Finite Group Databasebeta

← #34308108

Subgroups — #34308108

Order 20224.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098673C211yesmaximal, characteristic
2#23766371C211yesmaximal, characteristic, complemented
3#23764275C211yesmaximal, characteristic
4#23764265C211yesmaximal, characteristic
5#23764027C211yesmaximal, characteristic, complemented
6#23764024C211yesmaximal, characteristic, complemented
7#23764020C211yesmaximal, characteristic, complemented
8#23764017C211yesmaximal, characteristic, complemented
9#23763566C211yesmaximal, characteristic, complemented
10#23763410C211yesmaximal, characteristic, complemented
11#23763397C211yesmaximal, characteristic, complemented
12#23763298C211yesmaximal, characteristic, complemented
13C4×#14579092C211yesmaximal, characteristic, complemented
14#23763275C211yesmaximal, characteristic, complemented
15C79×#2354C211yesmaximal, characteristic, complemented
16C22×#11929609C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18#19098673C418yescomplemented
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30C2×#14579081C2211yescharacteristic
31C2×#14579072C2211yescharacteristic
32C2×#14579072C2211yescharacteristic
33C2×C4×C79⋊C8C2211yescharacteristic
34#19098465C2211yescharacteristic
35#19098465C2211yescharacteristic
36#19098453—21no—
37#19098453—21no—
38#19098429C2211yescharacteristic
39#19098425—21no—
40C4×C4⋊Dic79C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42#19098410—21no—
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49C2×C4×D4×C79C2211yescharacteristic
50C2×#117×C79—21no—
51#390×C79C2211yescharacteristic
52#390×C79C2211yescharacteristic
53#388×C79—21no—
54#388×C79—21no—
55#384×C79C2211yescharacteristic
56#380×C79—21no—
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59C4×C4⋊C4×C79C2211yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C2×#11929609—22no—
63C2×#11929609—22no—
64C2×#11929609C2×C414yes—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69C4×D316—22no—
70C4×D316C2×C414yes—
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—21no—
76C2×(C2×D79)⋊C4—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81#14579094—22no—
82#14579094—41no—
83#14579092—22no—
84#14579092—41no—
85#14579092C2×C414yes—
86C4⋊(C79⋊C8)—22no—
87#14579082—41no—
88#14579082—41no—
89#14579082C2×C414yes—
90#14579081—41no—
91#14579081—41no—
92#14579081C2×C414yes—
93#14579072—22no—
94#14579072C2×C414yes—
95C4×C79⋊C8—22no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C22×D4×C79C2311yescharacteristic
101C23×C4×C79—21no—
102C4×D4×C79—22no—
103C4×D4×C79C2×C414yes—
104#117×C79—44no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79D411yescharacteristic
108C2×C4⋊C4×C79D411yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79—21no—
115C2×C22⋊C4×C79—21no—
116C2×C22⋊C4×C79—21no—
117C2×C42×C79—21no—
118C2×C42×C79—21no—
119C2×C42×C79C2311yescharacteristic
120C2×C42×C79C2311yescharacteristic
121C2×C42×C79C2311yescharacteristic
122#95×C79—21no—
123#95×C79—21no—
124#95×C79D411yescharacteristic
125#95×C79D411yescharacteristic
126C23×D79—21no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316C22×C412yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C2×C4×D79—24no—
137C2×C4×D79—81no—
138C2×C4×D79—81no—
139#11929609—28no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8C22×C412yes—
149C2×C79⋊C8C22×C412yes—
150C2×C79⋊C8#4112yes—
151C2×C79⋊C8#4112yes—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79C22×C412yes—
158(C2×D79)⋊C4—24no—
159(C2×D79)⋊C4—42no—
160(C2×D79)⋊C4—42no—
161(C2×D79)⋊C4—81no—
162(C2×D79)⋊C4—81no—
163C4×Dic79—81no—
164C4×Dic79—81no—
165C24×C79—21no—
166C2×D4×C79—21no—
167C2×D4×C79—21no—
168C2×D4×C79—21no—
169C2×D4×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79C22×C412yes—
172C22×C4×C79—21no—
173C22×C4×C79—24no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—42no—
177C22×C4×C79—42no—
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2411yescharacteristic
185C4⋊C4×C79—22no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—41no—
192C4⋊C4×C79—42no—
193C4⋊C4×C79—81no—
194C4⋊C4×C79C22×C412yes—
195C22⋊C4×C79—24no—
196C22⋊C4×C79—42no—
197C22⋊C4×C79—42no—
198C22⋊C4×C79—81no—
199C22⋊C4×C79—81no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—42no—
205C42×C79—42no—
206C42×C79C22×C412yes—
207C42×C79#4112yes—
208C23×C79—22no—
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79—41no—
213C23×C79—41no—
214C23×C79—41no—
215C23×C79C22×D411yescharacteristic
216C22×D79—22no—
217C22×D79—41no—
218C22×D79—41no—
219C22×D79—41no—
220C22×D79—41no—
221C22×D79—41no—
222C22×D79—41no—
223D4×C79—22no—
224D4×C79—22no—
225D4×C79—41no—
226D4×C79—41no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—22no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—48no—
244C2×C4×C79—41no—
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×C79C4×D412yes—
253C2×C4×C79C4×D412yes—
254C2×C4×C79AGL(1,ℤ/8)12yes—
255C2×C4×C79C23×C411yescharacteristic
256C2×C4×C79C2×#4111yescharacteristic
257C2×C4×C79C2×#4111yescharacteristic
258C2×C4×C79C2×#4111yescharacteristic
259C2×C4×C79H(5,𝔽2)11yescharacteristic
260C2×C4×C792−1+411yescharacteristic
261C2×Dic79—22no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—41no—
269C2×Dic79—41no—
270D316—22no—
271D316—22no—
272D316—41no—
273D316—41no—
274C4×D79—84no—
275C79⋊C8—22no—
276C79⋊C8—41no—
277C79⋊C8—41no—
278C79⋊C8#11714yescomplemented
279C22×C79—42no—
280C22×C79—42no—
281C22×C79—42no—
282C22×C79—81no—
283C22×C79—81no—
284C22×C79—81no—
285C22×C79—81no—
286C22×C79C2×C4×D411yescharacteristic
287C22×C79#54711yescharacteristic
288C22×C79#55011yescharacteristic
289C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C79#57511yescharacteristic
291C22×C79#57611yescharacteristic
292C22×C79#57711yescharacteristic
293C2×D79—42no—
294C2×D79—42no—
295C2×D79—42no—
296C2×D79—81no—
297C2×D79—81no—
298C2×D79—81no—
299C2×D79—81no—
300C4×C79—22no—
301C4×C79—22no—
302C4×C79—22no—
303C4×C79—41no—
304C4×C79—41no—
305C4×C79—41no—
306C4×C79—42no—
307C4×C79—81no—
308C4×C79—81no—
309C4×C79C2×#11711yescharacteristic
310C4×C79#52011yescharacteristic
311C4×C79#55511yescharacteristic
312C4×C79#55611yescharacteristic
313Dic79—42no—
314Dic79—81no—
315Dic79—81no—
316#22817—791nomaximal, Sylow
317C2×C79—82no—
318C2×C79#235411yescharacteristic
319C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
320C2×C79#294611yescharacteristic
321C2×C79#334511yescharacteristic
322C2×C79#334711yescharacteristic
323C2×C79#339911yescharacteristic
324C2×C79#340111yescharacteristic
325D79—82no—
326C2×#442—791no—
327#2357—791no—
328#2354D7911yescharacteristic, complemented
329#1972—791no—
330#1970—791no—
331#1926—791no—
332#1926—791no—
333#1925—791no—
334#1925—791no—
335#1847—791no—
336#1804—791no—
337#1802—791no—
338#1776—791no—
339C4×#106—791no—
340#1771—791no—
341C79#2281711yescharacteristic, complemented, Sylow
342C22×D8—791no—
343C2×C4×D4—791no—
344C2×C4×D4C2×D7911yescharacteristic
345C2×#117—21no—
346#442—798no—
347C2×#107—791no—
348C2×#107—791no—
349C2×#106—791no—
350C2×#106—791no—
351C2×#106—791no—
352C2×C4⋊C8—791no—
353C2×#102—791no—
354C2×#102—791no—
355C2×#102—791no—
356C2×#102—791no—
357C2×#102—791no—
358C2×#102—791no—
359C2×#97—791no—
360C2×#97—791no—
361C2×C4×C8—791no—
362#390—791no—
363#390—791no—
364#390C2×D7911yescharacteristic
365#390C2×D7911yescharacteristic
366#388—21no—
367#388—21no—
368#388—1581no—
369#388—1581no—
370#384C2×D7911yescharacteristic
371#382—791no—
372#381—1581no—
373#380—21no—
374C4×C4⋊C4—791no—
375C4×C4⋊C4—791no—
376C4×C4⋊C4C2×D7911yescharacteristic
377C4×C4⋊C4C2×D7911yescharacteristic
378C4×C4⋊C4C2×D7911yescharacteristic
379C4×C22⋊C4—1581no—
380#340—791no—
381#340—791no—
382#340—791no—
383#340—791no—
384#336—791no—
385#336—791no—
386C22×D4—791no—
387C22×D4C22×D7911yescharacteristic
388C23×C4—21no—
389C23×C4—1581no—
390C2×D8—794no—
391C2×D8—1582no—
392C2×D8—1582no—
393C22×C8—791no—
394C22×C8—791no—
395C22×C8—791no—
396C22×C8—791no—
397C4×D4—22no—
398C4×D4—794no—
399C4×D4—1582no—
400C4×D4C4×D7914yes—
401#117—44no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—791no—
405C2×C4⋊C4—791no—
406C2×C4⋊C4—791no—
407C2×C4⋊C4C22⋊D7911yescharacteristic
408C2×C4⋊C4C22⋊D7911yescharacteristic
409C2×C4⋊C4C22×D7911yescharacteristic
410C2×C4⋊C4C22×D7911yescharacteristic
411C2×C4⋊C4C22×D7911yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1581no—
418C2×C22⋊C4—1581no—
419C2×C22⋊C4—1581no—
420C2×C22⋊C4—1581no—
421C2×C22⋊C4—1581no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—1581no—
425C2×C42—1581no—
426C2×C42C22×D7911yescharacteristic
427C2×C42C22×D7911yescharacteristic
428C2×C42C22×D7911yescharacteristic
429#107—3161no—
430#107—1582no—
431#106—3161no—
432#106—794no—
433#106—1582no—
434C4⋊C8—1582no—
435#102—3161no—
436#102—3161no—
437#102—3161no—
438#102—3161no—
439#102—794no—
440#102—794no—
441#97—794no—
442#97—1582no—
443C4×C8—1582no—
444#95—21no—
445#95—21no—
446#95—1581no—
447#95—1581no—
448#95—1581no—
449#95—1581no—
450#95C22⋊D7911yescharacteristic
451#95C22⋊D7911yescharacteristic
452C24—21no—
453C24—1581no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—792no—
460C2×D4—1581no—
461C2×D4—1581no—
462C2×D4—1581no—
463C2×D4—1581no—
464C2×D4—1581no—
465C2×D4C2×C4×D7912yes—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—6321no—
473C22×C4—6321no—
474C22×C4—1581no—
475C22×C4—1581no—
476C22×C4—1581no—
477C22×C4—1584no—
478C22×C4C2×C22⋊D7911yescharacteristic
479C22×C4C2×C22⋊D7911yescharacteristic
480C22×C4C2×C22⋊D7911yescharacteristic
481C22×C4C2×C22⋊D7911yescharacteristic
482C22×C4C2×C22⋊D7911yescharacteristic
483C22×C4C2×C22⋊D7911yescharacteristic
484C22×C4C23×D7911yescharacteristic
485D8—1588no—
486C2×C8—792no—
487C2×C8—792no—
488C2×C8—792no—
489C2×C8—792no—
490C2×C8—1581no—
491C2×C8—1581no—
492C2×C8—1581no—
493C2×C8—1581no—
494C2×C8—1581no—
495C2×C8—1581no—
496C2×C8—1581no—
497C2×C8—1581no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—792no—
508C4⋊C4—1581no—
509C4⋊C4—1581no—
510C4⋊C4—1581no—
511C4⋊C4—1581no—
512C4⋊C4—1581no—
513C4⋊C4C2×C4×D7912yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—3162no—
520C22⋊C4—3162no—
521C22⋊C4—6321no—
522C22⋊C4—6321no—
523C22⋊C4—1584no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C42—6321no—
531C42—6321no—
532C42C2×C4×D7912yes—
533C42#1192962412yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—3161no—
542C23—3161no—
543C23—3161no—
544C23—3161no—
545C23—3161no—
546C23—3161no—
547C23—1582no—
548C23C22×C22⋊D7911yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—3161no—
554D4—3161no—
555D4—1582no—
556D4—1582no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
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—6324no—
591C2×C4—1582no—
592C2×C4C4×C22⋊D7912yes—
593C2×C4C4×C22⋊D7912yes—
594C2×C4#1457918712yes—
595C2×C4C22×C4×D7911yescharacteristic
596C2×C4#41×D7911yescharacteristic
597C2×C4#41×D7911yescharacteristic
598C2×C4C2×#1192962411yescharacteristic
599C2×C4#1457920711yescharacteristic
600C2×C4#1457920811yescharacteristic
601C8—3161no—
602C8—3161no—
603C8—794no—
604C8—1582no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—3162no—
613C22—3162no—
614C22—3162no—
615C22—6321no—
616C22—6321no—
617C22—6321no—
618C22—6321no—
619C22C2×C4×C22⋊D7911yescharacteristic
620C22#1909921211yescharacteristic
621C22#1909923211yescharacteristic
622C22C2×#1457918711yescharacteristic
623C22#1909936111yescharacteristic
624C22#1909936611yescharacteristic
625C22#1909937111yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—3162no—
636C4—6321no—
637C4—6321no—
638C4#117×D7911yescharacteristic
639C4#1909904311yescharacteristic
640C4#1909927611yescharacteristic
641C4#1909928111yescharacteristic
642C2—82no—
643C2—6322no—
644C2#2376635011yescharacteristic
645C2C4×#1457918711yescharacteristic
646C2#2377222811yescharacteristic
647C2#2377587011yescharacteristic
648C2#2377590011yescharacteristic
649C2#2377644211yescharacteristic
650C2#2377646611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2354C2
socleC23×C79C22×D4
prime radicalC22×#11929609C22
cosocleC22×C4×C79C24
hypercenter#2354D79
hypocenterC79#22817
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #23766371C2semidirect
5. #23764027C2semidirect
6. #23764024C2semidirect
7. #23764020C2semidirect
8. #23764017C2semidirect
9. #23763566C2semidirect
10. #23763410C2semidirect
11. #23763397C2semidirect
12. #23763298C2semidirect
13. C4×#14579092C2semidirect
14. #23763275C2semidirect
15. C79×#2354C2semidirect
18. #19098673C4semidirect
278. C79⋊C8#117semidirect
328. #2354D79semidirect
341. C79#22817semidirect

Maximal subgroups

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

subgroupindex
C2×#190986732
#237663712
#237642752
#237642652
#237640272
#237640242
#237640202
#237640172
#237635662
#237634102
#237633972
#237632982
C4×#145790922
#237632752
C79×#23542
#2281779