Finite Group Databasebeta

← #34305607

Subgroups — #34305607

Order 20224.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098572C211yesmaximal, characteristic
2#23766287C211yesmaximal, characteristic
3Q8×C22⋊Dic79C211yesmaximal, characteristic
4#23764087C211yesmaximal, characteristic, complemented
5#23764073C211yesmaximal, characteristic, complemented
6#23764044C211yesmaximal, characteristic, complemented
7#23764037C211yesmaximal, characteristic, complemented
8#23763666C211yesmaximal, characteristic, complemented
9#23763652C211yesmaximal, characteristic, complemented
10#23763496C211yesmaximal, characteristic
11#23763431C211yesmaximal, characteristic
12#23763232C211yesmaximal, characteristic, complemented
13#103×Dic79C211yesmaximal, characteristic, complemented
14#23763186C211yesmaximal, characteristic
15C79×#1787C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×Q8×Dic79C2211yescharacteristic
18C2×Q8×Dic79C2211yescharacteristic
19C2×#14579120C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22#19098572C418yescomplemented
23C2×#14579085C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×#14579085C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29#19098470—21no—
30#19098470—21no—
31#19098470C2211yescharacteristic
32#19098470C2211yescharacteristic
33#19098459C2211yescharacteristic
34#19098459C2211yescharacteristic
35#19098457—21no—
36#19098457—21no—
37#19098427—21no—
38#19098423—21no—
39#19098423C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4×C22⋊Dic79—21no—
43C4×C22⋊Dic79C2211yescharacteristic
44#19098408—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C23×Q8×C79C2211yescharacteristic
54C2×#103×C79C2211yescharacteristic
55C2×#103×C79C2211yescharacteristic
56C2×#103×C79C2211yescharacteristic
57C2×#103×C79C2211yescharacteristic
58C2×C22⋊C8×C79C2211yescharacteristic
59#380×C79C2211yescharacteristic
60C2×#11929608—41no—
61C2×#11929608—41no—
62C2×#11929608C2×C414yes—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65Q8×Dic79—22no—
66Q8×Dic79—22no—
67Q8×Dic79—41no—
68Q8×Dic79—41no—
69#14579120—22no—
70#14579120C2×C414yes—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79D411yescharacteristic
82C2×C22⋊Dic79D411yescharacteristic
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79C2311yescharacteristic
89C2×C4×Dic79C2311yescharacteristic
90#14579094—41no—
91#14579092—41no—
92#14579085C2×C414yes—
93#14579085C2×C414yes—
94#14579085C2×C414yes—
95#14579085C2×C414yes—
96#14579071—41no—
97C8×Dic79—41no—
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—
107C79⋊#95D411yescharacteristic
108C79⋊#95D411yescharacteristic
109C22×Q8×C79—21no—
110C22×Q8×C79—22no—
111C22×Q8×C79—22no—
112C22×Q8×C79—22no—
113C22×Q8×C79—41no—
114C22×Q8×C79—41no—
115C22×Q8×C79C2×C414yes—
116C22×Q8×C79C2311yescharacteristic
117C22×Q8×C79C2311yescharacteristic
118C23×C4×C79—21no—
119C23×C4×C79C2311yescharacteristic
120C22×C8×C79C2311yescharacteristic
121C22×C8×C79C2311yescharacteristic
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C4⋊C4×C79C2311yescharacteristic
124C2×C22⋊C4×C79—21no—
125#103×C79—22no—
126#103×C79—22no—
127#103×C79—41no—
128#103×C79—41no—
129C22⋊C8×C79—22no—
130#95×C79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139#11929608—44no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8C22×C412yes—
145C2×C79⋊C8C22×C412yes—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—42no—
153C4⋊Dic79—42no—
154C4⋊Dic79C22×C412yes—
155C4⋊Dic79C22×C412yes—
156Dic79⋊C4—81no—
157Dic79⋊C4—81no—
158C22⋊Dic79—22no—
159C22⋊Dic79—24no—
160C22⋊Dic79—41no—
161C22⋊Dic79—42no—
162C4×Dic79—24no—
163C4×Dic79—21no—
164C4×Dic79—21no—
165C4×Dic79—21no—
166C4×Dic79—21no—
167C4×Dic79—42no—
168C4×Dic79—42no—
169C4×Dic79—42no—
170C4×Dic79—42no—
171C24×C79C2×D411yescharacteristic
172C2×Q8×C79—24no—
173C2×Q8×C79—21no—
174C2×Q8×C79—21no—
175C2×Q8×C79—21no—
176C2×Q8×C79—21no—
177C2×Q8×C79—28no—
178C2×Q8×C79—21no—
179C2×Q8×C79—21no—
180C2×Q8×C79—21no—
181C2×Q8×C79—21no—
182C2×Q8×C79—42no—
183C2×Q8×C79—42no—
184C2×Q8×C79—44no—
185C2×Q8×C79—44no—
186C2×Q8×C79—44no—
187C2×Q8×C79C22×C412yes—
188C2×Q8×C79C22×C412yes—
189C2×Q8×C79#4112yes—
190C2×Q8×C79#4112yes—
191C22×C4×C79—21no—
192C22×C4×C79—21no—
193C22×C4×C79—21no—
194C22×C4×C79—21no—
195C22×C4×C79—21no—
196C22×C4×C79—24no—
197C22×C4×C79—21no—
198C22×C4×C79—21no—
199C22×C4×C79—21no—
200C22×C4×C79—42no—
201C22×C4×C79—42no—
202C22×C4×C79C22×C412yes—
203C22×C4×C79C2×D411yescharacteristic
204C22×C4×C79#4112yes—
205C22×C4×C79C2411yescharacteristic
206C2×C8×C79—21no—
207C2×C8×C79—21no—
208C2×C8×C79—21no—
209C2×C8×C79—21no—
210C2×C8×C79—21no—
211C2×C8×C79—21no—
212C4⋊C4×C79—21no—
213C4⋊C4×C79—21no—
214C4⋊C4×C79—21no—
215C4⋊C4×C79—21no—
216C22⋊C4×C79—81no—
217C23×C79—21no—
218C23×C79—21no—
219C23×C79—21no—
220C23×C79—21no—
221C23×C79—21no—
222C23×C79—21no—
223C23×C79C4×D412yes—
224C23×C79C22×D411yescharacteristic
225Q8×C79—22no—
226Q8×C79—22no—
227Q8×C79—22no—
228Q8×C79—22no—
229Q8×C79—22no—
230Q8×C79—48no—
231Q8×C79—41no—
232Q8×C79—41no—
233Q8×C79#11714yes—
234C2×C4×C79—22no—
235C2×C4×C79—22no—
236C2×C4×C79—22no—
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—22no—
244C2×C4×C79—48no—
245C2×C4×C79—41no—
246C2×C4×C79—41no—
247C2×C4×C79—41no—
248C2×C4×C79—41no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79—41no—
253C2×C4×C79C4×D412yes—
254C2×C4×C79C23×C411yescharacteristic
255C2×C4×C79C2×#4111yescharacteristic
256C2×C4×C79C2×#4111yescharacteristic
257C2×C4×C79C2×#4111yescharacteristic
258C2×C4×C79H(5,𝔽2)11yescharacteristic
259C2×C4×C79H(5,𝔽2)11yescharacteristic
260C2×Dic79—21no—
261C2×Dic79—21no—
262C2×Dic79—22no—
263C2×Dic79—22no—
264C2×Dic79—21no—
265C2×Dic79—21no—
266C2×Dic79—21no—
267C2×Dic79—21no—
268C2×Dic79—21no—
269C2×Dic79—21no—
270C2×Dic79—21no—
271C2×Dic79—21no—
272C2×Dic79—21no—
273C2×Dic79—41no—
274C2×Dic79—41no—
275C2×Dic79—41no—
276C2×Dic79—41no—
277C2×Dic79—41no—
278C2×Dic79—41no—
279C2×Dic79—41no—
280C2×Dic79—41no—
281C2×Dic79—41no—
282C2×Dic79—41no—
283C2×Dic79#13712yes—
284C8×C79—41no—
285C8×C79—41no—
286C79⋊C8—22no—
287C79⋊C8—22no—
288C22×C79—22no—
289C22×C79—22no—
290C22×C79—22no—
291C22×C79—41no—
292C22×C79—41no—
293C22×C79—41no—
294C22×C79—41no—
295C22×C79C2×C4×D411yescharacteristic
296C22×C79#54611yescharacteristic
297C22×C79#54611yescharacteristic
298C22×C79C2×#13711yescharacteristic
299C22×C79#57511yescharacteristic
300C22×C79#57711yescharacteristic
301C22×C79#57811yescharacteristic
302C4×C79—22no—
303C4×C79—24no—
304C4×C79—42no—
305C4×C79—81no—
306C4×C79—81no—
307C4×C79C2×#11711yescharacteristic
308C4×C79#51811yescharacteristic
309C4×C79#55111yescharacteristic
310C4×C79#55111yescharacteristic
311Dic79—22no—
312Dic79—41no—
313Dic79—42no—
314Dic79—42no—
315Dic79—41no—
316Dic79—41no—
317Dic79—81no—
318Dic79—81no—
319#22585—791nomaximal, Sylow
320C2×C79—42no—
321C2×C79#234211yescharacteristic
322C2×C79C4×#13711yescharacteristic
323C2×C79#294711yescharacteristic
324C2×C79#331211yescharacteristic
325C2×C79#332111yescharacteristic
326C2×C79#332711yescharacteristic
327C2×C79#332911yescharacteristic
328C2×#419—791no—
329#2346—791no—
330Q8×C22⋊C4—791no—
331#1935—791no—
332#1933—791no—
333#1928—791no—
334#1927—791no—
335#1865—791no—
336#1863—791no—
337#1825—791no—
338#1809—791no—
339#1787D7911yescharacteristic
340#1763—791no—
341C4×#103—791no—
342#1753—791no—
343C79#2258511yescharacteristic, complemented, Sylow
344C23×Q8C2×D7911yescharacteristic
345C22×M16—791no—
346C2×C4×Q8—791no—
347C2×C4×Q8—791no—
348C2×#117—791no—
349C2×#107—791no—
350C2×#106—791no—
351#419—798no—
352C2×#103—791no—
353C2×#103—791no—
354C2×#103—791no—
355C2×#103—791no—
356C2×#103C2×D7911yescharacteristic
357C2×#103C2×D7911yescharacteristic
358C2×#103C2×D7911yescharacteristic
359C2×#103C2×D7911yescharacteristic
360C2×C22⋊C8C2×D7911yescharacteristic
361C2×#97—791no—
362C2×C4×C8—791no—
363#391—791no—
364#391—791no—
365#391—1581no—
366#391—1581no—
367#389—791no—
368#389—791no—
369#388—1581no—
370#388—1581no—
371#382—1581no—
372#380—791no—
373#380—1581no—
374#380C2×D7911yescharacteristic
375C4×C4⋊C4—791no—
376C4×C4⋊C4—791no—
377C4×C22⋊C4—791no—
378C4×C22⋊C4—1581no—
379#376—1581no—
380#340—791no—
381#340—791no—
382#340—791no—
383#340—791no—
384#340—791no—
385#340—791no—
386#336—791no—
387#336—791no—
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C4×D7914yes—
395C22×Q8C22×D7911yescharacteristic
396C22×Q8C22×D7911yescharacteristic
397C23×C4—21no—
398C23×C4C22×D7911yescharacteristic
399C2×M16—3161no—
400C2×M16—3161no—
401C2×M16—794no—
402C22×C8—791no—
403C22×C8—791no—
404C22×C8C22×D7911yescharacteristic
405C22×C8C22×D7911yescharacteristic
406C4×Q8—3161no—
407C4×Q8—3161no—
408C4×Q8—1582no—
409C4×Q8—1582no—
410#117—794no—
411#117—1582no—
412C2×C4⋊C4—791no—
413C2×C4⋊C4—791no—
414C2×C4⋊C4—791no—
415C2×C4⋊C4—791no—
416C2×C4⋊C4—1581no—
417C2×C4⋊C4—1581no—
418C2×C4⋊C4—1581no—
419C2×C4⋊C4—1581no—
420C2×C4⋊C4C22×D7911yescharacteristic
421C2×C4⋊C4C22×D7911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—791no—
424C2×C22⋊C4—791no—
425C2×C22⋊C4—1581no—
426C2×C22⋊C4—1581no—
427C2×C42—791no—
428C2×C42—791no—
429C2×C42—1581no—
430C2×C42—1581no—
431C2×C42—1581no—
432C2×C42—1581no—
433C2×C42—1581no—
434#107—3161no—
435#106—3161no—
436#103—22no—
437#103—22no—
438#103—41no—
439#103—41no—
440#103—794no—
441#103—794no—
442#103—794no—
443#103—794no—
444C22⋊C8—22no—
445#97—3161no—
446C4×C8—3161no—
447#95—21no—
448#95—791no—
449#95—791no—
450#95—1581no—
451#95—1581no—
452#95—1581no—
453#95—1581no—
454#95—1581no—
455#95—1581no—
456#95—1581no—
457#95—1581no—
458#95—1581no—
459C24D4×D7911yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—24no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C4×D7912yes—
476C2×Q8C2×C4×D7912yes—
477C2×Q8#1192962312yes—
478C2×Q8#1192962312yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—791no—
491C22×C4—791no—
492C22×C4—791no—
493C22×C4—791no—
494C22×C4—1581no—
495C22×C4—1581no—
496C22×C4—1581no—
497C22×C4—1581no—
498C22×C4C2×C4×D7912yes—
499C22×C4D4×D7911yescharacteristic
500C22×C4#1192962312yes—
501C22×C4C23×D7911yescharacteristic
502M16—3164no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—792no—
510C2×C8—792no—
511C2×C8—1581no—
512C2×C8—1581no—
513C2×C8—1581no—
514C2×C8—1581no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—3162no—
520C4⋊C4—3162no—
521C4⋊C4—792no—
522C4⋊C4—792no—
523C4⋊C4—6321no—
524C4⋊C4—6321no—
525C4⋊C4—1581no—
526C4⋊C4—1581no—
527C4⋊C4—1581no—
528C4⋊C4—1581no—
529C4⋊C4—1581no—
530C4⋊C4—1581no—
531C22⋊C4—81no—
532C22⋊C4—3162no—
533C22⋊C4—3161no—
534C22⋊C4—1582no—
535C22⋊C4—1584no—
536C42—3162no—
537C42—3162no—
538C42—3162no—
539C42—3162no—
540C42—1581no—
541C42—1581no—
542C42—1581no—
543C42—1581no—
544C42—1584no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23(C22⋊D79)⋊C412yes—
552C23C2×D4×D7911yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1457911714yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—3161no—
582C2×C4—3161no—
583C2×C4—3161no—
584C2×C4—3161no—
585C2×C4—3161no—
586C2×C4—3161no—
587C2×C4—3161no—
588C2×C4—3161no—
589C2×C4—3161no—
590C2×C4—3161no—
591C2×C4—792no—
592C2×C4—1581no—
593C2×C4—1581no—
594C2×C4—1581no—
595C2×C4—1582no—
596C2×C4—1582no—
597C2×C4—1581no—
598C2×C4—1581no—
599C2×C4—1581no—
600C2×C4—1581no—
601C2×C4—1581no—
602C2×C4—1581no—
603C2×C4—1581no—
604C2×C4—1581no—
605C2×C4(C22⋊D79)⋊C412yes—
606C2×C4C22×C4×D7911yescharacteristic
607C2×C4C2×#1192962311yescharacteristic
608C2×C4C2×#1192962311yescharacteristic
609C2×C4C2×#1192962311yescharacteristic
610C2×C4#1457920611yescharacteristic
611C2×C4#1457920611yescharacteristic
612C8—41no—
613C8—41no—
614C8—1582no—
615C8—1582no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×(C22⋊D79)⋊C411yescharacteristic
624C22#1909920011yescharacteristic
625C22#1909920011yescharacteristic
626C22#137×D7911yescharacteristic
627C22#1909936211yescharacteristic
628C22#1909936911yescharacteristic
629C22#1909937611yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—3162no—
636C4—3162no—
637C4—3161no—
638C4—3161no—
639C4—3161no—
640C4—6321no—
641C4—6321no—
642C4—1582no—
643C4C2×#1457911711yescharacteristic
644C4#1909903611yescharacteristic
645C4#1909924411yescharacteristic
646C4#1909924411yescharacteristic
647C2—42no—
648C2#2376625111yescharacteristic
649C2#2377221511yescharacteristic
650C2#2377223711yescharacteristic
651C2#2377544411yescharacteristic
652C2#2377557911yescharacteristic
653C2#2377567211yescharacteristic
654C2#2377568711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1787C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1787D79
hypocenterC79#22585
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
4. #23764087C2semidirect
5. #23764073C2semidirect
6. #23764044C2semidirect
7. #23764037C2semidirect
8. #23763666C2semidirect
9. #23763652C2semidirect
12. #23763232C2semidirect
13. #103×Dic79C2semidirect
22. #19098572C4semidirect
343. C79#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#190985722
#237662872
Q8×C22⋊Dic792
#237640872
#237640732
#237640442
#237640372
#237636662
#237636522
#237634962
#237634312
#237632322
#103×Dic792
#237631862
C79×#17872
#2258579