Finite Group Databasebeta

← #34214926

Subgroups — #34214926

Order 20224.

1414 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579151C211yesmaximal, characteristic
2C2×#19098430C211yesmaximal, characteristic
3C2×C4⋊C4×Dic79C211yesmaximal, characteristic
4#23761666C212yesmaximal
5#23761666C212yesmaximal
6#23761654C211yesmaximal, characteristic
7#23761653C211yesmaximal, characteristic
8#23761650C211yesmaximal, characteristic
9#23761634C212yesmaximal
10#23761631C212yesmaximal
11C79×#1441C211yesmaximal, characteristic
12C2×#14579151—24no—
13C2×#14579151—24no—
14C2×#14579151C418yescomplemented
15C22×C4⋊Dic79C2211yescharacteristic
16C22×C4⋊Dic79C2211yescharacteristic
17C22×Dic79⋊C4C2211yescharacteristic
18C22×Dic79⋊C4C2211yescharacteristic
19C22×Dic79⋊C4C2211yescharacteristic
20C22×Dic79⋊C4C2211yescharacteristic
21C22×C4×Dic79C2212yes—
22C22×C4×Dic79C2211yescharacteristic
23#19098430—24no—
24C4⋊C4×Dic79—24no—
25C2×C79⋊#95C2212yes—
26C2×C79⋊#95C2212yes—
27C2×C79⋊#95C2212yes—
28C2×C79⋊#95C2212yes—
29C2×C79⋊#95C2212yes—
30C2×C79⋊#95C2212yes—
31C2×C79⋊#95C2211yescharacteristic
32C2×C79⋊#95C2212yes—
33C2×C79⋊#95C2212yes—
34C2×C79⋊#95C2212yes—
35C22×C4⋊C4×C79C2211yescharacteristic
36C22×C4⋊C4×C79C2211yescharacteristic
37C2×#95×C79C2212yes—
38C2×#95×C79C2211yescharacteristic
39C2×#95×C79C2212yes—
40C23×Dic79C2312yes—
41C23×Dic79C2311yescharacteristic
42C23×Dic79C2311yescharacteristic
43C23×Dic79C2312yes—
44C23×Dic79C2311yescharacteristic
45C23×Dic79C2311yescharacteristic
46#14579151—232no—
47C2×C4⋊Dic79—22no—
48C2×C4⋊Dic79—22no—
49C2×C4⋊Dic79—22no—
50C2×C4⋊Dic79—24no—
51C2×C4⋊Dic79C2×C414yes—
52C2×Dic79⋊C4—22no—
53C2×Dic79⋊C4—22no—
54C2×Dic79⋊C4—22no—
55C2×Dic79⋊C4—22no—
56C2×Dic79⋊C4—22no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—22no—
59C2×Dic79⋊C4—22no—
60C2×Dic79⋊C4C2×C414yes—
61C2×Dic79⋊C4C2×C414yes—
62C2×Dic79⋊C4C2×C414yes—
63C2×Dic79⋊C4C2×C414yes—
64C2×C4×Dic79—24no—
65C2×C4×Dic79—42no—
66C2×C4×Dic79—42no—
67C2×C4×Dic79C2×C414yes—
68C2×C4×Dic79D414yes—
69C2×C4×Dic79D414yes—
70C79⋊#95—22no—
71C79⋊#95—24no—
72C79⋊#95—24no—
73C79⋊#95—24no—
74C79⋊#95—24no—
75C79⋊#95—24no—
76C79⋊#95—24no—
77C79⋊#95—42no—
78C79⋊#95—42no—
79C79⋊#95—42no—
80C23×C4×C79C2312yes—
81C23×C4×C79C2311yescharacteristic
82C23×C4×C79C2311yescharacteristic
83C23×C4×C79C2312yes—
84C23×C4×C79C2311yescharacteristic
85C2×C4⋊C4×C79—22no—
86C2×C4⋊C4×C79—22no—
87C2×C4⋊C4×C79—22no—
88C2×C4⋊C4×C79—24no—
89C2×C4⋊C4×C79C2×C414yes—
90#95×C79—22no—
91#95×C79—24no—
92#95×C79—42no—
93C22×Dic79—22no—
94C22×Dic79—21no—
95C22×Dic79—21no—
96C22×Dic79—21no—
97C22×Dic79—21no—
98C22×Dic79—22no—
99C22×Dic79—22no—
100C22×Dic79—22no—
101C22×Dic79—22no—
102C22×Dic79—22no—
103C22×Dic79—22no—
104C22×Dic79—21no—
105C22×Dic79—21no—
106C22×Dic79—21no—
107C22×Dic79—21no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—22no—
112C22×Dic79—21no—
113C22×Dic79—22no—
114C22×Dic79—21no—
115C22×Dic79—21no—
116C22×Dic79—22no—
117C22×Dic79—22no—
118C22×Dic79—21no—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—22no—
122C22×Dic79C22×C412yes—
123C22×Dic79C22×C412yes—
124C22×Dic79C22×C412yes—
125C22×Dic79C22×C412yes—
126C22×Dic79C2×D412yes—
127C22×Dic79C2×D412yes—
128C22×Dic79C2×D412yes—
129C22×Dic79C2×D412yes—
130C22×Dic79#4114yes—
131C4⋊Dic79—28no—
132C4⋊Dic79—28no—
133Dic79⋊C4—28no—
134Dic79⋊C4—28no—
135Dic79⋊C4—28no—
136Dic79⋊C4—28no—
137C4×Dic79—48no—
138C4×Dic79C22⋊C4116yes—
139C24×C79C2411yescharacteristic
140C22×C4×C79—22no—
141C22×C4×C79—22no—
142C22×C4×C79—22no—
143C22×C4×C79—21no—
144C22×C4×C79—21no—
145C22×C4×C79—21no—
146C22×C4×C79—21no—
147C22×C4×C79—22no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—21no—
151C22×C4×C79—22no—
152C22×C4×C79—21no—
153C22×C4×C79—22no—
154C22×C4×C79—21no—
155C22×C4×C79—21no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—22no—
159C22×C4×C79—22no—
160C22×C4×C79—22no—
161C22×C4×C79—22no—
162C22×C4×C79—22no—
163C22×C4×C79—22no—
164C22×C4×C79—22no—
165C22×C4×C79C22×C412yes—
166C22×C4×C79C22×C412yes—
167C22×C4×C79C22×C412yes—
168C22×C4×C79C2×D412yes—
169C22×C4×C79C2×D412yes—
170C22×C4×C79#4114yes—
171C4⋊C4×C79—28no—
172C4⋊C4×C79—28no—
173C23×C79C23×C411yescharacteristic
174C23×C79C22×D411yescharacteristic
175C23×C79C22×D411yescharacteristic
176C23×C79C2×#4112yes—
177C23×C79C2×#4112yes—
178C23×C79H(5,𝔽2)11yescharacteristic
179C23×C79H(5,𝔽2)11yescharacteristic
180C23×C792−1+412yes—
181C23×C792−1+411yescharacteristic
182C23×C792−1+412yes—
183C23×C792−1+411yescharacteristic
184C2×C4×C79—24no—
185C2×C4×C79—22no—
186C2×C4×C79—22no—
187C2×C4×C79—24no—
188C2×C4×C79—22no—
189C2×C4×C79—22no—
190C2×C4×C79—24no—
191C2×C4×C79—22no—
192C2×C4×C79—22no—
193C2×C4×C79—24no—
194C2×C4×C79—24no—
195C2×C4×C79—24no—
196C2×C4×C79—41no—
197C2×C4×C79—41no—
198C2×C4×C79—42no—
199C2×C4×C79—42no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—42no—
203C2×C4×C79—41no—
204C2×C4×C79—41no—
205C2×C4×C79—42no—
206C2×C4×C79—42no—
207C2×C4×C79—42no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—42no—
214C2×C4×C79C2×C22⋊C414yes—
215C2×Dic79—24no—
216C2×Dic79—22no—
217C2×Dic79—22no—
218C2×Dic79—22no—
219C2×Dic79—24no—
220C2×Dic79—22no—
221C2×Dic79—22no—
222C2×Dic79—22no—
223C2×Dic79—22no—
224C2×Dic79—22no—
225C2×Dic79—22no—
226C2×Dic79—22no—
227C2×Dic79—24no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—24no—
231C2×Dic79—24no—
232C2×Dic79—24no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—42no—
236C2×Dic79—42no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—42no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—42no—
243C2×Dic79—42no—
244C2×Dic79—42no—
245C2×Dic79—42no—
246C2×Dic79—42no—
247C2×Dic79—42no—
248C2×Dic79—42no—
249C2×Dic79—42no—
250C2×Dic79—42no—
251C2×Dic79C2×C22⋊C414yes—
252C2×Dic79C2×C22⋊C414yes—
253C22×C79C22×C22⋊C411yescharacteristic
254C22×C79C4×#4112yes—
255C22×C79#51912yes—
256C22×C79#52011yescharacteristic
257C22×C79#52011yescharacteristic
258C22×C79#52511yescharacteristic
259C22×C79#52511yescharacteristic
260C22×C79#53212yes—
261C22×C79#53511yescharacteristic
262C22×C79#53612yes—
263C22×C79#53611yescharacteristic
264C22×C79#53611yescharacteristic
265C22×C79#53611yescharacteristic
266C22×C79#53711yescharacteristic
267C22×C79#53711yescharacteristic
268C22×C79#53912yes—
269C22×C79#53912yes—
270C22×C79#53912yes—
271C22×C79#53912yes—
272C22×C79#54112yes—
273C22×C79#54112yes—
274C22×C79#54112yes—
275C22×C79#54112yes—
276C4×C79—24no—
277C4×C79—42no—
278C4×C79—44no—
279C4×C79—42no—
280C4×C79—82no—
281Dic79—24no—
282Dic79—24no—
283Dic79—42no—
284Dic79—44no—
285Dic79—42no—
286Dic79—82no—
287#14289—791nomaximal, Sylow
288C2×C79#229711yescharacteristic
289C2×C79#231011yescharacteristic
290C2×C79#231111yescharacteristic
291C2×C79#233612yes—
292C2×C79#233612yes—
293C2×C79#263311yescharacteristic
294C2×C79#263311yescharacteristic
295C2×C79#263411yescharacteristic
296C2×C79#263411yescharacteristic
297C2×C79#266412yes—
298C2×C79#266412yes—
299C22×#125—791no—
300C2×#383—791no—
301C2×C4×C4⋊C4—791no—
302#1445—792no—
303#1445—792no—
304#1441—791no—
305#1441—791no—
306#1441—791no—
307#1441D7911yescharacteristic
308#1436—792no—
309#1436—792no—
310C79#1428911yescharacteristic, complemented, Sylow
311C2×#125—798no—
312C2×#125—1584no—
313C2×#125—1584no—
314C22×C4⋊C4—791no—
315C22×C4⋊C4—791no—
316C22×C4⋊C4—791no—
317C22×C4⋊C4—791no—
318C22×C4⋊C4—791no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4C2×D7911yescharacteristic
321C22×C4⋊C4C2×D7911yescharacteristic
322C22×C42—792no—
323C22×C42—791no—
324#383—1584no—
325C4×C4⋊C4—1584no—
326C2×#95—792no—
327C2×#95—792no—
328C2×#95—792no—
329C2×#95—792no—
330C2×#95—792no—
331C2×#95—792no—
332C2×#95—792no—
333C2×#95—791no—
334C2×#95—792no—
335C2×#95—792no—
336C2×#95C2×D7912yes—
337C2×#95C2×D7911yescharacteristic
338C2×#95C2×D7912yes—
339C23×C4—791no—
340C23×C4—791no—
341C23×C4—791no—
342C23×C4—791no—
343C23×C4—792no—
344C23×C4—792no—
345C23×C4C22×D7912yes—
346C23×C4C22×D7911yescharacteristic
347C23×C4C22×D7911yescharacteristic
348C23×C4C22×D7912yes—
349C23×C4C22×D7911yescharacteristic
350#125—15832no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—794no—
356C2×C4⋊C4—794no—
357C2×C4⋊C4—794no—
358C2×C4⋊C4—794no—
359C2×C4⋊C4—794no—
360C2×C4⋊C4—1582no—
361C2×C4⋊C4—1582no—
362C2×C4⋊C4—1582no—
363C2×C4⋊C4—1582no—
364C2×C4⋊C4—1582no—
365C2×C4⋊C4—1582no—
366C2×C4⋊C4—1582no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1584no—
369C2×C4⋊C4—1582no—
370C2×C4⋊C4—1582no—
371C2×C4⋊C4—1582no—
372C2×C4⋊C4C4×D7914yes—
373C2×C42—3162no—
374C2×C42—3162no—
375C2×C42—794no—
376C2×C42—794no—
377C2×C42—794no—
378C2×C42—1584no—
379#95—22no—
380#95—24no—
381#95—42no—
382#95—3162no—
383#95—3162no—
384#95—3162no—
385#95—1584no—
386#95—1584no—
387#95—1584no—
388#95—1584no—
389#95—1584no—
390#95—1582no—
391#95—1584no—
392C24C23×D7911yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—792no—
419C22×C4—792no—
420C22×C4—792no—
421C22×C4—792no—
422C22×C4—792no—
423C22×C4—792no—
424C22×C4—794no—
425C22×C4—792no—
426C22×C4—792no—
427C22×C4—1582no—
428C22×C4—1582no—
429C22×C4—1582no—
430C22×C4—1582no—
431C22×C4—1582no—
432C22×C4—1582no—
433C22×C4—1582no—
434C22×C4—1582no—
435C22×C4—1582no—
436C22×C4—1582no—
437C22×C4—1581no—
438C22×C4—1582no—
439C22×C4—1581no—
440C22×C4—1582no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1582no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1582no—
454C22×C4—1582no—
455C22×C4—1581no—
456C22×C4C2×C4×D7912yes—
457C22×C4C2×C4×D7912yes—
458C22×C4C2×C4×D7912yes—
459C22×C4D4×D7912yes—
460C22×C4D4×D7912yes—
461C22×C4#1192962314yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—1588no—
465C4⋊C4—1588no—
466C4⋊C4—1588no—
467C4⋊C4—1588no—
468C4⋊C4—1588no—
469C4⋊C4—1588no—
470C42—3168no—
471C42—7916no—
472C23C22×C4×D7911yescharacteristic
473C23C2×D4×D7911yescharacteristic
474C23C2×D4×D7911yescharacteristic
475C23#41×D7912yes—
476C23C2×#1192962312yes—
477C23#1457920611yescharacteristic
478C23#1457920711yescharacteristic
479C23#1457920811yescharacteristic
480C23#1457920812yes—
481C23#1457920912yes—
482C23#1457920911yescharacteristic
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—3161no—
514C2×C4—3162no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3162no—
518C2×C4—3162no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3162no—
522C2×C4—3162no—
523C2×C4—3161no—
524C2×C4—3162no—
525C2×C4—3162no—
526C2×C4—3162no—
527C2×C4—3162no—
528C2×C4—3162no—
529C2×C4—3162no—
530C2×C4—3162no—
531C2×C4—794no—
532C2×C4—794no—
533C2×C4—1584no—
534C2×C4—1584no—
535C2×C4—1582no—
536C2×C4—1582no—
537C2×C4—1584no—
538C2×C4—1582no—
539C2×C4—1582no—
540C2×C4—1584no—
541C2×C4—1584no—
542C2×C4—1582no—
543C2×C4—1582no—
544C2×C4—1582no—
545C2×C4—1582no—
546C2×C4—1584no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4—1582no—
551C2×C4C22⋊C4×D7914yes—
552C22C2×C22⋊C4×D7911yescharacteristic
553C22C4×#1192962312yes—
554C22#1909903912yes—
555C22#1909904311yescharacteristic
556C22#1909904411yescharacteristic
557C22#1909906911yescharacteristic
558C22#1909907111yescharacteristic
559C22#1909910812yes—
560C22#1909912911yescharacteristic
561C22#1909913311yescharacteristic
562C22#1909913611yescharacteristic
563C22#1909913612yes—
564C22#1909913811yescharacteristic
565C22#1909914211yescharacteristic
566C22#1909914311yescharacteristic
567C22#1909915712yes—
568C22#1909915812yes—
569C22#1909915912yes—
570C22#1909916112yes—
571C22#1909917012yes—
572C22#1909917212yes—
573C22#1909917312yes—
574C22#1909917412yes—
575C4—24no—
576C4—42no—
577C4—44no—
578C4—42no—
579C4—82no—
580C4—3162no—
581C4—3164no—
582C4—3162no—
583C4—6322no—
584C4—1584no—
585C4—1584no—
586C2#2376588511yescharacteristic
587C2#2376598311yescharacteristic
588C2#2376598911yescharacteristic
589C2#2376619812yes—
590C2#2376620512yes—
591C2#2376926711yescharacteristic
592C2#2376927011yescharacteristic
593C2#2376927511yescharacteristic
594C2#2376927811yescharacteristic
595C2#2376955812yes—
596C2#2376956012yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC79×#1441C2
socle, prime radical, cosocleC24×C79C24
hypercenter#1441D79
hypocenterC79#14289
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
14. C2×#14579151C4semidirect
310. C79#14289semidirect

Maximal subgroups

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

subgroupindex
C22×#145791512
C2×#190984302
C2×C4⋊C4×Dic792
#237616662
#237616542
#237616532
#237616502
#237616342
#237616312
C79×#14412
#1428979