Finite Group Databasebeta

← #34295634

Subgroups — #34295634

Order 20224.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098901C211yesmaximal, characteristic
2#23766084C211yesmaximal, characteristic
3#23765757C211yesmaximal, characteristic
4#23764037C211yesmaximal, characteristic, complemented
5#23764037C211yesmaximal, characteristic, complemented
6#23763995C211yesmaximal, characteristic, complemented
7#23763981C211yesmaximal, characteristic, complemented
8#23763593C211yesmaximal, characteristic
9#23763496C211yesmaximal, characteristic
10#23763296C211yesmaximal, characteristic, complemented
11#23763288C211yesmaximal, characteristic, complemented
12#23763232C211yesmaximal, characteristic, complemented
13#103×Dic79C211yesmaximal, characteristic, complemented
14#23763168C211yesmaximal, characteristic
15C79×#1872C211yesmaximal, characteristic
16C2×Q8×Dic79C2211yescharacteristic
17C2×#14579120C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19#19098901C418yescomplemented
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×C22⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29#19098470C2211yescharacteristic
30#19098462—21no—
31#19098462C2211yescharacteristic
32#19098456—21no—
33#19098456—21no—
34#19098449—21no—
35#19098423C2211yescharacteristic
36C4⋊C4×Dic79—21no—
37C4⋊C4×Dic79—21no—
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4×C22⋊Dic79—21no—
44C4×C22⋊Dic79—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×C22⋊Q8×C79C2211yescharacteristic
54C22×C4⋊C4×C79C2211yescharacteristic
55C2×#103×C79C2211yescharacteristic
56C2×#103×C79C2211yescharacteristic
57C2×C22⋊C8×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62Q8×Dic79—22no—
63Q8×Dic79—22no—
64#14579120—22no—
65#14579120—22no—
66#14579120—22no—
67#14579120C2×C414yes—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79D411yescharacteristic
71C2×C4⋊Dic79D411yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79D411yescharacteristic
79C2×C22⋊Dic79D411yescharacteristic
80C2×C4×Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79C2311yescharacteristic
88C2×C4×Dic79C2311yescharacteristic
89#14579094C2×C414yescomplemented
90#14579092—22no—
91#14579092—22no—
92#14579092C2×C414yescomplemented
93#14579085C2×C414yescomplemented
94#14579085C2×C414yescomplemented
95C22⋊(C79⋊C8)C2×C414yes—
96#14579071—22no—
97C8×Dic79—22no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C22×Q8×C79C2311yescharacteristic
108C23×C4×C79—21no—
109C23×C4×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C22×C8×C79C2311yescharacteristic
112C22⋊Q8×C79—22no—
113C22⋊Q8×C79C2×C414yes—
114C2×C4⋊C4×C79—21no—
115C2×C4⋊C4×C79—21no—
116C2×C4⋊C4×C79—22no—
117C2×C4⋊C4×C79—22no—
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C22⋊C4×C79—21no—
122#103×C79—22no—
123#103×C79—22no—
124C22⋊C8×C79—22no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79C2×D411yescharacteristic
130C22×Dic79C2×D411yescharacteristic
131C22×Dic79C2×D411yescharacteristic
132C22×Dic79C2×D411yescharacteristic
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8C22×C412yes—
138C2×C79⋊C8C22×C412yes—
139C4⋊Dic79—22no—
140C4⋊Dic79—22no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—42no—
148C4⋊Dic79—81no—
149C4⋊Dic79C22×C412yes—
150C4⋊Dic79C22×C412yes—
151C22⋊Dic79—22no—
152C22⋊Dic79—22no—
153C22⋊Dic79—24no—
154C22⋊Dic79—81no—
155C4×Dic79—24no—
156C4×Dic79—21no—
157C4×Dic79—21no—
158C4×Dic79—42no—
159C4×Dic79—81no—
160C4×Dic79—81no—
161C4×Dic79—81no—
162C4×Dic79—81no—
163C4×Dic79—81no—
164C4×Dic79#4112yes—
165C4×Dic79#4112yes—
166C24×C79C2×D411yescharacteristic
167C2×Q8×C79—21no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79—21no—
172C2×Q8×C79C22×C412yes—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—24no—
182C22×C4×C79—81no—
183C22×C4×C79—81no—
184C22×C4×C79C22×C412yes—
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79#4112yes—
187C22×C4×C79C2411yescharacteristic
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C4⋊C4×C79—24no—
195C4⋊C4×C79—24no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79C22×C412yes—
200C4⋊C4×C79#4112yes—
201C4⋊C4×C79#4112yes—
202C22⋊C4×C79—24no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79C4×D412yes—
210C23×C79C22×D411yescharacteristic
211Q8×C79—22no—
212Q8×C79—22no—
213Q8×C79—41no—
214Q8×C79—41no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—22no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—84no—
234C2×C4×C79C4×D412yes—
235C2×C4×C79C23×C411yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—22no—
246C2×Dic79—22no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—21no—
250C2×Dic79—21no—
251C2×Dic79—21no—
252C2×Dic79—21no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79#12312yes—
264C2×Dic79#12312yes—
265C8×C79—41no—
266C8×C79—41no—
267C79⋊C8—22no—
268C79⋊C8—22no—
269C22×C79—22no—
270C22×C79—22no—
271C22×C79—22no—
272C22×C79—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79C2×C4×D411yescharacteristic
277C22×C79C2×#12311yescharacteristic
278C22×C79#52511yescharacteristic
279C22×C79#57711yescharacteristic
280C22×C79#57711yescharacteristic
281C22×C79#57711yescharacteristic
282C22×C79#57811yescharacteristic
283C4×C79—22no—
284C4×C79—42no—
285C4×C79—42no—
286C4×C79—42no—
287C4×C79—42no—
288C4×C79C4×#4111yescharacteristic
289C4×C79C4×#4111yescharacteristic
290C4×C79#52911yescharacteristic
291C4×C79#52911yescharacteristic
292Dic79—22no—
293Dic79—22no—
294Dic79—41no—
295Dic79—41no—
296Dic79—42no—
297Dic79—42no—
298Dic79—81no—
299Dic79—81no—
300#21591—791nomaximal, Sylow
301C2×C79—42no—
302C2×C79C4×#12311yescharacteristic
303C2×C79#294711yescharacteristic
304C2×C79#295211yescharacteristic
305C2×C79#309711yescharacteristic
306C2×C79#309911yescharacteristic
307C2×C79#312111yescharacteristic
308C2×C79#312311yescharacteristic
309C2×#485—791no—
310#2323—791no—
311#2274—791no—
312#1927—791no—
313#1927—791no—
314#1921—791no—
315#1919—791no—
316#1872D7911yescharacteristic
317#1853—791no—
318#1825—791no—
319#1776—791no—
320C4×#106—791no—
321#1763—791no—
322C4×#103—791no—
323#1750—791no—
324C79#2159111yescharacteristic, complemented, Sylow
325C2×C22⋊Q8C2×D7911yescharacteristic
326C2×C4×Q8—791no—
327C2×#117—791no—
328C2×#117—791no—
329C22×C4⋊C4C2×D7911yescharacteristic
330#485—798no—
331C2×#107—791no—
332C2×#106—791no—
333C2×#106—791no—
334C2×#106—791no—
335C2×#103—791no—
336C2×#103—791no—
337C2×#103C2×D7911yescharacteristic
338C2×#103C2×D7911yescharacteristic
339C2×C22⋊C8—791no—
340C2×C22⋊C8C2×D7911yescharacteristic
341C2×#97—791no—
342C2×C4×C8—791no—
343#391—791no—
344#389—791no—
345#389—1581no—
346#388—1581no—
347#388—1581no—
348#387—1581no—
349#380—791no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—791no—
355C4×C4⋊C4—1581no—
356C4×C4⋊C4—1581no—
357C4×C22⋊C4—1581no—
358C4×C22⋊C4—1581no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340C2×D7911yescharacteristic
366#340C2×D7911yescharacteristic
367#336—791no—
368#336—791no—
369C22×Q8C22×D7911yescharacteristic
370C23×C4—21no—
371C23×C4C22×D7911yescharacteristic
372C22×C8—791no—
373C22×C8—791no—
374C22×C8C22×D7911yescharacteristic
375C22×C8C22×D7911yescharacteristic
376C22⋊Q8—22no—
377C22⋊Q8C4×D7914yes—
378C4×Q8—1582no—
379C4×Q8—1582no—
380#117—794no—
381#117—1582no—
382#117—1582no—
383#117—1582no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—791no—
401C2×C22⋊C4—791no—
402C2×C22⋊C4—1581no—
403C2×C22⋊C4—1581no—
404C2×C42—791no—
405C2×C42—791no—
406C2×C42—1581no—
407C2×C42—1581no—
408C2×C42—1581no—
409C2×C42—1581no—
410C2×C42—1581no—
411C2×C42—1581no—
412C2×C42—1581no—
413#107—794no—
414#106—794no—
415#106—1582no—
416#106—1582no—
417#103—22no—
418#103—22no—
419#103—794no—
420#103—794no—
421C22⋊C8—22no—
422C22⋊C8—794no—
423#97—1582no—
424C4×C8—1582no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434C24D4×D7911yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D7912yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—791no—
453C22×C4—791no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4C2×C4×D7912yes—
461C22×C4D4×D7911yescharacteristic
462C22×C4#1192962312yes—
463C22×C4C23×D7911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—792no—
471C2×C8—792no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—3162no—
482C4⋊C4—792no—
483C4⋊C4—792no—
484C4⋊C4—6321no—
485C4⋊C4—1582no—
486C4⋊C4—1582no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4C2×C4×D7912yes—
494C4⋊C4#1192962312yes—
495C4⋊C4#1192962312yes—
496C22⋊C4—24no—
497C22⋊C4—6321no—
498C22⋊C4—1582no—
499C22⋊C4—1582no—
500C22⋊C4—1584no—
501C42—3162no—
502C42—792no—
503C42—792no—
504C42—6321no—
505C42—6321no—
506C42—6321no—
507C42—6321no—
508C42—6321no—
509C42—1581no—
510C42—1581no—
511C42—1584no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23(C22⋊D79)⋊C412yes—
519C23C2×D4×D7911yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
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—3161no—
553C2×C4—792no—
554C2×C4—792no—
555C2×C4—1582no—
556C2×C4—1582no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4—1581no—
563C2×C4—1581no—
564C2×C4—1581no—
565C2×C4—1581no—
566C2×C4—1581no—
567C2×C4(C22⋊D79)⋊C412yes—
568C2×C4C22×C4×D7911yescharacteristic
569C2×C4#41×D7911yescharacteristic
570C2×C4#41×D7911yescharacteristic
571C2×C4C2×#1192962311yescharacteristic
572C2×C4C2×#1192962311yescharacteristic
573C2×C4C2×#1192962311yescharacteristic
574C8—41no—
575C8—41no—
576C8—1582no—
577C8—1582no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×(C22⋊D79)⋊C411yescharacteristic
586C22#123×D7911yescharacteristic
587C22#1909906911yescharacteristic
588C22#1909936911yescharacteristic
589C22#1909937211yescharacteristic
590C22#1909937311yescharacteristic
591C22#1909937611yescharacteristic
592C4—22no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—3162no—
598C4—3162no—
599C4—3161no—
600C4—3161no—
601C4—6321no—
602C4—6321no—
603C4—1582no—
604C4—1582no—
605C4C4×#1192962311yescharacteristic
606C4C4×#1192962311yescharacteristic
607C4#1909909411yescharacteristic
608C4#1909909411yescharacteristic
609C2—42no—
610C2#2376592511yescharacteristic
611C2#2377223711yescharacteristic
612C2#2377227911yescharacteristic
613C2#2377336811yescharacteristic
614C2#2377338411yescharacteristic
615C2#2377363011yescharacteristic
616C2#2377364611yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1872C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1872D79
hypocenterC79#21591
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #23764037C2semidirect
5. #23764037C2semidirect
6. #23763995C2semidirect
7. #23763981C2semidirect
10. #23763296C2semidirect
11. #23763288C2semidirect
12. #23763232C2semidirect
13. #103×Dic79C2semidirect
19. #19098901C4semidirect
89. #14579094C2×C4semidirect
92. #14579092C2×C4semidirect
93. #14579085C2×C4semidirect
94. #14579085C2×C4semidirect
324. C79#21591semidirect

Maximal subgroups

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

subgroupindex
C2×#190989012
#237660842
#237657572
#237640372
#237639952
#237639812
#237635932
#237634962
#237632962
#237632882
#237632322
#103×Dic792
#237631682
C79×#18722
#2159179