C4C2214yes—
56C4×C4⋊Dic79C2212yes—
57#19098414C2214yes—
58C22⋊C4×Dic79C2214yes—
59C4×(C2×D79)⋊C4C2214yes—
60C4×(C2×D79)⋊C4C2214yes—
61#19098410C2214yes—
62#19098409C2214yes—
63#19098408C2214yes—
64#95×D79C2214yes—
65C2×#117×C79C2211yescharacteristic
66#388×C79C2214yes—
67#387×C79C2214yes—
68#384×C79C2211yescharacteristic
69#382×C79C2211yescharacteristic
70#380×C79C2212yes—
71C4×C4⋊C4×C79C2212yes—
72C2×#11929624—24no—
73C2×#11929624—24no—
74C2×#11929624C2×C418yes—
75C22×Dic158C2311yescharacteristic
76C22×D316C2311yescharacteristic
77C22×C22⋊D79C2311yescharacteristic
78C22×C22⋊D79C2311yescharacteristic
79C23×Dic79C2312yes—
80C22×C4×D79C2311yescharacteristic
81C22×C4×D79C2314yes—
82C22×C4×D79C2312yes—
83C22×C4×D79C2312yes—
84C22×C4×D79C2311yescharacteristic
85#14579126C2×C4116yescomplemented
86#14579123C2×C4116yes—
87(C22⋊D79)⋊C4C2×C4116yes—
88(C22⋊D79)⋊C4C2×C4116yescomplemented
89#14579119—28no—
90#14579118C2×C418yes—
91#14579118C2×C418yescomplemented
92C2×C4⋊Dic79C2312yes—
93C2×C4⋊Dic79C2312yes—
94C2×C4⋊Dic79C2314yes—
95C2×Dic79⋊C4C2314yes—
96C2×Dic79⋊C4C2312yes—
97C2×Dic79⋊C4C2312yes—
98C4⋊C4×D79—28no—
99C2×C22⋊Dic79C2314yes—
100C2×(C2×D79)⋊C4C2314yes—
101C2×(C2×D79)⋊C4C2314yes—
102C2×(C2×D79)⋊C4C2314yes—
103C2×(C2×D79)⋊C4C2312yes—
104C2×(C2×D79)⋊C4C2312yes—
105C2×C4×Dic79C2×C418yescomplemented
106C2×C4×Dic79C2314yes—
107C2×C4×Dic79C2314yes—
108C2×C4×Dic79C2314yes—
109C2×C4×Dic79C2314yes—
110C2×C4×Dic79C2312yes—
111C2×C4×Dic79C2312yes—
112C2×C4×Dic79C2311yescharacteristic
113C2×C4×Dic79C2312yes—
114C2×C4×Dic79C2311yescharacteristic
115C42×D79C2×C418yes—
116C42×D79C2×C418yescomplemented
117C79⋊#95C2314yes—
118C79⋊#95C2314yes—
119C79⋊#95C2314yes—
120C79⋊#95C2314yes—
121C79⋊#95C2314yes—
122C79⋊#95C2314yes—
123C79⋊#95C2314yes—
124C79⋊#95C2314yes—
125C79⋊#95C2314yes—
126C79⋊#95C2314yes—
127C79⋊#95C2314yes—
128C23×C4×C79C2311yescharacteristic
129#117×C79C2×C418yescomplemented
130C2×C4⋊C4×C79C2312yes—
131C2×C4⋊C4×C79C2312yes—
132C2×C4⋊C4×C79C2314yes—
133C2×C4⋊C4×C79C2312yes—
134C2×C22⋊C4×C79C2314yes—
135C2×C22⋊C4×C79C2312yes—
136C2×C42×C79C2314yes—
137C2×C42×C79C2311yescharacteristic
138C2×C42×C79C2311yescharacteristic
139#95×C79C2314yes—
140#95×C79C2314yes—
141#95×C79C2314yes—
142C23×D79C2411yescharacteristic
143C23×D79C2411yescharacteristic
144#11929624—232no—
145C2×Dic158—22no—
146C2×Dic158—22no—
147C2×Dic158C22×C414yes—
148C2×D316—22no—
149C2×D316—22no—
150C2×D316C22×C414yes—
151C2×C22⋊D79—22no—
152C2×C22⋊D79—22no—
153C2×C22⋊D79—22no—
154C2×C22⋊D79—22no—
155C2×C22⋊D79C22×C414yes—
156C2×C22⋊D79C22×C414yes—
157C22×Dic79—42no—
158C22×Dic79—42no—
159C22×Dic79C22×C418yes—
160C22×Dic79C2414yes—
161C22×Dic79C2414yes—
162C22×Dic79C2411yescharacteristic
163C22×Dic79C2412yes—
164C22×Dic79C2412yes—
165C22×Dic79C2411yescharacteristic
166C2×C4×D79—28no—
167C2×C4×D79—22no—
168C2×C4×D79—22no—
169C2×C4×D79—42no—
170C2×C4×D79—42no—
171C2×C4×D79—42no—
172C2×C4×D79—42no—
173C2×C4×D79—44no—
174C2×C4×D79—44no—
175C2×C4×D79C22×C414yes—
176C2×C4×D79C22×C418yes—
177C2×C4×D79C22×C418yes—
178C2×C4×D79C22×C414yes—
179C2×C4×D79#4114yes—
180C2×C4×D79#4114yes—
181C4⋊Dic79—24no—
182C4⋊Dic79—24no—
183C4⋊Dic79—44no—
184Dic79⋊C4—28no—
185Dic79⋊C4C22×C418yes—
186Dic79⋊C4C22×C418yes—
187C22⋊Dic79—44no—
188(C2×D79)⋊C4—28no—
189(C2×D79)⋊C4—44no—
190(C2×D79)⋊C4—44no—
191(C2×D79)⋊C4C22×C418yes—
192(C2×D79)⋊C4C22×C418yes—
193C4×Dic79—28no—
194C4×Dic79—44no—
195C4×Dic79—44no—
196C4×Dic79—44no—
197C4×Dic79C22×C414yes—
198C4×Dic79C22×C418yes—
199C4×Dic79C22×C418yes—
200C4×Dic79C22×C414yes—
201C4×Dic79C22×C418yes—
202C24×C79C2411yescharacteristic
203C22×C4×C79—22no—
204C22×C4×C79—22no—
205C22×C4×C79C22×C414yes—
206C22×C4×C79C2414yes—
207C22×C4×C79C2414yes—
208C22×C4×C79C2411yescharacteristic
209C22×C4×C79C2412yes—
210C22×C4×C79C2412yes—
211C22×C4×C79C2411yescharacteristic
212C4⋊C4×C79—24no—
213C4⋊C4×C79—24no—
214C4⋊C4×C79—44no—
215C4⋊C4×C79C22×C418yes—
216C22⋊C4×C79—44no—
217C22⋊C4×C79C22×C418yes—
218C42×C79—44no—
219C42×C79C22×C414yes—
220C42×C79C22×C414yes—
221C23×C79—21no—
222C23×C79—21no—
223C23×C79—22no—
224C23×C79—22no—
225C23×C79C23×C412yes—
226C23×C79C2511yescharacteristic
227C22×D79—22no—
228C22×D79—22no—
229C22×D79—21no—
230C22×D79—21no—
231C22×D79—22no—
232C22×D79—22no—
233C22×D79C23×C412yes—
234C22×D79C23×C412yes—
235C22×D79C2×#4112yes—
236C22×D79C2×#4112yes—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—22no—
240C2×C4×C79—22no—
241C2×C4×C79—22no—
242C2×C4×C79—22no—
243C2×C4×C79—28no—
244C2×C4×C79—24no—
245C2×C4×C79—24no—
246C2×C4×C79—24no—
247C2×C4×C79—24no—
248C2×C4×C79—24no—
249C2×C4×C79—24no—
250C2×C4×C79C23×C412yes—
251C2×C4×C79C23×C414yes—
252C2×C4×C79C23×C414yes—
253C2×C4×C79C23×C412yes—
254C2×C4×C79C2×#4112yes—
255C2×C4×C79C2×#4112yes—
256C22⋊D79—28no—
257C22⋊D79—28no—
258C2×Dic79—21no—
259C2×Dic79—21no—
260C2×Dic79—22no—
261C2×Dic79—22no—
262C2×Dic79—22no—
263C2×Dic79—22no—
264C2×Dic79—24no—
265C2×Dic79—24no—
266C2×Dic79—24no—
267C2×Dic79—24no—
268C2×Dic79—24no—
269C2×Dic79—24no—
270C2×Dic79—48no—
271C2×Dic79C23×C414yes—
272C2×Dic79C23×C414yes—
273C2×Dic79C23×C412yes—
274C2×Dic79C23×C412yes—
275C2×Dic79C2×#4112yes—
276C2×Dic79C2×#4112yes—
277D316—28no—
278C4×D79—28no—
279C4×D79—48no—
280C4×D79—48no—
281C4×D79—88no—
282C4×D79#117116yes—
283Dic158—28no—
284C22×C79—22no—
285C22×C79—24no—
286C22×C79—42no—
287C22×C79—41no—
288C22×C79—41no—
289C22×C79C24×C411yescharacteristic
290C22×C79C22×#4111yescharacteristic
291C22×C79C22×#4111yescharacteristic
292C22×C79#58512yes—
293C22×C79#58512yes—
294C2×D79—24no—
295C2×D79—24no—
296C2×D79—22no—
297C2×D79—22no—
298C2×D79—22no—
299C2×D79—24no—
300C2×D79—42no—
301C2×D79—41no—
302C2×D79—41no—
303C2×D79C2×#11714yes—
304C4×C79—22no—
305C4×C79—24no—
306C4×C79—24no—
307C4×C79—44no—
308C4×C79—44no—
309C4×C79C2×#11714yes—
310Dic79—22no—
311Dic79—24no—
312Dic79—24no—
313Dic79—44no—
314Dic79—44no—
315Dic79C2×#11714yes—
316#34189—791nomaximal, Sylow
317C2×C79—42no—
318C2×C79C22×#11711yescharacteristic
319C2×C79#342812yes—
320C2×C79#351012yes—
321C2×C79#351211yescharacteristic
322C2×C79#351211yescharacteristic
323D79—24no—
324D79—42no—
325C2×C4×#41—791no—
326#2357—794no—
327#2356—794no—
328#2353—794no—
329#2353—794no—
330#2314—791no—
331#2314—791no—
332#2314—791no—
333#2314D7911yescharacteristic, complemented
334#2311—792no—
335#2285—791no—
336#2283—792no—
337#2283—792no—
338C2×#382—791no—
339#2274—792no—
340C79#3418911yescharacteristic, complemented, Sylow
341C22×#41—791no—
342C4×#41—7916no—
343C2×C4×Q8—792no—
344C2×C4×D4—792no—
345C2×C4×D4—792no—
346C2×C4×D4—792no—
347C2×#117—792no—
348C2×#117—791no—
349C2×#117—791no—
350C2×#117C2×D7911yescharacteristic
351C22×C4⋊C4—792no—
352C22×C42—791no—
353C22×C42—791no—
354C22×C42—791no—
355#391—794no—
356#390—794no—
357#388—794no—
358#388—794no—
359#388—794no—
360#388C2×D7914yes—
361#387—794no—
362#387—794no—
363#387—794no—
364#387C2×D7914yes—
365#386—794no—
366#386—794no—
367#384—791no—
368#384C2×D7911yescharacteristic
369#383—794no—
370#383—794no—
371#382—792no—
372#382—792no—
373#382—791no—
374#382—794no—
375#382C2×D7911yescharacteristic
376#381—794no—
377#381—794no—
378#380—792no—
379#380—792no—
380#380—792no—
381#380—794no—
382#380C2×D7912yes—
383C4×C4⋊C4—792no—
384C4×C4⋊C4—792no—
385C4×C4⋊C4—794no—
386C4×C4⋊C4—792no—
387C4×C4⋊C4—794no—
388C4×C4⋊C4—794no—
389C4×C4⋊C4C2×D7912yes—
390C4×C22⋊C4—794no—
391C4×C22⋊C4—794no—
392C4×C22⋊C4—794no—
393C4×C22⋊C4—794no—
394#376—794no—
395#376—794no—
396C2×#95—794no—
397C2×#41—798no—
398C2×#41—1584no—
399C2×#41—1584no—
400C22×Q8—791no—
401C22×D4—791no—
402C22×D4—791no—
403C22×D4—791no—
404C23×C4—792no—
405C23×C4—792no—
406C23×C4—792no—
407C23×C4—794no—
408C23×C4—791no—
409C23×C4—791no—
410C23×C4C22×D7911yescharacteristic
411C4×Q8—7916no—
412C4×D4—7916no—
413C4×D4—7916no—
414C4×D4—7916no—
415#117—798no—
416#117—798no—
417#117—1588no—
418#117C4×D7918yescomplemented
419C2×C4⋊C4—794no—
420C2×C4⋊C4—792no—
421C2×C4⋊C4—792no—
422C2×C4⋊C4—792no—
423C2×C4⋊C4—794no—
424C2×C4⋊C4—792no—
425C2×C4⋊C4—1588no—
426C2×C4⋊C4C22×D7912yes—
427C2×C4⋊C4C22×D7912yes—
428C2×C4⋊C4C22×D7914yes—
429C2×C4⋊C4C22×D7912yes—
430C2×C22⋊C4—794no—
431C2×C22⋊C4—794no—
432C2×C22⋊C4—794no—
433C2×C22⋊C4—792no—
434C2×C22⋊C4—792no—
435C2×C22⋊C4—794no—
436C2×C22⋊C4C22×D7914yes—
437C2×C22⋊C4C22×D7912yes—
438C2×C42—794no—
439C2×C42—794no—
440C2×C42—794no—
441C2×C42—798no—
442C2×C42—791no—
443C2×C42—792no—
444C2×C42—792no—
445C2×C42—792no—
446C2×C42—791no—
447C2×C42—798no—
448C2×C42—798no—
449C2×C42—794no—
450C2×C42C22×D7914yes—
451C2×C42C22×D7911yescharacteristic
452C2×C42C22×D7911yescharacteristic
453#95—794no—
454#95—794no—
455#95—794no—
456#95—794no—
457#95—794no—
458#95—794no—
459#95—794no—
460#95—794no—
461#95—794no—
462#95—794no—
463#95—794no—
464#95C22×D7914yes—
465#95C22×D7914yes—
466#95C22×D7914yes—
467C24—791no—
468C24—791no—
469C24C23×D7911yescharacteristic
470#41—15832no—
471C2×Q8—794no—
472C2×Q8—1582no—
473C2×Q8—1582no—
474C2×D4—794no—
475C2×D4—794no—
476C2×D4—794no—
477C2×D4—1582no—
478C2×D4—1582no—
479C2×D4—1582no—
480C2×D4—1582no—
481C2×D4—1582no—
482C2×D4—1582no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—3162no—
486C22×C4—3162no—
487C22×C4—3162no—
488C22×C4—3162no—
489C22×C4—3164no—
490C22×C4—3164no—
491C22×C4—3162no—
492C22×C4—3162no—
493C22×C4—798no—
494C22×C4—798no—
495C22×C4—798no—
496C22×C4—791no—
497C22×C4—794no—
498C22×C4—794no—
499C22×C4—794no—
500C22×C4—792no—
501C22×C4—791no—
502C22×C4—792no—
503C22×C4—794no—
504C22×C4—794no—
505C22×C4—794no—
506C22×C4—1588no—
507C22×C4—1582no—
508C22×C4—1582no—
509C22×C4C2×C4×D7914yes—
510C22×C4C23×D7914yes—
511C22×C4C23×D7914yes—
512C22×C4C23×D7911yescharacteristic
513C22×C4C23×D7912yes—
514C22×C4C23×D7912yes—
515C22×C4C23×D7911yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—3164no—
520C4⋊C4—798no—
521C4⋊C4—798no—
522C4⋊C4—1588no—
523C4⋊C4—1584no—
524C4⋊C4—1584no—
525C4⋊C4C2×C4×D7918yes—
526C22⋊C4—44no—
527C22⋊C4—3164no—
528C22⋊C4—3164no—
529C22⋊C4—3164no—
530C22⋊C4—798no—
531C22⋊C4—798no—
532C22⋊C4—1588no—
533C22⋊C4C2×C4×D7918yes—
534C42—44no—
535C42—3164no—
536C42—3164no—
537C42—3164no—
538C42—794no—
539C42—794no—
540C42—798no—
541C42—798no—
542C42—798no—
543C42—1588no—
544C42C2×C4×D7914yes—
545C42C2×C4×D7914yes—
546C23—22no—
547C23—22no—
548C23—21no—
549C23—21no—
550C23—792no—
551C23—792no—
552C23—792no—
553C23—792no—
554C23—1582no—
555C23—1582no—
556C23—1581no—
557C23—1581no—
558C23—1582no—
559C23—1582no—
560C23C22×C4×D7912yes—
561C23C24×D7911yescharacteristic
562Q8—1588no—
563D4—1588no—
564D4—1588no—
565D4—1588no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—3168no—
580C2×C4—3168no—
581C2×C4—3168no—
582C2×C4—792no—
583C2×C4—792no—
584C2×C4—792no—
585C2×C4—7916no—
586C2×C4—794no—
587C2×C4—794no—
588C2×C4—792no—
589C2×C4—6328no—
590C2×C4—1584no—
591C2×C4—1584no—
592C2×C4—1584no—
593C2×C4—1584no—
594C2×C4—1584no—
595C2×C4—1584no—
596C2×C4—1582no—
597C2×C4—1582no—
598C2×C4—1582no—
599C2×C4—1582no—
600C2×C4—1581no—
601C2×C4—1581no—
602C2×C4—1588no—
603C2×C4C22×C4×D7912yes—
604C2×C4C22×C4×D7914yes—
605C2×C4C22×C4×D7914yes—
606C2×C4C22×C4×D7912yes—
607C2×C4#41×D7912yes—
608C2×C4#41×D7912yes—
609C22—22no—
610C22—24no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—3162no—
615C22—3161no—
616C22—3161no—
617C22—794no—
618C22—1582no—
619C22—1584no—
620C22—1584no—
621C22—1582no—
622C22—1582no—
623C22—1584no—
624C22C23×C4×D7911yescharacteristic
625C22C2×#41×D7911yescharacteristic
626C22C2×#41×D7911yescharacteristic
627C22#1909939812yes—
628C22#1909940012yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—3164no—
635C4—3164no—
636C4—794no—
637C4—1582no—
638C4—1584no—
639C4—1584no—
640C4#117×D7914yes—
641C2—42no—
642C2—3162no—
643C2—1584no—
644C2C2×#117×D7911yescharacteristic
645C2#2377660012yes—
646C2#2377729212yes—
647C2#2377731911yescharacteristic
648C2#2377732311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D79
commutatorC22×C79C24×C4
FittingC79×#2314C2
socle, cosocleC23×C79C25
prime radicalC22×#11929624C22
hypercenter#2314D79
hypocenterC79#34189
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #23766370C2semidirect
3. #23766362C2semidirect
4. #23766339C2semidirect
5. #23766335C2semidirect
6. #23766010C2semidirect
7. #23766009C2semidirect
8. #23766008C2semidirect
10. #23765805C2semidirect
11. #23765796C2semidirect
12. #23765795C2semidirect
13. #382×D79C2semidirect
14. #23765754C2semidirect
15. C79×#2314C2semidirect
17. #19099031C4semidirect
85. #14579126C2×C4semidirect
88. (C22⋊D79)⋊C4C2×C4semidirect
91. #14579118C2×C4semidirect
105. C2×C4×Dic79C2×C4semidirect
116. C42×D79C2×C4semidirect
129. #117×C79C2×C4semidirect
333. #2314D79semidirect
340. C79#34189semidirect
418. #117C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190990312
#237663702
#237663622
#237663392
#237663352
#237660102
#237660092
#237660082
#237659892
#237658052
#237657962
#237657952
#382×D792
#237657542
C79×#23142
#3418979