Finite Group Databasebeta

← #34317932

Subgroups — #34317932

Order 20224.

876 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098663C211yesmaximal, characteristic, complemented
2C2×#19098624C211yesmaximal, characteristic, complemented
3#23766082C211yesmaximal, characteristic, complemented
4#23764891C211yesmaximal, characteristic, complemented
5#23764831C211yesmaximal, characteristic, complemented
6#23764792C211yesmaximal, characteristic, complemented
7#23764774C211yesmaximal, characteristic, complemented
8#23764717C211yesmaximal, characteristic, complemented
9#23764596C211yesmaximal, characteristic, complemented
10#23764579C211yesmaximal, characteristic
11#23764403C211yesmaximal, characteristic
12#23763298C211yesmaximal, characteristic, complemented
13C4×#14579084C211yesmaximal, characteristic, complemented
14C4×#14579082C211yesmaximal, characteristic, complemented
15C79×#2324C211yesmaximal, characteristic
16C22×#11929612C2211yescharacteristic
17C2×#14579140C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19#19098663—24no—
20#19098624—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×#14579084C2211yescharacteristic
28C2×#14579084C2211yescharacteristic
29C2×#14579082C2211yescharacteristic
30C2×#14579082C2211yescharacteristic
31C2×#14579082C2211yescharacteristic
32C2×C22⋊(C79⋊C8)C2211yescharacteristic
33C2×C22⋊(C79⋊C8)C2211yescharacteristic
34C2×#14579072C2211yescharacteristic
35C2×C4×C79⋊C8C2211yescharacteristic
36#19098469C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098458—21no—
39#19098457—21no—
40#19098451—21no—
41#19098444—21no—
42#19098423C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50C2×#121×C79C2211yescharacteristic
51C2×C4×D4×C79C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53#390×C79—21no—
54#390×C79C2211yescharacteristic
55#389×C79C2211yescharacteristic
56#388×C79—21no—
57#386×C79—21no—
58#385×C79—21no—
59#380×C79C2211yescharacteristic
60C4×C4⋊C4×C79C2211yescharacteristic
61C22×Dic158C2311yescharacteristic
62C2×#11929612—22no—
63C2×#11929612—22no—
64C2×#11929612—22no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69#14579140—22no—
70#14579140—22no—
71C4×Dic158—22no—
72C4×Dic158—22no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83#14579094—22no—
84#14579094—22no—
85#14579092—22no—
86C4⋊(C79⋊C8)—22no—
87C4⋊(C79⋊C8)—22no—
88#14579084—22no—
89#14579084—22no—
90#14579084—22no—
91#14579082—22no—
92#14579082—22no—
93#14579082—22no—
94C22⋊(C79⋊C8)—22no—
95C22⋊(C79⋊C8)—22no—
96#14579072D414yescomplemented
97C4×C79⋊C8—22no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C22×D4×C79—21no—
104C22×D4×C79C2311yescharacteristic
105C23×C4×C79—21no—
106C23×C4×C79C2311yescharacteristic
107#121×C79—22no—
108#121×C79—22no—
109C4×D4×C79—22no—
110C4×D4×C79—22no—
111#117×C79—22no—
112#117×C79—22no—
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79D411yescharacteristic
115C2×C4⋊C4×C79D411yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C22⋊C4×C79—21no—
120C2×C22⋊C4×C79—21no—
121C2×C22⋊C4×C79—21no—
122C2×C22⋊C4×C79—21no—
123C2×C22⋊C4×C79—21no—
124C2×C22⋊C4×C79D411yescharacteristic
125C2×C22⋊C4×C79D411yescharacteristic
126C2×C42×C79—21no—
127C2×C42×C79C2311yescharacteristic
128C2×C42×C79C2311yescharacteristic
129#95×C79—21no—
130#95×C79—21no—
131#95×C79—21no—
132C2×Dic158—21no—
133C2×Dic158—21no—
134C2×Dic158—21no—
135C2×Dic158—21no—
136C2×Dic158—21no—
137C2×Dic158#4112yes—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79—21no—
142#11929612—28no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8C2×D412yes—
154C2×C79⋊C8C2×D412yes—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79#4112yes—
161Dic79⋊C4—24no—
162Dic79⋊C4—42no—
163Dic79⋊C4—81no—
164C22⋊Dic79—24no—
165C22⋊Dic79—81no—
166C4×Dic79—42no—
167C4×Dic79—81no—
168C24×C79—21no—
169C24×C79C2×D411yescharacteristic
170C2×D4×C79—24no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79—21no—
174C2×D4×C79—21no—
175C2×D4×C79—21no—
176C2×D4×C79—81no—
177C2×D4×C79—81no—
178C2×D4×C79#4112yes—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79—21no—
187C22×C4×C79—21no—
188C22×C4×C79—42no—
189C22×C4×C79—81no—
190C22×C4×C79—81no—
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79C2×D411yescharacteristic
193C22×C4×C79C2×D411yescharacteristic
194C22×C4×C79C2×D411yescharacteristic
195C22×C4×C79C2×D411yescharacteristic
196C22×C4×C79C2411yescharacteristic
197C4⋊C4×C79—22no—
198C4⋊C4×C79—22no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79—21no—
204C4⋊C4×C79—81no—
205C4⋊C4×C79#4112yes—
206C22⋊C4×C79—24no—
207C22⋊C4×C79—22no—
208C22⋊C4×C79—22no—
209C22⋊C4×C79—42no—
210C22⋊C4×C79—81no—
211C22⋊C4×C79—81no—
212C22⋊C4×C79—81no—
213C42×C79—21no—
214C42×C79—21no—
215C42×C79—81no—
216C42×C79C2×D412yes—
217C42×C79#4112yes—
218C23×C79—21no—
219C23×C79—21no—
220C23×C79—22no—
221C23×C79—21no—
222C23×C79—21no—
223C23×C79—21no—
224C23×C79—21no—
225C23×C79—21no—
226C23×C79—41no—
227C23×C79—41no—
228C23×C79—41no—
229C23×C79—41no—
230C23×C79—41no—
231C23×C79—41no—
232C23×C79C22×D411yescharacteristic
233D4×C79—22no—
234D4×C79—22no—
235D4×C79—41no—
236D4×C79—41no—
237D4×C79—84no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—22no—
248C2×C4×C79—21no—
249C2×C4×C79—21no—
250C2×C4×C79—21no—
251C2×C4×C79—21no—
252C2×C4×C79—41no—
253C2×C4×C79—41no—
254C2×C4×C79—41no—
255C2×C4×C79—41no—
256C2×C4×C79—41no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79—41no—
261C2×C4×C79—41no—
262C2×C4×C79—41no—
263C2×C4×C79—41no—
264C2×C4×C79—84no—
265C2×C4×C79#12112yes—
266C2×C4×C79#12112yes—
267C2×C4×C79C22×D411yescharacteristic
268C2×C4×C79C2×#4111yescharacteristic
269C2×C4×C79C2×#4111yescharacteristic
270C2×C4×C79C2×#4111yescharacteristic
271C2×C4×C79C2×#4111yescharacteristic
272C2×C4×C79C2×#4111yescharacteristic
273C2×Dic79—22no—
274C2×Dic79—22no—
275C2×Dic79—41no—
276C2×Dic79—41no—
277C2×Dic79—41no—
278C2×Dic79—41no—
279C2×Dic79—41no—
280C2×Dic79—41no—
281C2×Dic79—41no—
282C2×Dic79—41no—
283C2×Dic79—41no—
284C2×Dic79—41no—
285Dic158—22no—
286Dic158—22no—
287Dic158—41no—
288Dic158—41no—
289C79⋊C8—22no—
290C79⋊C8—22no—
291C79⋊C8—41no—
292C79⋊C8—41no—
293C22×C79—41no—
294C22×C79—42no—
295C22×C79—41no—
296C22×C79—42no—
297C22×C79—41no—
298C22×C79—41no—
299C22×C79—41no—
300C22×C79—42no—
301C22×C79—41no—
302C22×C79—41no—
303C22×C79—81no—
304C22×C79—81no—
305C22×C79—81no—
306C22×C79—81no—
307C22×C79C2×#12111yescharacteristic
308C22×C79#52511yescharacteristic
309C22×C79#52511yescharacteristic
310C22×C79#57611yescharacteristic
311C22×C79#57711yescharacteristic
312C22×C79#57711yescharacteristic
313C22×C79#57811yescharacteristic
314C4×C79—22no—
315C4×C79—22no—
316C4×C79—42no—
317C4×C79—41no—
318C4×C79—41no—
319C4×C79—41no—
320C4×C79—81no—
321C4×C79—81no—
322C4×C79#52511yescharacteristic
323C4×C79#52911yescharacteristic
324C4×C79#52911yescharacteristic
325C4×C79#53211yescharacteristic
326Dic79—42no—
327Dic79—42no—
328Dic79—81no—
329Dic79—81no—
330#23715—791nomaximal, Sylow
331C2×C79—81no—
332C2×C79—82no—
333C2×C79#239211yescharacteristic
334C2×C79#306211yescharacteristic
335C2×C79#306311yescharacteristic
336C2×C79#311811yescharacteristic
337C2×C79#312011yescharacteristic
338C2×C79#312411yescharacteristic
339C2×C79#312611yescharacteristic
340C2×#440—791no—
341C2×#433—791no—
342#2324D7911yescharacteristic
343#2323—791no—
344#2086—791no—
345#2076—791no—
346#2068—791no—
347#2065—791no—
348#2055—791no—
349#2035—791no—
350#2033—791no—
351#2000—791no—
352#1776—791no—
353C4×#103—791no—
354C4×#102—791no—
355C79#2371511yescharacteristic, complemented, Sylow
356C22×SD16—791no—
357C2×C22⋊Q8—791no—
358C2×#121C2×D7911yescharacteristic
359C2×C4×Q8—791no—
360C2×C4×D4C2×D7911yescharacteristic
361C2×#117C2×D7911yescharacteristic
362#440—1584no—
363#433—1584no—
364C2×#107—791no—
365C2×#107—791no—
366C2×#106—791no—
367C2×C4⋊C8—791no—
368C2×C4⋊C8—791no—
369C2×#103—791no—
370C2×#103—791no—
371C2×#103—791no—
372C2×#102—791no—
373C2×#102—791no—
374C2×#102—791no—
375C2×C22⋊C8—791no—
376C2×C22⋊C8—791no—
377C2×#97—791no—
378C2×C4×C8—791no—
379#391—791no—
380#390—21no—
381#390C2×D7911yescharacteristic
382#389—791no—
383#389—1581no—
384#389C2×D7911yescharacteristic
385#388—21no—
386#388—1581no—
387#388—1581no—
388#387—1581no—
389#386—21no—
390#385—21no—
391#380—791no—
392#380C2×D7911yescharacteristic
393C4×C4⋊C4—791no—
394C4×C4⋊C4C2×D7911yescharacteristic
395#340—791no—
396#340—791no—
397#340—791no—
398#340—791no—
399#340—791no—
400#340—791no—
401C22×Q8—791no—
402C22×D4—21no—
403C22×D4C22×D7911yescharacteristic
404C23×C4—21no—
405C23×C4C22×D7911yescharacteristic
406C2×SD16—1582no—
407C2×SD16—1582no—
408C2×SD16—1582no—
409C22×C8—791no—
410C22×C8—791no—
411C22×C8—791no—
412C22×C8—791no—
413C22⋊Q8—1582no—
414C22⋊Q8—1582no—
415#121—22no—
416#121—22no—
417C4×Q8—1582no—
418C4×Q8—1582no—
419C4×D4—22no—
420C4×D4—22no—
421#117—22no—
422#117—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—791no—
425C2×C4⋊C4—791no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—1581no—
428C2×C4⋊C4—1581no—
429C2×C4⋊C4—1581no—
430C2×C4⋊C4C22⋊D7911yescharacteristic
431C2×C4⋊C4C22⋊D7911yescharacteristic
432C2×C4⋊C4C22×D7911yescharacteristic
433C2×C4⋊C4C22×D7911yescharacteristic
434C2×C4⋊C4C22×D7911yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1581no—
441C2×C22⋊C4—1581no—
442C2×C22⋊C4C22⋊D7911yescharacteristic
443C2×C22⋊C4C22⋊D7911yescharacteristic
444C2×C42—21no—
445C2×C42—1581no—
446C2×C42—1581no—
447C2×C42C22×D7911yescharacteristic
448C2×C42C22×D7911yescharacteristic
449#107—1582no—
450#107—1582no—
451#106—1582no—
452C4⋊C8—1582no—
453C4⋊C8—1582no—
454#103—1582no—
455#103—1582no—
456#103—1582no—
457#102—1582no—
458#102—1582no—
459#102—1582no—
460C22⋊C8—1582no—
461C22⋊C8—1582no—
462#97—794no—
463C4×C8—1582no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1581no—
468#95—1581no—
469#95—1581no—
470#95—1581no—
471#95—1581no—
472C24—21no—
473C24C2×C22⋊D7911yescharacteristic
474C2×Q8—792no—
475C2×Q8—1581no—
476C2×Q8—1581no—
477C2×Q8—1581no—
478C2×Q8—1581no—
479C2×Q8—1581no—
480C2×D4—24no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4#1192962412yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—1581no—
502C22×C4—1581no—
503C22×C4—1581no—
504C22×C4—1581no—
505C22×C4C2×C22⋊D7911yescharacteristic
506C22×C4C2×C22⋊D7911yescharacteristic
507C22×C4C2×C22⋊D7911yescharacteristic
508C22×C4C2×C22⋊D7911yescharacteristic
509C22×C4C2×C22⋊D7911yescharacteristic
510C22×C4C23×D7911yescharacteristic
511SD16—1588no—
512C2×C8—792no—
513C2×C8—792no—
514C2×C8—1581no—
515C2×C8—1581no—
516C2×C8—1581no—
517C2×C8—1581no—
518C2×C8—1581no—
519C2×C8—1581no—
520C2×C8—1581no—
521C2×C8—1581no—
522C2×C8—1581no—
523C2×C8—1581no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—81no—
532C4⋊C4—3162no—
533C4⋊C4—792no—
534C4⋊C4—6321no—
535C4⋊C4—1584no—
536C4⋊C4—1581no—
537C4⋊C4—1581no—
538C4⋊C4—1581no—
539C4⋊C4—1581no—
540C4⋊C4—1581no—
541C4⋊C4#1192962412yes—
542C22⋊C4—24no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—42no—
546C22⋊C4—81no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—6321no—
550C22⋊C4—1584no—
551C42—21no—
552C42—21no—
553C42—81no—
554C42—3162no—
555C42—6321no—
556C42D4×D7912yes—
557C42#1192962312yes—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C22×C22⋊D7911yescharacteristic
573Q8—3161no—
574Q8—3161no—
575Q8—1582no—
576Q8—1582no—
577D4—22no—
578D4—22no—
579D4—41no—
580D4—41no—
581D4—84no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—3161no—
610C2×C4—3161no—
611C2×C4—3161no—
612C2×C4—3161no—
613C2×C4—3161no—
614C2×C4—3161no—
615C2×C4—3161no—
616C2×C4—3161no—
617C2×C4—3161no—
618C2×C4—3161no—
619C2×C4—1582no—
620C2×C4—1582no—
621C2×C4#1457913312yes—
622C2×C4#1457913312yes—
623C2×C4C2×D4×D7911yescharacteristic
624C2×C4#41×D7911yescharacteristic
625C2×C4#41×D7911yescharacteristic
626C2×C4C2×#1192962311yescharacteristic
627C2×C4C2×#1192962411yescharacteristic
628C2×C4C2×#1192962411yescharacteristic
629C8—3161no—
630C8—3161no—
631C8—1582no—
632C8—1582no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#1457913311yescharacteristic
648C22#1909907011yescharacteristic
649C22#1909907411yescharacteristic
650C22#1909936611yescharacteristic
651C22#1909937411yescharacteristic
652C22#1909937411yescharacteristic
653C22#1909937811yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4—81no—
661C4—81no—
662C4—3162no—
663C4—3162no—
664C4—6321no—
665C4—6321no—
666C4#1909907211yescharacteristic
667C4#1909909311yescharacteristic
668C4#1909909511yescharacteristic
669C4#1909911011yescharacteristic
670C2—81no—
671C2—82no—
672C2#2376664211yescharacteristic
673C2#2377307611yescharacteristic
674C2#2377309311yescharacteristic
675C2#2377358411yescharacteristic
676C2#2377361411yescharacteristic
677C2#2377365611yescharacteristic
678C2#2377368611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2324C2
socleC23×C79C22×D4
prime radicalC2×#121×C79C22
hypercenter#2324D79
hypocenterC79#23715
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C2×#19098663C2semidirect
2. C2×#19098624C2semidirect
3. #23766082C2semidirect
4. #23764891C2semidirect
5. #23764831C2semidirect
6. #23764792C2semidirect
7. #23764774C2semidirect
8. #23764717C2semidirect
9. #23764596C2semidirect
12. #23763298C2semidirect
13. C4×#14579084C2semidirect
14. C4×#14579082C2semidirect
96. #14579072D4semidirect
355. C79#23715semidirect

Maximal subgroups

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

subgroupindex
C2×#190986632
C2×#190986242
#237660822
#237648912
#237648312
#237647922
#237647742
#237647172
#237645962
#237645792
#237644032
#237632982
C4×#145790842
C4×#145790822
C79×#23242
#2371579