Finite Group Databasebeta

← #34351685

Subgroups — #34351685

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098870C211yesmaximal, characteristic, complemented
2C2×#19098661C211yesmaximal, characteristic
3#23767272C211yesmaximal, characteristic
4#23764924C211yesmaximal, characteristic
5#23764882C211yesmaximal, characteristic, complemented
6#23764771C211yesmaximal, characteristic
7#23764738C211yesmaximal, characteristic, complemented
8#23764140C211yesmaximal, characteristic, complemented
9#23763966C211yesmaximal, characteristic, complemented
10#23763755C211yesmaximal, characteristic
11#23763755C211yesmaximal, characteristic
12#23763668C211yesmaximal, characteristic, complemented
13#23763668C211yesmaximal, characteristic, complemented
14#23763660C211yesmaximal, characteristic
15C79×#2351C211yesmaximal, characteristic, complemented
16C22×#11929611C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19#19098870—24no—
20#19098661—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579085C2211yescharacteristic
28C2×#14579085C2211yescharacteristic
29C2×#14579085C2211yescharacteristic
30C2×#14579081C2211yescharacteristic
31C2×#14579081C2211yescharacteristic
32C2×#14579081C2211yescharacteristic
33C2×#14579072C2211yescharacteristic
34#19098498—21no—
35#19098487—21no—
36#19098465C2211yescharacteristic
37#19098465C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098434—21no—
41#19098422—21no—
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×C4×Q8×C79C2211yescharacteristic
51C2×C4×Q8×C79C2211yescharacteristic
52C2×C4×Q8×C79C2211yescharacteristic
53#389×C79—21no—
54#389×C79—21no—
55#389×C79C2211yescharacteristic
56#389×C79C2211yescharacteristic
57#382×C79—21no—
58#382×C79C2211yescharacteristic
59C4×C4⋊C4×C79—21no—
60C4×C4⋊C4×C79C2211yescharacteristic
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C2×#11929611—22no—
64C2×#11929611—22no—
65C2×#11929611Q814yes—
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70#14579153—22no—
71#14579153—41no—
72C4×D316—22no—
73C4×D316—41no—
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×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×(C2×D79)⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86#14579094—22no—
87#14579094—22no—
88#14579092—22no—
89#14579092—22no—
90#14579092—41no—
91C4⋊(C79⋊C8)—22no—
92#14579085—22no—
93#14579085—22no—
94#14579085D414yes—
95#14579081—22no—
96#14579081—22no—
97#14579081—41no—
98#14579072—22no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C22×Q8×C79C2311yescharacteristic
102C4×Q8×C79—22no—
103C4×Q8×C79—22no—
104C4×Q8×C79—22no—
105C4×Q8×C79—22no—
106C4×Q8×C79—41no—
107C4×Q8×C79—41no—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79—21no—
112C2×C4⋊C4×C79—21no—
113C2×C4⋊C4×C79D411yescharacteristic
114C2×C4⋊C4×C79D411yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C42×C79—21no—
119C2×C42×C79—21no—
120C2×C42×C79—21no—
121C2×C42×C79C2311yescharacteristic
122C2×C42×C79C2311yescharacteristic
123C2×C42×C79C2311yescharacteristic
124#95×C79—21no—
125#95×C79—21no—
126#95×C79—21no—
127#95×C79—21no—
128#95×C79D411yescharacteristic
129#95×C79D411yescharacteristic
130C23×D79—21no—
131C2×D316—21no—
132C2×D316—21no—
133C2×D316—21no—
134C2×D316—21no—
135C2×D316—21no—
136C2×D316C2×Q812yes—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C2×C4×D79—42no—
141C2×C4×D79—42no—
142C2×C4×D79—81no—
143#11929611—28no—
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⋊C8—21no—
154C2×C79⋊C8C2×D412yes—
155C2×C79⋊C8C2×Q812yes—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79C2×D412yes—
162Dic79⋊C4—42no—
163Dic79⋊C4—42no—
164Dic79⋊C4—81no—
165Dic79⋊C4—81no—
166(C2×D79)⋊C4—42no—
167(C2×D79)⋊C4—42no—
168(C2×D79)⋊C4—81no—
169(C2×D79)⋊C4—81no—
170(C2×D79)⋊C4—81no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79C2×D412yes—
175C2×Q8×C79C2×Q812yes—
176C2×Q8×C79#4112yes—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2411yescharacteristic
188C4⋊C4×C79—24no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—41no—
195C4⋊C4×C79—41no—
196C4⋊C4×C79—42no—
197C4⋊C4×C79—42no—
198C4⋊C4×C79—42no—
199C4⋊C4×C79—42no—
200C4⋊C4×C79#4112yes—
201C42×C79—24no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79—21no—
206C42×C79—21no—
207C42×C79—21no—
208C42×C79—42no—
209C42×C79—42no—
210C23×C79C22×D411yescharacteristic
211C22×D79—22no—
212C22×D79—41no—
213C22×D79—41no—
214C22×D79—41no—
215C22×D79—41no—
216C22×D79—41no—
217C22×D79—41no—
218Q8×C79—22no—
219Q8×C79—22no—
220Q8×C79—22no—
221Q8×C79C22⋊Q814yes—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—22no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
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—21no—
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×C79C22×D411yescharacteristic
253C2×C4×C79C22×Q811yescharacteristic
254C2×C4×C79C2×#4111yescharacteristic
255C2×C4×C79C2×#4111yescharacteristic
256C2×C4×C79H(5,𝔽2)11yescharacteristic
257C2×C4×C792−1+411yescharacteristic
258C2×Dic79—22no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267D316—22no—
268D316—22no—
269D316—41no—
270D316—41no—
271C4×D79—84no—
272C79⋊C8—22no—
273C79⋊C8—22no—
274C79⋊C8—41no—
275C79⋊C8—41no—
276C22×C79#52511yescharacteristic
277C22×C79D4211yescharacteristic
278C22×C79D4×Q811yescharacteristic
279C22×C79#57511yescharacteristic
280C22×C79#57511yescharacteristic
281C22×C79#57711yescharacteristic
282C22×C79#57711yescharacteristic
283C2×D79—42no—
284C2×D79—42no—
285C2×D79—42no—
286C2×D79—81no—
287C2×D79—81no—
288C2×D79—81no—
289C2×D79—81no—
290C4×C79—24no—
291C4×C79—41no—
292C4×C79—42no—
293C4×C79—41no—
294C4×C79—41no—
295C4×C79—41no—
296C4×C79—41no—
297C4×C79—41no—
298C4×C79—81no—
299C4×C79—81no—
300C4×C79C2×C22⋊Q811yescharacteristic
301C4×C79#53911yescharacteristic
302C4×C79#54711yescharacteristic
303C4×C79#55411yescharacteristic
304Dic79—42no—
305Dic79—81no—
306Dic79—81no—
307#26399—791nomaximal, Sylow
308C2×C79#244911yescharacteristic
309C2×C79#311111yescharacteristic
310C2×C79#312611yescharacteristic
311C2×C79#327111yescharacteristic
312C2×C79#327511yescharacteristic
313C2×C79#338811yescharacteristic
314C2×C79#338811yescharacteristic
315D79—82no—
316C2×#479—791no—
317C2×#440—791no—
318#2454—791no—
319#2351D7911yescharacteristic, complemented
320#2092—791no—
321#2085—791no—
322#2065—791no—
323#2059—791no—
324#1944—791no—
325#1917—791no—
326#1880—791no—
327#1880—791no—
328#1865—791no—
329#1865—791no—
330#1864—791no—
331C79#2639911yescharacteristic, complemented, Sylow
332C22×SD16—791no—
333C2×#125—791no—
334C2×C4×Q8C2×D7911yescharacteristic
335C2×C4×Q8C2×D7911yescharacteristic
336C2×C4×Q8C2×D7911yescharacteristic
337C2×C4×D4—791no—
338#479—1584no—
339#440—1584no—
340C2×#107—791no—
341C2×#107—791no—
342C2×#106—791no—
343C2×#106—791no—
344C2×#106—791no—
345C2×C4⋊C8—791no—
346C2×#103—791no—
347C2×#103—791no—
348C2×#103—791no—
349C2×#102—791no—
350C2×#102—791no—
351C2×#102—791no—
352C2×#97—791no—
353#399—1581no—
354#396—1581no—
355#390—791no—
356#390—791no—
357#389—21no—
358#389—21no—
359#389—791no—
360#389—791no—
361#389C2×D7911yescharacteristic
362#389C2×D7911yescharacteristic
363#385—1581no—
364#382—21no—
365#382C2×D7911yescharacteristic
366#380—1581no—
367C4×C4⋊C4—21no—
368C4×C4⋊C4C2×D7911yescharacteristic
369#340—791no—
370#340—791no—
371#340—791no—
372#340—791no—
373#336—791no—
374#336—791no—
375#336—791no—
376#336—791no—
377C22×Q8C22×D7911yescharacteristic
378C22×D4—791no—
379C23×C4—1581no—
380C2×SD16—794no—
381C2×SD16—1582no—
382C2×SD16—1582no—
383C22×C8—791no—
384C22×C8—791no—
385C22×C8—791no—
386C22×C8—791no—
387#125—3161no—
388#125—1582no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—41no—
394C4×Q8—41no—
395C4×D4—3161no—
396C4×D4—1582no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—791no—
403C2×C4⋊C4—791no—
404C2×C4⋊C4—791no—
405C2×C4⋊C4—1581no—
406C2×C4⋊C4—1581no—
407C2×C4⋊C4—1581no—
408C2×C4⋊C4—1581no—
409C2×C4⋊C4C22⋊D7911yescharacteristic
410C2×C4⋊C4C22⋊D7911yescharacteristic
411C2×C4⋊C4C22×D7911yescharacteristic
412C2×C4⋊C4C22×D7911yescharacteristic
413C2×C4⋊C4C22×D7911yescharacteristic
414C2×C22⋊C4—1581no—
415C2×C22⋊C4—1581no—
416C2×C22⋊C4—1581no—
417C2×C22⋊C4—1581no—
418C2×C22⋊C4—1581no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42C22×D7911yescharacteristic
423C2×C42C22×D7911yescharacteristic
424C2×C42C22×D7911yescharacteristic
425#107—1582no—
426#107—1582no—
427#106—3161no—
428#106—1582no—
429#106—1582no—
430C4⋊C8—1582no—
431#103—794no—
432#103—1582no—
433#103—1582no—
434#102—3161no—
435#102—1582no—
436#102—1582no—
437#97—1582no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1581no—
443#95—1581no—
444#95C22⋊D7911yescharacteristic
445#95C22⋊D7911yescharacteristic
446C24—1581no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8D4×D7912yes—
451C2×Q8Q8×D7912yes—
452C2×Q8#1192962412yes—
453C2×D4—792no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4—1581no—
457C2×D4—1581no—
458C2×D4—1581no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—3162no—
464C22×C4—3162no—
465C22×C4—6321no—
466C22×C4—1581no—
467C22×C4—1581no—
468C22×C4—1581no—
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C2×C22⋊D7911yescharacteristic
471C22×C4C2×C22⋊D7911yescharacteristic
472C22×C4C2×C22⋊D7911yescharacteristic
473C22×C4C2×C22⋊D7911yescharacteristic
474C22×C4C2×C22⋊D7911yescharacteristic
475C22×C4C23×D7911yescharacteristic
476SD16—1588no—
477C2×C8—792no—
478C2×C8—792no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8—1581no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—3162no—
502C4⋊C4—3162no—
503C4⋊C4—792no—
504C4⋊C4—6321no—
505C4⋊C4—6321no—
506C4⋊C4—1581no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4—1581no—
510C4⋊C4—1581no—
511C4⋊C4#1192962412yes—
512C22⋊C4—3162no—
513C22⋊C4—3162no—
514C22⋊C4—6321no—
515C22⋊C4—6321no—
516C22⋊C4—6321no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—42no—
526C23—3161no—
527C23—3161no—
528C23—3161no—
529C23—3161no—
530C23—3161no—
531C23—3161no—
532C23—1582no—
533C23C22×C22⋊D7911yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1457913914yes—
538D4—3161no—
539D4—3161no—
540D4—1582no—
541D4—1582no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—3161no—
573C2×C4—3161no—
574C2×C4—3161no—
575C2×C4—3161no—
576C2×C4—3161no—
577C2×C4—3161no—
578C2×C4—3161no—
579C2×C4—3161no—
580C2×C4—6324no—
581C2×C4—1582no—
582C2×C4C2×D4×D7911yescharacteristic
583C2×C4C2×Q8×D7911yescharacteristic
584C2×C4C2×#1192962411yescharacteristic
585C2×C4C2×#1192962411yescharacteristic
586C2×C4#1457920611yescharacteristic
587C2×C4#1457920911yescharacteristic
588C8—3161no—
589C8—3161no—
590C8—1582no—
591C8—1582no—
592C22—3162no—
593C22—3162no—
594C22—3162no—
595C22—6321no—
596C22—6321no—
597C22—6321no—
598C22—6321no—
599C22#1909907411yescharacteristic
600C22D4×C22⋊D7911yescharacteristic
601C22Q8×C22⋊D7911yescharacteristic
602C22#1909936111yescharacteristic
603C22#1909936111yescharacteristic
604C22#1909937111yescharacteristic
605C22#1909937111yescharacteristic
606C4—24no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—81no—
616C4—3162no—
617C4—6321no—
618C4—6321no—
619C4C2×#1457913911yescharacteristic
620C4#1909916011yescharacteristic
621C4#1909921411yescharacteristic
622C4#1909926811yescharacteristic
623C2—6322no—
624C2#2376720911yescharacteristic
625C2#2377350911yescharacteristic
626C2#2377369111yescharacteristic
627C2#2377498611yescharacteristic
628C2#2377503211yescharacteristic
629C2#2377633911yescharacteristic
630C2#2377633911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2351C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2351D79
hypocenterC79#26399
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#19098870C2semidirect
5. #23764882C2semidirect
7. #23764738C2semidirect
8. #23764140C2semidirect
9. #23763966C2semidirect
12. #23763668C2semidirect
13. #23763668C2semidirect
15. C79×#2351C2semidirect
319. #2351D79semidirect
331. C79#26399semidirect

Maximal subgroups

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

subgroupindex
C2×#190988702
C2×#190986612
#237672722
#237649242
#237648822
#237647712
#237647382
#237641402
#237639662
#237637552
#237636682
#237636602
C79×#23512
#2639979