Finite Group Databasebeta

← #34219318

Subgroups — #34219318

Order 20224.

1334 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×#19098428C211yesmaximal, characteristic
3C2×#19098427C211yesmaximal, characteristic
4#23761666C212yesmaximal
5#23761666C212yesmaximal
6#23761654C211yesmaximal, characteristic
7#23761653C211yesmaximal, characteristic
8#23761652C212yesmaximal
9#23761651C212yesmaximal
10#23761650C211yesmaximal, characteristic
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×Dic79⋊C4C2212yes—
22C22×C4×Dic79C2211yescharacteristic
23#19098428—24no—
24#19098427—24no—
25C2×C79⋊#95C2212yes—
26C2×C79⋊#95C2212yes—
27C2×C79⋊#95C2211yescharacteristic
28C2×C79⋊#95C2212yes—
29C2×C79⋊#95C2212yes—
30C2×C79⋊#95C2212yes—
31C2×C79⋊#95C2212yes—
32C2×C79⋊#95C2212yes—
33C2×C79⋊#95C2212yes—
34C2×C79⋊#95C2212yes—
35C22×C4⋊C4×C79C2211yescharacteristic
36C22×C4⋊C4×C79C2211yescharacteristic
37C2×#95×C79C2211yescharacteristic
38C2×#95×C79C2212yes—
39C2×#95×C79C2212yes—
40C23×Dic79C2312yes—
41C23×Dic79C2312yes—
42C23×Dic79C2311yescharacteristic
43C23×Dic79C2311yescharacteristic
44C23×Dic79C2311yescharacteristic
45C23×Dic79C2311yescharacteristic
46#14579151—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—22no—
55C2×Dic79⋊C4—22no—
56C2×Dic79⋊C4—22no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—22no—
59C2×Dic79⋊C4—22no—
60C2×Dic79⋊C4—24no—
61C2×Dic79⋊C4—42no—
62C2×Dic79⋊C4—42no—
63C2×Dic79⋊C4C2×C414yes—
64C2×Dic79⋊C4C2×C414yes—
65C2×Dic79⋊C4C2×C414yes—
66C2×Dic79⋊C4C2×C414yes—
67C2×C4×Dic79C2×C414yes—
68C2×C4×Dic79D414yes—
69C2×C4×Dic79D414yes—
70C79⋊#95—22no—
71C79⋊#95—24no—
72C79⋊#95—24no—
73C79⋊#95—24no—
74C79⋊#95—42no—
75C79⋊#95—42no—
76C79⋊#95—42no—
77C79⋊#95—42no—
78C79⋊#95—42no—
79C79⋊#95—42no—
80C23×C4×C79C2312yes—
81C23×C4×C79C2312yes—
82C23×C4×C79C2311yescharacteristic
83C23×C4×C79C2311yescharacteristic
84C23×C4×C79C2311yescharacteristic
85C2×C4⋊C4×C79—22no—
86C2×C4⋊C4×C79—22no—
87C2×C4⋊C4×C79—22no—
88C2×C4⋊C4×C79—42no—
89C2×C4⋊C4×C79C2×C414yes—
90#95×C79—22no—
91#95×C79—42no—
92#95×C79—42no—
93C22×Dic79—21no—
94C22×Dic79—21no—
95C22×Dic79—21no—
96C22×Dic79—21no—
97C22×Dic79—22no—
98C22×Dic79—22no—
99C22×Dic79—21no—
100C22×Dic79—21no—
101C22×Dic79—22no—
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—22no—
111C22×Dic79—21no—
112C22×Dic79—22no—
113C22×Dic79—21no—
114C22×Dic79—22no—
115C22×Dic79—21no—
116C22×Dic79—21no—
117C22×Dic79—22no—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—22no—
122C22×Dic79—22no—
123C22×Dic79C22×C412yes—
124C22×Dic79C22×C412yes—
125C22×Dic79C22×C412yes—
126C22×Dic79C22×C412yes—
127C22×Dic79C2×D412yes—
128C22×Dic79C2×D412yes—
129C22×Dic79C2×D412yes—
130C22×Dic79C2×D412yes—
131C4⋊Dic79—28no—
132C4⋊Dic79—44no—
133Dic79⋊C4—28no—
134Dic79⋊C4—28no—
135Dic79⋊C4—28no—
136Dic79⋊C4—28no—
137Dic79⋊C4—48no—
138C4×Dic79C22⋊C4116yes—
139C24×C79C2411yescharacteristic
140C22×C4×C79—22no—
141C22×C4×C79—22no—
142C22×C4×C79—21no—
143C22×C4×C79—21no—
144C22×C4×C79—21no—
145C22×C4×C79—21no—
146C22×C4×C79—21no—
147C22×C4×C79—22no—
148C22×C4×C79—21no—
149C22×C4×C79—21no—
150C22×C4×C79—22no—
151C22×C4×C79—21no—
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×C79—22no—
162C22×C4×C79—22no—
163C22×C4×C79—22no—
164C22×C4×C79—22no—
165C22×C4×C79—22no—
166C22×C4×C79C22×C412yes—
167C22×C4×C79C22×C412yes—
168C22×C4×C79C22×C412yes—
169C22×C4×C79C2×D412yes—
170C22×C4×C79C2×D412yes—
171C4⋊C4×C79—28no—
172C4⋊C4×C79—44no—
173C23×C79C23×C411yescharacteristic
174C23×C79C22×D411yescharacteristic
175C23×C79C22×D411yescharacteristic
176C23×C79H(5,𝔽2)12yes—
177C23×C79H(5,𝔽2)11yescharacteristic
178C23×C79H(5,𝔽2)11yescharacteristic
179C23×C792−1+412yes—
180C23×C792−1+411yescharacteristic
181C23×C792−1+412yes—
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—41no—
194C2×C4×C79—41no—
195C2×C4×C79—41no—
196C2×C4×C79—41no—
197C2×C4×C79—42no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—42no—
201C2×C4×C79—42no—
202C2×C4×C79—42no—
203C2×C4×C79—42no—
204C2×C4×C79—42no—
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—41no—
231C2×Dic79—41no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—42no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—42no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—42no—
241C2×Dic79—42no—
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×C79#51912yes—
255C22×C79#52012yes—
256C22×C79#52011yescharacteristic
257C22×C79#52011yescharacteristic
258C22×C79#53511yescharacteristic
259C22×C79#53511yescharacteristic
260C22×C79#53612yes—
261C22×C79#53611yescharacteristic
262C22×C79#53611yescharacteristic
263C22×C79#53611yescharacteristic
264C22×C79#53611yescharacteristic
265C22×C79#53712yes—
266C22×C79#53711yescharacteristic
267C22×C79#53711yescharacteristic
268C22×C79#56212yes—
269C22×C79#56212yes—
270C22×C79#56312yes—
271C22×C79#56312yes—
272C22×C79#56312yes—
273C22×C79#56312yes—
274C22×C79#56412yes—
275C22×C79#56412yes—
276C4×C79—24no—
277C4×C79—42no—
278C4×C79—42no—
279C4×C79—82no—
280C4×C79—82no—
281Dic79—24no—
282Dic79—24no—
283Dic79—42no—
284Dic79—42no—
285Dic79—82no—
286Dic79—82no—
287#14664—791nomaximal, Sylow
288C2×C79#231011yescharacteristic
289C2×C79#231111yescharacteristic
290C2×C79#231111yescharacteristic
291C2×C79#238212yes—
292C2×C79#238212yes—
293C2×C79#282012yes—
294C2×C79#282011yescharacteristic
295C2×C79#282011yescharacteristic
296C2×C79#282212yes—
297C2×C79#282211yescharacteristic
298C2×C79#282211yescharacteristic
299C22×#125—791no—
300C2×#382—791no—
301C2×#382—791no—
302#1445—792no—
303#1445—792no—
304#1441—792no—
305#1441—792no—
306#1441—791no—
307#1441—791no—
308#1441—791no—
309#1441D7911yescharacteristic
310C79#1466411yescharacteristic, complemented, Sylow
311C2×#125—798no—
312C2×#125—1584no—
313C2×#125—1584no—
314C22×C4⋊C4—792no—
315C22×C4⋊C4—791no—
316C22×C4⋊C4—791no—
317C22×C4⋊C4—791no—
318C22×C4⋊C4—791no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4—791no—
321C22×C4⋊C4C2×D7911yescharacteristic
322C22×C4⋊C4C2×D7911yescharacteristic
323C22×C42—791no—
324#382—1584no—
325#382—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×D7911yescharacteristic
337C2×#95C2×D7912yes—
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×D7912yes—
347C23×C4C22×D7911yescharacteristic
348C23×C4C22×D7911yescharacteristic
349C23×C4C22×D7911yescharacteristic
350#125—15832no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—3162no—
356C2×C4⋊C4—3162no—
357C2×C4⋊C4—3162no—
358C2×C4⋊C4—794no—
359C2×C4⋊C4—794no—
360C2×C4⋊C4—794no—
361C2×C4⋊C4—794no—
362C2×C4⋊C4—794no—
363C2×C4⋊C4—1584no—
364C2×C4⋊C4—1582no—
365C2×C4⋊C4—1582no—
366C2×C4⋊C4—1582no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1582no—
369C2×C4⋊C4—1582no—
370C2×C4⋊C4—1582no—
371C2×C4⋊C4—1582no—
372C2×C4⋊C4—1582no—
373C2×C4⋊C4—1582no—
374C2×C4⋊C4—1582no—
375C2×C4⋊C4C4×D7914yes—
376C2×C42—794no—
377C2×C42—794no—
378C2×C42—794no—
379#95—22no—
380#95—42no—
381#95—42no—
382#95—3162no—
383#95—3162no—
384#95—3162no—
385#95—3162no—
386#95—3162no—
387#95—3162no—
388#95—1584no—
389#95—1584no—
390#95—1584no—
391#95—1582no—
392C24C23×D7911yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—792no—
420C22×C4—792no—
421C22×C4—792no—
422C22×C4—792no—
423C22×C4—792no—
424C22×C4—792no—
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—1582no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1582no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
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—1581no—
454C22×C4—1582no—
455C22×C4—1582no—
456C22×C4—1581no—
457C22×C4C2×C4×D7912yes—
458C22×C4C2×C4×D7912yes—
459C22×C4C2×C4×D7912yes—
460C22×C4D4×D7912yes—
461C22×C4D4×D7912yes—
462C4⋊C4—28no—
463C4⋊C4—44no—
464C4⋊C4—3164no—
465C4⋊C4—3168no—
466C4⋊C4—1588no—
467C4⋊C4—1588no—
468C4⋊C4—1588no—
469C4⋊C4—1588no—
470C4⋊C4—1588no—
471C42—7916no—
472C23C22×C4×D7911yescharacteristic
473C23C2×D4×D7911yescharacteristic
474C23C2×D4×D7911yescharacteristic
475C23#1457920612yes—
476C23#1457920611yescharacteristic
477C23#1457920711yescharacteristic
478C23#1457920811yescharacteristic
479C23#1457920812yes—
480C23#1457920812yes—
481C23#1457920912yes—
482C23#1457920911yescharacteristic
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3162no—
521C2×C4—3162no—
522C2×C4—3162no—
523C2×C4—3162no—
524C2×C4—3162no—
525C2×C4—3162no—
526C2×C4—3162no—
527C2×C4—3162no—
528C2×C4—3162no—
529C2×C4—3162no—
530C2×C4—3162no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—3162no—
534C2×C4—794no—
535C2×C4—794no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4—1584no—
539C2×C4—1582no—
540C2×C4—1582no—
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
553C22#1909903912yes—
554C22#1909904212yes—
555C22#1909904311yescharacteristic
556C22#1909904411yescharacteristic
557C22#1909912811yescharacteristic
558C22#1909912911yescharacteristic
559C22#1909913311yescharacteristic
560C22#1909913511yescharacteristic
561C22#1909913612yes—
562C22#1909913611yescharacteristic
563C22#1909913811yescharacteristic
564C22#1909914012yes—
565C22#1909914211yescharacteristic
566C22#1909914311yescharacteristic
567C22#1909930812yes—
568C22#1909930912yes—
569C22#1909931212yes—
570C22#1909931412yes—
571C22#1909931512yes—
572C22#1909931612yes—
573C22C79⋊#56412yes—
574C22C79⋊#56412yes—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4—82no—
580C4—3162no—
581C4—3162no—
582C4—6322no—
583C4—6322no—
584C4—1584no—
585C4—1584no—
586C2#2376598311yescharacteristic
587C2#2376598711yescharacteristic
588C2#2376598911yescharacteristic
589C2#2376658212yes—
590C2#2376658412yes—
591C2#2377144912yes—
592C2#2377145311yescharacteristic
593C2#2377145511yescharacteristic
594C2#2377146612yes—
595C2#2377146911yescharacteristic
596C2#2377147011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC79×#1441C2
socle, prime radical, cosocleC24×C79C24
hypercenter#1441D79
hypocenterC79#14664
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#14664semidirect

Maximal subgroups

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

subgroupindex
C22×#145791512
C2×#190984282
C2×#190984272
#237616662
#237616542
#237616532
#237616522
#237616512
#237616502
C79×#14412
#1466479