Finite Group Databasebeta

← #34310291

Subgroups — #34310291

Order 20224.

850 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×#19098844C211yesmaximal, characteristic
2#23766489C211yesmaximal, characteristic, complemented
3#23766308C211yesmaximal, characteristic
4#23764372C211yesmaximal, characteristic, complemented
5#23764136C211yesmaximal, characteristic, complemented
6#23764131C211yesmaximal, characteristic, complemented
7#23763991C211yesmaximal, characteristic, complemented
8#23763970C211yesmaximal, characteristic, complemented
9#23763658C211yesmaximal, characteristic
10#23763644C211yesmaximal, characteristic
11#23763277C211yesmaximal, characteristic, complemented
12#23763228C211yesmaximal, characteristic
13#23763224C211yesmaximal, characteristic
14#23763214C211yesmaximal, characteristic
15C79×#1949C211yesmaximal, characteristic, complemented
16C2×#14579161C2211yescharacteristic
17C2×C4×D316C2211yescharacteristic
18C2×#14579123C2211yescharacteristic
19C2×#14579123C2211yescharacteristic
20#19098844C418yescomplemented
21C2×#14579095C2211yescharacteristic
22C2×#14579095C2211yescharacteristic
23C2×#14579083C2211yescharacteristic
24C2×#14579083C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579072C2211yescharacteristic
28C2×#14579071C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098467—21no—
31#19098467—21no—
32#19098461—21no—
33#19098459C2211yescharacteristic
34#19098459C2211yescharacteristic
35#19098459C2211yescharacteristic
36#19098459C2211yescharacteristic
37#19098437—21no—
38#19098437—21no—
39#19098422—21no—
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4×C4⋊Dic79—21no—
43C4×C4⋊Dic79C2211yescharacteristic
44#95×D79—21no—
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54C2×#106×C79C2211yescharacteristic
55C2×C4⋊C8×C79C2211yescharacteristic
56#382×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×Dic158—21no—
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C22×C4×D79—21no—
64C22×C4×D79—21no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579161—22no—
68#14579161C2×C414yescomplemented
69C4×D316—22no—
70C4×D316C2×C414yes—
71#14579123—22no—
72#14579123—22no—
73#14579123—41no—
74#14579123—41no—
75C2×C4⋊Dic79—21no—
76C2×C4⋊Dic79D411yescharacteristic
77C2×C4⋊Dic79D411yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×C4⋊Dic79C2311yescharacteristic
80C2×C4⋊Dic79C2311yescharacteristic
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—21no—
84C2×Dic79⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4—21no—
88C2×C4×Dic79—21no—
89C2×C4×Dic79—21no—
90C2×C4×Dic79C2311yescharacteristic
91C2×C4×Dic79C2311yescharacteristic
92#14579095C2×C414yescomplemented
93#14579095C2×C414yescomplemented
94#14579083C2×C414yes—
95#14579083C2×C414yes—
96#14579081—22no—
97#14579081Q814yescomplemented
98#14579072—22no—
99#14579071—41no—
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×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79C2311yescharacteristic
117#107×C79—41no—
118#106×C79—41no—
119C4⋊C8×C79C2×C414yescomplemented
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123C23×D79—21no—
124C2×Dic158—24no—
125C2×Dic158—81no—
126C2×Dic158—81no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316C22×C412yes—
131C2×D316C2×Q812yes—
132C2×D316#4112yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C2×C4×D79—24no—
141C2×C4×D79—42no—
142C2×C4×D79—42no—
143C2×C4×D79—42no—
144C2×C4×D79—42no—
145C2×C4×D79—81no—
146C2×C4×D79—81no—
147C2×C4×D79—81no—
148C2×C4×D79—81no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8C2×Q812yes—
155C4⋊Dic79—22no—
156C4⋊Dic79—24no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—41no—
161C4⋊Dic79C22×C412yes—
162C4⋊Dic79C22×C412yes—
163C4⋊Dic79C22×C412yes—
164Dic79⋊C4—81no—
165Dic79⋊C4—81no—
166Dic79⋊C4—81no—
167Dic79⋊C4—81no—
168(C2×D79)⋊C4—24no—
169(C2×D79)⋊C4—42no—
170(C2×D79)⋊C4—42no—
171C4×Dic79—21no—
172C4×Dic79—21no—
173C4×Dic79—21no—
174C4×Dic79—21no—
175C4×Dic79—81no—
176C4×Dic79—81no—
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×C79C2411yescharacteristic
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79C22×C412yes—
189C2×C8×C79C22×C412yes—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79C2×Q812yes—
198C42×C79—42no—
199C42×C79C22×C412yes—
200C42×C79#4112yes—
201C23×C79C22×D411yescharacteristic
202C22×D79—22no—
203C22×D79—22no—
204C22×D79—22no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—22no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79C4×D412yes—
226C2×C4×C79C4×D412yes—
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×Q811yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79H(5,𝔽2)11yescharacteristic
232C2×C4×C792−1+411yescharacteristic
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—22no—
236C2×Dic79—22no—
237C2×Dic79—22no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79#13712yes—
254D316—22no—
255D316—22no—
256D316—22no—
257D316C4×Q814yes—
258C4×D79—84no—
259C4×D79—84no—
260C4×D79—84no—
261Dic158—84no—
262C8×C79—22no—
263C8×C79—22no—
264C79⋊C8—22no—
265C79⋊C8—41no—
266C22×C79C2×C4×D411yescharacteristic
267C22×C79#54611yescharacteristic
268C22×C79D4×Q811yescharacteristic
269C22×C79C2×#13711yescharacteristic
270C22×C79#57511yescharacteristic
271C22×C79#57611yescharacteristic
272C22×C79#57611yescharacteristic
273C2×D79—24no—
274C2×D79—42no—
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—42no—
279C2×D79—42no—
280C4×C79—22no—
281C4×C79—22no—
282C4×C79—42no—
283C4×C79—81no—
284C4×C79—81no—
285C4×C79—81no—
286C4×C79C2×C4×Q811yescharacteristic
287C4×C79#52011yescharacteristic
288C4×C79#55211yescharacteristic
289C4×C79#55411yescharacteristic
290Dic79—22no—
291Dic79—42no—
292Dic79—42no—
293Dic79—42no—
294Dic79—41no—
295Dic79—41no—
296Dic79—41no—
297#22979—791nomaximal, Sylow
298C2×C79#236011yescharacteristic
299C2×C79C4×#13711yescharacteristic
300C2×C79#294811yescharacteristic
301C2×C79#331411yescharacteristic
302C2×C79#331911yescharacteristic
303C2×C79#338511yescharacteristic
304C2×C79#338911yescharacteristic
305D79—44no—
306C2×#474—791no—
307#2369—791no—
308#2349—791no—
309#1993—791no—
310#1949D7911yescharacteristic, complemented
311#1943—791no—
312#1942—791no—
313#1921—791no—
314#1918—791no—
315#1864—791no—
316#1862—791no—
317#1771—791no—
318#1762—791no—
319#1762—791no—
320C4×#102—791no—
321C79#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8—791no—
323C2×C4×D4—791no—
324C2×C4×D4—791no—
325C2×C4×D4—791no—
326#474—798no—
327C2×#107—791no—
328C2×#107—791no—
329C2×#107C2×D7911yescharacteristic
330C2×#106C2×D7911yescharacteristic
331C2×C4⋊C8C2×D7911yescharacteristic
332C2×#102—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#97—791no—
337C2×#97—791no—
338C2×C4×C8—791no—
339#391—1581no—
340#391—1581no—
341#389—791no—
342#389—791no—
343#389—791no—
344#389—791no—
345#389—1581no—
346#385—1581no—
347#385—1581no—
348#382C2×D7911yescharacteristic
349#380—1581no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—1581no—
354C4×C4⋊C4C2×D7911yescharacteristic
355C2×#95—1581no—
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×Q8—1581no—
367C22×D4—791no—
368C23×C4—1581no—
369C23×C4—1581no—
370C23×C4—1581no—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375C4⋊Q8—794no—
376C4⋊Q8—1582no—
377C4×D4—3161no—
378C4×D4—3161no—
379C4×D4—794no—
380C4×D4—1582no—
381C4×D4—1582no—
382C4×D4—1582no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—791no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—1581no—
389C2×C4⋊C4—1581no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4C22×D7911yescharacteristic
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C22⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C42—21no—
401C2×C42—791no—
402C2×C42—791no—
403C2×C42—1581no—
404C2×C42—1581no—
405C2×C42C22×D7911yescharacteristic
406#107—41no—
407#107—794no—
408#107—794no—
409#106—41no—
410C4⋊C8C4×D7914yescomplemented
411#102—794no—
412#102—794no—
413#102—794no—
414#102—1582no—
415#97—3161no—
416#97—1582no—
417C4×C8—3161no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—791no—
422#95—791no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428#95—1581no—
429C24—1581no—
430C2×Q8—6321no—
431C2×Q8—6321no—
432C2×Q8—1584no—
433C2×D4—792no—
434C2×D4—792no—
435C2×D4—792no—
436C2×D4—1581no—
437C2×D4—1581no—
438C2×D4—1581no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
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—6321no—
453C22×C4—6321no—
454C22×C4—6321no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4—1581no—
458C22×C4—1584no—
459C22×C4D4×D7911yescharacteristic
460C22×C4D4×D7911yescharacteristic
461C22×C4C23×D7911yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—792no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8C2×C4×D7912yes—
473C2×C8C2×C4×D7912yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—3161no—
482C4⋊C4—792no—
483C4⋊C4—792no—
484C4⋊C4—792no—
485C4⋊C4—6321no—
486C4⋊C4—6321no—
487C4⋊C4—6321no—
488C4⋊C4—6321no—
489C4⋊C4—1582no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1584no—
494C4⋊C4Q8×D7912yes—
495C22⋊C4—3162no—
496C22⋊C4—3162no—
497C22⋊C4—1584no—
498C42—42no—
499C42—6321no—
500C42—6321no—
501C42—1581no—
502C42—1581no—
503C42—1581no—
504C42—1581no—
505C42C2×C4×D7912yes—
506C42#1192962512yes—
507C23—3161no—
508C23—3161no—
509C23—3161no—
510C23—3161no—
511C23—1582no—
512C23—1582no—
513C23—1582no—
514C23C2×D4×D7911yescharacteristic
515Q8—6324no—
516D4—794no—
517D4—1582no—
518D4—1582no—
519D4—1582no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—792no—
543C2×C4—6324no—
544C2×C4—6324no—
545C2×C4—6324no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1582no—
550C2×C4—1582no—
551C2×C4—1582no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4#1457912312yes—
561C2×C4#1457912312yes—
562C2×C4C22×C4×D7911yescharacteristic
563C2×C4C2×Q8×D7911yescharacteristic
564C2×C4#41×D7911yescharacteristic
565C2×C4C2×#1192962511yescharacteristic
566C2×C4#1457920611yescharacteristic
567C2×C4#1457920811yescharacteristic
568C8—22no—
569C8—22no—
570C8—3161no—
571C8—1582no—
572C22—3162no—
573C22—3162no—
574C22—3162no—
575C22—3162no—
576C22—3162no—
577C22—3162no—
578C22—1584no—
579C22C2×#1457912311yescharacteristic
580C22#1909921011yescharacteristic
581C22#1909922611yescharacteristic
582C22#137×D7911yescharacteristic
583C22#1909935911yescharacteristic
584C22#1909936511yescharacteristic
585C22#1909936511yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—3161no—
593C4—3161no—
594C4—3161no—
595C4—3162no—
596C4—3162no—
597C4—3162no—
598C4—1582no—
599C4C4×Q8×D7911yescharacteristic
600C4#1909904211yescharacteristic
601C4#1909925111yescharacteristic
602C4#1909927411yescharacteristic
603C2—3164no—
604C2#2376640411yescharacteristic
605C2#2377221811yescharacteristic
606C2#2377225111yescharacteristic
607C2#2377548811yescharacteristic
608C2#2377556311yescharacteristic
609C2#2377631311yescharacteristic
610C2#2377635711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1949C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#1949D79
hypocenterC79#22979
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. #23766489C2semidirect
4. #23764372C2semidirect
5. #23764136C2semidirect
6. #23764131C2semidirect
7. #23763991C2semidirect
8. #23763970C2semidirect
11. #23763277C2semidirect
15. C79×#1949C2semidirect
20. #19098844C4semidirect
68. #14579161C2×C4semidirect
92. #14579095C2×C4semidirect
93. #14579095C2×C4semidirect
97. #14579081Q8semidirect
119. C4⋊C8×C79C2×C4semidirect
310. #1949D79semidirect
321. C79#22979semidirect
410. C4⋊C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190988442
#237664892
#237663082
#237643722
#237641362
#237641312
#237639912
#237639702
#237636582
#237636442
#237632772
#237632282
#237632242
#237632142
C79×#19492
#2297979