Finite Group Databasebeta

← #34219006

Subgroups — #34219006

Order 20224.

1326 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098462C211yesmaximal, characteristic, complemented
2C2×#19098456C211yesmaximal, characteristic
3#23761668C211yesmaximal, characteristic, complemented
4#23761665C211yesmaximal, characteristic, complemented
5#23761665C212yesmaximal, complemented
6#23761662C212yesmaximal, complemented
7#23761645C211yesmaximal, characteristic
8#23761645C212yesmaximal
9#23761645C212yesmaximal
10#23761625C211yesmaximal, characteristic, complemented
11C2×#397×C79C211yesmaximal, characteristic
12C22×C4⋊Dic79C2211yescharacteristic
13C22×C4⋊Dic79C2212yes—
14C22×C4⋊Dic79C2211yescharacteristic
15C22×C4⋊Dic79C2212yes—
16C22×C22⋊Dic79C2211yescharacteristic
17C22×C22⋊Dic79C2211yescharacteristic
18C22×C22⋊Dic79C2212yes—
19C22×C4×Dic79C2211yescharacteristic
20#19098462—24no—
21#19098456—24no—
22C2×C79⋊#95C2211yescharacteristic
23C2×C79⋊#95C2211yescharacteristic
24C2×C79⋊#95C2211yescharacteristic
25C2×C79⋊#95C2212yes—
26C2×C79⋊#95C2212yes—
27C2×C79⋊#95C2212yes—
28C2×C79⋊#95C2212yes—
29C2×C79⋊#95C2212yes—
30C2×C79⋊#95C2212yes—
31C2×C79⋊#95C2212yes—
32C22×C4⋊C4×C79C2211yescharacteristic
33C22×C22⋊C4×C79C2211yescharacteristic
34C22×C22⋊C4×C79C2212yes—
35#397×C79C418yescomplemented
36C2×#95×C79C2211yescharacteristic
37C2×#95×C79C2212yes—
38C23×Dic79C2311yescharacteristic
39C23×Dic79C2312yes—
40C23×Dic79C2311yescharacteristic
41C23×Dic79C2312yes—
42C23×Dic79C2311yescharacteristic
43C23×Dic79C2311yescharacteristic
44C2×C4⋊Dic79—22no—
45C2×C4⋊Dic79—22no—
46C2×C4⋊Dic79—24no—
47C2×C4⋊Dic79—42no—
48C2×C4⋊Dic79—42no—
49C2×C4⋊Dic79—42no—
50C2×C4⋊Dic79—42no—
51C2×C4⋊Dic79—42no—
52C2×C4⋊Dic79—42no—
53C2×C4⋊Dic79—42no—
54C2×C22⋊Dic79—22no—
55C2×C22⋊Dic79—22no—
56C2×C22⋊Dic79—42no—
57C2×C22⋊Dic79—42no—
58C2×C22⋊Dic79—42no—
59C2×C22⋊Dic79—42no—
60C2×C22⋊Dic79—42no—
61C2×C4×Dic79—42no—
62C2×C4×Dic79D414yes—
63C79⋊#95—22no—
64C79⋊#95—22no—
65C79⋊#95—22no—
66C79⋊#95—24no—
67C79⋊#95—42no—
68C79⋊#95—42no—
69C79⋊#95—42no—
70C79⋊#95—42no—
71C79⋊#95—42no—
72C79⋊#95—42no—
73C25×C79C2311yescharacteristic
74C23×C4×C79C2312yes—
75C23×C4×C79C2311yescharacteristic
76C23×C4×C79C2312yes—
77C23×C4×C79C2311yescharacteristic
78C2×C4⋊C4×C79—22no—
79C2×C4⋊C4×C79—22no—
80C2×C4⋊C4×C79C2×C414yes—
81C2×C22⋊C4×C79—22no—
82C2×C22⋊C4×C79—22no—
83C2×C22⋊C4×C79—42no—
84C2×C22⋊C4×C79—42no—
85C2×C22⋊C4×C79C2×C414yes—
86C2×C22⋊C4×C79C2×C418yes—
87#95×C79C2×C418yes—
88#95×C79C2×C414yes—
89C22×Dic79—21no—
90C22×Dic79—21no—
91C22×Dic79—22no—
92C22×Dic79—22no—
93C22×Dic79—21no—
94C22×Dic79—22no—
95C22×Dic79—22no—
96C22×Dic79—22no—
97C22×Dic79—21no—
98C22×Dic79—21no—
99C22×Dic79—21no—
100C22×Dic79—22no—
101C22×Dic79—21no—
102C22×Dic79—22no—
103C22×Dic79—22no—
104C22×Dic79—22no—
105C22×Dic79—22no—
106C22×Dic79—22no—
107C22×Dic79—22no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—22no—
112C22×Dic79—21no—
113C22×Dic79—22no—
114C22×Dic79—22no—
115C22×Dic79—22no—
116C22×Dic79—21no—
117C22×Dic79—22no—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—21no—
121C22×Dic79C2×D412yes—
122C22×Dic79C2×D412yes—
123C4⋊Dic79—44no—
124C4⋊Dic79—44no—
125C4⋊Dic79—48no—
126C4⋊Dic79—48no—
127C22⋊Dic79—44no—
128C22⋊Dic79—44no—
129C22⋊Dic79—48no—
130C4×Dic79—44no—
131C24×C79—21no—
132C24×C79—21no—
133C24×C79—21no—
134C24×C79—21no—
135C24×C79—22no—
136C24×C79—22no—
137C24×C79—21no—
138C24×C79—22no—
139C24×C79—22no—
140C24×C79—21no—
141C24×C79C22×C412yes—
142C24×C79C2411yescharacteristic
143C22×C4×C79—22no—
144C22×C4×C79—22no—
145C22×C4×C79—21no—
146C22×C4×C79—22no—
147C22×C4×C79—21no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—22no—
151C22×C4×C79—22no—
152C22×C4×C79—22no—
153C22×C4×C79—22no—
154C22×C4×C79—22no—
155C22×C4×C79—22no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—22no—
159C22×C4×C79—22no—
160C22×C4×C79—22no—
161C22×C4×C79C22×C412yes—
162C22×C4×C79C22×C414yes—
163C22×C4×C79C22×C414yes—
164C22×C4×C79C22×C412yes—
165C22×C4×C79C2×D412yes—
166C22×C4×C79#4112yes—
167C4⋊C4×C79—28no—
168C22⋊C4×C79—28no—
169C22⋊C4×C79—48no—
170C23×C79—22no—
171C23×C79—22no—
172C23×C79—24no—
173C23×C79—22no—
174C23×C79—24no—
175C23×C79—41no—
176C23×C79—41no—
177C23×C79—41no—
178C23×C79—41no—
179C23×C79—41no—
180C23×C79—41no—
181C23×C79—42no—
182C23×C79—41no—
183C23×C79—41no—
184C23×C79—42no—
185C23×C79—42no—
186C23×C79—42no—
187C23×C79—42no—
188C23×C79—42no—
189C23×C79—42no—
190C23×C79—42no—
191C23×C79—42no—
192C23×C79—42no—
193C23×C79C23×C411yescharacteristic
194C23×C79C22×D411yescharacteristic
195C23×C79C2×#4111yescharacteristic
196C23×C79H(5,𝔽2)11yescharacteristic
197C23×C79H(5,𝔽2)11yescharacteristic
198C23×C79H(5,𝔽2)12yes—
199C23×C79H(5,𝔽2)11yescharacteristic
200C23×C79H(5,𝔽2)12yes—
201C23×C792−1+412yes—
202C23×C792−1+412yes—
203C23×C792−1+411yescharacteristic
204C2×C4×C79—24no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—24no—
208C2×C4×C79—22no—
209C2×C4×C79—24no—
210C2×C4×C79—24no—
211C2×C4×C79—24no—
212C2×C4×C79—24no—
213C2×C4×C79—24no—
214C2×C4×C79—24no—
215C2×C4×C79—24no—
216C2×C4×C79—42no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—42no—
220C2×C4×C79—42no—
221C2×C4×C79—42no—
222C2×C4×C79—42no—
223C2×C4×C79—42no—
224C2×C4×C79—42no—
225C2×C4×C79—42no—
226C2×C4×C79—42no—
227C2×C4×C79C4×D414yes—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—42no—
235C2×Dic79—42no—
236C2×Dic79—42no—
237C2×Dic79—42no—
238C2×Dic79—41no—
239C2×Dic79—42no—
240C2×Dic79—41no—
241C2×Dic79—42no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—42no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—42no—
248C2×Dic79—42no—
249C2×Dic79—42no—
250C2×Dic79—42no—
251C2×Dic79—42no—
252C2×Dic79—42no—
253C2×Dic79—42no—
254C2×Dic79—42no—
255C2×Dic79—42no—
256C2×Dic79—42no—
257C2×Dic79—42no—
258C2×Dic79—42no—
259C2×Dic79—42no—
260C2×Dic79—42no—
261C2×Dic79—42no—
262C22×C79—42no—
263C22×C79—42no—
264C22×C79—42no—
265C22×C79—44no—
266C22×C79—44no—
267C22×C79—81no—
268C22×C79—81no—
269C22×C79—82no—
270C22×C79—82no—
271C22×C79—81no—
272C22×C79—81no—
273C22×C79C2×C4×D411yescharacteristic
274C22×C79#51811yescharacteristic
275C22×C79#52012yes—
276C22×C79#52011yescharacteristic
277C22×C79#52012yes—
278C22×C79#53411yescharacteristic
279C22×C79#53511yescharacteristic
280C22×C79#53511yescharacteristic
281C22×C79#53611yescharacteristic
282C22×C79#53611yescharacteristic
283C22×C79#53711yescharacteristic
284C22×C79#53912yes—
285C22×C79#54011yescharacteristic
286C22×C79#54211yescharacteristic
287C22×C79#54212yes—
288C22×C79#56012yes—
289C22×C79#56012yes—
290C22×C79#56212yes—
291C22×C79#56212yes—
292C22×C79#56312yes—
293C22×C79#56312yes—
294C22×C79#56312yes—
295C22×C79#56312yes—
296C4×C79—24no—
297C4×C79—44no—
298C4×C79—42no—
299C4×C79—44no—
300Dic79—42no—
301Dic79—42no—
302Dic79—81no—
303Dic79—82no—
304Dic79—81no—
305Dic79—82no—
306#14630—791nomaximal, Sylow
307C2×C79—82no—
308C2×C79#232211yescharacteristic
309C2×C79#232411yescharacteristic
310C2×C79#232611yescharacteristic
311C2×C79#237612yes—
312C2×C79#238312yes—
313C2×C79#281411yescharacteristic
314C2×C79#281511yescharacteristic
315C2×C79#282011yescharacteristic
316C2×C79#282111yescharacteristic
317C2×C79#283712yes—
318C2×C79#283812yes—
319C2×#397D7911yescharacteristic
320C2×#389—791no—
321C2×#388—791no—
322#1445—791no—
323#1444—791no—
324#1444—792no—
325#1443—792no—
326#1439—791no—
327#1439—792no—
328#1439—792no—
329#1434—791no—
330C79#1463011yescharacteristic, complemented, Sylow
331C22×C4⋊C4—792no—
332C22×C4⋊C4—792no—
333C22×C4⋊C4—791no—
334C22×C4⋊C4—791no—
335C22×C4⋊C4C2×D7911yescharacteristic
336C22×C22⋊C4—792no—
337C22×C22⋊C4—791no—
338C22×C22⋊C4—791no—
339C22×C22⋊C4C2×D7911yescharacteristic
340C22×C22⋊C4C2×D7912yes—
341C22×C42—791no—
342#397Dic7918yescomplemented
343#389—1584no—
344#388—1584no—
345C2×#95—792no—
346C2×#95—792no—
347C2×#95—792no—
348C2×#95—792no—
349C2×#95—792no—
350C2×#95—792no—
351C2×#95—791no—
352C2×#95—792no—
353C2×#95—791no—
354C2×#95—791no—
355C2×#95C2×D7911yescharacteristic
356C2×#95C2×D7912yes—
357C25C22×D7911yescharacteristic
358C23×C4—791no—
359C23×C4—791no—
360C23×C4—792no—
361C23×C4—791no—
362C23×C4—792no—
363C23×C4—791no—
364C23×C4C22×D7912yes—
365C23×C4C22×D7911yescharacteristic
366C23×C4C22×D7912yes—
367C23×C4C22×D7911yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—3162no—
371C2×C4⋊C4—3162no—
372C2×C4⋊C4—3162no—
373C2×C4⋊C4—3162no—
374C2×C4⋊C4—3162no—
375C2×C4⋊C4—3162no—
376C2×C4⋊C4—3162no—
377C2×C4⋊C4—1582no—
378C2×C4⋊C4—1584no—
379C2×C4⋊C4—1582no—
380C2×C4⋊C4C2×Dic7914yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—3162no—
386C2×C22⋊C4—3162no—
387C2×C22⋊C4—3162no—
388C2×C22⋊C4—3162no—
389C2×C22⋊C4—3162no—
390C2×C22⋊C4—1582no—
391C2×C22⋊C4—1582no—
392C2×C22⋊C4C2×Dic7914yes—
393C2×C22⋊C4C2×Dic7918yes—
394C2×C42—3162no—
395C2×C42—794no—
396#95—3162no—
397#95—3162no—
398#95—3162no—
399#95—3162no—
400#95—3162no—
401#95—3162no—
402#95—1582no—
403#95—1584no—
404#95—1582no—
405#95—1582no—
406#95C2×Dic7918yes—
407#95C2×Dic7914yes—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—21no—
413C24—22no—
414C24—21no—
415C24—22no—
416C24—22no—
417C24—21no—
418C24C22×Dic7912yes—
419C24C23×D7911yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—792no—
439C22×C4—792no—
440C22×C4—1582no—
441C22×C4—1581no—
442C22×C4—1582no—
443C22×C4—1581no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1581no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4—1582no—
450C22×C4—1582no—
451C22×C4—1582no—
452C22×C4—1582no—
453C22×C4—1582no—
454C22×C4—1582no—
455C22×C4—1582no—
456C22×C4—1582no—
457C22×C4—1581no—
458C22×C4—1582no—
459C22×C4—1582no—
460C22×C4—1582no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1582no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1582no—
467C22×C4—1582no—
468C22×C4—1582no—
469C22×C4—1581no—
470C22×C4—1582no—
471C22×C4—1581no—
472C22×C4C22×Dic7912yes—
473C22×C4C22×Dic7914yes—
474C22×C4C22×Dic7914yes—
475C22×C4C22×Dic7912yes—
476C22×C4D4×D7912yes—
477C22×C4#1192962312yes—
478C4⋊C4—28no—
479C4⋊C4—3164no—
480C4⋊C4—3168no—
481C4⋊C4—3168no—
482C4⋊C4—3164no—
483C22⋊C4—28no—
484C22⋊C4—48no—
485C22⋊C4—3168no—
486C22⋊C4—3164no—
487C22⋊C4—3164no—
488C42—3164no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C23×Dic7911yescharacteristic
513C23C2×D4×D7911yescharacteristic
514C23C2×#1192962311yescharacteristic
515C23#1457920611yescharacteristic
516C23#1457920611yescharacteristic
517C23#1457920712yes—
518C23#1457920711yescharacteristic
519C23#1457920712yes—
520C23#1457920811yescharacteristic
521C23#1457920812yes—
522C23#1457920812yes—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—3162no—
547C2×C4—3162no—
548C2×C4—3162no—
549C2×C4—3162no—
550C2×C4—3161no—
551C2×C4—3162no—
552C2×C4—3162no—
553C2×C4—3162no—
554C2×C4—3162no—
555C2×C4—3162no—
556C2×C4—3162no—
557C2×C4—3162no—
558C2×C4—3162no—
559C2×C4—3162no—
560C2×C4—3162no—
561C2×C4—3162no—
562C2×C4—3162no—
563C2×C4—3162no—
564C2×C4—3162no—
565C2×C4—3162no—
566C2×C4—3161no—
567C2×C4—3162no—
568C2×C4—3162no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3162no—
573C2×C4—3161no—
574C2×C4—1582no—
575C2×C4—1582no—
576C2×C4—1582no—
577C2×C4—1582no—
578C2×C4—1582no—
579C2×C4—1582no—
580C2×C4D4×Dic7914yes—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22C2×D4×Dic7911yescharacteristic
593C22#1909903811yescharacteristic
594C22#1909904612yes—
595C22#1909904612yes—
596C22#1909904611yescharacteristic
597C22#1909912411yescharacteristic
598C22#1909912911yescharacteristic
599C22#1909913111yescharacteristic
600C22#1909913311yescharacteristic
601C22#1909913511yescharacteristic
602C22#1909914211yescharacteristic
603C22#1909915712yes—
604C22#1909916611yescharacteristic
605C22#1909918011yescharacteristic
606C22#1909918212yes—
607C22#1909930312yes—
608C22#1909930412yes—
609C22#1909931012yes—
610C22#1909931012yes—
611C22#1909931512yes—
612C22#1909931512yes—
613C22#1909931512yes—
614C22#1909931612yes—
615C4—24no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—3162no—
620C4—3162no—
621C4—6321no—
622C4—6322no—
623C4—6321no—
624C4—6322no—
625C2—82no—
626C2#2376607211yescharacteristic
627C2#2376609511yescharacteristic
628C2#2376611311yescharacteristic
629C2#2376654812yes—
630C2#2376659212yes—
631C2#2377141411yescharacteristic
632C2#2377142311yescharacteristic
633C2#2377145611yescharacteristic
634C2#2377145711yescharacteristic
635C2#2377161812yes—
636C2#2377163312yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC2×#397×C79C2
socle, cosocleC24×C79C24
prime radicalC25×C79C23
hypercenterC2×#397D79
hypocenterC79#14630
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C2×#19098462C2semidirect
3. #23761668C2semidirect
4. #23761665C2semidirect
5. #23761665C2semidirect
6. #23761662C2semidirect
10. #23761625C2semidirect
35. #397×C79C4semidirect
330. C79#14630semidirect
342. #397Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190984622
C2×#190984562
#237616682
#237616652
#237616622
#237616452
#237616252
C2×#397×C792
#1463079