Finite Group Databasebeta

← #34214944

Subgroups — #34214944

Order 20224.

1610 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579154C211yesmaximal, characteristic
2C2×#19098428C211yesmaximal, characteristic, complemented
3C2×#19098414C211yesmaximal, characteristic, complemented
4#23761654C211yesmaximal, characteristic, complemented
5#23761651C212yesmaximal, complemented
6#23761650C211yesmaximal, characteristic, complemented
7#23761645C211yesmaximal, characteristic
8#23761643C212yesmaximal
9#23761637C212yesmaximal
10#23761634C212yesmaximal, complemented
11C79×#1439C211yesmaximal, characteristic
12C2×#14579154—24no—
13C2×#14579154—24no—
14C2×#14579154C418yescomplemented
15C22×C4⋊Dic79C2211yescharacteristic
16C22×C4⋊Dic79C2211yescharacteristic
17C22×Dic79⋊C4C2212yes—
18C22×Dic79⋊C4C2211yescharacteristic
19C22×Dic79⋊C4C2211yescharacteristic
20C22×C22⋊Dic79C2211yescharacteristic
21C22×C22⋊Dic79C2211yescharacteristic
22C22×C22⋊Dic79C2212yes—
23C22×C4×Dic79C2212yes—
24C22×C4×Dic79C2211yescharacteristic
25#19098428—24no—
26#19098414—24no—
27C2×C79⋊#95C2212yes—
28C2×C79⋊#95C2212yes—
29C2×C79⋊#95C2212yes—
30C2×C79⋊#95C2211yescharacteristic
31C2×C79⋊#95C2212yes—
32C2×C79⋊#95C2212yes—
33C2×C79⋊#95C2212yes—
34C2×C79⋊#95C2212yes—
35C22×C4⋊C4×C79C2211yescharacteristic
36C22×C22⋊C4×C79C2211yescharacteristic
37C22×C22⋊C4×C79C2212yes—
38C2×#95×C79C2212yes—
39C2×#95×C79C2211yescharacteristic
40C23×Dic79C2312yes—
41C23×Dic79C2311yescharacteristic
42C23×Dic79C2311yescharacteristic
43C23×Dic79C2312yes—
44C23×Dic79C2311yescharacteristic
45C23×Dic79C2311yescharacteristic
46#14579154—232no—
47C2×C4⋊Dic79—22no—
48C2×C4⋊Dic79—22no—
49C2×C4⋊Dic79—22no—
50C2×C4⋊Dic79—42no—
51C2×C4⋊Dic79C2×C414yes—
52C2×Dic79⋊C4—22no—
53C2×Dic79⋊C4—22no—
54C2×Dic79⋊C4—24no—
55C2×Dic79⋊C4—24no—
56C2×Dic79⋊C4—22no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—24no—
59C2×Dic79⋊C4C2×C414yes—
60C2×Dic79⋊C4C2×C414yes—
61C2×C22⋊Dic79—22no—
62C2×C22⋊Dic79—22no—
63C2×C22⋊Dic79—22no—
64C2×C22⋊Dic79—22no—
65C2×C22⋊Dic79—24no—
66C2×C22⋊Dic79—42no—
67C2×C22⋊Dic79—42no—
68C2×C22⋊Dic79C2×C414yes—
69C2×C22⋊Dic79C2×C414yes—
70C2×C4×Dic79—24no—
71C2×C4×Dic79—42no—
72C2×C4×Dic79—42no—
73C2×C4×Dic79C2×C414yes—
74C2×C4×Dic79D414yes—
75C2×C4×Dic79D414yes—
76C79⋊#95—22no—
77C79⋊#95—24no—
78C79⋊#95—24no—
79C79⋊#95—24no—
80C79⋊#95—42no—
81C79⋊#95—42no—
82C79⋊#95—42no—
83C79⋊#95—42no—
84C25×C79C2311yescharacteristic
85C23×C4×C79C2312yes—
86C23×C4×C79C2311yescharacteristic
87C23×C4×C79C2312yes—
88C23×C4×C79C2311yescharacteristic
89C2×C4⋊C4×C79—22no—
90C2×C4⋊C4×C79—42no—
91C2×C22⋊C4×C79—22no—
92C2×C22⋊C4×C79—22no—
93C2×C22⋊C4×C79—24no—
94C2×C22⋊C4×C79—42no—
95C2×C22⋊C4×C79—42no—
96C2×C22⋊C4×C79C2×C414yes—
97#95×C79—22no—
98#95×C79—24no—
99C22×Dic79—22no—
100C22×Dic79—21no—
101C22×Dic79—21no—
102C22×Dic79—21no—
103C22×Dic79—21no—
104C22×Dic79—22no—
105C22×Dic79—22no—
106C22×Dic79—22no—
107C22×Dic79—22no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—21no—
111C22×Dic79—21no—
112C22×Dic79—21no—
113C22×Dic79—22no—
114C22×Dic79—21no—
115C22×Dic79—22no—
116C22×Dic79—22no—
117C22×Dic79—22no—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—22no—
128C22×Dic79C22×C412yes—
129C22×Dic79C22×C412yes—
130C22×Dic79C22×C412yes—
131C22×Dic79C22×C412yes—
132C22×Dic79C2×D412yes—
133C22×Dic79C2×D412yes—
134C22×Dic79C2×D412yes—
135C22×Dic79C2×D412yes—
136C22×Dic79#4114yes—
137C4⋊Dic79—28no—
138C4⋊Dic79—44no—
139Dic79⋊C4—216no—
140Dic79⋊C4—28no—
141Dic79⋊C4—28no—
142C22⋊Dic79—28no—
143C22⋊Dic79—28no—
144C22⋊Dic79—48no—
145C4×Dic79—48no—
146C4×Dic79C22⋊C4116yescomplemented
147C24×C79—21no—
148C24×C79—21no—
149C24×C79—22no—
150C24×C79—22no—
151C24×C79—21no—
152C24×C79—22no—
153C24×C79—21no—
154C24×C79—22no—
155C24×C79—21no—
156C24×C79—21no—
157C24×C79C22×C412yes—
158C24×C79C2411yescharacteristic
159C22×C4×C79—22no—
160C22×C4×C79—22no—
161C22×C4×C79—22no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—22no—
165C22×C4×C79—22no—
166C22×C4×C79—22no—
167C22×C4×C79—22no—
168C22×C4×C79—22no—
169C22×C4×C79—22no—
170C22×C4×C79—22no—
171C22×C4×C79—22no—
172C22×C4×C79—22no—
173C22×C4×C79—22no—
174C22×C4×C79—22no—
175C22×C4×C79—22no—
176C22×C4×C79—22no—
177C22×C4×C79—22no—
178C22×C4×C79C22×C412yes—
179C22×C4×C79C22×C412yes—
180C22×C4×C79C2×D412yes—
181C22×C4×C79C2×D412yes—
182C22×C4×C79#4114yes—
183C4⋊C4×C79—44no—
184C22⋊C4×C79—28no—
185C22⋊C4×C79—48no—
186C23×C79—22no—
187C23×C79—24no—
188C23×C79—24no—
189C23×C79—22no—
190C23×C79—22no—
191C23×C79—42no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79—42no—
199C23×C79—41no—
200C23×C79—41no—
201C23×C79—42no—
202C23×C79—42no—
203C23×C79—42no—
204C23×C79—42no—
205C23×C79—42no—
206C23×C79—42no—
207C23×C79—42no—
208C23×C79—42no—
209C23×C79C23×C411yescharacteristic
210C23×C79C22×D411yescharacteristic
211C23×C79C22×D411yescharacteristic
212C23×C79C2×#4112yes—
213C23×C79C2×#4112yes—
214C23×C79H(5,𝔽2)11yescharacteristic
215C23×C79H(5,𝔽2)12yes—
216C23×C79H(5,𝔽2)11yescharacteristic
217C23×C792−1+412yes—
218C23×C792−1+411yescharacteristic
219C23×C792−1+411yescharacteristic
220C2×C4×C79—24no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—24no—
224C2×C4×C79—22no—
225C2×C4×C79—24no—
226C2×C4×C79—24no—
227C2×C4×C79—24no—
228C2×C4×C79—24no—
229C2×C4×C79—42no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—42no—
233C2×C4×C79—42no—
234C2×C4×C79—42no—
235C2×C4×C79—42no—
236C2×C4×C79—42no—
237C2×C4×C79—42no—
238C2×C4×C79—42no—
239C2×C4×C79—42no—
240C2×C4×C79—42no—
241C2×C4×C79—42no—
242C2×C4×C79—42no—
243C2×C4×C79C2×C22⋊C414yes—
244C2×Dic79—24no—
245C2×Dic79—22no—
246C2×Dic79—22no—
247C2×Dic79—24no—
248C2×Dic79—22no—
249C2×Dic79—22no—
250C2×Dic79—22no—
251C2×Dic79—22no—
252C2×Dic79—22no—
253C2×Dic79—22no—
254C2×Dic79—22no—
255C2×Dic79—22no—
256C2×Dic79—24no—
257C2×Dic79—22no—
258C2×Dic79—22no—
259C2×Dic79—24no—
260C2×Dic79—24no—
261C2×Dic79—24no—
262C2×Dic79—42no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—42no—
268C2×Dic79—42no—
269C2×Dic79—41no—
270C2×Dic79—41no—
271C2×Dic79—42no—
272C2×Dic79—42no—
273C2×Dic79—42no—
274C2×Dic79—42no—
275C2×Dic79—42no—
276C2×Dic79—42no—
277C2×Dic79—42no—
278C2×Dic79—42no—
279C2×Dic79—42no—
280C2×Dic79C2×C22⋊C414yes—
281C2×Dic79C2×C22⋊C414yes—
282C22×C79—42no—
283C22×C79—42no—
284C22×C79—44no—
285C22×C79—44no—
286C22×C79—42no—
287C22×C79—82no—
288C22×C79—82no—
289C22×C79—81no—
290C22×C79—81no—
291C22×C79—81no—
292C22×C79—81no—
293C22×C79C22×C22⋊C411yescharacteristic
294C22×C79C4×#4112yes—
295C22×C79#52012yes—
296C22×C79#52011yescharacteristic
297C22×C79#52011yescharacteristic
298C22×C79#52512yes—
299C22×C79#53211yescharacteristic
300C22×C79#53211yescharacteristic
301C22×C79#53511yescharacteristic
302C22×C79#53511yescharacteristic
303C22×C79#53611yescharacteristic
304C22×C79#53611yescharacteristic
305C22×C79#53712yes—
306C22×C79#53711yescharacteristic
307C22×C79#53711yescharacteristic
308C22×C79#53812yes—
309C22×C79#53812yes—
310C22×C79#53912yes—
311C22×C79#53912yes—
312C22×C79#54112yes—
313C22×C79#54112yes—
314C22×C79#54212yes—
315C22×C79#54212yes—
316C4×C79—24no—
317C4×C79—44no—
318C4×C79—42no—
319C4×C79—82no—
320Dic79—24no—
321Dic79—24no—
322Dic79—42no—
323Dic79—42no—
324Dic79—44no—
325Dic79—82no—
326#14291—791nomaximal, Sylow
327C2×C79—82no—
328C2×C79#229711yescharacteristic
329C2×C79#231111yescharacteristic
330C2×C79#231111yescharacteristic
331C2×C79#233412yes—
332C2×C79#233412yes—
333C2×C79#263612yes—
334C2×C79#263712yes—
335C2×C79#266211yescharacteristic
336C2×C79#266211yescharacteristic
337C2×C79#266411yescharacteristic
338C2×C79#266411yescharacteristic
339C22×#126—791no—
340C2×#382—791no—
341C2×C4×C4⋊C4—791no—
342#1441—792no—
343#1441—791no—
344#1441—791no—
345#1439—792no—
346#1439—791no—
347#1439D7911yescharacteristic
348#1437—792no—
349#1436—792no—
350C79#1429111yescharacteristic, complemented, Sylow
351C2×#126—798no—
352C2×#126—1584no—
353C2×#126—1584no—
354C22×C4⋊C4—791no—
355C22×C4⋊C4—791no—
356C22×C4⋊C4—791no—
357C22×C4⋊C4—791no—
358C22×C4⋊C4—792no—
359C22×C4⋊C4C2×D7911yescharacteristic
360C22×C22⋊C4—792no—
361C22×C22⋊C4—791no—
362C22×C22⋊C4—791no—
363C22×C22⋊C4C2×D7911yescharacteristic
364C22×C22⋊C4C2×D7912yes—
365C22×C42—792no—
366C22×C42—791no—
367#382—1584no—
368C4×C4⋊C4—1584no—
369C2×#95—792no—
370C2×#95—792no—
371C2×#95—792no—
372C2×#95—792no—
373C2×#95—792no—
374C2×#95—792no—
375C2×#95—791no—
376C2×#95—792no—
377C2×#95C2×D7912yes—
378C2×#95C2×D7911yescharacteristic
379C25C22×D7911yescharacteristic
380C23×C4—791no—
381C23×C4—791no—
382C23×C4—791no—
383C23×C4—791no—
384C23×C4—792no—
385C23×C4—792no—
386C23×C4C22×D7912yes—
387C23×C4C22×D7911yescharacteristic
388C23×C4C22×D7912yes—
389C23×C4C22×D7911yescharacteristic
390#126—15832no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—3162no—
394C2×C4⋊C4—794no—
395C2×C4⋊C4—794no—
396C2×C4⋊C4—794no—
397C2×C4⋊C4—1582no—
398C2×C4⋊C4—1582no—
399C2×C4⋊C4—1582no—
400C2×C4⋊C4—1582no—
401C2×C4⋊C4—1582no—
402C2×C4⋊C4—1584no—
403C2×C4⋊C4—1584no—
404C2×C4⋊C4—1582no—
405C2×C4⋊C4—1582no—
406C2×C4⋊C4—1584no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—3162no—
413C2×C22⋊C4—3162no—
414C2×C22⋊C4—794no—
415C2×C22⋊C4—794no—
416C2×C22⋊C4—1584no—
417C2×C22⋊C4—1582no—
418C2×C22⋊C4—1582no—
419C2×C22⋊C4—1582no—
420C2×C22⋊C4—1582no—
421C2×C22⋊C4C4×D7914yes—
422C2×C42—3162no—
423C2×C42—3162no—
424C2×C42—794no—
425C2×C42—794no—
426C2×C42—794no—
427C2×C42—1584no—
428#95—22no—
429#95—24no—
430#95—3162no—
431#95—3162no—
432#95—3162no—
433#95—3162no—
434#95—1584no—
435#95—1584no—
436#95—1584no—
437#95—1582no—
438C24—22no—
439C24—22no—
440C24—21no—
441C24—22no—
442C24—21no—
443C24—22no—
444C24—21no—
445C24—21no—
446C24—21no—
447C24—21no—
448C24C2×C4×D7912yes—
449C24C23×D7911yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—792no—
470C22×C4—792no—
471C22×C4—792no—
472C22×C4—792no—
473C22×C4—792no—
474C22×C4—792no—
475C22×C4—794no—
476C22×C4—792no—
477C22×C4—792no—
478C22×C4—1582no—
479C22×C4—1582no—
480C22×C4—1582no—
481C22×C4—1582no—
482C22×C4—1582no—
483C22×C4—1582no—
484C22×C4—1582no—
485C22×C4—1582no—
486C22×C4—1582no—
487C22×C4—1581no—
488C22×C4—1582no—
489C22×C4—1582no—
490C22×C4—1581no—
491C22×C4—1582no—
492C22×C4—1581no—
493C22×C4—1582no—
494C22×C4—1582no—
495C22×C4—1581no—
496C22×C4—1582no—
497C22×C4—1581no—
498C22×C4—1581no—
499C22×C4—1581no—
500C22×C4—1581no—
501C22×C4—1582no—
502C22×C4—1582no—
503C22×C4—1581no—
504C22×C4—1581no—
505C22×C4—1581no—
506C22×C4—1581no—
507C22×C4C2×C4×D7912yes—
508C22×C4C2×C4×D7912yes—
509C22×C4D4×D7912yes—
510C22×C4D4×D7912yes—
511C22×C4#1192962314yes—
512C4⋊C4—44no—
513C4⋊C4—3164no—
514C4⋊C4—1588no—
515C4⋊C4—1588no—
516C4⋊C4—15816no—
517C4⋊C4—1588no—
518C22⋊C4—28no—
519C22⋊C4—48no—
520C22⋊C4—3168no—
521C22⋊C4—1588no—
522C22⋊C4—1588no—
523C42—3168no—
524C42—7916no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C4×D7911yescharacteristic
549C23C2×D4×D7911yescharacteristic
550C23C2×D4×D7911yescharacteristic
551C23#41×D7912yes—
552C23C2×#1192962312yes—
553C23#1457920612yes—
554C23#1457920611yescharacteristic
555C23#1457920711yescharacteristic
556C23#1457920811yescharacteristic
557C23#1457920812yes—
558C23#1457920911yescharacteristic
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—3161no—
583C2×C4—3162no—
584C2×C4—3162no—
585C2×C4—3161no—
586C2×C4—3161no—
587C2×C4—3162no—
588C2×C4—3162no—
589C2×C4—3161no—
590C2×C4—3162no—
591C2×C4—3161no—
592C2×C4—3161no—
593C2×C4—3162no—
594C2×C4—3162no—
595C2×C4—3162no—
596C2×C4—3162no—
597C2×C4—3162no—
598C2×C4—3162no—
599C2×C4—3162no—
600C2×C4—794no—
601C2×C4—794no—
602C2×C4—1584no—
603C2×C4—1584no—
604C2×C4—1582no—
605C2×C4—1582no—
606C2×C4—1584no—
607C2×C4—1582no—
608C2×C4—1582no—
609C2×C4—1584no—
610C2×C4—1582no—
611C2×C4—1582no—
612C2×C4—1582no—
613C2×C4—1584no—
614C2×C4—1582no—
615C2×C4—1582no—
616C2×C4—1584no—
617C2×C4—1582no—
618C2×C4—1582no—
619C2×C4—1582no—
620C2×C4C22⋊C4×D7914yes—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22C2×C22⋊C4×D7911yescharacteristic
633C22C4×#1192962312yes—
634C22#1909904212yes—
635C22#1909904311yescharacteristic
636C22#1909904411yescharacteristic
637C22#1909907312yes—
638C22#1909911111yescharacteristic
639C22#1909911211yescharacteristic
640C22#1909912811yescharacteristic
641C22#1909912911yescharacteristic
642C22#1909913511yescharacteristic
643C22#1909913611yescharacteristic
644C22#1909914012yes—
645C22#1909914211yescharacteristic
646C22#1909914311yescharacteristic
647C22#1909914712yes—
648C22#1909914812yes—
649C22#1909915612yes—
650C22#1909915712yes—
651C22#1909917012yes—
652C22#1909917312yes—
653C22#1909918012yes—
654C22#1909918312yes—
655C4—24no—
656C4—42no—
657C4—44no—
658C4—82no—
659C4—3164no—
660C4—3162no—
661C4—3162no—
662C4—6322no—
663C4—1584no—
664C4—1584no—
665C2—82no—
666C2#2376588711yescharacteristic
667C2#2376598711yescharacteristic
668C2#2376598911yescharacteristic
669C2#2376617012yes—
670C2#2376618212yes—
671C2#2376930212yes—
672C2#2376931312yes—
673C2#2376954211yescharacteristic
674C2#2376954611yescharacteristic
675C2#2376956411yescharacteristic
676C2#2376956811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC79×#1439C2
socle, cosocleC24×C79C24
prime radicalC25×C79C23
hypercenter#1439D79
hypocenterC79#14291
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#19098428C2semidirect
3. C2×#19098414C2semidirect
4. #23761654C2semidirect
5. #23761651C2semidirect
6. #23761650C2semidirect
10. #23761634C2semidirect
14. C2×#14579154C4semidirect
146. C4×Dic79C22⋊C4semidirect
350. C79#14291semidirect

Maximal subgroups

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

subgroupindex
C22×#145791542
C2×#190984282
C2×#190984142
#237616542
#237616512
#237616502
#237616452
#237616432
#237616372
#237616342
C79×#14392
#1429179