Finite Group Databasebeta

← #34310069

Subgroups — #34310069

Order 20224.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098841C211yesmaximal, characteristic
2#23766459C211yesmaximal, characteristic, complemented
3D4×Dic79⋊C4C211yesmaximal, characteristic
4#23764359C211yesmaximal, characteristic, complemented
5#23764157C211yesmaximal, characteristic, complemented
6#23764133C211yesmaximal, characteristic, complemented
7#23764132C211yesmaximal, characteristic, complemented
8#23763971C211yesmaximal, characteristic, complemented
9#23763970C211yesmaximal, characteristic, complemented
10#23763649C211yesmaximal, characteristic
11#23763647C211yesmaximal, characteristic
12C8×Dic79⋊C4C211yesmaximal, characteristic, complemented
13#102×Dic79C211yesmaximal, characteristic
14#102×Dic79C211yesmaximal, characteristic
15C4×#102×C79C211yesmaximal, characteristic
16C2×#14579160C2211yescharacteristic
17C2×C4×Dic158—21no—
18C2×D4×Dic79C2211yescharacteristic
19C2×D4×Dic79C2211yescharacteristic
20C22×Dic79⋊C4—21no—
21#19098841C418yescomplemented
22C2×#14579095C2211yescharacteristic
23C2×#14579095C2211yescharacteristic
24C2×#14579094C2211yescharacteristic
25C2×#14579094C2211yescharacteristic
26C2×Dic79⋊C8C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30C2×C8×Dic79C2211yescharacteristic
31#19098467—21no—
32#19098462C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098461C2211yescharacteristic
35#19098459C2211yescharacteristic
36#19098450—21no—
37#19098447—21no—
38#19098436—21no—
39#19098435—21no—
40#19098432C2211yescharacteristic
41#19098421—21no—
42C4⋊C4×Dic79C2211yescharacteristic
43C4⋊C4×Dic79C2211yescharacteristic
44C4×Dic79⋊C4C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098318C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54C2×#102×C79C2211yescharacteristic
55C2×#102×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158—21no—
61C23×Dic79—21no—
62C23×Dic79—21no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579160—22no—
66#14579160C2×C414yescomplemented
67C4×Dic158—44no—
68D4×Dic79—22no—
69D4×Dic79—22no—
70D4×Dic79—41no—
71D4×Dic79C2×C414yes—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—44no—
84C2×Dic79⋊C4D411yescharacteristic
85C2×Dic79⋊C4D411yescharacteristic
86C2×C22⋊Dic79—21no—
87C2×C22⋊Dic79—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79—21no—
90C2×C4×Dic79C2311yescharacteristic
91C2×C4×Dic79C2311yescharacteristic
92#14579095—41no—
93#14579095C2×C414yescomplemented
94#14579094—41no—
95#14579094C2×C414yescomplemented
96Dic79⋊C8C2×C414yescomplemented
97#14579082—22no—
98#14579082C2×C414yes—
99C8×Dic79—22no—
100C8×Dic79—41no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95D411yescharacteristic
108C79⋊#95D411yescharacteristic
109C22×D4×C79C2311yescharacteristic
110C23×C4×C79—21no—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C4×D4×C79—22no—
114C4×D4×C79—41no—
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C22⋊C4×C79—21no—
119C2×C42×C79—21no—
120C2×C42×C79C2311yescharacteristic
121#102×C79C2×C414yes—
122#102×C79Q814yescomplemented
123C4×C8×C79—41no—
124#95×C79—21no—
125C2×Dic158—24no—
126C2×Dic158—42no—
127C2×Dic158—42no—
128C22×Dic79—24no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—42no—
134C22×Dic79—42no—
135C22×Dic79—42no—
136C22×Dic79—42no—
137C22×Dic79—81no—
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C22×Dic79C2×D411yescharacteristic
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8C22×C412yes—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—42no—
155C4⋊Dic79C22×C412yes—
156C4⋊Dic79C22×C412yes—
157Dic79⋊C4—24no—
158Dic79⋊C4—22no—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161Dic79⋊C4—42no—
162Dic79⋊C4—42no—
163Dic79⋊C4—81no—
164Dic79⋊C4D814yes—
165C22⋊Dic79—24no—
166C22⋊Dic79—42no—
167C4×Dic79—21no—
168C4×Dic79—21no—
169C4×Dic79—42no—
170C4×Dic79—42no—
171C4×Dic79C22×C412yes—
172C4×Dic79#4112yes—
173C24×C79—21no—
174C2×D4×C79—21no—
175C2×D4×C79—21no—
176C2×D4×C79—21no—
177C2×D4×C79C22×C412yes—
178C2×D4×C79C2×Q812yes—
179C2×D4×C79#4112yes—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—42no—
184C22×C4×C79—42no—
185C22×C4×C79—81no—
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2411yescharacteristic
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79C22×C412yes—
194C2×C8×C79C2×Q812yes—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79C22×C412yes—
200C4⋊C4×C79C2×Q812yes—
201C22⋊C4×C79—42no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—42no—
205C23×C79—22no—
206C23×C79—22no—
207C23×C79—22no—
208C23×C79—41no—
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—41no—
212C23×C79C22×D411yescharacteristic
213D4×C79—22no—
214D4×C79—22no—
215D4×C79—22no—
216D4×C79C4×Q814yescomplemented
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—22no—
220C2×C4×C79—21no—
221C2×C4×C79—22no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—84no—
232C2×C4×C79C2×D812yes—
233C2×C4×C79C23×C411yescharacteristic
234C2×C4×C79C22×Q811yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79H(5,𝔽2)11yescharacteristic
238C2×C4×C792−1+411yescharacteristic
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—48no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—84no—
260C2×Dic79C4×D412yes—
261C2×Dic79C4×D412yes—
262C2×Dic79C2×D812yes—
263C2×Dic79C2×D812yes—
264C2×Dic79#13512yes—
265Dic158—48no—
266C8×C79—22no—
267C8×C79—22no—
268C79⋊C8—22no—
269C79⋊C8—41no—
270C22×C79—24no—
271C22×C79—42no—
272C22×C79—42no—
273C22×C79—42no—
274C22×C79—42no—
275C22×C79—42no—
276C22×C79—42no—
277C22×C79C2×C4×D411yescharacteristic
278C22×C79#54611yescharacteristic
279C22×C79D4×Q811yescharacteristic
280C22×C79C22×D811yescharacteristic
281C22×C79C2×#13511yescharacteristic
282C22×C79#57711yescharacteristic
283C22×C79#57811yescharacteristic
284C4×C79—22no—
285C4×C79—42no—
286C4×C79—42no—
287C4×C79—41no—
288C4×C79—81no—
289C4×C79C2×C4×Q811yescharacteristic
290C4×C79#52011yescharacteristic
291C4×C79#55211yescharacteristic
292C4×C79#55411yescharacteristic
293Dic79—22no—
294Dic79—22no—
295Dic79—42no—
296Dic79—42no—
297Dic79—41no—
298Dic79—81no—
299Dic79—81no—
300Dic79C4×D814yes—
301#22962—791nomaximal, Sylow
302C2×C79—44no—
303C2×C79#236011yescharacteristic
304C2×C79C2×C4×D811yescharacteristic
305C2×C79#294711yescharacteristic
306C2×C79#329411yescharacteristic
307C2×C79#330211yescharacteristic
308C2×C79Q8×D811yescharacteristic
309C2×C79#338111yescharacteristic
310C2×#474—791no—
311#2365—791no—
312D4×C4⋊C4—791no—
313#1990—791no—
314#1947—791no—
315#1942—791no—
316#1942—791no—
317#1918—791no—
318#1918—791no—
319#1862—791no—
320#1862—791no—
321C8×C4⋊C4—791no—
322C4×#102—791no—
323C4×#102—791no—
324C4×#102D7911yescharacteristic
325C79#2296211yescharacteristic, complemented, Sylow
326C2×C4⋊Q8—791no—
327C2×C4×Q8—1581no—
328C2×C4×D4—791no—
329C2×C4×D4—791no—
330C2×C4×D4C2×D7911yescharacteristic
331C22×C4⋊C4—1581no—
332#474—798no—
333C2×#107—791no—
334C2×#107—791no—
335C2×#107—791no—
336C2×#107—791no—
337C2×C4⋊C8—791no—
338C2×#102—791no—
339C2×#102—791no—
340C2×#102C2×D7911yescharacteristic
341C2×#102C2×D7911yescharacteristic
342C2×C4×C8—791no—
343C2×C4×C8—791no—
344C2×C4×C8C2×D7911yescharacteristic
345#391—1581no—
346#389—791no—
347#389—791no—
348#389—791no—
349#389—791no—
350#387—1581no—
351#387—1581no—
352#385—1581no—
353#385—1581no—
354#384—791no—
355#380—1581no—
356C4×C4⋊C4—791no—
357C4×C4⋊C4—791no—
358C4×C4⋊C4—791no—
359C4×C4⋊C4C2×D7911yescharacteristic
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340C2×D7911yescharacteristic
367#340C2×D7911yescharacteristic
368#336—791no—
369#336—791no—
370C22×Q8—1581no—
371C22×D4C22×D7911yescharacteristic
372C23×C4—21no—
373C23×C4—1581no—
374C23×C4—1581no—
375C22×C8—791no—
376C22×C8—791no—
377C22×C8C22×D7911yescharacteristic
378C22×C8C22×D7911yescharacteristic
379C4⋊Q8—794no—
380C4⋊Q8—1582no—
381C4×Q8—3164no—
382C4×D4—22no—
383C4×D4—41no—
384C4×D4—3161no—
385C4×D4—794no—
386C4×D4—1582no—
387C4×D4—1582no—
388C2×C4⋊C4—3164no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4—1581no—
402C2×C4⋊C4C22×D7911yescharacteristic
403C2×C4⋊C4C22×D7911yescharacteristic
404C2×C4⋊C4C22×D7911yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1581no—
407C2×C22⋊C4—1581no—
408C2×C42—21no—
409C2×C42—791no—
410C2×C42—791no—
411C2×C42—1581no—
412C2×C42—1581no—
413C2×C42C22×D7911yescharacteristic
414#107—3161no—
415#107—3161no—
416#107—794no—
417#107—794no—
418C4⋊C8—794no—
419#102—794no—
420#102—1582no—
421#102Dic15814yescomplemented
422#102C4×D7914yes—
423C4×C8—41no—
424C4×C8—3161no—
425C4×C8—1582no—
426#95—21no—
427#95—791no—
428#95—791no—
429#95—1581no—
430#95—1581no—
431#95—1581no—
432#95—1581no—
433#95—1581no—
434#95—1581no—
435C24—21no—
436C2×Q8—3162no—
437C2×Q8—3162no—
438C2×Q8—1584no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D7912yes—
443C2×D4C2×Dic15812yes—
444C2×D4#1192962412yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—3162no—
452C22×C4—3162no—
453C22×C4—3162no—
454C22×C4—3162no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—6321no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1584no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4D4×D7911yescharacteristic
466C22×C4D4×D7911yescharacteristic
467C22×C4C23×D7911yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—792no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8C2×C4×D7912yes—
479C2×C8C2×Dic15812yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—3162no—
485C4⋊C4—3162no—
486C4⋊C4—3162no—
487C4⋊C4—3162no—
488C4⋊C4—3162no—
489C4⋊C4—794no—
490C4⋊C4—792no—
491C4⋊C4—792no—
492C4⋊C4—6321no—
493C4⋊C4—1582no—
494C4⋊C4—1584no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4C2×C4×D7912yes—
502C4⋊C4C2×Dic15812yes—
503C22⋊C4—42no—
504C22⋊C4—3162no—
505C22⋊C4—1584no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—3162no—
510C42—3162no—
511C42—792no—
512C42—792no—
513C42—1581no—
514C42—1581no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D7911yescharacteristic
523Q8—3168no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4C4×Dic15814yescomplemented
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3168no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—792no—
555C2×C4—792no—
556C2×C4—792no—
557C2×C4—792no—
558C2×C4—792no—
559C2×C4—6324no—
560C2×C4—1581no—
561C2×C4—1582no—
562C2×C4—1582no—
563C2×C4—1581no—
564C2×C4—1581no—
565C2×C4—1581no—
566C2×C4—1581no—
567C2×C4—1581no—
568C2×C4—1581no—
569C2×C4D8×D7912yes—
570C2×C4C22×C4×D7911yescharacteristic
571C2×C4C22×Dic15811yescharacteristic
572C2×C4#41×D7911yescharacteristic
573C2×C4C2×#1192962411yescharacteristic
574C2×C4#1457920611yescharacteristic
575C2×C4#1457920811yescharacteristic
576C8—22no—
577C8—22no—
578C8—3161no—
579C8—1582no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D7911yescharacteristic
588C22#1909920311yescharacteristic
589C22D4×Dic15811yescharacteristic
590C22C2×D8×D7911yescharacteristic
591C22#135×D7911yescharacteristic
592C22#1909936911yescharacteristic
593C22#1909937711yescharacteristic
594C4—22no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—3161no—
600C4—3162no—
601C4—3162no—
602C4—794no—
603C4—6321no—
604C4—6321no—
605C4—1582no—
606C4—1582no—
607C4C2×C4×Dic15811yescharacteristic
608C4#1909904211yescharacteristic
609C4#1909924611yescharacteristic
610C4#1909927211yescharacteristic
611C2—44no—
612C2#2376640611yescharacteristic
613C2C4×D8×D7911yescharacteristic
614C2#2377223611yescharacteristic
615C2#2377520911yescharacteristic
616C2#2377531311yescharacteristic
617C2D8×Dic15811yescharacteristic
618C2#2377627611yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC4×#102×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenterC4×#102D79
hypocenterC79#22962
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #23766459C2semidirect
4. #23764359C2semidirect
5. #23764157C2semidirect
6. #23764133C2semidirect
7. #23764132C2semidirect
8. #23763971C2semidirect
9. #23763970C2semidirect
12. C8×Dic79⋊C4C2semidirect
21. #19098841C4semidirect
66. #14579160C2×C4semidirect
93. #14579095C2×C4semidirect
95. #14579094C2×C4semidirect
96. Dic79⋊C8C2×C4semidirect
122. #102×C79Q8semidirect
216. D4×C79C4×Q8semidirect
325. C79#22962semidirect
421. #102Dic158semidirect
527. D4C4×Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190988412
#237664592
D4×Dic79⋊C42
#237643592
#237641572
#237641332
#237641322
#237639712
#237639702
#237636492
#237636472
C8×Dic79⋊C42
#102×Dic792
C4×#102×C792
#2296279