Finite Group Databasebeta

← #34310283

Subgroups — #34310283

Order 20224.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098861C211yesmaximal, characteristic
2#23766492C211yesmaximal, characteristic, complemented
3D4×Dic79⋊C4C211yesmaximal, characteristic
4#23764367C211yesmaximal, characteristic, complemented
5#23764162C211yesmaximal, characteristic, complemented
6#23764138C211yesmaximal, characteristic, complemented
7#23764127C211yesmaximal, characteristic, complemented
8#23763985C211yesmaximal, characteristic, complemented
9#23763970C211yesmaximal, characteristic, complemented
10#23763663C211yesmaximal, characteristic
11#23763661C211yesmaximal, characteristic
12#23763276C211yesmaximal, characteristic, complemented
13#23763227C211yesmaximal, characteristic
14#102×Dic79C211yesmaximal, characteristic
15C79×#1762C211yesmaximal, characteristic
16C2×#14579152C2211yescharacteristic
17C2×D4×Dic79C2211yescharacteristic
18C2×D4×Dic79C2211yescharacteristic
19C22×Dic79⋊C4—21no—
20#19098861C418yescomplemented
21C2×#14579095C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×Dic79⋊C8C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579071C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098462C2211yescharacteristic
31#19098461—21no—
32#19098461C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098459C2211yescharacteristic
35#19098450—21no—
36#19098446—21no—
37#19098436—21no—
38#19098435—21no—
39#19098432C2211yescharacteristic
40#19098421—21no—
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4×Dic79⋊C4—21no—
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×#97×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C23×Dic79—21no—
61C23×Dic79—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579152—22no—
65#14579152C2×C414yescomplemented
66D4×Dic79—22no—
67D4×Dic79—22no—
68D4×Dic79—41no—
69D4×Dic79C2×C414yes—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—44no—
83C2×Dic79⋊C4D411yescharacteristic
84C2×Dic79⋊C4D411yescharacteristic
85C2×C22⋊Dic79—21no—
86C2×C22⋊Dic79—21no—
87C2×C4×Dic79—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79C2311yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91#14579095C2×C414yescomplemented
92#14579093—41no—
93#14579092—41no—
94#14579092C2×C414yescomplemented
95Dic79⋊C8C2×C414yescomplemented
96#14579082—22no—
97#14579082C2×C414yes—
98#14579071—22no—
99C8×Dic79—41no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95D411yescharacteristic
107C79⋊#95D411yescharacteristic
108C22×D4×C79C2311yescharacteristic
109C23×C4×C79—21no—
110C22×C8×C79C2311yescharacteristic
111C22×C8×C79C2311yescharacteristic
112C4×D4×C79—22no—
113C4×D4×C79—41no—
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C22⋊C4×C79—21no—
118C2×C42×C79—21no—
119C2×C42×C79C2311yescharacteristic
120#102×C79C2×C414yes—
121#102×C79Q814yes—
122#97×C79—41no—
123#95×C79—21no—
124C22×Dic79—24no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—42no—
130C22×Dic79—42no—
131C22×Dic79—42no—
132C22×Dic79—42no—
133C22×Dic79—81no—
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8C22×C412yes—
144C4⋊Dic79—24no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—81no—
152C4⋊Dic79C22×C412yes—
153C4⋊Dic79C22×C412yes—
154Dic79⋊C4—24no—
155Dic79⋊C4—22no—
156Dic79⋊C4—41no—
157Dic79⋊C4—42no—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161Dic79⋊C4—81no—
162C22⋊Dic79—24no—
163C22⋊Dic79—42no—
164C4×Dic79—21no—
165C4×Dic79—21no—
166C4×Dic79—42no—
167C4×Dic79—81no—
168C4×Dic79C22×C412yes—
169C4×Dic79#4112yes—
170C24×C79—21no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79—21no—
174C2×D4×C79C22×C412yes—
175C2×D4×C79C2×Q812yes—
176C2×D4×C79#4112yes—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—42no—
181C22×C4×C79—42no—
182C22×C4×C79—81no—
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2411yescharacteristic
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79C22×C412yes—
191C2×C8×C79C2×Q812yes—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79C22×C412yes—
197C4⋊C4×C79C2×Q812yes—
198C22⋊C4×C79—42no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—81no—
202C23×C79—22no—
203C23×C79—22no—
204C23×C79—22no—
205C23×C79—41no—
206C23×C79—41no—
207C23×C79—41no—
208C23×C79—41no—
209C23×C79C22×D411yescharacteristic
210D4×C79—22no—
211D4×C79—22no—
212D4×C79—22no—
213D4×C79C4×Q814yes—
214C2×C4×C79—22no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—22no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—84no—
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C22×Q811yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79H(5,𝔽2)11yescharacteristic
235C2×C4×C792−1+411yescharacteristic
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—48no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—84no—
259C2×Dic79C4×D412yes—
260C2×Dic79C4×D412yes—
261C2×Dic79AGL(1,ℤ/8)12yes—
262C8×C79—22no—
263C8×C79—22no—
264C79⋊C8—22no—
265C79⋊C8—41no—
266C22×C79—24no—
267C22×C79—42no—
268C22×C79—42no—
269C22×C79—42no—
270C22×C79—42no—
271C22×C79—42no—
272C22×C79—42no—
273C22×C79C2×C4×D411yescharacteristic
274C22×C79#54611yescharacteristic
275C22×C79D4×Q811yescharacteristic
276C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C79#57511yescharacteristic
278C22×C79#57711yescharacteristic
279C22×C79#57711yescharacteristic
280C4×C79—42no—
281C4×C79—42no—
282C4×C79—41no—
283C4×C79—41no—
284C4×C79—81no—
285C4×C79C2×C4×Q811yescharacteristic
286C4×C79#52011yescharacteristic
287C4×C79#55211yescharacteristic
288C4×C79#55411yescharacteristic
289Dic79—22no—
290Dic79—22no—
291Dic79—22no—
292Dic79—42no—
293Dic79—42no—
294Dic79—41no—
295Dic79—81no—
296Dic79—81no—
297#22978—791nomaximal, Sylow
298C2×C79—44no—
299C2×C79#236011yescharacteristic
300C2×C79C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C79#295211yescharacteristic
302C2×C79#330711yescharacteristic
303C2×C79#331011yescharacteristic
304C2×C79#338411yescharacteristic
305C2×C79#338811yescharacteristic
306C2×#478—791no—
307#2370—791no—
308D4×C4⋊C4—791no—
309#1992—791no—
310#1948—791no—
311#1943—791no—
312#1941—791no—
313#1920—791no—
314#1918—791no—
315#1864—791no—
316#1864—791no—
317#1771—791no—
318#1762—791no—
319#1762D7911yescharacteristic
320C4×#102—791no—
321C79#2297811yescharacteristic, complemented, Sylow
322C2×#125—791no—
323C2×C4×D4—791no—
324C2×C4×D4—791no—
325C2×C4×D4C2×D7911yescharacteristic
326C22×C4⋊C4—1581no—
327#478—798no—
328C2×#107—791no—
329C2×#106—791no—
330C2×#106—791no—
331C2×#106—791no—
332C2×C4⋊C8—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102C2×D7911yescharacteristic
336C2×#102C2×D7911yescharacteristic
337C2×#97—791no—
338C2×#97C2×D7911yescharacteristic
339C2×C4×C8—791no—
340#389—791no—
341#389—791no—
342#389—791no—
343#389—791no—
344#389—1581no—
345#387—1581no—
346#387—1581no—
347#385—1581no—
348#385—1581no—
349#384—791no—
350#380—1581no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—791no—
354C4×C4⋊C4—1581no—
355C4×C4⋊C4C2×D7911yescharacteristic
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340C2×D7911yescharacteristic
363#340C2×D7911yescharacteristic
364#336—791no—
365#336—791no—
366C22×D4C22×D7911yescharacteristic
367C23×C4—21no—
368C23×C4—1581no—
369C23×C4—1581no—
370C22×C8—791no—
371C22×C8—791no—
372C22×C8C22×D7911yescharacteristic
373C22×C8C22×D7911yescharacteristic
374#125—794no—
375#125—1582no—
376C4×D4—22no—
377C4×D4—41no—
378C4×D4—3161no—
379C4×D4—794no—
380C4×D4—1582no—
381C4×D4—1582no—
382C2×C4⋊C4—3164no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C4⋊C4C22×D7911yescharacteristic
399C2×C4⋊C4C22×D7911yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C42—21no—
404C2×C42—791no—
405C2×C42—791no—
406C2×C42—1581no—
407C2×C42—1581no—
408C2×C42C22×D7911yescharacteristic
409#107—794no—
410#106—3161no—
411#106—3161no—
412#106—794no—
413C4⋊C8—794no—
414#102—794no—
415#102—1582no—
416#102Dic15814yes—
417#102C4×D7914yes—
418#97—41no—
419#97—1582no—
420C4×C8—3161no—
421#95—21no—
422#95—791no—
423#95—791no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429#95—1581no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D7912yes—
435C2×D4C2×Dic15812yes—
436C2×D4#1192962412yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—81no—
443C22×C4—3162no—
444C22×C4—3162no—
445C22×C4—3162no—
446C22×C4—3162no—
447C22×C4—791no—
448C22×C4—791no—
449C22×C4—791no—
450C22×C4—791no—
451C22×C4—6321no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1584no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4D4×D7911yescharacteristic
458C22×C4D4×D7911yescharacteristic
459C22×C4C23×D7911yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—792no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8C2×C4×D7912yes—
471C2×C8C2×Dic15812yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—3162no—
477C4⋊C4—3162no—
478C4⋊C4—3161no—
479C4⋊C4—3162no—
480C4⋊C4—3162no—
481C4⋊C4—792no—
482C4⋊C4—792no—
483C4⋊C4—6321no—
484C4⋊C4—6321no—
485C4⋊C4—1582no—
486C4⋊C4—1584no—
487C4⋊C4—1584no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4C2×C4×D7912yes—
495C4⋊C4C2×Dic15812yes—
496C22⋊C4—42no—
497C22⋊C4—3162no—
498C22⋊C4—1584no—
499C42—21no—
500C42—21no—
501C42—81no—
502C42—3162no—
503C42—792no—
504C42—792no—
505C42—6321no—
506C42—1581no—
507C42—1581no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D7911yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic15814yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3168no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—792no—
548C2×C4—792no—
549C2×C4—792no—
550C2×C4—6324no—
551C2×C4—1581no—
552C2×C4—1582no—
553C2×C4—1582no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4C22×C4×D7911yescharacteristic
563C2×C4C22×Dic15811yescharacteristic
564C2×C4#41×D7911yescharacteristic
565C2×C4C2×#1192962411yescharacteristic
566C2×C4#1457920611yescharacteristic
567C2×C4#1457920811yescharacteristic
568C8—22no—
569C8—22no—
570C8—3161no—
571C8—1582no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D7911yescharacteristic
580C22#1909920311yescharacteristic
581C22D4×Dic15811yescharacteristic
582C22AGL(1,ℤ/8)×D7911yescharacteristic
583C22#1909936211yescharacteristic
584C22#1909937311yescharacteristic
585C22#1909937311yescharacteristic
586C4—42no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—3162no—
592C4—3162no—
593C4—3161no—
594C4—6321no—
595C4—6321no—
596C4—1582no—
597C4—1582no—
598C4—1582no—
599C4C2×C4×Dic15811yescharacteristic
600C4#1909904211yescharacteristic
601C4#1909924611yescharacteristic
602C4#1909927211yescharacteristic
603C2—44no—
604C2#2376640611yescharacteristic
605C2#2377220511yescharacteristic
606C2#2377228211yescharacteristic
607C2#2377537711yescharacteristic
608C2#2377542211yescharacteristic
609C2#2377630011yescharacteristic
610C2#2377634611yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1762C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1762D79
hypocenterC79#22978
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #23766492C2semidirect
4. #23764367C2semidirect
5. #23764162C2semidirect
6. #23764138C2semidirect
7. #23764127C2semidirect
8. #23763985C2semidirect
9. #23763970C2semidirect
12. #23763276C2semidirect
20. #19098861C4semidirect
65. #14579152C2×C4semidirect
91. #14579095C2×C4semidirect
94. #14579092C2×C4semidirect
95. Dic79⋊C8C2×C4semidirect
321. C79#22978semidirect

Maximal subgroups

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

subgroupindex
C2×#190988612
#237664922
D4×Dic79⋊C42
#237643672
#237641622
#237641382
#237641272
#237639852
#237639702
#237636632
#237636612
#237632762
#237632272
#102×Dic792
C79×#17622
#2297879